**Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques**

Mónica Corea, Jean-Pierre E. Grolier and José Manuel del Río

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69706

## Abstract

Thermodynamic experimental techniques using titration are usually employed to study the interaction between solutes in a diluted solution. This chapter deals with the underlying thermodynamic framework when titration technique is applied with densimetry, sound speed measurement and isothermal titration calorimetry. In the case of partial volumes and partial adiabatic compressibilities, a physical interpretation is proposed based upon atomic, free volume and hydration contributions.

Keywords: thermodynamics, molar partial volumes, molar partial adiabatic compressibilities, molar partial enthalpies, densimetry, sound velocity, isothermal titration calorimetry

## 1. Introduction

The purposes of this chapter are twofold. First, the thermodynamics fundaments are studied in detail to determine experimentally, calculate and interpret thermodynamic partial molar properties using different titration techniques. Second, the postgraduate students are provided with the necessary thermodynamic background to extract behavioural trends from experimental techniques including densimetry, sound speed measurement and isothermal titration calorimetry.

The first concept introduced in this chapter is "thermodynamic description". It is defined as a set of variables employed to define thermodynamically the studied system. For example, a description by components of a multicomponent system is:

$$J = J(n\_1, n\_2, n\_3) \tag{1}$$

where J is an extensive thermodynamic property; n1, n2 and n3 are the number of moles of components 1, 2 and 3. Other type of thermodynamic description is in terms of the concept of "fraction of a system". A fraction of a system is a thermodynamic entity, with internal composition, which groups several components. For example, the above-mentioned system can be considered as being composed of the component 1, and a fraction F grouping components 2 and 3. In this way, J can be written as:

$$J = J(n\_1, n\_F, x\_{f^3}) \tag{2}$$

where nF is the total number of moles of the fraction F and xf3 is a variable related to the composition of the fraction. Depending on the system, one can choose the more adequate description. For example, in a liquid mixture, a description by components (Eq. (1)) can be suitable. Other systems as those shown in Figure 1 could be better described in terms of fractions.

Figure 1A shows a system composed of the solvent (component 1), solute A (component 2) and solute B (component 3). This system will be described in this chapter using a description by fractions representing a "complex solute" composed of solutes A and B (see Figure 1B). This description is appropriate to use in conditions of infinite dilution and dilute solutions. Other example (see Figure 1C and D) is a functionalized latex particle. A latex is a system composed of polymeric particles dispersed in a solvent. In a functionalized latex, particles are composed of nonpolar groups and functional groups (usually polar groups). In this case, a description by components expressed in Eq. (1) and visualized in Figure 1C is very difficult to use and it is more convenient to consider a fraction (polymeric particle) composed of non-polar groups (component 2) and polar groups (component 3). Figure 1D shows a sketch of this description.

When different descriptions are considered for a system, we have to reconsider the relation between the description and the thermodynamic object studied. In principle, one might think that all descriptions are equivalent. But this is not true because not all descriptions can retain all

Figure 1. Examples of different descriptions in two systems. (A) and (B) are several solutes in a solvent. (C) and (D) are a functionalized latex with polar groups.

features of a thermodynamic system. For example, it is not possible to speak about thermodynamic partial properties at infinite dilution in multicomponent systems. This fact should not be surprising because in differential geometry [1], there is the same problem associated with the relation between a parametrization and a geometric object. Let's consider, for example, the sphere of radius equal to one, and a parametrization is:

$$\mathbf{X\_{I}(x,y)} = \left(\mathbf{x}, y, \sqrt{1 - (\mathbf{x}^{2} + y^{2})}\right) \tag{3}$$

The problem with this parametrization is that it only covers the top half of the sphere. In addition, it is not differentiable in the points of the sphere's equator. Other possibility is:

$$\mathbf{X\_2}(\mathbf{x}, y) = \left(\mathbf{x}, y, -\sqrt{1 - (\mathbf{x}^2 + y^2)}\right) \tag{4}$$

But, it only covers the lower half of the sphere and neither is differentiable in points of the sphere's equator. Even if we consider a combination of X<sup>1</sup> and X2, we have the problem of the lack of differentiability in the points of the sphere's equator. Another possible parametrization is:

$$\mathbf{X}\_3(\theta,\varphi) = (\sin\theta\cos\varphi, \sin\theta\sin\varphi, \cos\theta) \tag{5}$$

where θ is the colatitude (the complement of the latitude) and ϕ the longitude. X<sup>3</sup> covers the whole surface of the sphere and it is also differentiable in all points. For this reason, it contains more information about the sphere (geometric object) than X<sup>1</sup> and X2. Backing to thermodynamics, in the same case than for X3, the partial molar properties at infinite dilution cannot be obtained and manipulated using the description by components, and it is necessary to use the description by fractions.

The other concept also introduced in this chapter is the "interaction between components of a system". The first principle of thermodynamics establishes the way, in which systems interact between them and/or with surroundings. In this case, we are interested in the interaction inside the systems and this cannot be interpreted macroscopically using the first principle of thermodynamics. With the concept of interaction between components, we can define mathematically a dilute solution and characterize its thermodynamic behaviour in terms of molar partial properties. In addition to this, we will consider the partial molar properties at infinite dilution. These properties are essential in studies of polymeric particles because they contain the information about the interactions inside the particles. These interactions determine the architecture and final application of the particle.

### 2. Mathematical fundaments

J ¼ Jðn1, n2, n3Þ (1)

J ¼ Jðn1, nF, xf <sup>3</sup>Þ (2)

where J is an extensive thermodynamic property; n1, n2 and n3 are the number of moles of components 1, 2 and 3. Other type of thermodynamic description is in terms of the concept of "fraction of a system". A fraction of a system is a thermodynamic entity, with internal composition, which groups several components. For example, the above-mentioned system can be considered as being composed of the component 1, and a fraction F grouping components 2

where nF is the total number of moles of the fraction F and xf3 is a variable related to the composition of the fraction. Depending on the system, one can choose the more adequate description. For example, in a liquid mixture, a description by components (Eq. (1)) can be suitable. Other

Figure 1A shows a system composed of the solvent (component 1), solute A (component 2) and solute B (component 3). This system will be described in this chapter using a description by fractions representing a "complex solute" composed of solutes A and B (see Figure 1B). This description is appropriate to use in conditions of infinite dilution and dilute solutions. Other example (see Figure 1C and D) is a functionalized latex particle. A latex is a system composed of polymeric particles dispersed in a solvent. In a functionalized latex, particles are composed of nonpolar groups and functional groups (usually polar groups). In this case, a description by components expressed in Eq. (1) and visualized in Figure 1C is very difficult to use and it is more convenient to consider a fraction (polymeric particle) composed of non-polar groups (component 2)

When different descriptions are considered for a system, we have to reconsider the relation between the description and the thermodynamic object studied. In principle, one might think that all descriptions are equivalent. But this is not true because not all descriptions can retain all

Figure 1. Examples of different descriptions in two systems. (A) and (B) are several solutes in a solvent. (C) and (D) are a

systems as those shown in Figure 1 could be better described in terms of fractions.

and polar groups (component 3). Figure 1D shows a sketch of this description.

and 3. In this way, J can be written as:

100 Advances in Titration Techniques

functionalized latex with polar groups.

In this section, some mathematical tools are presented such as changes of variable, changes of size, the Euler theorem and limits in multivariable functions. Variable changes will allow us to relate partial properties of different descriptions. Changes of size are the processes underlying the extensivity and non-extensivity of thermodynamic properties, which will be mathematically implemented by the concept of homogeneity. The Euler's theorem will be treated in the more general form, and in its demonstration we will avoid some aspects, which remain unclear in the versions of the textbooks of Callen [2] and Klotz and Rosenberg [3].

#### 2.1. Changes of variable

Let f be the function defined as:

$$f = f(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) \tag{6}$$

The gradient of f with respect to the variables x1, x2 and x3 is the vector:

$$\nabla f(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) = \begin{bmatrix} \left(\frac{\partial f}{\partial \mathbf{x}\_1}\right)\_{\mathbf{x}\_2, \mathbf{x}\_3} \\ \left(\frac{\partial f}{\partial \mathbf{x}\_2}\right)\_{\mathbf{x}\_1, \mathbf{x}\_3} \\ \left(\frac{\partial f}{\partial \mathbf{x}\_3}\right)\_{\mathbf{x}\_1, \mathbf{x}\_2} \end{bmatrix} \tag{7}$$

If we consider the change of variable:

$$\begin{cases} \mathbf{x}\_1 = \mathbf{x}\_1(y\_{1'}, y\_{2'}, y\_3) \\ \mathbf{x}\_2 = \mathbf{x}\_2(y\_{1'}, y\_{2'}, y\_3) \\ \mathbf{x}\_3 = \mathbf{x}\_3(y\_{1'}, y\_{2'}, y\_3) \end{cases} \tag{8}$$

the function f will take the form:

$$f = f(y\_{1'}, y\_{2'}, y\_3) \tag{9}$$

where its gradient will be:

$$\nabla f(y\_1, y\_2, y\_3) = \begin{bmatrix} \left(\frac{\partial f}{\partial y\_1}\right)\_{y\_2, y\_3} \\\\ \left(\frac{\partial f}{\partial y\_2}\right)\_{y\_1, y\_3} \\\\ \left(\frac{\partial f}{\partial y\_3}\right)\_{y\_1, y\_2} \end{bmatrix} \tag{10}$$

Our interest is to relate the partial derivatives with respect to the variables x1, x2 and x3 given in Eq. (7) with the partial properties with respect to y1, y2 and y3 given in Eq. (10). From Eq. (8), the total differential of x1 is:

$$d\mathbf{x}\_1 = \left(\frac{\partial \mathbf{x}\_1}{\partial y\_1}\right)\_{y\_2, y\_3} dy\_1 + \left(\frac{\partial \mathbf{x}\_1}{\partial y\_2}\right)\_{y\_1, y\_3} dy\_2 + \left(\frac{\partial \mathbf{x}\_1}{\partial y\_3}\right)\_{y\_1, y\_2} dy\_3 \tag{11}$$

Using dx1 given by Eq. (11) and similarly with equations for dx2 and dx3, we can write:

Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques http://dx.doi.org/10.5772/intechopen.69706 103

$$
\begin{bmatrix} dx\_1 \\ dx\_2 \\ dx\_3 \end{bmatrix} = \mathbf{T} \begin{pmatrix} \mathbf{x}\_1 \ \mathbf{x}\_2 \ \mathbf{x}\_3 \\ \mathbf{y}\_1 \ \mathbf{y}\_2 \ \mathbf{y}\_3 \end{pmatrix} \begin{bmatrix} dy\_1 \\ dy\_2 \\ dy\_3 \end{bmatrix} \tag{12}
$$

where the matrix T is:

the extensivity and non-extensivity of thermodynamic properties, which will be mathematically implemented by the concept of homogeneity. The Euler's theorem will be treated in the more general form, and in its demonstration we will avoid some aspects, which remain unclear

f ¼ fðx1, x2, x3Þ (6)

f ¼ fðy1, y2, y3Þ (9)

(7)

(8)

(10)

in the versions of the textbooks of Callen [2] and Klotz and Rosenberg [3].

The gradient of f with respect to the variables x1, x2 and x3 is the vector:

∇fðx1, x2, x3Þ ¼

8 < :

∇fðy1, y2, y3Þ ¼

∂f ∂x<sup>1</sup> � �

∂f ∂y<sup>1</sup> � �

Our interest is to relate the partial derivatives with respect to the variables x1, x2 and x3 given in Eq. (7) with the partial properties with respect to y1, y2 and y3 given in Eq. (10). From Eq. (8),

> ∂x<sup>1</sup> ∂y<sup>2</sup> � �

y1, y<sup>3</sup>

∂f ∂y<sup>2</sup> � �

∂f ∂y<sup>3</sup> � �

y2,y<sup>3</sup>

y1,y<sup>3</sup>

y1,y<sup>2</sup>

dy<sup>2</sup> þ

∂x<sup>1</sup> ∂y<sup>3</sup> � �

y1,y<sup>2</sup>

dy<sup>3</sup> (11)

x<sup>1</sup> ¼ x1ðy1, y2, y3Þ x<sup>2</sup> ¼ x2ðy1, y2, y3Þ x<sup>3</sup> ¼ x3ðy1, y2, y3Þ

x2,x<sup>3</sup> ∂f ∂x<sup>2</sup> � �

x1,x<sup>3</sup> ∂f ∂x<sup>3</sup> � �

x1,x<sup>2</sup>

2.1. Changes of variable

102 Advances in Titration Techniques

Let f be the function defined as:

If we consider the change of variable:

the function f will take the form:

where its gradient will be:

the total differential of x1 is:

dx<sup>1</sup> <sup>¼</sup> <sup>∂</sup>x<sup>1</sup> ∂y<sup>1</sup> � �

y2,y<sup>3</sup>

dy<sup>1</sup> þ

Using dx1 given by Eq. (11) and similarly with equations for dx2 and dx3, we can write:

$$\mathbf{T}\begin{pmatrix}x\_{1}\ x\_{2}\ x\_{3}\end{pmatrix}=\begin{bmatrix}\begin{pmatrix}\operatorname{\mathbf{d}x\_{1}}\\\operatorname{\mathbf{d}y\_{1}}\end{pmatrix}\_{y\_{2},y\_{3}}\quad\begin{pmatrix}\operatorname{\mathbf{d}x\_{1}}\\\operatorname{\mathbf{d}y\_{2}}\end{pmatrix}\_{y\_{1},y\_{3}}\quad\begin{pmatrix}\operatorname{\mathbf{d}x\_{1}}\\\operatorname{\mathbf{d}y\_{3}}\end{pmatrix}\_{y\_{1},y\_{3}}\\\begin{pmatrix}\operatorname{\mathbf{d}y\_{1}}\\\operatorname{\mathbf{d}y\_{1}}\end{pmatrix}\_{y\_{2},y\_{3}}\quad\begin{pmatrix}\operatorname{\mathbf{d}x\_{2}}\\\operatorname{\mathbf{d}y\_{2}}\end{pmatrix}\_{y\_{1},y\_{3}}\quad\begin{pmatrix}\operatorname{\mathbf{d}x\_{2}}\\\operatorname{\mathbf{d}y\_{3}}\end{pmatrix}\_{y\_{1},y\_{2}}\end{pmatrix}\_{y\_{1},y\_{2}}\tag{13}$$

From (6) and using (7), the total differential of f can be expressed as:

$$df = \left(\frac{\partial f}{\partial \mathbf{x}\_1}\right)\_{\mathbf{x}\_2, \mathbf{x}\_3} d\mathbf{x}\_1 + \left(\frac{\partial f}{\partial \mathbf{x}\_2}\right)\_{\mathbf{x}\_1, \mathbf{x}\_3} d\mathbf{x}\_2 + \left(\frac{\partial f}{\partial \mathbf{x}\_3}\right)\_{\mathbf{x}\_1, \mathbf{x}\_2} d\mathbf{x}\_3 = \left[\nabla f(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3)\right]^\mathrm{T} \begin{bmatrix} d\mathbf{x}\_1\\ d\mathbf{x}\_2\\ d\mathbf{x}\_3 \end{bmatrix} \tag{14}$$

where the symbol "T" indicates "transpose". From Eq. (9) using Eq. (10), the differential of f can be written as:

$$df = \left(\frac{\partial f}{\partial y\_1}\right)\_{y\_2, y\_3} dy\_1 + \left(\frac{\partial f}{\partial y\_2}\right)\_{y\_1, y\_3} dy\_2 + \left(\frac{\partial f}{\partial y\_3}\right)\_{y\_1, y\_2} dy\_3 = \left[\nabla f(y\_1, y\_2, y\_3)\right]^\Gamma \begin{bmatrix} dy\_1 \\ dy\_2 \\ dy\_3 \end{bmatrix} \tag{15}$$

Equaling (15) to (14) and using (12):

$$\left[\nabla f(\mathbf{y}\_1, \mathbf{y}\_2, \mathbf{y}\_3)\right]^\mathrm{T} = \left[\nabla f(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3)\right]^\mathrm{T} \mathbf{T} \begin{pmatrix} \mathbf{x}\_1 \ \mathbf{x}\_2 \ \mathbf{x}\_3\\ \mathbf{y}\_1 \ \mathbf{y}\_2 \ \mathbf{y}\_3 \end{pmatrix} \tag{16}$$

Remembering that x being a vector and A a matrix, then (x<sup>T</sup> A)<sup>T</sup> = AT x, and taking the transpose in both sides of (16):

$$\nabla f(y\_1, y\_2, y\_3) = \left[\mathbf{T}\begin{pmatrix} \mathbf{x}\_1 \ \mathbf{x}\_2 \ \mathbf{x}\_3\\ \mathbf{y}\_1 \ \mathbf{y}\_2 \ \mathbf{y}\_3 \end{pmatrix}\right]^\top \nabla f(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) \tag{17}$$

Eq. (17) relates the vector gradient with respect to the variables (x1, x2, x3) to the vector gradient with respect to the variables (y1, y2, y3), and it will allow us to express the partial properties in two different descriptions.

#### 2.2. Changes of size

In this paragraph, the process of size change in thermodynamic systems is analyzed. The behaviour of systems in a size change has consequences on the behaviour or nature of the thermodynamic properties as well as on the form of the thermodynamic equations of the system. Figure 2 shows a visualization of this process in both directions: increasing and reduction.

Figure 2. Sketch of the change of size (increasing and reduction) of a system with volume V.

From Figure 2, it is clear that being V the volume, N the number of moles and U the internal energy, the configuration of this system is under increasing size λ times:

$$\begin{aligned} V \xrightarrow{\lambda-\text{times}} V' &= \lambda V\\ N \xrightarrow{\lambda-\text{times}} N' &= \lambda N\\ U \xrightarrow{\lambda-\text{times}} U' &= \lambda U \end{aligned} \tag{18}$$

Thermodynamic properties, which transform accordingly to (18), depend on the size of the system and are named extensive variables. Not all thermodynamic variables transform according to Eq. (18). An example is the molar fraction of the component 2 (x2) in a two-component system. We can see this formally in the following way. For a two-component system:

$$\begin{aligned} N\_1 \xrightarrow{\lambda - \text{times}} N'\_1 &= \lambda N\_1\\ N\_2 \xrightarrow{\lambda - \text{times}} N'\_2 &= \lambda N\_2 \end{aligned} \tag{19}$$

and x<sup>2</sup> transforms as:

$$\mathbf{x}\_{2} \xrightarrow{\lambda - \text{times}} \mathbf{x}'\_{2} = \frac{N'\_{2}}{N'\_{1} + N'\_{2}} = \frac{\lambda N\_{2}}{\lambda N\_{1} + \lambda N\_{2}} = \frac{N\_{2}}{N\_{1} + N\_{2}} = \mathbf{x}\_{2} \tag{20}$$

That is, the molar fraction of the component 2 is independent of the system size. Properties, which remain constant upon size change, are named intensive properties. Other thermodynamic properties with such characteristics are temperature, pressure, pH and concentration c<sup>2</sup> (c<sup>2</sup> = N2/V). It is also interesting to look at the behaviour of functions, which depend on thermodynamic variables (intensive and/or extensive), in a size change. Let, for example, the function f be given by f = f(T, P, N1, N2, …). For particular values of the variables T0, P0, N01, N02, …, the function f takes the value f0, and in a change of size:

#### Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques http://dx.doi.org/10.5772/intechopen.69706 105

$$\begin{aligned} T\_0 &\xrightarrow{\lambda-\text{times}} T'\_0 = T\_0\\ P\_0 &\xrightarrow{\lambda-\text{times}} P'\_0 = P\_0\\ N\_{01} &\xrightarrow{\lambda-\text{times}} N'\_{01} = \lambda N\_{01}\\ N\_{02} &\xrightarrow{\lambda-\text{times}} N'\_{02} = \lambda N\_{02} \\ \cdots\\ f\_0 &\xrightarrow{\lambda-\text{times}} f'\_0 \end{aligned} \tag{21}$$

If f 0 <sup>0</sup> ¼ lf <sup>0</sup>, the function will behave as an extensive property. This concept is mathematically implemented as:

$$f(T, P, \lambda N\_1, \lambda N\_2, \dots) = \lambda f(T, P, N\_1, N\_2, \dots) \tag{22}$$

In this case, f is a homogeneous function of one degree. If f 0 <sup>0</sup> ¼ f <sup>0</sup>, f will behave as an intensive property. Mathematically, f is expressed as a homogeneous function of zero degree as:

$$f(T, P, \lambda N\_1, \lambda N\_2, \dots) = f(T, P, N\_1, N\_2, \dots) \tag{23}$$

#### 2.3. Euler´s theorem

Let f = f(x1, x2,…; y1, y2,…) be a function, which is a homogeneous function of one degree with respect to the variables y1, y2,…:

$$f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_1, \lambda y\_2, \dots) = \lambda f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; y\_{1'}, y\_{2'}, \dots) \tag{24}$$

Then,

(18)

(19)

¼ x<sup>2</sup> (20)

From Figure 2, it is clear that being V the volume, N the number of moles and U the internal

<sup>V</sup>! <sup>λ</sup>�times <sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>λ</sup><sup>V</sup> <sup>N</sup>! <sup>λ</sup>�times <sup>N</sup><sup>0</sup> <sup>¼</sup> <sup>λ</sup><sup>N</sup> <sup>U</sup>! <sup>λ</sup>�times <sup>U</sup><sup>0</sup> <sup>¼</sup> <sup>λ</sup><sup>U</sup>

Thermodynamic properties, which transform accordingly to (18), depend on the size of the system and are named extensive variables. Not all thermodynamic variables transform according to Eq. (18). An example is the molar fraction of the component 2 (x2) in a two-component system.

<sup>1</sup> ¼ λN<sup>1</sup>

<sup>2</sup> ¼ λN<sup>2</sup>

<sup>¼</sup> <sup>N</sup><sup>2</sup> N<sup>1</sup> þ N<sup>2</sup>

<sup>¼</sup> <sup>λ</sup>N<sup>2</sup> λN<sup>1</sup> þ λN<sup>2</sup>

That is, the molar fraction of the component 2 is independent of the system size. Properties, which remain constant upon size change, are named intensive properties. Other thermodynamic properties with such characteristics are temperature, pressure, pH and concentration c<sup>2</sup> (c<sup>2</sup> = N2/V). It is also interesting to look at the behaviour of functions, which depend on thermodynamic variables (intensive and/or extensive), in a size change. Let, for example, the function f be given by f = f(T, P, N1, N2, …). For particular values of the variables T0, P0, N01,

energy, the configuration of this system is under increasing size λ times:

Figure 2. Sketch of the change of size (increasing and reduction) of a system with volume V.

We can see this formally in the following way. For a two-component system:

<sup>2</sup> <sup>¼</sup> <sup>N</sup><sup>0</sup>

N0 <sup>1</sup> þ N<sup>0</sup> 2

and x<sup>2</sup> transforms as:

104 Advances in Titration Techniques

<sup>x</sup><sup>2</sup>! <sup>λ</sup>�times

x0

N02, …, the function f takes the value f0, and in a change of size:

<sup>N</sup><sup>1</sup>! <sup>λ</sup>�times <sup>N</sup><sup>0</sup>

<sup>N</sup><sup>2</sup>! <sup>λ</sup>�times <sup>N</sup><sup>0</sup>

2

$$f = \left(\frac{\partial f}{\partial y\_1}\right)\_{x\_1, x\_2, \dots, x\_{\mathcal{Y}\_2}, y\_3, \dots} \quad y\_1 + \left(\frac{\partial f}{\partial y\_2}\right)\_{x\_1, x\_2, \dots, x\_{\mathcal{Y}\_1}, y\_3, \dots} \quad y\_2 + \dots \tag{25}$$

The demonstration is as follows. The differential with respect to λ in the left side of (24) is:

$$\frac{d f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_1, \lambda y\_2, \dots)}{d \lambda} = \left(\frac{\partial f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_1, \lambda y\_2, \dots)}{\partial \mathbf{x}\_1}\right)\_{\mathbf{x}\_2, \mathbf{x}\_3, \dots; \lambda y\_1, \lambda y\_2, \dots}$$

$$\frac{d \mathbf{x}\_1}{d \lambda} + \left(\frac{\partial f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_1, \lambda y\_2, \dots)}{\partial \mathbf{x}\_2}\right)\_{\mathbf{x}\_1, \mathbf{x}\_3, \dots; \lambda y\_1, \lambda y\_2, \dots}$$

$$\frac{d \mathbf{x}\_2}{d \lambda} + \left(\frac{\partial f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_1, \lambda y\_2, \dots)}{\partial (\lambda y\_1)}\right)\_{\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_2, \lambda y\_3, \dots}$$

$$\frac{d(\lambda y\_1)}{d \lambda} + \left(\frac{\partial f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_1, \lambda y\_2, \dots)}{\partial (\lambda y\_2)}\right)\_{\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_1, \lambda y\_2, \dots} \frac{d(\lambda y\_2)}{d \lambda} + \dots \tag{26}$$

For the sets of variables x1, x2,… and y1,y2,…, we obtain respectively that:

$$\frac{d\mathbf{x}\_1}{d\lambda} = \frac{d\mathbf{x}\_2}{d\lambda} = \dots = \mathbf{0} \tag{27}$$

$$\frac{d(\lambda y\_1)}{d\lambda} = y\_{1'} \qquad \frac{d(\lambda y\_2)}{d\lambda} = y\_{2'} \quad \dots \tag{28}$$

The following step in this demonstration is different from the step proposed in other textbooks [2, 3]. The partial derivative of f with respect to (λy1) can be expressed as:

$$\left(\frac{\partial f(\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{A} \mathbf{y}\_1, \lambda \mathbf{y}\_2, \dots)}{\partial (\lambda \mathbf{y}\_1)}\right)\_{\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda \mathbf{y}\_2, \lambda \mathbf{y}\_3, \dots}$$

$$= \lim\_{\Delta \to 0} \frac{f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda \mathbf{y}\_1 + \Delta, \lambda \mathbf{y}\_2, \dots) - f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda \mathbf{y}\_1, \lambda \mathbf{y}\_2, \dots)}{\Delta} \tag{29}$$

Considering that f is a homogeneous function of one degree with respect to the variables y1,y2, … and making Δ'= Δ/λ in (29),

$$\left(\frac{\partial f(\mathbf{x}\_1, \mathbf{x}\_2, \dots \boldsymbol{\lambda} \mathbf{y}\_1, \lambda \mathbf{y}\_2, \dots)}{\partial(\lambda \mathbf{y}\_1)}\right)\_{\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_{\mathbf{y}\_2, \lambda \mathbf{y}\_3, \dots}}$$

$$\mathbf{x} = \lim\_{\Delta' \to 0} \frac{f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \mathbf{y}\_1 + \Delta', \mathbf{y}\_2, \dots) - f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \mathbf{y}\_1, \mathbf{y}\_2, \dots)}{\Delta'} = \left(\frac{\partial f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \mathbf{y}\_1, \mathbf{y}\_2, \dots)}{\partial \mathbf{y}\_1}\right)\_{\mathbf{x}\_1, \mathbf{x}\_2, \dots; \mathbf{y}\_2, \mathbf{y}\_3, \dots} \tag{30}$$

The differential of f with respect to λ in the right side of (24) is:

$$\frac{d[\lambda f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; y\_{1'}, y\_{2'}, \dots)]}{d\lambda} = f(\mathbf{x}\_1, \mathbf{x}\_2, \dots; y\_{1'}, y\_{2'}, \dots) \tag{31}$$

Eq. (25) is obtained by substituting Eqs. (27), (28), (30), (31) in Eq. (26). In addition, it is interesting to see that, defining f<sup>1</sup> as f<sup>1</sup> = (∂f/∂x1) and using (30), f<sup>1</sup> is a homogeneous function of zero degree with respect to the variables y1, y2,…:

$$f\_1(\mathbf{x}\_1, \mathbf{x}\_2, \dots; \lambda y\_{1'} \lambda y\_{2'} \dots) = f\_1(\mathbf{x}\_1, \mathbf{x}\_2, \dots; y\_{1'}, y\_{2'} \dots) \tag{32}$$

## 3. Thermodynamic descriptions

#### 3.1. Description by components

Let it be a three-component system (e.g., as those of Figure 1A and C). Being J an extensive property, a description by components is:

$$J = \mathcal{J}(n\_1, n\_2, n\_3) \tag{33}$$

where n1, n<sup>2</sup> and n<sup>3</sup> are the number of moles of components 1, 2 and 3. The partial property of 1 is defined as:

Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques http://dx.doi.org/10.5772/intechopen.69706 107

$$(j\_{1;2,3}(n\_1, n\_2, n\_3) = \left(\frac{\partial f(n\_1, n\_2, n\_3)}{\partial n\_1}\right)\_{n\_2, n\_3} \tag{34}$$

From the above section, we know that j1;2,3 is homogeneous function of zero degree with respect to n1, n<sup>2</sup> and n3. With this and considering λ = 1/(n1+n2+n3),

$$\begin{split} \mathbf{j}\_{1;2,3}(\mathbf{n}\_1, \mathbf{n}\_2, \mathbf{n}\_3) &= \mathbf{j}\_{1;2,3} \left( \frac{\mathbf{n}\_1}{\mathbf{n}\_1 + \mathbf{n}\_2 + \mathbf{n}\_3}, \frac{\mathbf{n}\_2}{\mathbf{n}\_1 + \mathbf{n}\_2 + \mathbf{n}\_3}, \frac{\mathbf{n}\_3}{\mathbf{n}\_1 + \mathbf{n}\_2 + \mathbf{n}\_3} \right) = \\ &= \mathbf{j}\_{1;2,3}(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3) = \mathbf{j}\_{1;2,3}(\mathbf{x}\_2, \mathbf{x}\_3) \end{split} \tag{35}$$

where we have considered that x<sup>1</sup> is a function of x<sup>2</sup> and x<sup>3</sup> because x<sup>1</sup> = 1�x2�x3. From (35), we see that the partial molar properties depend only on the composition of the system. Alternatively to (35), we could use other scales of composition/concentration to express j1;2,3.

The equation of Gibbs is obtained by differentiating J in (33) and using Eq. (34) and similar definitions for components 2 and 3:

$$d\!\!/ = j\_{1;2,3}dn\_1 + j\_{2;1,3}dn\_2 + j\_{3;1,2}dn\_3 \tag{36}$$

The Euler equation is obtained by considering that J is a homogeneous function with respect to n1, n2 and n3 and applying the Euler's theorem:

$$J = n\_1 j\_{1;2,3} + n\_2 j\_{2;1,3} + n\_3 j\_{3;1,2} \tag{37}$$

The Gibbs-Duhem equation is obtained by differentiating in Eq. (37), equalling to Eq. (36) and cancelling common terms:

$$0 = n\_1 d\dot{j}\_{1;2,3} + n\_2 d\dot{j}\_{2;1,3} + n\_3 d\dot{j}\_{3;1,2} \tag{38}$$

If we consider that partial molar properties are function of n1, n<sup>2</sup> and n3, Eq. (38) would be the Gibbs-Duhem equation in the representation of variables n1, n<sup>2</sup> and n3. The representation in the variables x<sup>2</sup> and x<sup>3</sup> is as follows. Dividing (38) by the total number of moles,

$$0 = \mathbf{x}\_1 d\mathbf{j}\_{1;2,3} + \mathbf{x}\_2 d\mathbf{j}\_{2;1,3} + \mathbf{x}\_3 d\mathbf{j}\_{3;1,2} \tag{39}$$

Calculating the differentials by considering that partial molar properties depend on x<sup>2</sup> and x3, and bearing in mind that x<sup>2</sup> and x<sup>3</sup> are independent variables, (38) can be written in an alternative way as:

$$\begin{cases} \mathbf{x}\_{1} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{1;2,3}}{\partial \mathbf{x}\_{2}} \right)\_{\mathbf{x}\_{3}} + \mathbf{x}\_{2} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{2;1,3}}{\partial \mathbf{x}\_{2}} \right)\_{\mathbf{x}\_{3}} + \mathbf{x}\_{3} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{3;1,2}}{\partial \mathbf{x}\_{2}} \right)\_{\mathbf{x}\_{3}} = \mathbf{0} \\\ \mathbf{x}\_{1} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{1;2,3}}{\partial \mathbf{x}\_{3}} \right)\_{\mathbf{x}\_{2}} + \mathbf{x}\_{2} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{2;1,3}}{\partial \mathbf{x}\_{3}} \right)\_{\mathbf{x}\_{2}} + \mathbf{x}\_{3} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{3;1,2}}{\partial \mathbf{x}\_{3}} \right)\_{\mathbf{x}\_{2}} = \mathbf{0} \end{cases} \tag{40}$$

#### 3.2. Description by fractions

dx<sup>1</sup> <sup>d</sup><sup>λ</sup> <sup>¼</sup> dx<sup>2</sup>

books [2, 3]. The partial derivative of f with respect to (λy1) can be expressed as:

∂fðx1,x2,…;λy1,λy2,…Þ ∂ðλy1Þ 

∂fðx1,x2,…;λy1,λy2,…Þ ∂ðλy1Þ 

The differential of f with respect to λ in the right side of (24) is:

of zero degree with respect to the variables y1, y2,…:

3. Thermodynamic descriptions

property, a description by components is:

3.1. Description by components

is defined as:

d½λfðx1, x2, …; y1, y2,…Þ�

, y2,…Þ �fðx1, x2,…; y1, y2,…Þ

<sup>d</sup><sup>λ</sup> <sup>¼</sup> <sup>y</sup>1, <sup>d</sup>ðλy2<sup>Þ</sup>

The following step in this demonstration is different from the step proposed in other text-

Considering that f is a homogeneous function of one degree with respect to the variables y1,y2,

Eq. (25) is obtained by substituting Eqs. (27), (28), (30), (31) in Eq. (26). In addition, it is interesting to see that, defining f<sup>1</sup> as f<sup>1</sup> = (∂f/∂x1) and using (30), f<sup>1</sup> is a homogeneous function

Let it be a three-component system (e.g., as those of Figure 1A and C). Being J an extensive

where n1, n<sup>2</sup> and n<sup>3</sup> are the number of moles of components 1, 2 and 3. The partial property of 1

fðx1, x2,…; λy<sup>1</sup> þ Δ, λy2, …Þ � fðx1, x2,…; λy1, λy2, …Þ

dðλy1Þ

¼ lim<sup>Δ</sup>!<sup>0</sup>

… and making Δ'= Δ/λ in (29),

106 Advances in Titration Techniques

fðx1, x2,…; y<sup>1</sup> þΔ<sup>0</sup>

¼ lim<sup>Δ</sup><sup>0</sup> !0 <sup>d</sup><sup>λ</sup> <sup>¼</sup> … <sup>¼</sup> <sup>0</sup> (27)

x1,x2,…;λy2,λy3,…

x1,x2,…;λy2,λy3,…

<sup>d</sup><sup>λ</sup> <sup>¼</sup> <sup>f</sup>ðx1, x2, …; y1, y2, …<sup>Þ</sup> (31)

J ¼ Jðn1, n2, n3Þ (33)

f <sup>1</sup>ðx1, x2, …; λy1, λy2, …Þ ¼ f <sup>1</sup>ðx1, x2, …; y1, y2, …Þ (32)

<sup>Δ</sup><sup>0</sup> <sup>¼</sup> <sup>∂</sup>fðx1,x2,…;y1,y2,…<sup>Þ</sup>

<sup>d</sup><sup>λ</sup> <sup>¼</sup> <sup>y</sup>2, … (28)

<sup>Δ</sup> (29)

∂y<sup>1</sup> 

x1,x2,…;y2,y3,…

(30)

In a description by fractions, we consider the three-component system as composed of a component 1 and a group (or fraction) composed of components 2 and 3. Figure 1B shows the example when two solutes are grouped in a "complex solute", and Figure 1D shows the example in which a polymeric particle composed of polar and non-polar groups is considered as a fraction of the system. In this case, the extensive property J is expressed as:

$$J = J(n\_1, n\_F, x\_{f^3}) \tag{41}$$

where

$$n\_F = n\_2 + n\_3 \tag{42}$$

$$\mathbf{x}\_{f3} = \frac{n\_3}{n\_2 + n\_3} \tag{43}$$

The variable nF is the total number of moles of the fraction F, and xf3 is a variable related to its internal composition. The partial molar properties of J in this description are:

$$j\_{1; \text{F}}(n\_1, n\_{\text{F}}, \mathbf{x}\_{f3}) = \left(\frac{\text{\eth}l(n\_1, n\_{\text{F}}, \mathbf{x}\_{f3})}{\text{\eth}n\_1}\right)\_{n\_{\text{F}}, \mathbf{x}\_{f3}}\tag{44}$$

$$j\_{\mathbb{F};1}(n\_1, n\_{\mathbb{F}}, \mathbf{x}\_{\mathbb{f}^3}) = \left(\frac{\eth \mathbf{J}(n\_1, n\_{\mathbb{F}}, \mathbf{x}\_{\mathbb{f}^3})}{\eth n\_{\mathbb{F}}}\right)\_{n\_{\mathbb{f}^3}}\tag{45}$$

Because J is a homogeneous function of n<sup>1</sup> and nF, the partial properties j1;<sup>F</sup> and jF;1 will be homogeneous functions of zero degree with respect to the variables n<sup>1</sup> and nF. In this way and similarly to Eq. (35):

$$j\_{1\circ F}(\mathbf{n}\_1, \mathbf{n}\_F, \mathbf{x}\_{f3}) = j\_{1\circ F}(\mathbf{x}\_F, \mathbf{x}\_{f3}) \tag{46}$$

where xF = nF/(n1+nF). Now, we will see the relation between both descriptions. From (42) and (43), the change of variable of Eq. (8) is in this case:

$$\begin{cases} n\_1(n\_1, n\_F, \mathbf{x}\_{f3}) = n\_1 \\ n\_2(n\_1, n\_F, \mathbf{x}\_{f3}) = (1 - \mathbf{x}\_{f3}) n\_F \\ n\_3(n\_1, n\_F, \mathbf{x}\_{f3}) = \mathbf{x}\_{f3} n\_F \end{cases} \tag{47}$$

Substituting (47) in (13) and the result in (17), one obtains that:

$$j\_{1;F} = j\_{1;2,3} \tag{48}$$

$$j\_{\mathbb{F};1} = \mathbf{x}\_{f2} j\_{\mathbb{2};1,3} + \mathbf{x}\_{f3} j\_{\mathbb{3};1,2} \tag{49}$$

$$\left(\frac{\partial \mathbf{j}}{\partial \mathbf{x}\_{f^3}}\right)\_{n\_1, n\_F} = n\_F(j\_{3; 1, 2} - j\_{2; 1, 3})\tag{50}$$

The equations of Gibbs, Euler and Gibbs-Duhem in this description are as follows. The Gibbs equation is obtained by differentiating in (41) and considering the definitions given in (44) and (45): Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques http://dx.doi.org/10.5772/intechopen.69706 109

$$d\mathcal{J} = j\_{1/F}d\mathfrak{n}\_1 + j\_{\mathcal{F};1}d\mathfrak{n}\_{\mathcal{F}} + \left(\frac{\partial \mathcal{J}}{\partial \mathfrak{x}\_{f^3}}\right)\_{\mathfrak{n}\_1,\mathfrak{n}\_{\mathcal{F}}}d\mathfrak{x}\_{f^3} \tag{51}$$

The Euler equation is obtained by remembering that J is a homogeneous function of degree one of n<sup>1</sup> and nF and using the Euler's theorem:

the example when two solutes are grouped in a "complex solute", and Figure 1D shows the example in which a polymeric particle composed of polar and non-polar groups is considered

> xf <sup>3</sup> <sup>¼</sup> <sup>n</sup><sup>3</sup> n<sup>2</sup> þ n<sup>3</sup>

The variable nF is the total number of moles of the fraction F, and xf3 is a variable related to its

∂n<sup>1</sup> � �

∂nF � �

<sup>1</sup>;Fðn1, nF, xf <sup>3</sup>Þ ¼ <sup>∂</sup>Jðn1,nF, xf <sup>3</sup><sup>Þ</sup>

F; <sup>1</sup>ðn1, nF, xf <sup>3</sup>Þ ¼ <sup>∂</sup>Jðn1,nF, xf <sup>3</sup><sup>Þ</sup>

<sup>1</sup>;Fðn1, nF, xf <sup>3</sup>Þ ¼ j

n1ðn1, nF, xf <sup>3</sup>Þ ¼ n<sup>1</sup>

j <sup>1</sup>;F ¼ j

n1,nF

Because J is a homogeneous function of n<sup>1</sup> and nF, the partial properties j1;<sup>F</sup> and jF;1 will be homogeneous functions of zero degree with respect to the variables n<sup>1</sup> and nF. In this way and

where xF = nF/(n1+nF). Now, we will see the relation between both descriptions. From (42) and

n2ðn1, nF, xf <sup>3</sup>Þ¼ð1 � xf <sup>3</sup>ÞnF n3ðn1, nF, xf <sup>3</sup>Þ ¼ xf <sup>3</sup>nF

<sup>2</sup>; <sup>1</sup>,<sup>3</sup> þ xf <sup>3</sup>j

<sup>3</sup>;1,<sup>2</sup> � j

¼ nFðj

The equations of Gibbs, Euler and Gibbs-Duhem in this description are as follows. The Gibbs equation is obtained by differentiating in (41) and considering the definitions given in (44) and (45):

J ¼ Jðn1, nF, xf <sup>3</sup>Þ (41)

nF ¼ n<sup>2</sup> þ n<sup>3</sup> (42)

nF,xf <sup>3</sup>

n1,xf <sup>3</sup>

<sup>1</sup>;FðxF, xf <sup>3</sup>Þ (46)

<sup>1</sup>;2, <sup>3</sup> (48)

<sup>3</sup>; <sup>1</sup>, <sup>2</sup> (49)

<sup>2</sup>;1,3Þ (50)

(43)

(44)

(45)

(47)

as a fraction of the system. In this case, the extensive property J is expressed as:

internal composition. The partial molar properties of J in this description are:

j

j

(43), the change of variable of Eq. (8) is in this case:

j

8 < :

Substituting (47) in (13) and the result in (17), one obtains that:

j F; <sup>1</sup> ¼ xf <sup>2</sup>j

∂J ∂xf <sup>3</sup> � �

where

108 Advances in Titration Techniques

similarly to Eq. (35):

$$J = n\_1 j\_{1; \mathcal{F}} + n\_{\mathcal{F}} j\_{\mathcal{F}; 1} \tag{52}$$

The Gibbs-Duhem equation in the representation of variables xF and xf3 is obtained by differentiating in (52), equalling to (51) and cancelling common terms, and dividing by the total number of moles:

$$\mathbf{x}\_1 dj\_{1; \mathbf{F}} + \mathbf{x}\_{\mathbf{F}} dj\_{\mathbf{F}; 1} = \mathbf{x}\_{\mathbf{F}} (j\_{3; 1, 2} - j\_{2; 1, 3}) d\mathbf{x}\_{f3} \tag{53}$$

Considering that j1;F and jF;<sup>1</sup> are functions of the independent variables xF and xf3, then (53) will take the form:

$$\begin{cases} \mathbf{x}\_{1} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{1;F}}{\partial \mathbf{x}\_{F}} \right)\_{\mathbf{x}\_{\mathcal{I}3}} + \mathbf{x}\_{F} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{F;1}}{\partial \mathbf{x}\_{F}} \right)\_{\mathbf{x}\_{\mathcal{I}3}} = \mathbf{0} \\\ \mathbf{x}\_{1} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{1;F}}{\partial \mathbf{x}\_{f}} \right)\_{\mathbf{x}\_{F}} + \mathbf{x}\_{F} \left( \frac{\partial \dot{\boldsymbol{\eta}}\_{F;1}}{\partial \mathbf{x}\_{f}} \right)\_{\mathbf{x}\_{F}} = \mathbf{x}\_{F} (\dot{\boldsymbol{\eta}}\_{3;1,2} - \dot{\boldsymbol{\eta}}\_{2;1,3}) \end{cases} \tag{54}$$

Calculating the partial derivative of jF;1 with respect to xf<sup>3</sup> in Eq. (49) and substituting in Eq. (54), we obtain:

$$\begin{cases} \mathbf{x}\_{1} \left( \frac{\partial \mathbf{j}\_{1:F}}{\partial \mathbf{x}\_{\mathcal{F}}} \right)\_{\mathbf{x}\_{\mathcal{I}3}} + \mathbf{x}\_{\mathcal{F}} \left( \frac{\partial \mathbf{j}\_{\mathcal{F}1}}{\partial \mathbf{x}\_{\mathcal{F}}} \right)\_{\mathbf{x}\_{\mathcal{I}3}} = \mathbf{0} \\\\ \mathbf{x}\_{1} \left( \frac{\partial \mathbf{j}\_{1:F}}{\partial \mathbf{x}\_{\mathcal{I}3}} \right)\_{\mathbf{x}\_{\mathcal{I}}} + \mathbf{x}\_{\mathcal{F}} \left[ \mathbf{x}\_{\mathcal{I}2} \left( \frac{\partial \mathbf{j}\_{2:1,3}}{\partial \mathbf{x}\_{\mathcal{I}3}} \right)\_{\mathbf{x}\_{\mathcal{I}}} + \mathbf{x}\_{\mathcal{I}3} \left( \frac{\partial \mathbf{j}\_{3:1,2}}{\partial \mathbf{x}\_{\mathcal{I}3}} \right)\_{\mathbf{x}\_{\mathcal{I}}} \right]\_{\mathbf{x}\_{\mathcal{I}}} = \mathbf{0} \end{cases} \tag{55}$$

It is interesting to observe that considering constant composition (dxf<sup>3</sup> = 0) in Eqs. (51)–(53), then the system behaves as a two-component system. This fact cannot be obtained using the description by components.

## 4. Partial properties in diluted solutions of multicomponent systems

We consider intuitively a diluted solution when the properties of the solution are similar to those of its solvent in pure state. In this section, we will study the thermodynamic behaviour of the partial molar properties in this region of concentrations.

#### 4.1. Thermodynamic concept of interaction between components

In this paragraph, we will define the concept of non-interaction and prove that when applying it to a system, the system behaves as an ideal mixing. From a thermodynamic point of view, the components of a system are not interacting if both following points hold simultaneously.


Mathematically, the first point can be written as:

$$\left(\frac{\partial \dot{\mathbf{j}}\_{1;2,3}(\mathbf{x}\_2, \mathbf{x}\_3)}{\partial \mathbf{x}\_2}\right)\_{\mathbf{x}\_3} = \left(\frac{\partial \dot{\mathbf{j}}\_{1;2,3}(\mathbf{x}\_2, \mathbf{x}\_3)}{\partial \mathbf{x}\_3}\right)\_{\mathbf{x}\_2} = \mathbf{0} \Leftrightarrow \dot{\mathbf{j}}\_{1;2,3} = \dot{\mathbf{j}}\_{1;2,3}(\mathbf{x}\_1) \tag{56}$$

Substituting (56) in (40) and considering also that:

$$\begin{cases} \begin{pmatrix} \frac{\partial \dot{\boldsymbol{\eta}}\_{1;2,3}}{\partial \mathbf{x}\_{2}} \end{pmatrix}\_{\boldsymbol{x}\_{3}} = \begin{pmatrix} \frac{\partial \dot{\boldsymbol{\eta}}\_{1;2,3}}{\partial \mathbf{x}\_{1}} \end{pmatrix}\_{\boldsymbol{x}\_{3}} \begin{pmatrix} \frac{\partial \boldsymbol{x}\_{1}}{\partial \mathbf{x}\_{2}} \end{pmatrix}\_{\boldsymbol{x}\_{3}} = -\begin{pmatrix} \frac{\partial \dot{\boldsymbol{\eta}}\_{1;2,3}}{\partial \mathbf{x}\_{1}} \end{pmatrix}\_{\boldsymbol{x}\_{3}}\\\begin{pmatrix} \frac{\partial \dot{\boldsymbol{\eta}}\_{1;2,3}}{\partial \mathbf{x}} \end{pmatrix}\_{\boldsymbol{x}\_{2}} = \begin{pmatrix} \frac{\partial \dot{\boldsymbol{\eta}}\_{1;2,3}}{\partial \mathbf{x}\_{1}} \end{pmatrix}\_{\boldsymbol{x}\_{2}} \begin{pmatrix} \frac{\partial \dot{\boldsymbol{\eta}}\_{1}}{\partial \mathbf{x}\_{1}} \end{pmatrix}\_{\boldsymbol{x}\_{2}} \end{cases} \tag{57}$$

it is obtained that:

$$\begin{cases} -\varkappa\_1 \left( \frac{\partial j\_{1;2,3}}{\partial x\_1} \right)\_{x\_3} + \varkappa\_2 \left( \frac{\partial j\_{2;1,3}}{\partial x\_2} \right)\_{x\_3} = 0\\ -\varkappa\_1 \left( \frac{\partial j\_{1;2,3}}{\partial x\_1} \right)\_{x\_2} + \varkappa\_3 \left( \frac{\partial j\_{2;1,3}}{\partial x\_3} \right)\_{x\_2} = 0 \end{cases} \tag{58}$$

Because j1;2,3 depends only on x1:

$$
\left(\frac{\partial j\_{1;2,3}}{\partial \mathbf{x}\_1}\right)\_{\mathbf{x}\_2} = \left(\frac{\partial j\_{1;2,3}}{\partial \mathbf{x}\_1}\right)\_{\mathbf{x}\_3} \tag{59}
$$

and then (57) yields:

$$\mathbf{x}\_1 \left( \frac{\partial \mathbf{j}\_{1;2,3}(\mathbf{x}\_1)}{\partial \mathbf{x}\_1} \right)\_{\mathbf{x}\_3} = \mathbf{x}\_2 \left( \frac{\partial \mathbf{j}\_{2;1,3}(\mathbf{x}\_2)}{\partial \mathbf{x}\_2} \right)\_{\mathbf{x}\_3} = \mathbf{x}\_3 \left( \frac{\partial \mathbf{j}\_{3;1,2}(\mathbf{x}\_3)}{\partial \mathbf{x}\_3} \right)\_{\mathbf{x}\_3} \tag{60}$$

Because the first term depends only on x<sup>1</sup> and the second and third terms depend only on x<sup>2</sup> and x3, respectively, from (60), we have that:

$$\mathbf{x}\_1 \frac{d\dot{j}\_{1;2,3}}{d\mathbf{x}\_1} = k\_{\parallel}(T, P) \tag{61}$$

where kJ is a function, which only depends on temperature T and pressure P. Similar equations to (61) are obtained for j2;1,3 and j3;1,2. Integrating in (61) between x<sup>0</sup> <sup>1</sup> ¼ 1 and x1,

$$j\_{1;2,3}(\mathbf{x}\_1) = j\_1 + k\_l(T, P)\ln(\mathbf{x}\_1) \tag{62}$$

For components 2 and 3, similar equations to (62) are obtained. Now, we will apply the second point of the above definition of non-interaction. The zero cost of energy for the system and surroundings is equivalent to:

$$Q\_{\rm mix} = W\_{\rm mix} = 0 \Rightarrow \Delta U\_{\rm mix} = 0 \Rightarrow \begin{cases} u\_{1;2,3} = u\_1 \\ h\_{1;2,3} = h\_1 \\ v\_{1;2,3} = v\_1 \end{cases} \tag{63}$$

Considering u1; 2,3, h1; 2,3 and v1; 2,3 as (62) and bearing in mind (63):

$$k\_{\mathcal{U}}(T, P) = k\_{\mathcal{H}}(T, P) = k\_{\mathcal{V}}(T, P) = 0 \tag{64}$$

In addition to this g1;2,3 (free energy of Gibbs),

$$\left(\sigma\_{1;2,3} = \left(\frac{\partial g\_{1;2,3}}{\partial P}\right)\_{T,x\_2,x\_3} \Rightarrow k\sqrt{T,P}\right)\_T = \left(\frac{\partial k\_G(T,P)}{\partial P}\right)\_T \tag{65}$$

$$\frac{h\_{1;2,3}}{T^2} = -\left(\frac{\partial}{\partial T}\left(\frac{\mathbf{g\_{1;2,3}}}{T}\right)\right)\_{P,x\_2,x\_3} \Rightarrow \frac{k\_H(T,P)}{T^2} = -\left(\frac{\partial}{\partial T}\left(\frac{k\_G(T,P)}{T}\right)\right)\_T \tag{66}$$

Combining Eqs. (64)–(66), we have that:

$$k\_G(T, P) = kT \tag{67}$$

where k is a constant. For the entropy, one gets:

$$-s\_{1;2,3} = \left(\frac{\partial \mathbf{g}\_{1;2,3}}{\partial T}\right)\_{P, x\_2, x\_3} \Rightarrow -k\_S(T, P) = \left(\frac{\partial k\_G(T, P)}{\partial T}\right)\_P \tag{68}$$

With this,

1. The state of each component in the system, expressed in terms of its partial molar properties, does not vary by changes of composition of the other components. It means each

2. The formation of the system from its pure components is carried out with any cost of

<sup>1</sup>; <sup>2</sup>, <sup>3</sup>ðx2, x3Þ ∂x<sup>3</sup> � �

> x3 ∂x<sup>1</sup> ∂x<sup>2</sup> � � x3

> x2 ∂x<sup>1</sup> ∂x<sup>3</sup> � � x2

x3 þ x<sup>2</sup> ∂j <sup>2</sup>; <sup>1</sup>,<sup>3</sup> ∂x<sup>2</sup> � �

x2 þ x<sup>3</sup> ∂j <sup>2</sup>; <sup>1</sup>,<sup>3</sup> ∂x<sup>3</sup> � �

x2

∂j <sup>2</sup>; <sup>1</sup>, <sup>3</sup>ðx2Þ ∂x<sup>2</sup> � �

Because the first term depends only on x<sup>1</sup> and the second and third terms depend only on x<sup>2</sup>

where kJ is a function, which only depends on temperature T and pressure P. Similar equations

<sup>¼</sup> <sup>∂</sup><sup>j</sup> 1;2,3 ∂x<sup>1</sup> � �

x2

¼ 0 ⇔ j

¼ � <sup>∂</sup><sup>j</sup>

¼ � <sup>∂</sup><sup>j</sup>

x3 ¼ 0

x2 ¼ 0

x3

∂j <sup>3</sup>;1,2ðx3Þ ∂x<sup>3</sup> � �

x3

¼ kJðT,PÞ (61)

<sup>1</sup> þ kJðT,PÞlnðx1Þ (62)

<sup>1</sup> ¼ 1 and x1,

x3 ¼ x<sup>3</sup> <sup>1</sup>; <sup>2</sup>, <sup>3</sup> ¼ j

<sup>1</sup>;2, <sup>3</sup> ∂x<sup>1</sup> � �

<sup>1</sup>;2, <sup>3</sup> ∂x<sup>1</sup> � �

x3

x2

<sup>1</sup>;2, <sup>3</sup>ðx1Þ (56)

(57)

(58)

(59)

(60)

component does not detect the presence of the other components.

<sup>¼</sup> <sup>∂</sup><sup>j</sup>

energy, neither for the system nor for the surroundings.

x3

x3 <sup>¼</sup> <sup>∂</sup><sup>j</sup> <sup>1</sup>; <sup>2</sup>, <sup>3</sup> ∂x<sup>1</sup> � �

x2 <sup>¼</sup> <sup>∂</sup><sup>j</sup> <sup>1</sup>; <sup>2</sup>, <sup>3</sup> ∂x<sup>1</sup> � �

�x<sup>1</sup> ∂j <sup>1</sup>; <sup>2</sup>, <sup>3</sup> ∂x<sup>1</sup> � �

8 ><

>:

�x<sup>1</sup> ∂j <sup>1</sup>; <sup>2</sup>, <sup>3</sup> ∂x<sup>1</sup> � �

> x3 ¼ x<sup>2</sup>

to (61) are obtained for j2;1,3 and j3;1,2. Integrating in (61) between x<sup>0</sup>

j

x1 dj1; <sup>2</sup>, <sup>3</sup> dx<sup>1</sup>

<sup>1</sup>;2, <sup>3</sup>ðx1Þ ¼ j

∂j 1;2,3 ∂x<sup>1</sup> � �

∂j <sup>1</sup>; <sup>2</sup>,<sup>3</sup> ∂x<sup>2</sup> � �

∂j <sup>1</sup>; <sup>2</sup>,<sup>3</sup> ∂x<sup>3</sup> � �

Mathematically, the first point can be written as:

<sup>1</sup>;2,3ðx2, x3Þ ∂x<sup>2</sup> � �

Substituting (56) in (40) and considering also that:

8 ><

>:

∂j

it is obtained that:

110 Advances in Titration Techniques

and then (57) yields:

Because j1;2,3 depends only on x1:

x1 ∂j <sup>1</sup>; <sup>2</sup>,3ðx1Þ ∂x<sup>1</sup> � �

and x3, respectively, from (60), we have that:

$$\mathbf{g}\_{1;2,3} = \mathbf{g}\_1 + kT \ln(\mathbf{x}\_1) \tag{69}$$

$$\mathbf{s}\_{1;2,3} = \mathbf{s}\_1 - k \ln(\mathbf{x}\_1) \tag{70}$$

and we have demonstrated that a system holding the non-interaction definition proposed is an ideal mixing.

#### 4.2. Diluted solutions

In this section, we will define the thermodynamic concept of diluted solutions and study the behaviour of the partial molar properties of these solutions. Commonly and intuitively, we consider a solution as diluted when its properties are similar to those of the pure solvent. We can implement mathematically this concept in the following way. When we remove all solutes from a solution, we have that:

$$\lim\_{\mathbf{x}\_2 + \mathbf{x}\_3 \to 0} j(\mathbf{x}\_2, \mathbf{x}\_3) = j\_1 \tag{71}$$

where j is the molar property of the extensive thermodynamic property J. In addition, the partial derivatives must vanish:

$$\lim\_{\mathbf{x}\downarrow \mathbf{x}\_{2}\rightarrow\mathbf{x}\_{3}\rightarrow 0} \left(\frac{\partial \mathbf{j}\_{1;2,3}(\mathbf{x}\_{2},\mathbf{x}\_{3})}{\partial \mathbf{x}\_{2}}\right)\_{\mathbf{x}\_{3}} = \lim\_{\mathbf{x}\_{3}\rightarrow\mathbf{x}\_{3}\rightarrow 0} \left(\frac{\partial \mathbf{j}\_{1;2,3}(\mathbf{x}\_{2},\mathbf{x}\_{3})}{\partial \mathbf{x}\_{3}}\right)\_{\mathbf{x}\_{2}} = \mathbf{0} \tag{72}$$

Otherwise, we would have memory effects and we can see this with an example. If we purify water, the pure substance obtained does not depend on the initial diluted solution employed. Actually, pure water is commonly used as a standard because it does not depend on the part of world, in which it is obtained. The Taylor's expansion of j1;2,3 is:

$$\mathbf{j}\_{1;2,3}(\mathbf{x}\_2, \mathbf{x}\_3) = j\_1(0, 0) + \left[\nabla j\_{1;2,3}(0, 0)\right]^\mathrm{T} \begin{bmatrix} \mathbf{x}\_2\\ \mathbf{x}\_3 \end{bmatrix} + \frac{1}{2}(\mathbf{x}\_2, \mathbf{x}\_3) \mathbf{H} j\_{1;2,3}(0, 0) \begin{bmatrix} \mathbf{x}\_2\\ \mathbf{x}\_3 \end{bmatrix} + \dots \tag{73}$$

where ∇j1;2,3(0,0) and Hj1;2,3(0,0) are, respectively, the vector gradient and the Hessian of j1;2,3 matrix at (0,0). Considering (71) and (72) in (73) and that all partial derivatives mush vanish at (0,0), we have that for diluted solutions:

$$j\_{1;2,3}(\mathbf{x}\_2, \mathbf{x}\_3) \approx j\_1 + \dots \tag{74}$$

From Eq. (74), we have for diluted solutions:

$$\left(\frac{\partial \mathbf{j}\_{1;2,3}}{\partial \mathbf{x}\_2}\right)\_{\mathbf{x}\_3} \approx \left(\frac{\partial \mathbf{j}\_{1;2,3}}{\partial \mathbf{x}\_3}\right)\_{\mathbf{x}\_2} \approx \mathbf{0} \tag{75}$$

The behaviour of molar partial properties of solutes is as follows. Considering a "complex solute" S composed of 2 and 3 (as in Figure 1B),

$$j\_{1;S}(\mathbf{x}\_{\mathcal{S}}, \mathbf{x}\_{\mathcal{S}}) \approx j\_1 \tag{76}$$

and substituting Eq. (74) in the first equation of (55),

$$\left(\frac{\partial \mathbf{j}\_{\mathbf{S};1}}{\partial \mathbf{x}\_{\mathbf{S}}}\right)\_{\mathbf{x}\_{\mathbf{S}}} \simeq \mathbf{0} \Leftrightarrow \mathbf{j}\_{\mathbf{S};1}(\mathbf{x}\_{\mathbf{S}}, \mathbf{x}\_{\mathbf{S}3}) \approx \mathbf{j}\_{\mathbf{S};1}(\mathbf{x}\_{\mathbf{S}3}) \tag{77}$$

Inserting Eq. (76) in the second equation of (55), it is obtained that:

$$\left(\mathbf{x}\_{s2}\left(\frac{\partial \mathbf{j}\_{2\cdot 1,3}}{\partial \mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} + \mathbf{x}\_{s3}\left(\frac{\partial \mathbf{j}\_{3\cdot 1,2}}{\partial \mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} \approx \mathbf{0} \tag{78}$$

Until now, we have seen the effect of the dilution in the capacity of detecting the presence of other components in a diluted solution. In order to gain an insight into the interactions, we have to study the process of mixing in diluted solutions. From (71), we can write:

$$\begin{cases} \lim\_{\mathbf{x}\_2 + \mathbf{x}\_3 \to 0} h = h\_1 \Rightarrow \lim\_{\mathbf{x}\_2 + \mathbf{x}\_3 \to 0} q\_{\min} = \lim\_{\mathbf{x}\_2 + \mathbf{x}\_3 \to 0} \Delta\_{\min} h = 0\\ \lim\_{\mathbf{x}\_2 + \mathbf{x}\_3 \to 0} \boldsymbol{\upsilon} = \boldsymbol{\upsilon}\_1 \Rightarrow \lim\_{\mathbf{x}\_2 + \mathbf{x}\_3 \to 0} \boldsymbol{\upsilon}\_{\min} = \lim\_{\mathbf{x}\_2 + \mathbf{x}\_3 \to 0} -P\Delta\_{\min} \boldsymbol{\upsilon} = 0 \end{cases} \tag{79}$$

It indicates that in the limit of infinite dilution, components do not interact because the process of mixture does not have any energy cost. This result implicates that in diluted solutions, according to the asymptotic approach given by Eq. (74), the interaction between solvent and solutes is weak and it can be neglected.

#### 4.3. Partial molar properties of interaction in diluted solutions

The molar property j of a diluted solution can be written as:

$$j \approx \mathbf{x}\_1 \mathbf{j}\_1 + \mathbf{x}\_S \mathbf{j}\_{S;1} \tag{80}$$

where we are considering the interaction between components 2 and 3 since

$$j\_{S;1} = \mathbf{x}\_{s2} j\_{2;1,3} + \mathbf{x}\_{s3} j\_{3;1,2} \tag{81}$$

In a diluted solution without interaction between 2 and 3, the property j <sup>Ø</sup> can be written as:

$$\mathbf{j}^{\mathcal{D}} = \mathbf{x}\_1 \mathbf{j}\_1 + \mathbf{x}\_{\mathcal{S}} (\mathbf{x}\_{\mathcal{S}2} \mathbf{j}\_{2 \cdot 1} + \mathbf{x}\_{\mathcal{S}3} \mathbf{j}\_{3 \cdot 1}) \tag{82}$$

In this way, we can calculate the interaction contributions to j as:

$$
\Delta \mathbf{j}^{\text{int}} = j - \mathbf{j}^{\text{\mathcal{D}}} \approx \mathbf{x}\_{\text{\mathcal{S}}} \Delta \mathbf{j}\_{\text{\mathcal{S}};1} \tag{83}
$$

where

where j is the molar property of the extensive thermodynamic property J. In addition, the

Otherwise, we would have memory effects and we can see this with an example. If we purify water, the pure substance obtained does not depend on the initial diluted solution employed. Actually, pure water is commonly used as a standard because it does not depend on the part of

> x3

where ∇j1;2,3(0,0) and Hj1;2,3(0,0) are, respectively, the vector gradient and the Hessian of j1;2,3 matrix at (0,0). Considering (71) and (72) in (73) and that all partial derivatives mush vanish at

<sup>1</sup>; <sup>2</sup>, <sup>3</sup>ðx2, x3Þ ≈ j

x3

j

≈ 0 ⇔ j

xS þ xs<sup>3</sup>

have to study the process of mixing in diluted solutions. From (71), we can write:

Until now, we have seen the effect of the dilution in the capacity of detecting the presence of other components in a diluted solution. In order to gain an insight into the interactions, we

> lim<sup>x</sup>2þx3!<sup>0</sup> h ¼ h<sup>1</sup> ) lim<sup>x</sup>2þx3!<sup>0</sup> qmix ¼ lim<sup>x</sup>2þx3!<sup>0</sup> Δmixh ¼ 0 lim<sup>x</sup>2þx3!<sup>0</sup> v ¼ v<sup>1</sup> ) lim<sup>x</sup>2þx3!<sup>0</sup> wmix ¼ lim<sup>x</sup>2þx3!<sup>0</sup> � PΔmixv ¼ 0

xs<sup>3</sup>

<sup>≈</sup> <sup>∂</sup><sup>j</sup> 1; 2, 3 ∂x<sup>3</sup> 

The behaviour of molar partial properties of solutes is as follows. Considering a "complex

<sup>1</sup>;SðxS, xs3Þ ≈ j

S;1ðxS, xs3Þ ≈ j

∂j 3; 1, 2 ∂xs<sup>3</sup> 

xS

x2

þ 1 2

¼ lim<sup>x</sup>2þx3!<sup>0</sup>

∂j

ðx2, x3ÞHj

<sup>1</sup>; <sup>2</sup>, <sup>3</sup>ðx2,x3Þ ∂x<sup>3</sup> 

x2

<sup>1</sup>; <sup>2</sup>,3ð0, <sup>0</sup><sup>Þ</sup> <sup>x</sup><sup>2</sup>

<sup>1</sup> þ … (74)

≈ 0 (75)

<sup>1</sup> (76)

S;1ðxs3Þ (77)

≈ 0 (78)

(79)

x3 

¼ 0 (72)

þ … (73)

x3

<sup>1</sup>;2,3ð0, <sup>0</sup>Þ�<sup>T</sup> <sup>x</sup><sup>2</sup>

j

∂j 1; 2, 3 ∂x<sup>2</sup> 

partial derivatives must vanish:

112 Advances in Titration Techniques

j

<sup>1</sup>;2,3ðx2, x3Þ ¼ j

(0,0), we have that for diluted solutions:

From Eq. (74), we have for diluted solutions:

solute" S composed of 2 and 3 (as in Figure 1B),

and substituting Eq. (74) in the first equation of (55),

∂j S; 1 ∂xS 

Inserting Eq. (76) in the second equation of (55), it is obtained that:

xs<sup>2</sup> ∂j 2; 1,3 ∂xs<sup>3</sup> 

lim<sup>x</sup>2þx3!<sup>0</sup>

∂j

world, in which it is obtained. The Taylor's expansion of j1;2,3 is:

<sup>1</sup>ð0, 0Þþ½∇j

<sup>1</sup>;2,3ðx2, x3Þ ∂x<sup>2</sup> 

$$
\Delta \mathbf{j}\_{\mathbf{S};1} = \mathbf{x}\_{\mathbf{s}2} \Delta \mathbf{j}\_{\mathbf{2};1,3} + \mathbf{x}\_{\mathbf{s}3} \Delta \mathbf{j}\_{\mathbf{3};1,2} \tag{84}
$$

is the partial molar property of interaction of the complex solute and

$$\begin{cases} \Delta \dot{j}\_{2;1,3} = \dot{j}\_{2;1,3} - \dot{j}\_{2;1} \\ \Delta \dot{j}\_{3;1,2} = \dot{j}\_{3;1,2} - \dot{j}\_{3;1} \end{cases} \tag{85}$$

are the partial molar properties of interaction of the components 2 and 3, respectively. These properties are not independent as we will see as follows. Combining (78) and (85),

$$\left(\mathbf{x}\_{s2}\left(\frac{\partial\Delta\mathbf{j}\_{2:1,3}}{\partial\mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} + \mathbf{x}\_{s3}\left(\frac{\partial\Delta\mathbf{j}\_{3:1,2}}{\partial\mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} + \mathbf{x}\_{s2}\left(\frac{\partial\mathbf{j}\_{2:1}}{\partial\mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} + \mathbf{x}\_{s3}\left(\frac{\partial\mathbf{j}\_{3:1}}{\partial\mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} \approx 0\tag{86}$$

In Eq. (85), Δj2;1,3 and Δj3;1,2 are evaluated when using concentrations xS and xs3. Accordingly, j2,1 is evaluated using the concentration x<sup>2</sup> given by x<sup>2</sup> = xS (1�xs3), and then,

$$
\left(\frac{\partial \dot{\jmath}\_{2\cdot 1}}{\partial \mathbf{x}\_{\circ 3}}\right)\_{\mathbf{x}\_{\circ 3}} = \frac{d \dot{\jmath}\_{2\cdot 1}}{d \mathbf{x}\_2} \left(\frac{\partial \mathbf{x}\_2}{\partial \mathbf{x}\_{\circ 3}}\right)\_{\mathbf{x}\_{\circ 3}} = -\mathbf{x}\_{\circ} \frac{d \dot{\jmath}\_{2\cdot 1}}{d \mathbf{x}\_2} \tag{87}
$$

Considering the Gibbs-Duhem equation for a two-component system:

$$\mathbf{x}\_1 \frac{d\dot{\mathbf{j}}\_{1;2}}{d\mathbf{x}\_2} + \mathbf{x}\_2 \frac{d\dot{\mathbf{j}}\_{2;1}}{d\mathbf{x}\_2} = \mathbf{0} \tag{88}$$

in Eq. (87) and bearing in mind that solutions are diluted,

$$\mathbf{x}\_{s2} \left(\frac{\partial \dot{j}\_{2\cdot 1}}{\partial \mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} = \mathbf{x}\_1 \frac{d \dot{j}\_{1\cdot 2}}{d \mathbf{x}\_2} \approx \mathbf{0} \tag{89}$$

Similarly for component 3,

$$\left(\mathbf{x}\_{\rm s3}\left(\frac{\partial j\_{3;1}}{\partial \mathbf{x}\_{\rm s3}}\right)\_{\mathbf{x}\_{\rm s3}} = \mathbf{x}\_1 \frac{dj\_{1;3}}{d\mathbf{x}\_3} \approx \mathbf{0} \tag{90}$$

Substituting (89) and (90) in (86), we obtain:

$$\left(\mathbf{x}\_{s2}\left(\frac{\partial \Delta \dot{j}\_{2;1,3}}{\partial \mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} + \mathbf{x}\_{s3}\left(\frac{\partial \Delta \dot{j}\_{3;1,2}}{\partial \mathbf{x}\_{s3}}\right)\_{\mathbf{x}\_{\mathcal{S}}} \approx \mathbf{0} \tag{91}$$

Eq. (91) indicates that in a diluted solution, the interaction between components 2 and 3 is not vanished. The partial molar property of interaction of the complex solute can be calculated experimentally as:

$$\Delta \mathbf{j}\_{\mathbf{S};1} \approx \mathbf{j}\_{\mathbf{S};1} - (\mathbf{x}\_{\mathbf{S}} \mathbf{j}\_{\mathbf{2};1} + \mathbf{x}\_{\mathbf{S}} \mathbf{j}\_{\mathbf{3};1}) \tag{92}$$

and the partial properties of interaction of components 2 and 3 can be obtained from (92) using the equations:

$$\begin{cases} \Delta \dot{j}\_{2;1,3} \approx \Delta \dot{j}\_{S;1} - \chi\_{\rm s3} \frac{d \Delta \dot{j}\_{S;1}}{d \mathbf{x}\_{\rm s3}}\\ \Delta \dot{j}\_{3;1,2} \approx \Delta \dot{j}\_{S;1} + (1 - \chi\_{\rm s3}) \frac{d \Delta \dot{j}\_{S;1}}{d \mathbf{x}\_{\rm s3}} \end{cases} \tag{93}$$

Eq. (93) is obtained by differentiating in Eq. (92) with respect to xs3, using Eq. (91) and combining the result with Eq. (92). As we will see below, Eq. (93) will allow us to obtain the interaction partial properties of 2 and 3 from experimental data.

#### 4.4. Experimental determination of partial molar properties of interaction in diluted solutions

### 4.4.1. Partial specific volumes of interaction and partial specific adiabatic compressibility of interaction

As an example, we will consider the interaction between functionalized polymeric particles and an electrolyte at 30�C [4]. For that, polymeric particles synthesized of poly(n-butyl acrylate-co-methyl methacrylate) functionalized with different concentrations of acrylic acid were used in this study. The electrolyte was NaOH. Similarly to Figure 1A, water (solvent) was considered as component 1, polymeric particles as component 2 and electrolyte as component 3. And similarly to Figure 1B, the system was fractionalized in component 1 and a complex solute composed of polymeric particles and electrolyte. The experimental measurements were carried out using a Density and Sound Analyzer DSA 5000 from Anton-Paar connected to a titration cell. It is of full cell type, which is usually employed in isothermal titration calorimetry. Polymeric particles were located in the titration cell, and electrolyte was located in the syringe. Concentrations of polymeric particles (c2) and electrolyte (c3) after each titration were calculated as [4, 5]:

x1 dj1;<sup>2</sup> dx<sup>2</sup>

xs<sup>2</sup> ∂j 2;1 ∂xs<sup>3</sup> � �

xs<sup>3</sup> ∂j 3;1 ∂xs<sup>3</sup> � �

∂Δj 2; 1,3 ∂xs<sup>3</sup> � �

Δj S;<sup>1</sup> ≈ j

Δj

8 >><

>>:

partial properties of 2 and 3 from experimental data.

Δj

<sup>2</sup>;1,<sup>3</sup> ≈Δj

<sup>3</sup>;1,<sup>2</sup> ≈Δj

xs<sup>2</sup>

in Eq. (87) and bearing in mind that solutions are diluted,

Similarly for component 3,

114 Advances in Titration Techniques

experimentally as:

the equations:

solutions

of interaction

Substituting (89) and (90) in (86), we obtain:

þ x<sup>2</sup>

xS ¼ x<sup>1</sup>

xS ¼ x<sup>1</sup>

xS þ xs<sup>3</sup>

Eq. (91) indicates that in a diluted solution, the interaction between components 2 and 3 is not vanished. The partial molar property of interaction of the complex solute can be calculated

S; <sup>1</sup> � ðxs2j

and the partial properties of interaction of components 2 and 3 can be obtained from (92) using

S;<sup>1</sup> � xs<sup>3</sup>

Eq. (93) is obtained by differentiating in Eq. (92) with respect to xs3, using Eq. (91) and combining the result with Eq. (92). As we will see below, Eq. (93) will allow us to obtain the interaction

As an example, we will consider the interaction between functionalized polymeric particles and an electrolyte at 30�C [4]. For that, polymeric particles synthesized of poly(n-butyl acrylate-co-methyl methacrylate) functionalized with different concentrations of acrylic acid were used in this study. The electrolyte was NaOH. Similarly to Figure 1A, water (solvent) was considered as component 1, polymeric particles as component 2 and electrolyte as component

4.4. Experimental determination of partial molar properties of interaction in diluted

4.4.1. Partial specific volumes of interaction and partial specific adiabatic compressibility

S;<sup>1</sup> þ ð1 � xs3Þ

dj2;<sup>1</sup> dx<sup>2</sup>

> dj1; <sup>2</sup> dx<sup>2</sup>

> dj1; <sup>3</sup> dx<sup>3</sup>

∂Δj 3; 1, 2 ∂xs<sup>3</sup> � �

<sup>2</sup>; <sup>1</sup> þ xs3j

dΔj S; 1 dxs<sup>3</sup>

dΔj S;1 dxs<sup>3</sup>

xS

¼ 0 (88)

≈ 0 (89)

≈ 0 (90)

≈ 0 (91)

<sup>3</sup>; <sup>1</sup>Þ (92)

(93)

$$\begin{cases} c\_2^{i+1} = c\_2^i \ e^{-\frac{v}{\nabla}}\\ c\_3^{i+1} = c\_3^s - (c\_3^s - c\_3^i) \ e^{-\frac{v}{\nabla}} \end{cases} \tag{94}$$

where V is the effective volume of the titration cell, v is the titration volume and cs <sup>3</sup> is the stock concentration of electrolyte in the syringe. Figure 3A and B shows, respectively, data of density (r) and sound speed (u) as function of the electrolyte concentration. The specific volume (v) and the specific adiabatic compressibility (ks) were calculated as:

$$v = \frac{1}{\rho} \tag{95}$$

$$k\_s = \left(\frac{10}{\rho u}\right)^2\tag{96}$$

Considering the solution in the cell as diluted, the partial specific volume (and similarly the partial specific adiabatic compressibility) of the complex solute can be calculated as:

$$v\_{\rm S;1} = \frac{v - t\_1 v\_1}{t\_{\rm S}} \tag{97}$$

where t1 and tS are the mass fraction of the water and of the complex solute, respectively. Figure 3C and D shows the partial specific volume and partial specific adiabatic compressibility as function of tf3 (mass fraction of the electrolyte in the complex solute). The term of interaction ΔvS;1 is calculated by Eq. (92), where v2;1 is obtained by considering that:

$$
\upsilon\_{2^\circ 1} = \lim\_{t\_{33} \to 0} \upsilon\_{3^\circ 1} \tag{98}
$$

in Figure 3C. The term v3;1 is calculated by extrapolating the linear part of vS;1 in Figure 3C as:

$$
\sigma\_{3;1} = \lim\_{t\_{\mathbb{S}^1} \to 1} \sigma\_{\mathbb{S};1} \tag{99}
$$

The partial specific volume of interaction of the polymeric particles (Δv2;1,3) and the partial specific volume of interaction of the electrolyte (Δv3;1,2) were obtained using Eq. (93). The numerical method employed to calculate the derivatives is shown elsewhere [4]. Figure 4 shows the values of ΔvS;1, Δv2;1,3 and Δv3;1,2, and Figure 5 shows the values of Δks S;1, Δks 2;1,3 and Δks 3;1,2 obtained in a similar way than for volumes.

Partial volume of polymeric particles (v2;1) can be broken down in the following contributions [6–11]:

$$
\upsilon\_{2;1} = \upsilon\_{2;1/atm} + \upsilon\_{2;1/free} + \upsilon\_{2;1/hyd} \tag{100}
$$

which are shown in Figure 6. The atomic volume contribution (v2;1/atom) is the sum of all volumes of the atoms, which make up polymeric chains. The free volume contribution (v2;1/free) is consequence of the imperfect packing of the polymeric chains. The atomic volume contribution and free volume contribution are both positive contributions. The hydration contribution (v2;1/hyd) is negative, as a consequence of that the specific volume of water molecules in bulk is larger than the specific volume in the hydration shell. The contributions to the partial specific adiabatic compressibility are the free volume and hydration because the effect of the pressure on the atomic volume is neglected [10, 12–21]:

$$k\_{T\cdot 2;1} = -\left(\frac{\partial v\_{2;1}}{\partial P}\right)\_T = -\left(\frac{\partial v\_{2;1/f\text{ref}}}{\partial P}\right)\_T - \left(\frac{\partial v\_{2;1/\text{hyd}}}{\partial P}\right)\_T = k\_{T\cdot 2;1/f\text{ref}} + k\_{T\cdot 2;1/\text{hyd}}\tag{101}$$

The contribution kT 2;1/free is positive, and the contribution kT 2;1/hyd is negative [4, 8]. In this chapter, we will take the adiabatic compressibility as an approximation of the isothermal

Figure 3. (A) Density as function of the electrolyte concentration c<sup>3</sup> (g/L). (B) Sound speed as function of the electrolyte concentration. (C) Partial specific volume of the complex solute composed of polymeric particles and electrolyte as function of the mass fraction of the electrolyte in the complex solute (ts3). (D) Partial specific adiabatic compressibility of the complex solute as function of tf3.

Figure 4. (A) Partial specific volume of interaction of the complex solute (polymeric particles + electrolyte) as function of the mass fraction of the electrolyte in the complex solute (ts3). (B) Partial specific volume of interaction of the polymeric particles as function of ts3. (C) Partial specific volume of interaction of the electrolyte as function of ts3.

Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques http://dx.doi.org/10.5772/intechopen.69706 117

Figure 5. (A) Partial specific adiabatic compressibility of interaction of the complex solute (polymeric particles + electrolyte) as function of the mass fraction of the electrolyte in the complex solute (ts3). (B) Partial specific adiabatic compressibility of interaction of the polymeric particles as function of ts3. (C) Partial specific adiabatic compressibility of interaction of the electrolyte as function of ts3.

Figure 6. Contributions to the partial volume in a polymeric particle.

v2;<sup>1</sup> ¼ v2; <sup>1</sup>=atom þ v2;1=f ree þ v2;1=hyd (100)

which are shown in Figure 6. The atomic volume contribution (v2;1/atom) is the sum of all volumes of the atoms, which make up polymeric chains. The free volume contribution (v2;1/free) is consequence of the imperfect packing of the polymeric chains. The atomic volume contribution and free volume contribution are both positive contributions. The hydration contribution (v2;1/hyd) is negative, as a consequence of that the specific volume of water molecules in bulk is larger than the specific volume in the hydration shell. The contributions to the partial specific adiabatic compressibility are the free volume and hydration because the effect of the pressure on the

T

The contribution kT 2;1/free is positive, and the contribution kT 2;1/hyd is negative [4, 8]. In this chapter, we will take the adiabatic compressibility as an approximation of the isothermal

Figure 3. (A) Density as function of the electrolyte concentration c<sup>3</sup> (g/L). (B) Sound speed as function of the electrolyte concentration. (C) Partial specific volume of the complex solute composed of polymeric particles and electrolyte as function of the mass fraction of the electrolyte in the complex solute (ts3). (D) Partial specific adiabatic compressibility of

0.0 0.1 0.2 0.3

t s3

Figure 4. (A) Partial specific volume of interaction of the complex solute (polymeric particles + electrolyte) as function of the mass fraction of the electrolyte in the complex solute (ts3). (B) Partial specific volume of interaction of the polymeric

particles as function of ts3. (C) Partial specific volume of interaction of the electrolyte as function of ts3.

*v*2;1,3 (10

Δ


**B**

� <sup>∂</sup>v2;1=hyd ∂P 

T

¼ kT <sup>2</sup>;1=f ree þ kT <sup>2</sup>; <sup>1</sup>=hyd (101)

**C**

0.0 0.1 0.2 0.3

t s3

0.0

0.1

*v*3;1,2 (cm 3/g)

Δ

0.2

0.3

atomic volume is neglected [10, 12–21]:

116 Advances in Titration Techniques

kT 2;1 ¼ � <sup>∂</sup>v2; <sup>1</sup>

0.0 0.1 0.2 0.3

t s3

0

4

*v*S;1 (10

Δ


8

12

**A**

the complex solute as function of tf3.

∂P 

T

¼ � <sup>∂</sup>v2; <sup>1</sup>=f ree ∂P 

> compressibility. For the electrolyte, the free volume contribution is null, and then, v3;1 and kT 3;1 will take the following form:

$$
\boldsymbol{\sigma}\_{\text{3;1}} = \boldsymbol{\sigma}\_{\text{3;1/atom}} + \boldsymbol{\sigma}\_{\text{3;1/hyd}} \tag{102}
$$

$$k\_{T3;1} = -\left(\frac{\partial v\_{3;1}}{\partial P}\right)\_T = -\left(\frac{\partial v\_{3;1/hyd}}{\partial P}\right)\_T = k\_{T3;1/hyd} \tag{103}$$

For the complex solute, we can write a similar breakdown:

$$
\sigma\_{\rm S;1} = \sigma\_{\rm S;1/atom} + \sigma\_{\rm S;1/free} + \sigma\_{\rm S;1/hyd} \tag{104}
$$

Inserting Eqs. (100), (102) and (104) in Eq. (92) and neglecting the variation in the atomic contributions, the following equation for the interaction specific partial volume is obtained:

$$
\Delta \mathbf{v}\_{\rm S,1} = \Delta \mathbf{v}\_{\rm S,1/free} + \Delta \mathbf{v}\_{\rm S,1/hyd} \tag{105}
$$

where

$$
\Delta v\_{\text{S};1/free} = v\_{\text{S};1/free} - t\_{\text{s2}} v\_{\text{2};1/free} \tag{106}
$$

$$
\Delta \mathbf{v}\_{\mathbf{S};1/\text{hyd}} = \mathbf{v}\_{\mathbf{S};1/\text{hyd}} - \left( t\_{\mathbf{s2}\mathbf{\mathcal{D}};1/\text{hyd}} + t\_{\mathbf{s3}\mathbf{\mathcal{D}}\_{\mathbf{S};1/\text{hyd}}} \right) \tag{107}
$$

Substituting (105) in (93), we get

$$\begin{cases} \Delta \mathbf{v}\_{2;1,3} = \left(\Delta \mathbf{v}\_{\mathbf{S};1/f\text{re}} - t\_{\text{s3}} \frac{d \Delta \mathbf{v}\_{\mathbf{S};1/f\text{re}}}{dt\_{\text{s3}}}\right) + \left(\Delta \mathbf{v}\_{\mathbf{S};1/\text{hyd}} - t\_{\text{s3}} \frac{d \Delta \mathbf{v}\_{\mathbf{S};1/\text{hyd}}}{dt\_{\text{s3}}}\right) \\\\ \Delta \mathbf{v}\_{3;1,2} = \left(\Delta \mathbf{v}\_{\mathbf{S};1/f\text{re}} + (1 - t\_{\text{s3}}) \frac{d \Delta \mathbf{v}\_{\mathbf{S};1/f\text{re}}}{dt\_{\text{s3}}}\right) + \left(\Delta \mathbf{v}\_{\mathbf{S};1/\text{hyd}} + (1 - t\_{\text{s3}}) \frac{d \Delta \mathbf{v}\_{\mathbf{S};1/\text{hyd}}}{dt\_{\text{s3}}}\right) \end{cases} \tag{108}$$

Defining now:

$$\begin{split} \Delta v\_{2;1,3/free} &= \Delta v\_{S;1/free} - t\_{s3} \frac{d \Delta v\_{S;1/free}}{dt\_{s3}} \\ \Delta v\_{2;1,3/hyd} &= \Delta v\_{S;1/hyd} - t\_{s3} \frac{d \Delta v\_{S;1/hyd}}{dt\_{s3}} \\ \Delta v\_{3;1,2/free} &= \Delta v\_{S;1/free} + (1 - t\_{s3}) \frac{d \Delta v\_{S;1/free}}{dt\_{s3}} \\ \Delta v\_{3;1,2/hyd} &= \Delta v\_{S;1/hyd} + (1 - t\_{s3}) \frac{d \Delta v\_{S;1/hyd}}{dt\_{s3}} \end{split} \tag{109}$$

One arrives at the following result:

$$
\Delta\upsilon\_{2;1,3} = \Delta\upsilon\_{2;1,3/free} + \Delta\upsilon\_{3;1/free}
$$

$$
\Delta\upsilon\_{3;1,2} = \Delta\upsilon\_{3;1,2/free} + \Delta\upsilon\_{3;1,2/hyd}
\tag{110}
$$

where similar equations are obtained for the interaction partial specific compressibilities.

Considering these contributions, the interpretation of the partial specific volumes of interaction of the particle as function of the electrolyte concentration is as follows. From tf3 = 0 to around 0.05 (see Figure 4B), there is an increment in Δv2,1,3 which can be interpreted as a gain of free volume by the disentanglement of the polymeric chains. This increment of free volume is accompanied by an increment in the hydrodynamic radius [4]. From around tf3 = 0.05 to around 0.1, there is a decrement in Δv2,1,3 due to hydration. In this region of compositions, the separation of polymeric chains allows the entrance of water molecules in the polymeric particle. As a result, the hydrodynamic radius of the particle increases [4]. From around ts3 = 0.1 to 0.15, Δv2,1,3 increases sharply. This fact can be interpreted as an increment of the dehydration. Beyond ts3 = 0.15, Δv2,1,3 becomes constant, indicating that the interaction between particles and the electrolyte is saturated. Similar regions with similar interpretations are obtained for the partial specific adiabatic compressibility (see Figure 5B).

#### 4.4.2. Partial specific enthalpies of interaction

ΔvS, <sup>1</sup> ¼ ΔvS, <sup>1</sup>=f ree þ ΔvS,<sup>1</sup>=hyd (105)

ΔvS;<sup>1</sup>=f ree ¼ vS; <sup>1</sup>=f ree � ts2v2;1=f ree (106)

dΔvS;<sup>1</sup>=hyd dts<sup>3</sup>

� �

dΔvS; <sup>1</sup>=hyd dts<sup>3</sup>

(108)

(109)

(110)

ΔvS;<sup>1</sup>=hyd ¼ vS;<sup>1</sup>=hyd � ðts2v2; <sup>1</sup>=hyd þ ts3v3; <sup>1</sup>=hydÞ (107)

� �

þ ΔvS;<sup>1</sup>=hyd þ ð1 � ts3Þ

dΔvS; <sup>1</sup>=f ree dts<sup>3</sup>

dΔvS;<sup>1</sup>=hyd dts<sup>3</sup>

> dΔvS; <sup>1</sup>=f ree dts<sup>3</sup>

dΔvS;<sup>1</sup>=hyd dts<sup>3</sup>

þ ΔvS; <sup>1</sup>=hyd � ts<sup>3</sup>

where

118 Advances in Titration Techniques

8 >>><

>>>:

Defining now:

Substituting (105) in (93), we get

Δv2; <sup>1</sup>, <sup>3</sup> ¼ ΔvS;<sup>1</sup>=f ree � ts<sup>3</sup>

One arrives at the following result:

Δv3; <sup>1</sup>, <sup>2</sup> ¼ ΔvS;<sup>1</sup>=f ree þ ð1 � ts3Þ

dΔvS; <sup>1</sup>=f ree dts<sup>3</sup>

> dΔvS;<sup>1</sup>=f ree dts<sup>3</sup>

Δv2;1,3=f ree ¼ ΔvS; <sup>1</sup>=f ree � ts<sup>3</sup>

Δv2;1,3=hyd ¼ ΔvS; <sup>1</sup>=hyd � ts<sup>3</sup>

Δv3;1,2=f ree ¼ ΔvS; <sup>1</sup>=f ree þ ð1 � ts3Þ

Δv3;1,2=hyd ¼ ΔvS; <sup>1</sup>=hyd þ ð1 � ts3Þ

Δv2;1,<sup>3</sup> ¼ Δv2; <sup>1</sup>,3=f ree þ ΔvS; <sup>1</sup>=f ree Δv3;1,<sup>2</sup> ¼ Δv3; <sup>1</sup>,2=f ree þ Δv3;1, <sup>2</sup>=hyd

Considering these contributions, the interpretation of the partial specific volumes of interaction of the particle as function of the electrolyte concentration is as follows. From tf3 = 0 to around 0.05 (see Figure 4B), there is an increment in Δv2,1,3 which can be interpreted as a gain of free volume by the disentanglement of the polymeric chains. This increment of free volume is accompanied by an increment in the hydrodynamic radius [4]. From around tf3 = 0.05 to around 0.1, there is a decrement in Δv2,1,3 due to hydration. In this region of compositions, the separation of polymeric chains allows the entrance of water molecules in the polymeric particle. As a result, the hydrodynamic radius of the particle increases [4]. From around ts3 = 0.1 to 0.15, Δv2,1,3 increases sharply. This fact can be interpreted as an increment of the dehydration. Beyond ts3 = 0.15, Δv2,1,3 becomes constant, indicating that the interaction between particles and the

where similar equations are obtained for the interaction partial specific compressibilities.

� �

� �

This section deals with the determination of the partial specific enthalpies of interaction of the same system than in the latest example [4]. Partial specific enthalpy of interaction of polymeric particles is:

$$
\Delta h\_{2\div1,3} = h\_{2\div1,3} - h\_{2\div1} \tag{111}
$$

and the partial specific enthalpy interaction of the electrolyte is:

$$
\Delta h\_{3;1,2} = h\_{3;1,2} - h\_{3;1} \tag{112}
$$

The partial specific enthalpy of interaction of the electrolyte can be measured by isothermal titration calorimetry using the combination of two experiments [6, 7]. The first experiment is locating the polymeric particles in the cell and the electrolyte in the syringe. The heat per unit of titration volume in an infinitesimal titration is:

$$\frac{d\mathbb{Q}^{st}}{dv} = (\rho^s - c\_3^s)h\_{1;2,3} + c\_3^s h\_{3;1,2} - h\_v(c\_3^s) \tag{113}$$

where <sup>ρ</sup><sup>s</sup> is the density of the stock solution and hvðcs <sup>3</sup>Þ is the enthalpy of the stock solution per unit volume. The second experiment consists of titrating water with the above stock solution, and its heat per unit of titration volume in an infinitesimal titration is:

$$\frac{dQ^c}{dv} = (\rho^s - c\_3^s)h\_{1;3} + c\_3^s h\_{1;3} - h\_v(c\_3^s) \tag{114}$$

The partial specific enthalpy of interaction of the electrolyte is obtained by subtracting (114) from (113), considering Eq. (112), diluted solutions and bearing in mind that dn<sup>2</sup> <sup>3</sup> <sup>¼</sup> cs <sup>3</sup>dv:

$$\frac{dQ^{cd}}{dn\_3^s} - \frac{dQ^c}{dn\_3^s} = \Delta h\_{3;1,2} \tag{115}$$

Figure 7A shows the experimental values Δh3;1,2. The partial specific enthalpy of interaction of polymeric particles was calculated by integrating Eq. (91) [7]:

$$
\Delta h\_{2;1,3}(t\_{f3}) = -\int\_0^{t\_{f3}} \frac{t'\_{f3}}{1 - t'\_{f3}} \left(\frac{d\Delta h\_{3;1,2}}{dt'\_{f3}}\right) dt'\_{f3} \tag{116}
$$

and the values of Δh2;1,3 are shown in Figure 7B. It is very interesting to observe in Figure 7B that Δh2;1,3 is zero from ts<sup>3</sup> = 0 to around ts<sup>3</sup> = 0.1. This fact indicates that the changes, which take place in the first two regions in Figures 4B and 5B, are entropic in origin.

Figure 7. (A) Partial specific enthalpy of interaction of the electrolyte as function of the mass fraction of the electrolyte in the complex solute (ts3). (B) Partial specific enthalpy of interaction of the polymeric particles as function of ts3.

## 5. Partial molar properties at infinite dilution

First, we will discuss the case of the two-component system and then make the extension to three-component system. In this section, J can be U, H, V or their derivatives Cv = (∂H/∂T)V, Cp = (∂H/∂T)<sup>P</sup> or E = (∂V/∂T)P, KT = (∂V/∂P)T and KS = (∂V/∂P)S.

#### 5.1. Two-component systems

In a two-component system, we only have one way to calculate limits at infinite dilution and it is to take a component as solvent (component 1) and the other as solute (component 2). For a two-component system, j takes the form:

$$j(\mathbf{x}\_2) = \mathbf{x}\_1 j\_{1;2}(\mathbf{x}\_2) + \mathbf{x}\_2 j\_{2;1}(\mathbf{x}\_2) \tag{117}$$

Because

$$\lim\_{\mathbf{x}\_2 \to 0} \mathbf{j}(\mathbf{x}\_2) = j\_1 \tag{118}$$

and using Eq. (117), we have:

$$\lim\_{\mathbf{x}\_2 \to 0} j\_{1;2}(\mathbf{x}\_2) = j\_1 \tag{119}$$

For the solute, we have:

$$\text{lim}\_{\mathbf{x}\_2 \to 0} \mathbf{j}\_{\mathbf{2};1}(\mathbf{x}\_2) = \mathbf{j}\_{\mathbf{2};1}^\rho \tag{120}$$

We can obtain experimentally the value of j o <sup>2</sup>; <sup>1</sup> as follows. The Taylor's expansion of j(x2) around x<sup>2</sup> = 0 is:

Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques http://dx.doi.org/10.5772/intechopen.69706 121

$$j(\mathbf{x}\_2) = j(0) + \frac{dj(0)}{d\mathbf{x}\_2}\mathbf{x}\_2 + \dots \tag{121}$$

Differentiating (117) with respect to x2, considering the Gibbs-Duhem equation for a 2-component system and combining the results with equations (117), (120) and (121):

$$j(\mathbf{x}\_2) = j\_1 + (j\_{2;1}^\circ - j\_1)\mathbf{x}\_2 \tag{122}$$

For this reason, we can obtain experimentally j o <sup>2</sup>; <sup>1</sup> from a linear fit in a plot of j(x2) as function of x2.

#### 5.2. Three-component systems

In three-component systems, we have two ways to calculate limits at infinite dilution. The first way is to group two components in a "complex solvent" and to calculate the limit at infinite dilution of the other component in this complex solvent (type I). The other way is considering a component as solvent, to group the other two components in a complex solute, and to calculate the limit at infinite dilution of the complex solute in the solvent (type II).

#### 5.2.1. Limits of type I

5. Partial molar properties at infinite dilution

Cp = (∂H/∂T)<sup>P</sup> or E = (∂V/∂T)P, KT = (∂V/∂P)T and KS = (∂V/∂P)S.

5.1. Two-component systems

120 Advances in Titration Techniques

and using Eq. (117), we have:

For the solute, we have:

around x<sup>2</sup> = 0 is:

Because

two-component system, j takes the form:

We can obtain experimentally the value of j

First, we will discuss the case of the two-component system and then make the extension to three-component system. In this section, J can be U, H, V or their derivatives Cv = (∂H/∂T)V,

Figure 7. (A) Partial specific enthalpy of interaction of the electrolyte as function of the mass fraction of the electrolyte in

the complex solute (ts3). (B) Partial specific enthalpy of interaction of the polymeric particles as function of ts3.

In a two-component system, we only have one way to calculate limits at infinite dilution and it is to take a component as solvent (component 1) and the other as solute (component 2). For a

<sup>1</sup>;2ðx2Þ þ x2j

lim<sup>x</sup>2!<sup>0</sup>jðx2Þ ¼ j

<sup>1</sup>; <sup>2</sup>ðx2Þ ¼ j

<sup>2</sup>;1ðx2Þ ¼ j

o

o

lim<sup>x</sup>2!<sup>0</sup>j

lim<sup>x</sup>2!<sup>0</sup>j

<sup>2</sup>; <sup>1</sup>ðx2Þ (117)

<sup>1</sup> (118)

<sup>1</sup> (119)

<sup>2</sup>;<sup>1</sup> (120)

<sup>2</sup>; <sup>1</sup> as follows. The Taylor's expansion of j(x2)

jðx2Þ ¼ x1j

In this case, we consider a complex solvent B composed of components 1 and 2 and a solute (component 3). For this system,

$$J = J(n\_{\rm 3\prime} n\_{\rm 3\prime} x\_{\rm 62}) \tag{123}$$

where nB = n1+n<sup>2</sup> and xb<sup>2</sup> = n2/(n1+n2). With this, j can be written as:

$$j(\mathbf{x}\_{3\prime}, \mathbf{x}\_{b2}) = \mathbf{x}\_{\mathcal{B}} j\_{\mathbf{B};3}(\mathbf{x}\_{3\prime}, \mathbf{x}\_{b2}) + \mathbf{x}\_{\mathcal{B}} j\_{\mathbf{3};\mathcal{B}}(\mathbf{x}\_{3\prime}, \mathbf{x}\_{b2}) \tag{124}$$

where x<sup>3</sup> is the mole fraction of the component 3. At infinite dilution, we have:

$$\lim\_{\substack{\mathbf{x}\_{\mathcal{B}} \to 0 \\ \mathbf{x}\_{\mathcal{b}2} \text{ constant}}} j(\mathbf{x}\_{\mathcal{B}}, \mathbf{x}\_{\mathcal{b}2}) = j\_{\mathcal{B}}(\mathbf{x}\_{\mathcal{b}2}) \tag{125}$$

and then combining Eq. (124) with (125), one gets for the solvent:

$$\lim\_{\substack{\mathbf{x}\_{\mathcal{B}} \rightarrow 0 \\ \mathbf{x}\_{\mathcal{B}2} \text{ constant}}} j\_{\mathcal{B};3}(\mathbf{x}\_{\mathcal{B}}, \mathbf{x}\_{\mathcal{b}2}) = j\_{\mathcal{B}}(\mathbf{x}\_{\mathcal{b}2}) \tag{126}$$

For the solute, it is obtained that:

$$\lim\_{\substack{\mathbf{x}\_{3}\rightarrow 0\\\mathbf{x}\_{b2}\text{ constant}}} j\_{\mathbf{3};\mathbf{3}}(\mathbf{x}\_{3\prime},\mathbf{x}\_{b2}) = j\_{\mathbf{3};\mathbf{3}}^{\boldsymbol{\rho}}(\mathbf{x}\_{b2}) \equiv j\_{\mathbf{3};1,2}^{\boldsymbol{\rho}}(\mathbf{x}\_{b2}) \tag{127}$$

where we have used Eq. (48). Similarly to the case of two-component systems, the amount j o 3;1, 2 can be obtained experimentally by using the equation:

$$j(\mathbf{x}\_{3\prime}, \mathbf{x}\_{b2}) = j\_B(\mathbf{x}\_{b2}) + \left(j\_{3;1,2}^{\flat} - j\_B(\mathbf{x}\_{b2})\right) \mathbf{x}\_3 \tag{128}$$

This equation is obtained by using the first-order Taylor's expansion of j(x3,xb2) around x<sup>3</sup> = 0, the partial derivative of j(x3, xb2) with respect to x3, the Gibbs-Duhem equation of the fractionalized system considering the composition of the fraction as constant and Eqs. (126) and (127).

#### 5.2.2. Limits of type II

In this case (see Figure 1A and B), we will consider the component 1 as solvent and a "complex solute" S composed of 2 and 3 and then:

$$J = J(n\_1, n\_{\rm S}, \mathbf{x}\_{\rm s3}) \tag{129}$$

where n<sup>S</sup> = n2+n<sup>3</sup> and xs<sup>3</sup> = n3/(n<sup>2</sup> + n3). The molar property j is:

$$j(\mathbf{x}\_{\rm S}, \mathbf{x}\_{\rm s3}) = \mathbf{x}\_1 j\_{1; \rm S}(\mathbf{x}\_{\rm S}, \mathbf{x}\_{\rm s3}) + \mathbf{x}\_{\rm S} j\_{\rm S;1}(\mathbf{x}\_{\rm S}, \mathbf{x}\_{\rm s3}) \tag{130}$$

Similarly to the above cases, at infinite dilution we have for the solvent:

$$\lim\_{\substack{\mathbf{x}\_{\mathcal{S}} \to 0 \\ \mathbf{x}\_{\mathcal{S}} \text{ constant}}} j\_{\mathbf{i},\mathbf{\mathcal{S}}}(\mathbf{x}\_{\mathcal{S}},\mathbf{x}\_{\mathcal{S}}) = j\_1 \tag{131}$$

Accordingly to case of the two-component system, one gets for the complex solute:

$$\lim\_{\substack{\mathbf{x}\_{\mathcal{S}} \to 0 \\ \mathbf{x}\_{\mathcal{S}} \text{ constant}}} j\_{\mathcal{S};1}(\mathbf{x}\_{\mathcal{S}}, \mathbf{x}\_{\mathcal{S}3}) = j\_{\mathcal{S};1}^{\mathcal{o}}(\mathbf{x}\_{\mathcal{S}}) \tag{132}$$

and in a similar way than for the type I limits, j o S;<sup>1</sup> can be calculated as

$$j(\mathbf{x}\_{\\$}, \mathbf{x}\_{\\$3}) = j\_1 + \left( j\_{\\$;1}^p(\mathbf{x}\_{\\$3}) - j\_1 \right) \mathbf{x}\_{\\$} \tag{133}$$

In order to study the contributions of components 2 and 3 to j o S;1, we define the following limits an infinite dilution:

$$\begin{cases} \lim\_{\mathbf{x}\_{\ $} \to 0} & j\_{2;1,3}(\mathbf{x}\_{\$ }, \mathbf{x}\_{\ $3}) = j\_{2;1,3}^{\Delta}(\mathbf{x}\_{\$ 3})\\ \quad \text{x}\_{\ $3} \text{ constant} \\ \lim\_{\mathbf{x}\_{\$ } \to 0} & j\_{3;1,2}(\mathbf{x}\_{\ $}, \mathbf{x}\_{\$ 3}) = j\_{3;1,2}^{\Delta}(\mathbf{x}\_{\ $3})\\ \quad \text{x}\_{\$ 3} \text{ constant} \end{cases} \tag{134}$$

In this way, taking limits in both sides of Eq. (49), and bearing in mind Eqs. (132) and (134), we have that:

$$\mathbf{x}\_{\text{S};1}^{\circ}(\mathbf{x}\_{\text{s}3}) = \mathbf{x}\_{\text{s}2}\mathbf{j}\_{\text{2};1,3}^{\Lambda}(\mathbf{x}\_{\text{s}3}) + \mathbf{x}\_{\text{s}3}\mathbf{j}\_{\text{3};1,2}^{\Lambda}(\mathbf{x}\_{\text{s}3}) \tag{135}$$

Now, we will see some mathematical properties of limits of type II. One of them is for example:

$$\text{dim}\_{\mathbf{x}\_{\text{s}3}\to 0} \mathfrak{j}^{\Delta}\_{2;1,3}(\mathbf{x}\_{\text{s}3}) = \mathfrak{j}^{\boldsymbol{\rho}}\_{2;1} \tag{136}$$

This property is demonstrated by using iterated limits:

$$\lim\_{\mathbf{x}\_{3}\to 0} \mathbf{j}\_{2,1,3}^{\mathbf{A}}(\mathbf{x}\_{3}) = \lim\_{\mathbf{x}\_{3}\to 0} \mathbf{1}\_{\mathbf{x}\_{3}\to 0} \left[ \lim\_{\mathbf{x}\_{3}\to 0} \mathbf{j}\_{2,1,3}(\mathbf{x}\_{3}, \mathbf{x}\_{3}) \right]$$

$$\mathbf{j} = \lim\_{\mathbf{x}\_{3}\to 0} \mathbf{0} \left[ \lim\_{\mathbf{x}\_{3}\to 0} \mathbf{j}\_{2,1,3}(\mathbf{x}\_{3}, \mathbf{x}\_{3}) \right] = \lim\_{\mathbf{x}\_{3}\to 0} \mathbf{j}\_{2,1,3}(\mathbf{x}\_{3}, \mathbf{0}) = \lim\_{\mathbf{x}\_{3}\to 0} \mathbf{j}\_{2,1}(\mathbf{x}\_{3}) = \mathbf{j}\_{2,1}^{\rho} \tag{137}$$

The other mathematical property is:

jðx3, xb2Þ ¼ j

where n<sup>S</sup> = n2+n<sup>3</sup> and xs<sup>3</sup> = n3/(n<sup>2</sup> + n3). The molar property j is:

jðxS, xs3Þ ¼ x1j

limxS ! <sup>0</sup> xs3 constant

jðxS, xs3Þ ¼ j

In order to study the contributions of components 2 and 3 to j

8 >><

>>:

j o

limxS ! <sup>0</sup> xs3 constant

limxS ! <sup>0</sup> xs3 constant

S;1ðxs3Þ ¼ xs2j

and in a similar way than for the type I limits, j

Similarly to the above cases, at infinite dilution we have for the solvent:

limxS ! <sup>0</sup> xs3 constant

Accordingly to case of the two-component system, one gets for the complex solute:

j

<sup>1</sup> þ � j o

j

j

Δ

limxs3!<sup>0</sup>j

5.2.2. Limits of type II

122 Advances in Titration Techniques

an infinite dilution:

have that:

solute" S composed of 2 and 3 and then:

<sup>B</sup>ðxb2Þ þ

� j o <sup>3</sup>; <sup>1</sup>,<sup>2</sup> � j

This equation is obtained by using the first-order Taylor's expansion of j(x3,xb2) around x<sup>3</sup> = 0, the partial derivative of j(x3, xb2) with respect to x3, the Gibbs-Duhem equation of the fractionalized system considering the composition of the fraction as constant and Eqs. (126) and (127).

In this case (see Figure 1A and B), we will consider the component 1 as solvent and a "complex

<sup>1</sup>;SðxS, xs3Þ þ xSj

<sup>1</sup>;SðxS, xs3Þ ¼ j

o

S;<sup>1</sup> can be calculated as

1 �

Δ <sup>2</sup>;1,3ðxs3Þ

Δ <sup>3</sup>;1,2ðxs3Þ

Δ

o

o

S;1ðxS, xs3Þ ¼ j

S; <sup>1</sup>ðxs3Þ � j

<sup>2</sup>;1,3ðxS, xs3Þ ¼ j

<sup>3</sup>;1,2ðxS, xs3Þ ¼ j

In this way, taking limits in both sides of Eq. (49), and bearing in mind Eqs. (132) and (134), we

Now, we will see some mathematical properties of limits of type II. One of them is for example:

Δ

<sup>2</sup>; <sup>1</sup>, <sup>3</sup>ðxs3Þ þ xs3j

<sup>2</sup>; <sup>1</sup>, <sup>3</sup>ðxs3Þ ¼ j

o

j

<sup>B</sup>ðxb2Þ �

J ¼ Jðn1, nS, xs3Þ (129)

S;1ðxS, xs3Þ (130)

<sup>1</sup> (131)

S;1ðxs3Þ (132)

xS (133)

S;1, we define the following limits

<sup>3</sup>;1,2ðxs3Þ (135)

<sup>2</sup>;<sup>1</sup> (136)

(134)

x<sup>3</sup> (128)

$$\mathbf{lim}\_{\mathbf{x}\_{s3}\to 1} \mathbf{j}\_{3;1,2}^{\Lambda}(\mathbf{x}\_{s3}) = \mathbf{j}\_{2;1,3}^{\nu}(\mathbf{0}) \tag{138}$$

where its demonstration is as follows:

$$\begin{aligned} \lim\_{\mathbf{x}\_{\mathcal{S}}\to 1} \mathbf{j}\_{\mathbf{3};1,2}^{\mathbf{A}}(\mathbf{x}\_{\mathcal{S}}) &= \lim\_{\mathbf{x}\_{\mathcal{S}}\to \mathbf{0}} \mathbf{1}\_{\mathbf{x}\_{\mathcal{S}}\to 1} \left[ \lim\_{\mathbf{x}\_{\mathcal{S}}\to \mathbf{0}} \mathbf{j}\_{\mathbf{3};1,2}(\mathbf{x}\_{\mathcal{S}}, \mathbf{x}\_{\mathcal{S}}) \right] \\ &= \lim\_{\mathbf{x}\_{\mathcal{S}}\to \mathbf{0}} \left[ \lim\_{\mathbf{x}\_{\mathcal{S}}\to \mathbf{0}} \mathbf{j}\_{\mathbf{3};1,2}(\mathbf{x}\_{\mathcal{S}}, \mathbf{x}\_{\mathcal{S}}) \right] \end{aligned} \tag{139}$$

Now, it is necessary to consider other way to fractionalize the system. For convenience, we will consider a complex solvent B composed of 1 and 3, and a solute 2 where the variable xB represents the molar fraction of B and xb<sup>3</sup> = n3/(n1+n3). With this,

$$\begin{cases} \lim\_{\mathbf{x}\_{\mathcal{S}} \to 1} \mathbf{x}\_{2} = \lim\_{\mathbf{x}\_{\mathcal{S}} \to 1} \mathbf{x}\_{\mathbf{x}\_{\mathcal{S}}} \to 1 & \mathbf{x}\_{\mathcal{S}}(1 - \mathbf{x}\_{\mathcal{S}}) = \mathbf{0} \\ \quad \mathbf{x}\_{\mathcal{S}} \text{ constant} & \mathbf{x}\_{\mathcal{S}} \text{ constant} \\ \lim\_{\mathbf{x}\_{\mathcal{S}} \to 1} \mathbf{x}\_{\mathcal{S}3} = \lim\_{\mathbf{x}\_{\mathcal{S}} \to 1} \mathbf{x}\_{\mathbf{x}\_{\mathcal{S}}} \mathbf{x}\_{\mathcal{S}} [1 - \mathbf{x}\_{\mathcal{S}}(1 - \mathbf{x}\_{\mathcal{S}})] = \mathbf{x}\_{\mathcal{S}} \\ \quad \quad \mathbf{x}\_{\mathcal{S}} \text{ constant} & \mathbf{x}\_{\mathcal{S}} \text{ constant} \end{cases} \tag{140}$$

and considering Eq. (140), (139) transforms into:

$$\begin{aligned} \lim\_{\mathbf{x}\_{\mathcal{S}}\to 1} \mathbf{j}\_{3;1,2}^{\Lambda}(\mathbf{x}\_{\mathcal{S}}) &= \lim\_{\mathbf{x}\_{\mathcal{S}}\to 0} \quad \left[ \lim\_{\mathbf{x}\_{\mathcal{S}}\to 0} \quad j\_{3;1,2}(\mathbf{x}\_{\mathcal{S}}, \mathbf{x}\_{\mathcal{U}}) \right] = \\ &= \lim\_{\mathbf{x}\_{\mathcal{S}}\to 0} \quad j\_{3;1,2}^{\mathcal{I}}(\mathbf{x}\_{\mathcal{U}}) = j\_{3;1,2}^{\mathcal{I}}(\mathbf{0}) \end{aligned} \tag{141}$$

Other interesting property of the limits of type II is that they are related to each other by the following equation:

$$\mathbf{x}\_{s2}\frac{d\dot{j}\_{2;1,3}^{\Lambda}}{d\mathbf{x}\_{s3}} + \mathbf{x}\_{s3}\frac{d\dot{j}\_{3;1,2}^{\Lambda}}{d\mathbf{x}\_{s3}} = 0\tag{142}$$

The demonstration of this equation is as follows. Both sides of the following equation:

$$\mathbf{x}\_{s2} \left( \frac{\partial \mathbf{j}\_{2;1,3}}{\partial \mathbf{x}\_{s3}} \right)\_{\mathbf{x}\_{\mathcal{S}}} + \mathbf{x}\_{s3} \left( \frac{\partial \mathbf{j}\_{3;1,2}}{\partial \mathbf{x}\_{s3}} \right)\_{\mathbf{x}\_{\mathcal{S}}} = \mathbf{x}\_{\mathcal{S}} (1 - \mathbf{x}\_{\mathcal{S}}) \left( \frac{\partial (\mathbf{j}\_{3;1,2} - \mathbf{j}\_{2;1,3})}{\partial \mathbf{x}\_{\mathcal{S}}} \right)\_{\mathbf{x}\_{\mathcal{S}}} \tag{143}$$

are calculated in the following way. The left-hand side is obtained by deriving partially Eq. (49) with respect to xs3. The right-hand side of (143) is calculated considering that:

$$\left(\frac{\partial \dot{l}\_{S;1}}{\partial \mathbf{x}\_{s3}}\right)\_{x\_S} = \frac{\partial l}{\partial \mathbf{x}\_{s3} \partial n\_S} = \frac{\partial l}{\partial n\_S \partial \mathbf{x}\_{s3}}\tag{144}$$

Using (50) in (144) and cancelling common terms, Eq. (143) is obtained. Taking the limit when xs approaches to zero when xs<sup>3</sup> is kept constant in both sides of Eq. (143) and considering that:

$$\lim\_{\substack{\mathbf{x}\_{\mathcal{S}} \to 0 \\ \mathbf{x}\_{\mathcal{S}} \text{ constant}}} \left( \frac{\partial \left( \dot{j}\_{\mathcal{S};1,2} (\mathbf{x}\_{\mathcal{S}}, \mathbf{x}\_{\mathcal{S}3}) - \dot{j}\_{\mathcal{Z};1,3} (\mathbf{x}\_{\mathcal{S}}, \mathbf{x}\_{\mathcal{S}3}) \right)}{\partial \mathbf{x}\_{\mathcal{S}}} \right)\_{\mathbf{x}\_{\mathcal{S}}} $$
 
$$= \left( \frac{\partial \dot{j}\_{\mathcal{Z};1,2} (0, \mathbf{x}\_{\mathcal{S}})}{\partial \mathbf{x}\_{\mathcal{S}}} \right)\_{\mathbf{x}\_{\mathcal{S}}} - \left( \frac{\partial \dot{j}\_{\mathcal{Z};1,3} (0, \mathbf{x}\_{\mathcal{S}})}{\partial \mathbf{x}\_{\mathcal{S}}} \right)\_{\mathbf{x}\_{\mathcal{S}}} = f(\mathbf{x}\_{\mathcal{S}}) \tag{145}$$

Eq. (142) is obtained.

From values of j o S;<sup>1</sup> it is possible to obtain j Δ <sup>2</sup>;1,<sup>3</sup> and j Δ <sup>3</sup>;1,<sup>2</sup> by using the following equations:

$$\begin{cases} \boldsymbol{j}\_{2;1,3}^{\Lambda} = \boldsymbol{j}\_{\boldsymbol{S};1}^{\boldsymbol{o}} - \mathbf{x}\_{\boldsymbol{s}3} \frac{d \boldsymbol{j}\_{\boldsymbol{S};1}^{\boldsymbol{o}}}{d \mathbf{x}\_{\boldsymbol{s}3}}\\ \boldsymbol{j}\_{3;1,2}^{\Lambda} = \boldsymbol{j}\_{\boldsymbol{S};1}^{\boldsymbol{o}} + (1 - \mathbf{x}\_{\boldsymbol{s}3}) \frac{d \boldsymbol{j}\_{\boldsymbol{S};1}^{\boldsymbol{o}}}{d \mathbf{x}\_{\boldsymbol{s}3}} \end{cases} \tag{146}$$

Eq. (146) was obtained by differentiating Eq. (135) with respect to xs3, considering Eq. (142) and combining the result with Eq. (135).

#### 5.2.3. Application of the limits of type II to the study of polymeric particles

The polymeric particles used were synthesized with a gradient of concentration of functional groups (acrylic acid) inside the particle [9]. In this system, the content of acrylic acid represents the polar groups, while poly(butyl acrylate-co-methylmethacrylate) is the non-polar groups. As seen in Figure 1C and D, component 1 is water, component 2 is non-polar groups and component 3 is polar groups. The polymeric particle (composed of polar and non-polar groups) is taken as a fraction "P" of the system where the variable tp<sup>3</sup> = n3/(n2+n3) will be the mass fraction of polar groups in the particle. In this study [9], the same experimental equipment than in Section 4.4.1 was used and measurements of density and sound speed were carried out by titrating water (in the cell) with latex of polymeric particles (in the syringe). Figure 7A and B shows the density ρ and u as functions of the concentration for several values of tp3. The density and sound speed were transformed into specific volumes and specific adiabatic compressibilities by using Eqs. (95) and (96), and results are shown in Figure 1C and D.

Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques http://dx.doi.org/10.5772/intechopen.69706 125

In this case, Eq. (133) will take the form:

xs<sup>2</sup> ∂j 2;1,3 ∂xs<sup>3</sup> � �

124 Advances in Titration Techniques

xS þ xs<sup>3</sup>

limxS ! <sup>0</sup> xs<sup>3</sup> constant

<sup>¼</sup> <sup>∂</sup><sup>j</sup>

S;<sup>1</sup> it is possible to obtain j

Eq. (142) is obtained.

o

combining the result with Eq. (135).

From values of j

∂j 3; 1, 2 ∂xs<sup>3</sup> � �

with respect to xs3. The right-hand side of (143) is calculated considering that:

xS

∂ � j

xs<sup>3</sup>

Δ <sup>2</sup>;1,<sup>3</sup> and j

0 @

<sup>3</sup>; <sup>1</sup>, <sup>2</sup>ð0,xs3Þ ∂xS � �

> j Δ <sup>2</sup>; <sup>1</sup>,<sup>3</sup> ¼ j o S; <sup>1</sup> � xs<sup>3</sup>

8 >><

>>:

5.2.3. Application of the limits of type II to the study of polymeric particles

ities by using Eqs. (95) and (96), and results are shown in Figure 1C and D.

j Δ <sup>3</sup>; <sup>1</sup>,<sup>2</sup> ¼ j o

∂j S;1 ∂xs<sup>3</sup> � � xS

are calculated in the following way. The left-hand side is obtained by deriving partially Eq. (49)

<sup>¼</sup> <sup>∂</sup><sup>J</sup> ∂xs3∂nS

Using (50) in (144) and cancelling common terms, Eq. (143) is obtained. Taking the limit when xs approaches to zero when xs<sup>3</sup> is kept constant in both sides of Eq. (143) and considering that:

<sup>3</sup>;1,2ðxS, xs3Þ � j

� <sup>∂</sup><sup>j</sup>

∂xS

Δ

djo S; 1 dxs<sup>3</sup>

djo S;1 dxs<sup>3</sup>

S; <sup>1</sup> þ ð1 � xs3Þ

Eq. (146) was obtained by differentiating Eq. (135) with respect to xs3, considering Eq. (142) and

The polymeric particles used were synthesized with a gradient of concentration of functional groups (acrylic acid) inside the particle [9]. In this system, the content of acrylic acid represents the polar groups, while poly(butyl acrylate-co-methylmethacrylate) is the non-polar groups. As seen in Figure 1C and D, component 1 is water, component 2 is non-polar groups and component 3 is polar groups. The polymeric particle (composed of polar and non-polar groups) is taken as a fraction "P" of the system where the variable tp<sup>3</sup> = n3/(n2+n3) will be the mass fraction of polar groups in the particle. In this study [9], the same experimental equipment than in Section 4.4.1 was used and measurements of density and sound speed were carried out by titrating water (in the cell) with latex of polymeric particles (in the syringe). Figure 7A and B shows the density ρ and u as functions of the concentration for several values of tp3. The density and sound speed were transformed into specific volumes and specific adiabatic compressibil-

<sup>2</sup>; <sup>1</sup>, <sup>3</sup>ð0,xs3Þ ∂xS � �

¼ xSð1 � xSÞ

<sup>¼</sup> <sup>∂</sup><sup>J</sup> ∂nS∂xs<sup>3</sup>

∂ðj

<sup>2</sup>;1,3ðxS, xs3Þ

xs<sup>3</sup>

<sup>3</sup>; <sup>1</sup>, <sup>2</sup> � j

∂xS � �

�

1 A xs<sup>3</sup>

<sup>3</sup>;1,<sup>2</sup> by using the following equations:

<sup>2</sup>; <sup>1</sup>, <sup>3</sup>Þ

xs<sup>3</sup>

¼ fðxs3Þ (145)

(143)

(144)

(146)

$$j = j\_1 + \left(j\_{P;1}^p - j\_1\right)t\_P \tag{147}$$

and considering that t<sup>1</sup> = 1 – tP, Eq. (147) transforms into:

$$j = j\_{P;1}^{\circ} + \left(j\_1 - j\_{P;1}^{\circ}\right)t\_1 \tag{148}$$

Using Eq. (148) as a fit function in Figure 8C and D, the partial specific volume at infinite dilution of the particles (v<sup>o</sup> P; <sup>1</sup>) and the partial specific adiabatic compressibility at infinite dilution of the particles (k<sup>o</sup> S P; <sup>1</sup>) were obtained from the independent term of Eq. (148) and the results are shown in Figure 9A and B as functions of tp3. In this case, Eqs. (100) and (101) take the form:

$$
\boldsymbol{\upsilon}^{\boldsymbol{\rho}}\_{\text{P};1} = \boldsymbol{\upsilon}^{\boldsymbol{\rho}}\_{\text{P};1/\text{atom}} + \boldsymbol{\upsilon}^{\boldsymbol{\rho}}\_{\text{P};1/\text{free}} + \boldsymbol{\upsilon}^{\boldsymbol{\rho}}\_{\text{P};1/\text{hyd}} \tag{149}
$$

$$k\_{\text{S }P;1}^{\rho} = k\_{\text{S }P;1/free}^{\rho} + k\_{\text{S }P;1/hyd}^{\rho} \tag{150}$$

The partial specific properties of polar (j Δ <sup>3</sup>; <sup>1</sup>, <sup>2</sup>) and non-polar (j Δ <sup>2</sup>; <sup>1</sup>, <sup>3</sup>) groups were calculated by using Eq. (146). The derivatives of Eq. (146) were calculated numerically by using schemes of finite differences. Figure 9C and D shows, as functions of the amount of polar groups (tp3), the values of specific partial volumes of non-polar and polar groups, respectively. Figure 9D and F shows, respectively, the specific partial adiabatic compressibility of non-polar and polar groups.

With similar arguments than in Section 4.4.1, we can get the following equations for the volumes:

$$
\sigma\_{2;1,3}^{\Delta} = \sigma\_{2;1,3/atom}^{\Delta} + \upsilon\_{2;1,3/free}^{\Delta} + \upsilon\_{2;1,3/hyd}^{\Delta} \tag{151}
$$

$$
\sigma\_{3;1,2}^{\Lambda} = \sigma\_{3;1,2/atom}^{\Lambda} + \upsilon\_{3;1,2/free}^{\Lambda} + \upsilon\_{3;1,2/hyd}^{\Lambda} \tag{152}
$$

and for the adiabatic compressibilities:

$$k\_{T\ 2;1,3}^{\Lambda} = k\_{T\ 2;1,3/free}^{\Lambda} + k\_{T\ 2;1,3/hyd}^{\Lambda} \tag{153}$$

$$k\_{T\ 3;1,2}^{\Lambda} = k\_{T\ 3;1,2/free}^{\Lambda} + k\_{T\ 3;1,2/hyd}^{\Lambda} \tag{154}$$

In addition to this, by combining Eqs. (135), (149), (151) and (152), one gets the following equations:

$$
\sigma^{\rho}\_{\text{P;1/atom}} = t\_{\text{p2}} \sigma^{\Lambda}\_{\text{2;1,3/atom}} + t\_{\text{p3}} \sigma^{\Lambda}\_{\text{3;1,2/atom}} \tag{155}
$$

$$\mathbf{v}\_{\text{p};1/fre}^{\rho} = \mathbf{t}\_{\text{p2}} \mathbf{v}\_{\text{2;1,3/fre}}^{\Lambda} + \mathbf{t}\_{\text{p3}} \mathbf{v}\_{\text{3;1,2/fre}}^{\Lambda} \tag{156}$$

$$
\boldsymbol{\sigma}^{\boldsymbol{\rho}}\_{\text{P};1/\text{hyd}} = \boldsymbol{t}\_{p2} \boldsymbol{\sigma}^{\boldsymbol{\Delta}}\_{\text{2};1,3/\text{hyd}} + \boldsymbol{t}\_{p3} \boldsymbol{\sigma}^{\boldsymbol{\Delta}}\_{\text{3};1,2/\text{hyd}} \tag{157}
$$

Figure 8. (A) Density of latex as function of polymeric particles concentration. (B) Sound speed as function of polymeric particles concentration. (C) Specific volume of latex as function of mass fraction of solvent (water). (D) Specific adiabatic compressibility as function of the mass fraction of solvent (water). In all figures (□) 0 wt%, (○) 5wt%, (Δ) 10 wt%, (◊) 15 wt %, (◁) 20 wt%, (⬢) 25wt%.

Figure 9. (A) Partial specific volume of the polymeric particles at infinite dilution as function of the polar group content. (B) Partial specific adiabatic compressibility of particles at infinite dilution as function of the polar group content. (C) Partial specific volume of non-polar groups at infinite dilution as function of the polar group content. (D) Partial specific adiabatic compressibility of non-polar groups at infinite dilution as function of the polar group content. (E) Partial specific volume of polar groups at infinite dilution as function of the polar group content. (F) Partial specific adiabatic compressibility of polar groups at infinite dilution as function of the polar group content.

where similar equations can be obtained for the adiabatic compressibilities. Figure 9A and B shows that vo P;<sup>1</sup> and k o sP;<sup>1</sup> decrease when the amount of polar groups increases. This fact indicates an increment of the hydration in the interior of the particle when the amount of polar groups increases. The distribution of this hydration is as follows. Figure 9C and D shows that v<sup>Δ</sup> <sup>2</sup>; <sup>1</sup>, <sup>3</sup> and k Δ <sup>s</sup>2; <sup>1</sup>, <sup>3</sup> decrease from 0 to 15% of polar groups, while Figure 9E and F shows that v<sup>Δ</sup> <sup>3</sup>;1, <sup>2</sup> and k<sup>Δ</sup> s3;1, 2 increase. This fact can be interpreted because the hydration is redistributed from the polar groups to the non-polar groups. In the region of 15–25%, this behaviour is reversed.

## 6. Conclusions

<sup>02468</sup> 0.9955

%, (◁) 20 wt%, (⬢) 25wt%.

cL (g/L **)** 02468

cL (g/L)

**B C D**

1.0030

1.0035

v **(**cm3/g**)**

Figure 8. (A) Density of latex as function of polymeric particles concentration. (B) Sound speed as function of polymeric particles concentration. (C) Specific volume of latex as function of mass fraction of solvent (water). (D) Specific adiabatic compressibility as function of the mass fraction of solvent (water). In all figures (□) 0 wt%, (○) 5wt%, (Δ) 10 wt%, (◊) 15 wt

Figure 9. (A) Partial specific volume of the polymeric particles at infinite dilution as function of the polar group content. (B) Partial specific adiabatic compressibility of particles at infinite dilution as function of the polar group content. (C) Partial specific volume of non-polar groups at infinite dilution as function of the polar group content. (D) Partial specific adiabatic compressibility of non-polar groups at infinite dilution as function of the polar group content. (E) Partial specific volume of polar groups at infinite dilution as function of the polar group content. (F) Partial specific adiabatic compress-

ibility of polar groups at infinite dilution as function of the polar group content.

1.0040

1.0045

0.993 0.996 0.999

0.993 0.996 0.999

**t 1**

4.404

4.412

kS

**(**10-5 cm3 /bar g**)**

4.420

4.428

**t 1**

1509.2

1509.9

1510.6

u (m/s)

1511.3

0.9960

0.9965

(g/cm**)**

p

0.9970

**A**

126 Advances in Titration Techniques

In this chapter, we have developed common thermodynamic bases for isothermal titration calorimetry, densimetry and measurement of sound speed in terms of thermodynamic partial properties (interaction partial enthalpies, partial volumes and partial adiabatic compressibilities). To build these common thermodynamic bases, it is necessary to introduce new concepts, i.e., the concept of fraction of a system and the concept of thermodynamic interaction between components of a system. An advantage of the proposed thermodynamic scheme is the possibility of including new thermodynamic partial properties as partial heat capacities.

## Author details

Mónica Corea<sup>1</sup> , Jean-Pierre E. Grolier<sup>2</sup> and José Manuel del Río3 \*


## References


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[5] Grolier J-PE, del Río JM. On the physical meaning of the isothermal titration calorimetry measurements in calorimeters with full cells. International Journal of Molecular Sciences.

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128 Advances in Titration Techniques


**Titrimetric Principles in Electrolytic Systems**

#### **Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES)** Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES)

DOI: 10.5772/intechopen.69248

Anna Maria Michałowska‐Kaczmarczyk, Aneta Spórna‐Kucab and Tadeusz Michałowski Anna Maria Michałowska-Kaczmarczyk, Aneta Spórna-Kucab and Tadeusz

Additional information is available at the end of the chapter Michałowski

http://dx.doi.org/10.5772/intechopen.69248 Additional information is available at the end of the chapter

### Abstract

The generalized equivalent mass (GEM) concept, based on firm algebraic foundations of the generalized approach to electrolytic systems (GATES), is considered and put against the equivalent "weight" concept, based on a "fragile" stoichiometric reaction notation still advocated by IUPAC. The GEM is formulated a priori, with no relevance to a stoichiometry. GEM is formulated in a unified manner, and referred to systems of any degree of complexity with special emphasis put on redox systems, where generalized electron balance (GEB) is involved. GEM is formulated on the basis of all attainable (and preselected) physicochemical knowledge on the system in question, and resolved with use of iterative computer programs. It is possible to calculate coordinates of the end points taken from the vicinity of equivalence point. This way, one can choose (among others) a proper indicator and the most appropriate (from analytical viewpoint) color change of the indicator. Some interpolative and extrapolative methods of equivalence volume Veq determination are recalled and discussed. The GATES realized for GEM purposes provides the basis for optimization of analytical procedures a priori. The GATES procedure realized for GEM purposes enables to foresee and optimize new analytical methods, or modify, improve, and optimize old analytical methods.

Keywords: equilibrium analysis, mathematical modeling, redox titration curves, equivalence volume, Gran methods

## 1. Introductory remarks

Titrimetry reckons to the oldest analytical methods, still widely used because of high precision, accuracy, convenience, and affordability [1]. Nowadays, according to Comité Consultatif pour la

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Quantité de la Matière (CCQM) opinion [2], it is considered as one of the primary methods of analysis i.e., it fulfills the demands of the highest metrological qualities. Titration is then perceived as a very simple and reliable technique, applied in different areas of chemical analysis. A physical chemist may perform a titration in order to determine equilibrium constants, whereas an analytical chemist performs a titration in order to determine the concentration of one or several components in a sample.

In a typical titration, V0 mL of titrand (D) containing the analyte A of an unknown (in principle) concentration C0 is titrated with V mL of titrant (T) containing the reagent B (C); V is the total volume of T added into D from the very beginning to a given point of the titration, where total volume of D þ T mixture is V0 þ V, if the volume additivity condition is fulfilled. Symbolically, the titration T ! D in such systems will be denoted as B(C,V) ! A(C0,V0). Potentiometric acid-base pH titrations are usually carried out by using combined (glass þ reference) electrode, responding to hydrogen-ion activity rather than hydrogen-ion concentration. Potentiometric titrations in redox systems are made with use of redox indicator electrodes (RIE) e.g., combined (Pt þ reference) electrode [3–5]. For detection of specific ions in a mixture, ion-selective electrodes (ISE) are also used [5]. The degree of advancement of the reaction between B and A is the fraction titrated [6], named also as the degree of titration, and expressed as the quotient Φ ¼ nB/nA of the numbers of mmoles: nB ¼ C�V of B and nA ¼ C0�V0 of A, i.e.,

$$
\Phi = \frac{\mathbf{C} \cdot \mathbf{V}}{\mathbf{C}\_0 \cdot \mathbf{V}\_0} \tag{1}
$$

We refer here to visual, pH, and potentiometric (E) titrations. The functional relationships between potential E or pH of a solution versus V or Φ, i.e., E ¼ E(V) or E ¼ E(Φ) and pH ¼ pH(V) or pH ¼ pH(Φ) functions, are expressed by continuous plots named as the related titration curves. The Φ provides a kind of normalization in visual presentation of the appropriate system. In the simplest case of acid-base systems, it is much easier to formulate the functional relationship Φ ¼ Φ(pH), not pH ¼ pH(Φ). In particular, the expression for Φ depends on the composition of D and T, see Appendix.

The detailed considerations in this chapter are based on principles of the generalized approach to electrolytic systems (GATES), formulated by Michałowski [9] and presented recently in a series of papers, related to redox [7–26] and nonredox systems [27–32] in aqueous and in mixed-solvent media [33–37]. The closed system separated from its environment by diathermal walls secure a heat exchange between the system and its environment, and realize dynamic processes in a quasistatic manner under isothermal conditions.

The mathematical description of electrolytic nonredox systems within GATES is based on general rules of charge and elements conservation. Nonredox systems are formulated with use of charge (ChB) and concentration balances f(Yg), for elements/cores Yg 6¼ H, O. The description of redox systems is complemented by generalized electron balance (GEB) concept, discovered by Michałowski as the Approach I to GEB (1992) and the Approach II to GEB (2006); GEB is considered as a law of a matter conservation, as the law of nature [7, 9, 11, 13, 25].

Formulation of redox systems according to GATES principles is denoted as GATES/GEB. Within the Approach II to GEB, based on linear combination 2�f(O) – f(H) of the balances: f(H) for H and f(O) for O, the prior knowledge of oxidation degrees of all elements constituting the system is not needed; oxidants and reductants are not indicated. Moreover, the linear independency or dependency of 2�f(O) – f(H) from other balances: ChB and f(Yg), is the general criterion distinguishing between redox and nonredox systems. Concentrations of the species within the balances are interrelated in a complete set of equations for equilibrium constants, formulated according to the mass action law principles. The GATES and GATES/GEB in particular, provide the best possible tool applicable for thermodynamic resolution of electrolytic systems of any degree of complexity, with the possibility of application of all physicochemical knowledge involved.

Several methods of equivalence volume (Veq) determination are also presented in terms of the generalized equivalence mass (GEM) [8] concept, suggested by Michałowski (1979), with an emphasis put on the Gran methods and their modifications. The GEM concept has no relevance to a chemical reaction notation. Within GATES, the chemical reaction notation is only the basis to formulate the expression for the related equilibrium constant.

## 2. Formulation of generalized equivalent mass (GEM)

Quantité de la Matière (CCQM) opinion [2], it is considered as one of the primary methods of analysis i.e., it fulfills the demands of the highest metrological qualities. Titration is then perceived as a very simple and reliable technique, applied in different areas of chemical analysis. A physical chemist may perform a titration in order to determine equilibrium constants, whereas an analytical chemist performs a titration in order to determine the concentra-

In a typical titration, V0 mL of titrand (D) containing the analyte A of an unknown (in principle) concentration C0 is titrated with V mL of titrant (T) containing the reagent B (C); V is the total volume of T added into D from the very beginning to a given point of the titration, where total volume of D þ T mixture is V0 þ V, if the volume additivity condition is fulfilled. Symbolically, the titration T ! D in such systems will be denoted as B(C,V) ! A(C0,V0). Potentiometric acid-base pH titrations are usually carried out by using combined (glass þ reference) electrode, responding to hydrogen-ion activity rather than hydrogen-ion concentration. Potentiometric titrations in redox systems are made with use of redox indicator electrodes (RIE) e.g., combined (Pt þ reference) electrode [3–5]. For detection of specific ions in a mixture, ion-selective electrodes (ISE) are also used [5]. The degree of advancement of the reaction between B and A is the fraction titrated [6], named also as the degree of titration, and expressed as the quotient Φ ¼ nB/nA of the numbers of mmoles: nB ¼ C�V of B and nA ¼ C0�V0

> <sup>Φ</sup> <sup>¼</sup> <sup>C</sup> � <sup>V</sup> C0 � V0

We refer here to visual, pH, and potentiometric (E) titrations. The functional relationships between potential E or pH of a solution versus V or Φ, i.e., E ¼ E(V) or E ¼ E(Φ) and pH ¼ pH(V) or pH ¼ pH(Φ) functions, are expressed by continuous plots named as the related titration curves. The Φ provides a kind of normalization in visual presentation of the appropriate system. In the simplest case of acid-base systems, it is much easier to formulate the functional relationship Φ ¼ Φ(pH), not pH ¼ pH(Φ). In particular, the expression for Φ depends on the composition of D and T, see

The detailed considerations in this chapter are based on principles of the generalized approach to electrolytic systems (GATES), formulated by Michałowski [9] and presented recently in a series of papers, related to redox [7–26] and nonredox systems [27–32] in aqueous and in mixed-solvent media [33–37]. The closed system separated from its environment by diathermal walls secure a heat exchange between the system and its environment, and realize dynamic processes in a

The mathematical description of electrolytic nonredox systems within GATES is based on general rules of charge and elements conservation. Nonredox systems are formulated with use of charge (ChB) and concentration balances f(Yg), for elements/cores Yg 6¼ H, O. The description of redox systems is complemented by generalized electron balance (GEB) concept, discovered by Michałowski as the Approach I to GEB (1992) and the Approach II to GEB (2006); GEB is considered as a law of a matter conservation, as the law of nature

ð1Þ

tion of one or several components in a sample.

134 Advances in Titration Techniques

quasistatic manner under isothermal conditions.

of A, i.e.,

Appendix.

[7, 9, 11, 13, 25].

The main task of titration is the estimation of the equivalence volume, Veq, corresponding to the volume V ¼ Veq of T, where the fraction titrated (1) assumes the value

$$
\Phi\_{\text{eq}} = \frac{\mathbf{C} \cdot \mathbf{V}\_{\text{eq}}}{\mathbf{C}\_0 \cdot \mathbf{V}\_0} \tag{2}
$$

In contradistinction to visual titrations, where the end volume Ve ffi Veq is registered, all instrumental titrations aim, in principle, to obtain the Veq value on the basis of experimental data {(Vj , yj )|j ¼ 1,…,N}, where y ¼ pH, E for potentiometric methods of analysis. Referring to Eq. (1), we have

$$\mathbf{C}\_0 \cdot \mathbf{V}\_0 = 10^3 \cdot \mathbf{m}\_{\rm A} / \mathbf{M}\_{\rm A} \tag{3}$$

where mA [g] and MA [g/mol] denote mass and molar mass of analyte (A), respectively. From Eqs. (1) and (3), we get

$$\mathbf{m\_A} = 10^{-3} \cdot \mathbb{C} \cdot \mathbf{M\_A} \cdot \mathbf{V} / \Phi \tag{4}$$

The value of the fraction V/Φ in Eq. (4), obtained from Eq. (1),

$$\mathbf{V}/\Phi = \mathbf{C}\_0 \cdot \mathbf{V}\_0/\mathbf{C} \tag{5}$$

is constant during the titration. Particularly, at the end (e) and equivalence (eq) points, we have

$$\mathbf{V}/\Phi = \mathbf{V}\_{\mathbf{e}}/\Phi\_{\mathbf{e}} = \mathbf{V}\_{\mathbf{eq}}/\Phi\_{\mathbf{eq}} \tag{6}$$

The Ve [mL] value is the volume of T consumed up to the end (e) point, where the titration is terminated (ended). The Ve value is usually determined in visual titration, when a preassumed color (or color change) of D þ T mixture is obtained. In a visual acid-base titration, pHe value corresponds to the volume Ve (mL) of T added from the start for the titration and

$$\Phi\_{\mathbf{e}} = \frac{\mathbf{C} \cdot \mathbf{V}\_{\mathbf{e}}}{\mathbf{C}\_0 \cdot \mathbf{V}\_0} \tag{7}$$

is the Φ-value related to the end point. From Eqs. (4) and (6), one obtains:

$$\mathbf{m\_A} = 10^{-3} \cdot \mathbf{C} \cdot \mathbf{V\_e} \cdot \frac{\mathbf{M\_A}}{\Phi\_e} \tag{8a}$$

$$\mathbf{m\_A} = 10^{-3} \cdot \mathbf{C} \cdot \mathbf{V\_{eq}} \cdot \frac{\mathbf{M\_A}}{\Phi\_{\text{eq}}} \tag{8b}$$

This does not mean that we may choose between the two formulas: (8a) and (8b), to calculate mA. Namely, Eq. (8a) cannot be applied for the evaluation of mA: Ve is known, but Φ<sup>e</sup> unknown; calculation of Φ<sup>e</sup> needs prior knowledge of C0 value; e.g., for the titration NaOH (C,V) ! HCl(C0,V0), see Appendix, we have

$$\boldsymbol{\Phi}\_{\text{e}} = \frac{\mathbb{C}}{\mathbb{C}\_{0}} \times \frac{\mathbb{C}\_{0} - \alpha\_{\text{e}}}{\mathbb{C} + \alpha\_{\text{e}}} \text{ where } \alpha \text{(Appendix), \text{and } \alpha\_{\text{e}} = \alpha (\text{pH}\_{\text{e}}) \tag{9}$$

However, C0 is unknown before the titration; otherwise, the titration would be purposeless. The approximate pHe value is known in visual titration. Also Eq. (8b) is useless: the "round" Φeq value is known exactly, but Veq is unknown; Ve (not Veq) is determined in visual titrations.

Because Eqs. (8a) and (8b) appear to be useless, the third, approximate formula for mA, has to be applied, namely:

$$\mathbf{m\_A}' \cong \mathbf{10^{-3}} \cdot \mathbf{C} \cdot \mathbf{V\_e} \cdot \mathbf{M\_A}/\Phi\_{\mathrm{eq}} = \mathbf{10^{-3}} \cdot \mathbf{C} \cdot \mathbf{V\_e} \cdot \mathbf{R\_A}^{\mathrm{eq}} \tag{10}$$

where Φeq is put for Φ<sup>e</sup> in Eq. (8a), and

$$\mathbf{R\_A}^{\text{eq}} = \frac{\mathbf{M\_A}}{\mathbf{0\_{eq}}} \tag{11}$$

is named as the equivalent mass. The relative error in accuracy, resulting from this substitution, equals to

$$\delta = (\mathbf{m}\_{\rm A} \,' - \mathbf{m}\_{\rm A}) / \mathbf{m}\_{\rm A} = \mathbf{m}\_{\rm A} \,' / \mathbf{m}\_{\rm A} - 1 = \mathbf{V}\_{\rm e} / \mathbf{V}\_{\rm eq} - 1 = \Phi\_{\rm e} / \Phi\_{\rm eq} - 1 \tag{12}$$

For Φ<sup>e</sup> ¼ Φeq, we get δ ¼ 0 and mA' ¼ mA; thus Φ<sup>e</sup> ffi Φeq (i.e., Ve ffi Veq) corresponds to mA' ffi mA. A conscious choice of an indicator and a pH-range of its color change during the titration is possible on the basis of analysis of the related titration curve. From Eqs. (10) and (8b), we get

$$\mathbf{m\_A} = \mathbf{m\_A}'/(1+\mathfrak{d}) = \mathbf{m\_A}' \cdot \left(1-\mathfrak{d}+\mathfrak{d}^2-\dots\right) \tag{13}$$

## 3. Accuracy and precision

V=Φ ¼ Ve=Φ<sup>e</sup> ¼ Veq=Φeq ð6Þ

ð7Þ

ð8aÞ

ð8bÞ

ð11Þ

The Ve [mL] value is the volume of T consumed up to the end (e) point, where the titration is terminated (ended). The Ve value is usually determined in visual titration, when a preassumed color (or color change) of D þ T mixture is obtained. In a visual acid-base titration, pHe value

> <sup>Φ</sup><sup>e</sup> <sup>¼</sup> <sup>C</sup> � Ve C0 � V0

mA <sup>¼</sup> <sup>10</sup>�<sup>3</sup> � <sup>C</sup> � Ve �

mA <sup>¼</sup> <sup>10</sup>�<sup>3</sup> � <sup>C</sup> � Veq �

This does not mean that we may choose between the two formulas: (8a) and (8b), to calculate mA. Namely, Eq. (8a) cannot be applied for the evaluation of mA: Ve is known, but Φ<sup>e</sup> unknown; calculation of Φ<sup>e</sup> needs prior knowledge of C0 value; e.g., for the titration NaOH

where α Appendix

However, C0 is unknown before the titration; otherwise, the titration would be purposeless. The approximate pHe value is known in visual titration. Also Eq. (8b) is useless: the "round" Φeq value is known exactly, but Veq is unknown; Ve (not Veq) is determined in visual titrations. Because Eqs. (8a) and (8b) appear to be useless, the third, approximate formula for mA, has to

> RAeq <sup>¼</sup> MA Φeq

is named as the equivalent mass. The relative error in accuracy, resulting from this substitu-

δ ¼ ð Þ mA' � mA =mA ¼ mA'=mA � 1 ¼ Ve=Veq � 1 ¼ Φe=Φeq � 1 ð12Þ

MA Φ<sup>e</sup>

> MA Φeq

mA' ffi <sup>10</sup>–<sup>3</sup> � <sup>C</sup> � Ve � MA=Φeq <sup>¼</sup> <sup>10</sup>–<sup>3</sup> � <sup>C</sup> � Ve � RAeq <sup>ð</sup>10<sup>Þ</sup>

, and <sup>α</sup><sup>e</sup> <sup>¼</sup> <sup>α</sup>ðpHeÞ ð9<sup>Þ</sup>

corresponds to the volume Ve (mL) of T added from the start for the titration and

is the Φ-value related to the end point. From Eqs. (4) and (6), one obtains:

(C,V) ! HCl(C0,V0), see Appendix, we have

<sup>Φ</sup><sup>e</sup> <sup>¼</sup> <sup>C</sup> C0 �

where Φeq is put for Φ<sup>e</sup> in Eq. (8a), and

be applied, namely:

136 Advances in Titration Techniques

tion, equals to

C0 � α<sup>e</sup> C þ α<sup>e</sup>

In everyday conversation, the terms "accuracy" and "precision" are often used interchangeably, but in science—and analytical chemistry, in particular—they have very specific, and different definitions [38].

Accuracy refers to how close a result of measurement, e.g., expressed by concentration x (as an intensive variable), agrees with a known/true value x0 of x in a sample tested. In N repeated trials made on this sample, we obtain xj (j <sup>¼</sup> 1, …, N) and then the mean value x and variance s<sup>2</sup> are obtained

$$\overline{\mathbf{x}} = \frac{1}{\mathbf{N}} \cdot \sum\_{\mathbf{j}=1}^{N} \mathbf{x}\_{\mathbf{j}} \cdot \mathbf{s}^2 = \frac{1}{\mathbf{N} - \mathbf{1}} \cdot \sum\_{\mathbf{j}=1}^{N} (\mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_0)^2 \tag{14}$$

The accuracy can be defined by the absolute value |x � x0|, whereas precision is defined by standard deviation, s <sup>¼</sup> (s<sup>2</sup> ) 1/2; the accuracy and precision are brought here into the same units.

Accuracy and precision are the terms of (nearly) equal importance (weights: 1 and (1 – 1/N) for the weighted sum of squares [39]) when involved in the relation [40, 41]

$$\frac{1}{N} \cdot \sum\_{j=1}^{N} \left(\mathbf{x}\_{j} - \mathbf{x}\_{0}\right)^{2} = 1 \cdot \left(\overline{\mathbf{x}} - \mathbf{x}\_{0}\right)^{2} + \left(1 - 1/N\right) \cdot \mathbf{s}^{2} \tag{15}$$

where xj — experimental (j <sup>¼</sup> 1, …, N) and true (x0) values for x, <sup>x</sup> — mean value, s2 — variance. The problem referred to accuracy and precision of different methods of Veq determination has been raised, e.g., in Refs. [42, 43].

Accuracy and precision of the results obtained from titrimetric analyses depend both on a nature of D þ T system considered and the method of Veq evaluation. Herein, the kinetics of chemical reactions and transportation phenomena are of paramount importance.

## 4. The E ¼ E(Φ) and/or pH ¼ pH(Φ) functions

Relatively simple, functional relationships for Φ ¼ Φ(pH), ascribed to acid-base D þ T systems, are specified in an elegant/compact form in Refs. [6, 27, 28, 30], see Appendix.

In acid-base systems occurred in aqueous media, pH is a monotonic function of V or Φ. From the relation,

$$\frac{\text{d}\text{pH}}{\text{d}\text{d}\text{O}} = \frac{\text{d}\text{pH}}{\text{dV}} \cdot \frac{\text{dV}}{\text{d}\text{O}} = \frac{\text{C}\_0 \cdot \text{V}\_0}{\text{C}} \cdot \frac{\text{d}\text{pH}}{\text{dV}}\tag{16}$$

it results that the Φ ¼ Φ(pH) and pH ¼ pH(Φ) relationships are mutually interchangeable, C0V0/C > 0. The relation (16) can be extended on other plots.

Explicit formulation of functional relationships: Φ ¼ Φ(pH) and E ¼ E(Φ), is impossible in complex systems, where two or more different kinds (acid-base, redox, complexation, precipitation, liquid-liquid phase equilibria [44, 45]) of chemical reactions occur sequentially or/and simultaneously [8]. The E values are referred to SHE scale.

Monotonicity of pH ¼ pH(Φ) and/or E ¼ E(Φ) is not a general property in electrolytic redox systems. In Figure 1, the monotonic growth of E ¼ E(Φ), i.e., dE/dΦ > 0, is accompanied by monotonic growth of pH ¼ pH(Φ), i.e., dpH/dΦ > 0 [20].

In Figure 2, the monotonic drops of E ¼ E(Φ), i.e., dE/dΦ < 0, are accompanied by nonmonotonic changes of pH ¼ pH(Φ) [9, 46, 47].

From inspection of Figure 2B, it results that the neighboring, quasi linear segments of the line (at CHg ¼ 0) intersect at the equivalent points Φeq1 ¼ 2.5 and Φeq2 ¼ 3.0. So, it might seem that the pH titration is an alternative to the potentiometric titration method for the Veq detection. It should be noted, however, that there are small changes within the pH range, where the characteristics of glass electrode is nonlinear, and an extended calibration procedure of this electrode is required. The opportunities arising from potential E measurement are here incomparably higher, so the choice of potentiometric titration is obvious.

In Figure 3, the nonmonotonic changes of E ¼ E(V) are accompanied by nonmonotonic changes of pH ¼ pH(V) [16].

The unusual shape of the respective plots for E ¼ E(Φ) and pH ¼ pH(Φ) is shown in Figure 4 [13].

Figure 1. The collected (A) E ¼ E(Φ) and (B) pH ¼ pH(Φ) curves plotted for D þ T system KMnO4 (C) ! FeSO4 (C0) þ H2SO4 (C01) at V0 ¼ 100, C0 ¼ 0.01, C ¼ 0.02, and different C01 values, indicated in Figures (B), (C), and (D) (in enlarged scales), before and after Φ ¼ Φeq ¼ 0.2.

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 139

dpH <sup>d</sup><sup>Φ</sup> <sup>¼</sup> dpH dV �

C0V0/C > 0. The relation (16) can be extended on other plots.

simultaneously [8]. The E values are referred to SHE scale.

monotonic growth of pH ¼ pH(Φ), i.e., dpH/dΦ > 0 [20].

parably higher, so the choice of potentiometric titration is obvious.

changes of pH ¼ pH(Φ) [9, 46, 47].

138 Advances in Titration Techniques

changes of pH ¼ pH(V) [16].

scales), before and after Φ ¼ Φeq ¼ 0.2.

dV

it results that the Φ ¼ Φ(pH) and pH ¼ pH(Φ) relationships are mutually interchangeable,

Explicit formulation of functional relationships: Φ ¼ Φ(pH) and E ¼ E(Φ), is impossible in complex systems, where two or more different kinds (acid-base, redox, complexation, precipitation, liquid-liquid phase equilibria [44, 45]) of chemical reactions occur sequentially or/and

Monotonicity of pH ¼ pH(Φ) and/or E ¼ E(Φ) is not a general property in electrolytic redox systems. In Figure 1, the monotonic growth of E ¼ E(Φ), i.e., dE/dΦ > 0, is accompanied by

In Figure 2, the monotonic drops of E ¼ E(Φ), i.e., dE/dΦ < 0, are accompanied by nonmonotonic

From inspection of Figure 2B, it results that the neighboring, quasi linear segments of the line (at CHg ¼ 0) intersect at the equivalent points Φeq1 ¼ 2.5 and Φeq2 ¼ 3.0. So, it might seem that the pH titration is an alternative to the potentiometric titration method for the Veq detection. It should be noted, however, that there are small changes within the pH range, where the characteristics of glass electrode is nonlinear, and an extended calibration procedure of this electrode is required. The opportunities arising from potential E measurement are here incom-

In Figure 3, the nonmonotonic changes of E ¼ E(V) are accompanied by nonmonotonic

The unusual shape of the respective plots for E ¼ E(Φ) and pH ¼ pH(Φ) is shown in Figure 4 [13].

Figure 1. The collected (A) E ¼ E(Φ) and (B) pH ¼ pH(Φ) curves plotted for D þ T system KMnO4 (C) ! FeSO4 (C0) þ H2SO4 (C01) at V0 ¼ 100, C0 ¼ 0.01, C ¼ 0.02, and different C01 values, indicated in Figures (B), (C), and (D) (in enlarged

<sup>d</sup><sup>Φ</sup> <sup>¼</sup> C0 � V0

C �

dpH

dV <sup>ð</sup>16<sup>Þ</sup>

Figure 2. The theoretical plots of (A) E ¼ E(Φ) and (B) pH ¼ pH(Φ) functions for the D þ T system, with KIO3 (C0 ¼ 0.01) þ HCl (C01 ¼ 0.02) þ H2SeO3 (CSe ¼ 0.02) þ HgCl2 (CHg) as D, and ascorbic acid C6H8O6 (C ¼ 0.1) as T; V0 ¼ 100, and (a) CHg ¼ 0, (b) CHg ¼ 0.07.

Figure 3. The theoretical plots of (A) E ¼ E(V) and (B) pH ¼ pH(V) functions for the system with V0 ¼ 100 mL of NaBr (C0 ¼ 0.01) þ Cl2 (C02) as D titrated with V mL of KBrO3 (C ¼ 0.1) as T, at indicated (a, b, c) C02 values.

Other examples of the nonmonotonicity were presented in Refs. [7, 9, 46–49]. The nonmonotonic pH versus V relationships were also stated in experimental pH titrations made in some binary-solvent media [33]. Then, the Gran's statement "all titration curves are monotonic" [50] is not true, in general.

Figure 4. The plots of (A) E ¼ E(Φ) and (B) pH ¼ pH(Φ) functions for the system HI (C ¼ 0.1) ! KIO3 (C<sup>0</sup> ¼ 0.01).

## 5. Location of inflection and equivalence points

Some of the E ¼ E(Φ) and/or pH ¼ pH(Φ) (or E ¼ E(V) and/or pH ¼ pH(V)) functions have inflection point(s), and characteristic S-shape (or reverse S-shape) is assumed within defined Φ (or V) range [51].

Generalizing, let us introduce the functions y ¼ y(V), where y ¼ E or pH and denote V ¼ VIP, with the volume referred to inflection point (IP) [52, 53], i.e., the point (VIP, yIP) of maximal slope |η|

$$\eta = \frac{\text{dy}}{\text{dV}} = \frac{1}{\text{dV/dy}} \tag{17}$$

on the related curve y ¼ y(V) (y ¼ E, pH), plotted in normal coordinates (V, y) or their derivatives: dy/dV <sup>¼</sup> y1(V) and d<sup>2</sup> y/dV<sup>2</sup> <sup>¼</sup> y2(V) on the ordinate. We have, by turns [54],

$$\frac{\text{d}^2 \text{y}}{\text{d} \text{V}^2} = -\frac{1}{\left(\text{dV/dy}\right)^3} \cdot \frac{\text{d}^2 \text{V}}{\text{dy}^2} \tag{18a}$$

$$\frac{\mathbf{d}^2 \mathbf{y}}{\mathbf{d} \mathbf{V}^2} + \eta^3 \cdot \frac{\mathbf{d}^2 \mathbf{V}}{\mathbf{d} \mathbf{y}^2} = 0 \tag{18b}$$

At <sup>η</sup> 6¼ 0, from Eq. (18b), we get d<sup>2</sup> V/dy<sup>2</sup> <sup>¼</sup> 0. Analogously to Eq. (16), we have

$$\frac{\mathbf{dE}}{\mathbf{d}\Phi} = \frac{\mathbf{C}\_0 \cdot \mathbf{V}\_0}{\mathbf{C}} \cdot \frac{\mathbf{dE}}{\mathbf{dV}}$$

At the inflection point on the curve y <sup>¼</sup> y(V), we have maxima for dy/d<sup>Φ</sup> and d<sup>2</sup> y/dV<sup>2</sup> <sup>¼</sup> 0, see Figure 5 for y ¼ E [55].

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 141

Figure 5. The function (A) E ¼ E(Φ) and the difference quotient DE/DΦ ¼ (Ejþ<sup>1</sup> � Ej)/(Φjþ<sup>1</sup> � Φ<sup>j</sup> ) versus (Φjþ<sup>1</sup> þ Φ<sup>j</sup> )/2 relationships in the vicinity of Φ ¼ 0.2 (B) and Φ ¼ 0.5 (C) plotted for the system KIO3 (C ¼ 0.1) –> KI (C<sup>0</sup> ¼ 0.01) þ HCl (C<sup>01</sup> ¼ 0.2).

Referring to examples presented in Figures 1A and 2A, we see that the inflection points (ΦIP, EIP) have the abscissas close to the related equivalence points (Φeq, Eeq), namely:

(0.2, 1.034)—see Table 1 and Figure 1A;

5. Location of inflection and equivalence points

(or V) range [51].

140 Advances in Titration Techniques

derivatives: dy/dV <sup>¼</sup> y1(V) and d<sup>2</sup>

At <sup>η</sup> 6¼ 0, from Eq. (18b), we get d<sup>2</sup>

Figure 5 for y ¼ E [55].

Some of the E ¼ E(Φ) and/or pH ¼ pH(Φ) (or E ¼ E(V) and/or pH ¼ pH(V)) functions have inflection point(s), and characteristic S-shape (or reverse S-shape) is assumed within defined Φ

Figure 4. The plots of (A) E ¼ E(Φ) and (B) pH ¼ pH(Φ) functions for the system HI (C ¼ 0.1) ! KIO3 (C<sup>0</sup> ¼ 0.01).

Generalizing, let us introduce the functions y ¼ y(V), where y ¼ E or pH and denote V ¼ VIP, with the volume referred to inflection point (IP) [52, 53], i.e., the point (VIP, yIP) of maximal slope |η|

dV <sup>¼</sup> <sup>1</sup>

on the related curve y ¼ y(V) (y ¼ E, pH), plotted in normal coordinates (V, y) or their

ðdV=dyÞ

d2 V

C �

3 � d2 V

V/dy<sup>2</sup> <sup>¼</sup> 0. Analogously to Eq. (16), we have

dE dV

dV=dy <sup>ð</sup>17<sup>Þ</sup>

dy<sup>2</sup> <sup>ð</sup>18a<sup>Þ</sup>

y/dV<sup>2</sup> <sup>¼</sup> 0, see

dy2 <sup>¼</sup> <sup>0</sup> <sup>ð</sup>18b<sup>Þ</sup>

y/dV<sup>2</sup> <sup>¼</sup> y2(V) on the ordinate. We have, by turns [54],

<sup>η</sup> <sup>¼</sup> dy

d2 y dV<sup>2</sup> ¼ � <sup>1</sup>

> d2 y dV<sup>2</sup> <sup>þ</sup> <sup>η</sup><sup>3</sup> �

dE

At the inflection point on the curve y <sup>¼</sup> y(V), we have maxima for dy/d<sup>Φ</sup> and d<sup>2</sup>

<sup>d</sup><sup>Φ</sup> <sup>¼</sup> C0 � V0

(2.5, 0.903), (3.0, 0.414)—see Table 2 and the curve a in Figure 2A;

(3.0, 0.652)—see Table 2 and the curve b in Figure 2A;

Then we can consider Φeq (Eq. (2)) as a ratio of small natural numbers: p and q, i.e.,

$$\Phi\_{\rm eq} = \frac{\mathbf{p}}{\mathbf{q}} \quad (\mathbf{p}, \ \mathbf{q} \in \mathcal{N}) \tag{19}$$

e.g., Φeq ¼ 1 (¼1/1) for titration in D þ T system with A ¼ HCl and B ¼ NaOH (see Eq. (9)); Φeq ¼ 1/5 ¼ 0.2 in Figure 1A (see Table 1); Φeq ¼ 5/2 ¼ 2.5 or Φeq ¼ 3/1 ¼ 3 in Figure 2A (see Table 2).


Table 1. The (Φ, E) values related to C<sup>01</sup> ¼ 0 and other data presented in legend for Figure 1A.


Table 2. The (Φ, E) values related to the data presented in legend for Figure 2A.

As we see (Eq. 12), the Φ<sup>e</sup> values are compared each time with the "round" Φeq ¼ p/q value for Φ<sup>e</sup> due to the fact that just Φeq is placed in the denominator of the expression for the equivalent mass, RA eq (Eq. (11)).

The Φ<sup>e</sup> values, presented in Tables 1 and 2 refer—in any case—to the close vicinity of the Φeq value(s), see e.g. Φeq1 ¼ 2.5 and Φeq2 ¼ 3.0.

Then from Figures 1A and 2A, it results that location of IP is an interpolative method and VIP ffi Veq [56], but in practice, this assumption may appear to be a mere fiction, especially in context with accuracy of measurements.

## 6. The case of diluted solutions

The Veq and VIP do not overlap in the systems of diluted solutions. For titration of V0 mL of HB (C0) with V mL of MOH (C), we have [6, 57]

$$\mathbf{V\_{eq}} - \mathbf{V\_{IP}} = \frac{\mathbf{x\_{IP}}}{1 + \mathbf{x\_{IP}}} \cdot (\mathbf{C\_0/C} + 1) \cdot \mathbf{V\_0} \tag{20}$$

where

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 143

$$\chi\_{\rm IP} = \frac{8\mathbf{K}\_W}{\mathbf{C}^2} + \left(\frac{8\mathbf{K}\_W}{\mathbf{C}^2}\right)^2 + \dots \tag{21}$$

and KW <sup>¼</sup> [Hþ<sup>1</sup> ][OH�<sup>1</sup> ]. Similar relationship occurs for AgNO3 (C,V) ! NaCl (C0,V0) system; in this case, the relations [57]: Eq. (20) and

$$\chi\_{\rm IP} = \frac{8\mathbf{K\_{sp}}}{\mathbf{C^2}} + \left(\frac{8\mathbf{K\_{sp}}}{\mathbf{C^2}}\right)^2 + \dots \tag{22}$$

where Ksp <sup>¼</sup> [Agþ<sup>1</sup> ][Cl�<sup>1</sup> ], are valid.

## 7. Some interpolative methods of Veq determination

#### 7.1. The Michałowski method

Two interpolative methods, not based on the IP location, were presented by Fortuin [58] and Michałowski [6, 57]. The Fortuin method is based on an nomogram; an extended form of Fortuin's nomogram was prepared by the author of Ref. [6]. The Michałowski and Fortuin methods are particularly applicable to NaOH (C,V) ! HCl (C0,V0) and NaOH (C,V) ! HCl (C0,V0) systems. However, the applicability of the Michałowski method is restricted to diluted D and T, where the Fortuin method is invalid. In the Michałowski method, Veq is the real and positive root of the equation

$$(1 - 2\mathbf{a}) \cdot \mathbf{V\_{eq}}^3 + (2 - 3\mathbf{a}) \cdot \mathbf{V\_0} \cdot \mathbf{V\_{eq}}^2 + \mathbf{V\_0}^2 \cdot \mathbf{V\_{eq}} - \mathbf{a} \cdot \mathbf{V\_0}^3 = \mathbf{0} \tag{23}$$

where

As we see (Eq. 12), the Φ<sup>e</sup> values are compared each time with the "round" Φeq ¼ p/q value for Φ<sup>e</sup> due to the fact that just Φeq is placed in the denominator of the expression for the equivalent

CHg ¼ 0 CHg ¼ 0.07

Φ E Φ E Φ E 2.45 1.004 2.95 0.632 2.95 0.97 2.475 1 2.975 0.62 2.975 0.96 2.49 0.995 2.99 0.607 2.99 0.947 2.492 0.994 2.992 0.604 2.992 0.944 2.494 0.992 2.994 0.6 2.994 0.94 2.496 0.989 2.996 0.595 2.996 0.935 2.498 0.983 2.998 0.586 2.998 0.926 2.5 0.903 3 0.414 3 0.652 2.502 0.809 3.002 0.38 3.002 0.379 2.504 0.791 3.004 0.371 3.004 0.371 2.506 0.781 3.006 0.365 3.006 0.365 2.508 0.774 3.008 0.362 3.008 0.362 2.51 0.768 3.01 0.359 3.01 0.359 2.525 0.744 3.03 0.345 3.03 0.345 2.55 0.727 3.06 0.336 3.06 0.336

The Φ<sup>e</sup> values, presented in Tables 1 and 2 refer—in any case—to the close vicinity of the Φeq

Then from Figures 1A and 2A, it results that location of IP is an interpolative method and VIP ffi Veq [56], but in practice, this assumption may appear to be a mere fiction, especially in

The Veq and VIP do not overlap in the systems of diluted solutions. For titration of V0 mL of HB

� ðC0=C þ 1Þ � V0 ð20Þ

1 þ xIP

Veq � VIP <sup>¼</sup> xIP

mass, RA

where

eq (Eq. (11)).

142 Advances in Titration Techniques

value(s), see e.g. Φeq1 ¼ 2.5 and Φeq2 ¼ 3.0.

Table 2. The (Φ, E) values related to the data presented in legend for Figure 2A.

context with accuracy of measurements.

6. The case of diluted solutions

(C0) with V mL of MOH (C), we have [6, 57]

$$\mathbf{a} = \frac{1}{3} \cdot \frac{3\mathbf{A}\_0 - 2\mathbf{V}\_0 \mathbf{A}\_1 + \mathbf{V}\_0^2 \mathbf{A}\_2}{\mathbf{A}\_0 - \mathbf{V}\_0 \mathbf{A}\_1 + \mathbf{V}\_0^3 \mathbf{A}\_3} \tag{24}$$

and A0, A1, A2, A3 are obtained from results {(Vj , Ej )|j ¼ 1,…, N} of potentiometric titration, after applying the least squares method (LSM) to the function

$$\left(\mathbf{1} + \frac{\mathbf{V}}{\mathbf{V}\_0}\right)^3 \cdot \mathbf{E} = \sum\_{i=0}^3 \mathbf{A}\_i \cdot \mathbf{V}^i \tag{25}$$

A useful criterion of validity of the Veq value are: pK ¼ – log K (K ¼ KW or Ksp) and standard redox potential (E0), calculated from the formulas [59]:

$$\mathbf{p}\mathbf{K} = \log\left(\frac{24}{\mathbf{C}^2}\right) + \log\left(-\frac{\mathbf{a}\_3}{\mathbf{a}\_1}\right);\ \mathbf{E}\_0 = \mathbf{a}\_0 + \frac{\mathbf{R}\mathbf{T}}{2\mathbf{F}} \cdot \ln 10 \cdot \mathbf{p}\mathbf{K} \tag{26}$$

where

$$\mathbf{a}\_{3} = \frac{\mathbf{V}\_{0}^{3}}{\mathfrak{N}\_{\mathrm{eq}}} \cdot \frac{3\mathbf{A}\_{0} - 2\mathbf{V}\_{0}\mathbf{A}\_{1} + \mathbf{V}\_{0}^{2}\mathbf{A}\_{2}}{\left(\mathbf{V}\_{0} + \mathbf{V}\_{\mathrm{eq}}\right)^{2}};\\\mathbf{a}\_{1} = \frac{3\mathbf{a}\_{3}\mathbf{V}\_{\mathrm{eq}}}{2\mathbf{V}\_{0}^{2}} \cdot \left(\mathbf{V}\_{0} - \mathbf{V}\_{\mathrm{eq}}\right) + \frac{\mathbf{V}\_{0}}{2} \cdot \frac{3\mathbf{A}\_{0} - \mathbf{A}\_{2}\mathbf{V}\_{0}^{2}}{\mathbf{V}\_{0} + \mathbf{V}\_{\mathrm{eq}}};\\\mathbf{a}\_{0} = \mathbf{V}\_{0}^{3} \cdot \mathbf{A}\_{3} + \mathbf{a}\_{1} + \mathbf{a}\_{3} \tag{27}$$

#### 7.2. The Fenwick–Yan method

The Yan method [59] is based on Newton's interpolation formula

$$\mathbf{f}(\mathbf{x}) = \mathbf{f}(\mathbf{x}\_0) + \sum\_{i=1}^{n} \mathbf{f}\_i(\mathbf{x}\_i) \cdot \prod\_{j=0}^{i-1} (\mathbf{x} - \mathbf{x}\_j) \tag{28}$$

where

$$\begin{aligned} \mathbf{f}\_{1}(\mathbf{x}\_{\mathbf{j}}) &= \frac{\mathbf{f}(\mathbf{x}\_{\mathbf{j}}) - \mathbf{f}(\mathbf{x}\_{0})}{\mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{0}} \text{ for } j = 1, 2, \dots, n \\\\ \mathbf{f}\_{i}(\mathbf{x}\_{\mathbf{j}}) &= \frac{\mathbf{f}\_{i-1}(\mathbf{x}\_{\mathbf{j}}) - \mathbf{f}\_{i-1}(\mathbf{x}\_{i-1})}{\mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{i-1}} \text{ for } j = i, \dots, n \end{aligned}$$

and on the assumption that Veq ffi VIP. Putting n <sup>¼</sup> 3 in Eq. (28) and setting d<sup>2</sup> <sup>f</sup>ðxÞ=dx2 <sup>¼</sup> 0 for IP, after rearranging the terms one obtains

$$\mathbf{x\_{IP}} = \frac{1}{3} \cdot \left(\mathbf{x\_0} + \mathbf{x\_1} + \mathbf{x\_2} - \frac{\mathbf{f\_2(x\_2)}}{\mathbf{f\_3(x\_3)}}\right) \tag{29}$$

Let xj ¼ Vkþ<sup>j</sup> , f(xj ) ¼ ykþ<sup>j</sup> , j ¼ 0, 1, 2, 3; y ¼ pH or E. According to Yan's suggestion, xIP ffiVeq. Then, on the basis of 4 experimental points (Vkþ<sup>j</sup> , ykþ<sup>j</sup> ) (j ¼ 0, 1, 2, 3) taken from the immediate vicinity of Veq, we get

$$\mathbf{V\_{eq}} = \frac{1}{3} \cdot \left(\mathbf{V\_{k}} + \mathbf{V\_{k+1}} + \mathbf{V\_{k+2}} - \frac{\mathbf{f\_2(V\_{k+2})}}{\mathbf{f\_3(V\_{k+3})}}\right) \tag{30}$$

Volumes Vkþ<sup>j</sup> of T added were chosen from the immediate vicinity of Veq. The best results are obtained if Vkþ<sup>1</sup> < Veq < Vkþ2. The error in accuracy may be significant if Vk < Veq < Vkþ<sup>1</sup> or Vkþ<sup>2</sup> < Veq < Vkþ3. Moreover, the following conditions are also necessary for obtaining the accurate results: (i) volume increments Vkþiþ<sup>1</sup> – Vkþ<sup>I</sup> (ca. 0.1 mL) are small and rather equal and (ii) concentrations of reagent in T and analyte in D are similar.

When the titrant is added in equal volume increments ΔV in the vicinity of the equivalence point, then Vkþ<sup>j</sup> – Vkþ<sup>i</sup> ¼ (j – i)�ΔV, and Eq. (30) assumes the form

$$\mathbf{V\_{eq}} = \mathbf{V\_{k+1}} + \frac{\mathbf{y\_k} - 2\mathbf{y\_{k+1}} + \mathbf{y\_{k+2}}}{\mathbf{y\_k} - 3\mathbf{y\_{k+1}} + 3\mathbf{y\_{k+2}} - \mathbf{y\_{k+3}}} \cdot \Delta \mathbf{V} \tag{31}$$

identical with one obtained earlier by Fenwick [60] on the basis of the polynomial function

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 145

$$\mathbf{y} = \mathbf{A}\_0 + \mathbf{A}\_1 \cdot \mathbf{V} + \mathbf{A}\_2 \cdot \mathbf{V}^2 + \mathbf{A}\_3 \cdot \mathbf{V}^3 \tag{32}$$

(compare it with Eq. (25)). In Ref. [6], it was stated that a simple equation for x ffi Veq can be obtained after setting n ¼ 4 in Eq. (28). Then one obtains the following equation

$$\mathbf{f}\mathbf{f}\mathbf{f}\_{4}(\mathbf{V}\_{k+4})\cdot\mathbf{V}\_{\mathbf{eq}}\,^{2}+\mathbf{3}\left(\mathbf{f}\_{3}(\mathbf{V}\_{k+3})-\mathbf{\mathcal{B}}\cdot\mathbf{f}\_{4}(\mathbf{V}\_{k+4})\right)\cdot\mathbf{V}\_{\mathbf{eq}}+\mathbf{f}\_{2}(\mathbf{V}\_{k+2})-\boldsymbol{\sigma}\cdot\mathbf{f}\_{3}(\mathbf{V}\_{k+3})+\boldsymbol{\gamma}\cdot\mathbf{f}\_{4}(\mathbf{V}\_{k+4})=\mathbf{0}\tag{33}$$

where the parameters:

a3 <sup>¼</sup> V0 3 3Veq �

144 Advances in Titration Techniques

where

Let xj ¼ Vkþ<sup>j</sup>

, f(xj

vicinity of Veq, we get

3A0 � 2V0A1 þ V0

7.2. The Fenwick–Yan method

ðV0 þ VeqÞ

2A2

<sup>2</sup> ; a1 <sup>¼</sup> 3a3Veq

The Yan method [59] is based on Newton's interpolation formula

f1ðxj Þ ¼ fðxj

fiðxj Þ ¼

IP, after rearranging the terms one obtains

) ¼ ykþ<sup>j</sup>

Then, on the basis of 4 experimental points (Vkþ<sup>j</sup>

Veq <sup>¼</sup> <sup>1</sup>

and (ii) concentrations of reagent in T and analyte in D are similar.

point, then Vkþ<sup>j</sup> – Vkþ<sup>i</sup> ¼ (j – i)�ΔV, and Eq. (30) assumes the form

2V0

<sup>f</sup>ðxÞ ¼ <sup>f</sup>ðx0Þ þX<sup>n</sup>

fi�<sup>1</sup>ðxj

and on the assumption that Veq ffi VIP. Putting n <sup>¼</sup> 3 in Eq. (28) and setting d<sup>2</sup>

xIP <sup>¼</sup> <sup>1</sup>

i¼1 fiðxi Þ � <sup>Y</sup> i�1

Þ � fðx0Þ xj � x0

Þ � fi�<sup>1</sup>ðxi�<sup>1</sup><sup>Þ</sup> xj � xi�<sup>1</sup>

<sup>3</sup> � x0 <sup>þ</sup> x1 <sup>þ</sup> x2 � f2ðx2<sup>Þ</sup>

� �

, ykþ<sup>j</sup>

� �

<sup>3</sup> � Vk <sup>þ</sup> Vkþ<sup>1</sup> <sup>þ</sup> Vkþ<sup>2</sup> � f2ðVkþ<sup>2</sup><sup>Þ</sup>

Volumes Vkþ<sup>j</sup> of T added were chosen from the immediate vicinity of Veq. The best results are obtained if Vkþ<sup>1</sup> < Veq < Vkþ2. The error in accuracy may be significant if Vk < Veq < Vkþ<sup>1</sup> or Vkþ<sup>2</sup> < Veq < Vkþ3. Moreover, the following conditions are also necessary for obtaining the accurate results: (i) volume increments Vkþiþ<sup>1</sup> – Vkþ<sup>I</sup> (ca. 0.1 mL) are small and rather equal

When the titrant is added in equal volume increments ΔV in the vicinity of the equivalence

yk � 3ykþ<sup>1</sup> <sup>þ</sup> 3ykþ<sup>2</sup> � ykþ<sup>3</sup>

Veq <sup>¼</sup> Vkþ<sup>1</sup> <sup>þ</sup> yk � 2ykþ<sup>1</sup> <sup>þ</sup> ykþ<sup>2</sup>

identical with one obtained earlier by Fenwick [60] on the basis of the polynomial function

<sup>2</sup> � ðV0 � VeqÞ þ V0

2 �

j¼0

for j ¼ 1, 2, …, n

for j ¼ i, …, n

f3ðx3Þ

, j ¼ 0, 1, 2, 3; y ¼ pH or E. According to Yan's suggestion, xIP ffiVeq.

f3ðVkþ<sup>3</sup><sup>Þ</sup>

3A0 � A2V0

V0 þ Veq

2

; a0 ¼ V0

ðx � xjÞ ð28Þ

) (j ¼ 0, 1, 2, 3) taken from the immediate

� ΔV ð31Þ

<sup>3</sup> � A3 <sup>þ</sup> a1 <sup>þ</sup> a3

<sup>f</sup>ðxÞ=dx2 <sup>¼</sup> 0 for

ð29Þ

ð30Þ

ð27Þ

$$
\sigma = \mathbf{V}\_{\mathbf{k}} + \mathbf{V}\_{\mathbf{k}+1} + \mathbf{V}\_{\mathbf{k}=2\prime} \\
\boldsymbol{\beta} = \sigma + \mathbf{V}\_{\mathbf{k}+3\prime} \\
\gamma = \sum\_{i>j=0}^{3} \mathbf{V}\_{\mathbf{k}+i} \cdot \mathbf{V}\_{\mathbf{k}+j}
$$

are obtained on the basis of 5 points {(Vkþ<sup>j</sup> , ykþ<sup>j</sup> )|j¼0,…,4} from the close vicinity of Veq.

#### 8. Standardization and titrimetric analyses

The amount of an analyte in titrimetric analysis is determined from the volume of a titrant T (standard or standardized solution) required to react completely with the analyte in D. Titrations are based on standardization and determination steps. During the standardization, the titrant T with unknown concentration C of the species B is added into titrand D containing the standard S (e.g., potassium hydrogen phthalate, borax) with mass the mS (g) known accurately. In this context, different effects involved with accuracy of visual titrations will be discussed.

Discussion on the formula 12 in context with Eq. (15) will be preceded by detailed considerations, associated with (1�) selection of an indicator (pHe), (2�) volume V0 of titrand D, (3�) concentration C0In of indicator in D, (4�) buffer effect, and (5�) drop error, being considered as a whole. These effects will be considered first in context with nonredox systems. One should also draw attention whether the indicator is present in D as the salt or in the acidic form [61]; e.g., methyl orange is in the form of sodium salt, NaIn ¼ C14H14N3NaO3S, more soluble than HIn ¼ C14H15N3O3S.

To explain the effects 1� and 2�, we consider first a simple example, where the primary standard sample S is taken as an analyte A, A ¼ S.

Example 1. We consider first the titration of nS ¼ 1 mmole of potassium hydrogen phthalate KHL solution with C ¼ 0.1 mol/L NaOH. The equation for the related titration curve

$$\Phi = \frac{\mathbb{C}}{\mathbb{C}\_0} \cdot \frac{(1 - \overline{\mathfrak{n}}) \cdot \mathbb{C}\_0 - a}{\mathbb{C} + a} \tag{34}$$

is valid here [62], where α is specified in Appendix,

$$\overline{\mathbf{m}} = \frac{2 \cdot [\mathbf{H}\_2 \mathbf{L}] + [\mathbf{H} \mathbf{L}^{-1}]}{[\mathbf{H}\_2 \mathbf{L}] + [\mathbf{H} \mathbf{L}^{-1}] + [\mathbf{L}^{-2}]} = \frac{2 \cdot 10^{7.68 - 2 \text{pH}} + 10^{4.92 - \text{pH}}}{10^{7.68 - 2 \text{pH}} + 10^{4.92 - \text{pH}} + 1} \tag{35}$$

and C0 ¼ 1/V0 (V0 in mL). The values for the corresponding equilibrium constants are: pKW ¼ 14 for H2O (in α), and pK1 ¼ 2.76, pK2 ¼ 4.92 for phthalic acid (H2L).

The Φ ¼ Φ<sup>e</sup> values in Table 3 are calculated from Eq. (34) at some particular pHe values, which denote limiting pH-values of color change for phenol red (6.4 ÷ 8.0), phenolphthalein (8.0 ÷ 10.0), and thymolphthalein (9.3 ÷ 10.5). A (unfavorable) dilution effect, expressed by different V0 values, is involved here in context with particular indicators; at pHe ¼ 6.4, the dilution effect is insignificant, but grows significantly at higher pHe values e.g., 10.5. As we see, at pHe ¼ 8.0, the Φ ¼ Φ<sup>e</sup> value is closest to 1, assumed as Φeq in this case. At pHe ¼ 6.4 and 10.5, the Φ<sup>e</sup> values differ significantly from 1. At V0 ¼ 100 and phenolphthalein used as indicator, at first appearance of pink color (pH ≈ 8.0), from Eq. (34) we have Φ<sup>e</sup> ¼ 0.9993 ) δ ¼ – 0.07%. The dilution practically does not affect the results of NaOH standardization against potassium hydrogen phthalate if pH titration is applied and titration is terminated at pHe ≈ 8.0 (Table 3).

A properly chosen indicator is one of the components of the D þ T system in visual titrations. As a component of D having acid-base properties, the indicator should be included in the related balances [6, 62, 63]. The indicator effect, involved with its concentration, is considered in Examples 2 and 3. Moreover, the buffer effect is considered in Example 3.

Example 2. The equation of the titration curve for titration of V0 mL of D containing nS ¼ 1 mmole of borax in the presence of C0In mol/l methyl red (pKIn ¼ 5.3) as an indicator with C ¼ 0.1 mol/L HCl as T, is as follows [49, 62]

$$\Phi = \frac{\mathbb{C}}{\mathbb{C}\_{\text{0S}}} \cdot \frac{(4\overline{n} - 10) \cdot \mathbb{C}\_{\text{0S}} + (1 - \overline{m}) \cdot \mathbb{C}\_{\text{0ln}} + \alpha}{\mathbb{C} - \alpha} \tag{36}$$

where α (Appendix), C0 ¼ C0S ¼1/V0, and

$$\overline{m} = \frac{3 \cdot [\text{H}\_3\text{BO}\_3] + 2 \cdot [\text{H}\_2\text{BO}\_3] + [\text{H}\text{BO}\_3]}{[\text{H}\_3\text{BO}\_3] + [\text{H}\_2\text{BO}\_3] + [\text{H}\text{BO}\_3] + [\text{BO}\_3]} = \frac{3 \cdot 10^{35.78 - 3 \text{pH}} + 2 \cdot 10^{26.54 - 2 \text{pH}} + 10^{13.80 - \text{pH}}}{10^{35.78 - 3 \text{pH}} + 10^{26.54 - 2 \text{pH}} + 10^{13.80 - \text{pH}} + 1} \quad (37)$$


Table 3. The Φ<sup>e</sup> values for different pH ¼ pHe, calculated from Eq. (34), at C0 and C values assumed in Example 1.

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 147

$$\overline{\text{m}} = \frac{[\text{HIn}]}{[\text{HIn}] + [\text{In}]} = \frac{1}{1 + 10^{\text{pH} - 5.3}} \tag{38}$$

It should be noted that the solution obtained after introducing 1 mmole of borax into water is equivalent to the solution containing a mixture of 2 mmoles of H3BO3 and 2 mmoles of NaH2BO3; Na2B4O7 þ 5H2O ¼ 2H3BO3 þ 2NaH2BO3, resulting from complete hydrolysis of borax [62]. The results of calculations are presented in Table 4.

<sup>n</sup> <sup>¼</sup> <sup>2</sup> � ½H2L�þ½HL�<sup>1</sup>

½H2L�þ½HL�<sup>1</sup>

for H2O (in α), and pK1 ¼ 2.76, pK2 ¼ 4.92 for phthalic acid (H2L).

�

and C0 ¼ 1/V0 (V0 in mL). The values for the corresponding equilibrium constants are: pKW ¼ 14

The Φ ¼ Φ<sup>e</sup> values in Table 3 are calculated from Eq. (34) at some particular pHe values, which denote limiting pH-values of color change for phenol red (6.4 ÷ 8.0), phenolphthalein (8.0 ÷ 10.0), and thymolphthalein (9.3 ÷ 10.5). A (unfavorable) dilution effect, expressed by different V0 values, is involved here in context with particular indicators; at pHe ¼ 6.4, the dilution effect is insignificant, but grows significantly at higher pHe values e.g., 10.5. As we see, at pHe ¼ 8.0, the Φ ¼ Φ<sup>e</sup> value is closest to 1, assumed as Φeq in this case. At pHe ¼ 6.4 and 10.5, the Φ<sup>e</sup> values differ significantly from 1. At V0 ¼ 100 and phenolphthalein used as indicator, at first appearance of pink color (pH ≈ 8.0), from Eq. (34) we have Φ<sup>e</sup> ¼ 0.9993 ) δ ¼ – 0.07%. The dilution practically does not affect the results of NaOH standardization against potassium hydrogen

A properly chosen indicator is one of the components of the D þ T system in visual titrations. As a component of D having acid-base properties, the indicator should be included in the related balances [6, 62, 63]. The indicator effect, involved with its concentration, is considered

Example 2. The equation of the titration curve for titration of V0 mL of D containing nS ¼ 1 mmole of borax in the presence of C0In mol/l methyl red (pKIn ¼ 5.3) as an indicator with C ¼ 0.1 mol/L

6.4 0.9679 0.9679 0.9678 8.0 0.9992 0.9993 0.9994 9.3 1.0012 1.0022 1.0051 10.0 1.0060 1.0010 1.0260 10.5 1.0190 1.0349 1.0825

Table 3. The Φ<sup>e</sup> values for different pH ¼ pHe, calculated from Eq. (34), at C0 and C values assumed in Example 1.

ð4n � 10Þ � C0S þ ð1 � mÞ � C0In þ α

<sup>¼</sup> <sup>2</sup> � <sup>10</sup><sup>7</sup>:68�2pH <sup>þ</sup> <sup>104</sup>:92�pH

<sup>107</sup>:68�2pH <sup>þ</sup> <sup>104</sup>:92�pH <sup>þ</sup> <sup>1</sup> <sup>ð</sup>35<sup>Þ</sup>

<sup>C</sup> � <sup>α</sup> <sup>ð</sup>36<sup>Þ</sup>

<sup>1035</sup>:78�3pH <sup>þ</sup> <sup>1026</sup>:54�2pH <sup>þ</sup> <sup>1013</sup>:80�pH <sup>þ</sup> <sup>1</sup> <sup>ð</sup>37<sup>Þ</sup>

<sup>¼</sup> <sup>3</sup> � <sup>1035</sup>:78�3pH <sup>þ</sup> <sup>2</sup> � <sup>1026</sup>:54�2pH <sup>þ</sup> <sup>1013</sup>:80�pH

V<sup>0</sup> ¼ 50 V<sup>0</sup> ¼ 100 V<sup>0</sup> ¼ 200

�þ½L�<sup>2</sup> �

phthalate if pH titration is applied and titration is terminated at pHe ≈ 8.0 (Table 3).

in Examples 2 and 3. Moreover, the buffer effect is considered in Example 3.

<sup>Φ</sup> <sup>¼</sup> <sup>C</sup> C0S �

where α (Appendix), C0 ¼ C0S ¼1/V0, and

<sup>n</sup> <sup>¼</sup> <sup>3</sup> � ½H3BO3� þ <sup>2</sup> � ½H2BO3�þ½HBO3� ½H3BO3�þ½H2BO3�þ½HBO3�þ½BO3�

HCl as T, is as follows [49, 62]

146 Advances in Titration Techniques

pHe Φ<sup>e</sup>

In context with Table 4, we refer to the one-drop error. For this purpose, let us assume that the end point was not attained after addition of V' mL of titrant T, and the analyst decided to add the next drop of volume ΔV mL of the T. If the end point is attained this time, i.e., Ve ¼ V' þ ΔV, the uncertainty in the T volume equals ΔV. Assuming ΔV ¼ 0.03 mL and applying Eq. (1), we have:

Φ' ¼ C � V'=ðC<sup>0</sup> � V0Þ, Φ<sup>e</sup> ¼ C � Ve=ðC<sup>0</sup> � V0Þ and then ΔΦ ¼ Φ<sup>e</sup> � Φ' ¼ C � Ve=ðC<sup>0</sup> �V0Þ � C � V'= ðC<sup>0</sup> � V0Þ ¼ C � ΔV=ðC<sup>0</sup> � V0Þ: At V0 ¼ 100 mL, C0 ¼ 0.01 mol/L, C ¼ 0.1 mol/L, and ΔV ¼ 0.03 mL, we have

$$
\Delta\Phi = \text{ } \text{C} \cdot \Delta V / (\text{C}\_0 \cdot V\_0) = 0.003 \tag{39}
$$

Taking the value Φ<sup>e</sup> ¼ 2.0048 in Table 4, which refers to V0 ¼ 100 mL, C0 ¼ 0.01 mol/L, <sup>C</sup> <sup>¼</sup> 0.1 mol/L, C0In <sup>¼</sup> <sup>10</sup>�<sup>5</sup> mol/L and pHe <sup>¼</sup> 4.4, we see that |2.0048 – 2| <sup>¼</sup> 0.0048 > 0.003 i.e., the discrepancy between Φeq and Φ<sup>e</sup> is greater than the one assumed for ΔΦ ¼ 0.003; it corresponds to ca. 1.5 drop of the titrant. At pHe ¼ 6.2 and other data chosen as previously, we get |1.9973 – 2| ¼ 0.0027 < 0.003 i.e., this uncertainty falls within one–drop error.

The indicator effect stated in Table 4, for V0 ¼ 100, C0 ¼ 0.01, C ¼ 0.1 and pHe ¼ 4.4 equals in <sup>Φ</sup>-units: |2.0048 – 2.0047| <sup>¼</sup> 0.0001 at C0In <sup>¼</sup> <sup>10</sup>�<sup>5</sup> or |2.0058 – 2.0047| <sup>¼</sup> 0.0011, i.e., it appears to be insignificant in comparison to ΔΦ ¼ 0.003, and can therefore be neglected.


Table 4. The Φ<sup>e</sup> values calculated from Eqs. (36) to (38) for different pH ¼ pHe, C0In and V<sup>0</sup> (mL) values assumed in Example 2. The pHe values are related to the pH-interval <4.4 ÷ 6.2> corresponding to the color change of methyl red (HIn).

Figure 6. The logy versus <sup>Φ</sup> relationships in the close vicinity of <sup>Φ</sup>eq <sup>¼</sup> 1, for <sup>C</sup>In <sup>¼</sup> <sup>p</sup> � <sup>10</sup>-5 mol/L (p <sup>¼</sup> 2, 4, 6, 8, 10); curves ap correspond to CNH3 ¼ 0.1 mol/L, curves bp correspond to CNH3 ¼ 1.0 mol/L; (A) refers to r ¼ 1, (B) refers to r ¼ 4.

Figure 7. The logy versus Φ relationships plotted at C<sup>N</sup> ¼ 1 mol/L and r ¼ 1 (curve 1b), and r ¼ 4 (curve 4b).

Example 3. The solution of ZnCl2 (C0 ¼ 0.01) buffered with NH4Cl (C1) and NH3 (C2), C1 þ C2 ¼ CN, r ¼ C2/C1, is titrated with EDTA (C ¼ 0.02) in presence of Eriochrome Black T (CIn <sup>¼</sup> <sup>p</sup>�10�<sup>5</sup> , p ¼ 2, 4, 6, 8, 10) as the indicator changes from wine red to blue color. The curves of logy versus Φ relationships, where

$$\mathbf{y} = \frac{\mathbf{x}\_2}{\mathbf{x}\_1} \text{ and } \mathbf{:}\\\mathbf{x}\_1 = \sum\_{i=0}^3 \left[\mathbf{H}\_i \text{In}\right] \\\mathbf{:}\\\mathbf{x}\_2 = \left[\mathbf{ZnIn}\right] + 2\left[\mathbf{ZnIn}\_2\right]$$

are plotted in Figure 6, where (A) refers to r ¼ 1, (B) refers to r ¼ 4. It is stated that at CN ¼ 0.1, the solution becomes violet (red þ blue) in the nearest vicinity of Φeq ¼ 1, and the color change occurs at this point. At CN ¼ 1.0, the solution has the mixed color from the very beginning of the titration (Figure 7). At CN > 1.0, the solution is blue from the start of the titration. This system was discussed in more details in Refs. [9, 37, 49, 62].

## 9. Intermediary comments

Example 3. The solution of ZnCl2 (C0 ¼ 0.01) buffered with NH4Cl (C1) and NH3 (C2), C1 þ C2 ¼ CN, r ¼ C2/C1, is titrated with EDTA (C ¼ 0.02) in presence of Eriochrome Black T

Figure 7. The logy versus Φ relationships plotted at C<sup>N</sup> ¼ 1 mol/L and r ¼ 1 (curve 1b), and r ¼ 4 (curve 4b).

Figure 6. The logy versus <sup>Φ</sup> relationships in the close vicinity of <sup>Φ</sup>eq <sup>¼</sup> 1, for <sup>C</sup>In <sup>¼</sup> <sup>p</sup> � <sup>10</sup>-5 mol/L (p <sup>¼</sup> 2, 4, 6, 8, 10); curves ap correspond to CNH3 ¼ 0.1 mol/L, curves bp correspond to CNH3 ¼ 1.0 mol/L; (A) refers to r ¼ 1, (B) refers to r ¼ 4.

, p ¼ 2, 4, 6, 8, 10) as the indicator changes from wine red to blue color. The

(CIn <sup>¼</sup> <sup>p</sup>�10�<sup>5</sup>

148 Advances in Titration Techniques

curves of logy versus Φ relationships, where

If a concentration C of the properly chosen reagent B in T is known accurately from the standardization, the B (C mol/L) solution can be used later as titrant T, applied for determination of the unknown mass mA of the analyte A in D. The B (C) reacts selectively with an analyte A (C0 mol/L) contained in the titrand (D). This way, NaOH is standardized as in Example 1, and HCl is standardized as in Example 2. In Example 3, the standard solution of EDTA can be prepared from accurately weighed portion of this preparation, without a need for standardization, if EDTA itself can be obtained in enough pure form.

The reaction between A and S, B and A, or S and A should be fast i.e., equilibrium is reached after each consecutive portion of T added in the titration made with use of calibrated measuring instrument and volumetric ware.

In pH or potentiometric (E) titration, the correct readout with use of the proper measuring instrument needs identical equilibrium conditions at the measuring electrode and in the bulk solution, after each consecutive portion of T added in a quasistatic a priori manner under isothermal conditions assumed in the D þ T system.

The quasistaticity assumption is fulfilled only approximately; however, the resulting error in accuracy is affected by a drift involved with retardation of processes occurred at the indicator electrode against ones in the bulk solution, where titrant T is supplied. Then, the methods based on the inflection point (IP) registration give biased results, as a rule. This discrepancy can be limited to a certain degree, after slowing down the titrant dosage. Otherwise, the end point lags behind the equivalence point because of a slow response of the electrode.

In modern chemical analysis, titrations are performed automatically and the titrant is introduced continuously. In this context, the transportation factors concerning the response of the indicating system are of paramount importance. At low concentration of analyte, the degree of incompleteness of the reaction is the highest around the equivalence point, and then the methods based on the inflection point registration give biased results, as a rule. The results like ones obtained with precision 0.02% within 5 min of the potentiometric titration performed with use of an ion–selective electrode or alike (according to some literature reports), can be considered only as a mere fiction.

In this context, for the reasons specified above, it is safer to apply extrapolative methods of titrimetric analyses. Such a requirement is fulfilled by some methods applied in potentiometric analysis; the best known ones are the Gran methods considered e.g., in Refs. [3, 6, 65, 77]. The Gran methods of Veq determination can replace the currently used first-derivative method in the potentiometric titration procedure.

In the mathematical model applied for Veq evaluation, it is tacitly assumed that activity coefficients and electrode junction potentials are invariable during the titration. The slope of indicator electrode should be known accurately; the statement that the slope should necessarily be Nernstian [66] is not correct. In reference to acid-base titrations, T and D should not be contaminated by carbonate; it particularly refers to a strong base solution used as T [67, 68].

## 10. The Gran methods

#### 10.1. Introductory remarks

The Gran methods is an eponym of the well–known methods of linearization of the S–shaped curves of potentiometric E or pH titration [69–71]. In principle, there are two original Gran methods, known as Gran I method (abbr. G(I)) [72] and Gran II (abbr. G(II)) method [73, 74].

In current laboratory practice, only G(II) is applied mainly in alkalinity [75] (referred to seawaters, as a rule) and acid–base titrations, in general. The presumable reasons of G(I) factual rejection (this statement was nowhere pronounced in literature) were clearly presented in the chapter [65], where G(I) and G(II) were thoroughly discussed. It was stated that the main reason of rejection was too high error, inherent in the simplified model that can be brought to the approximation

$$
\ln(1+\mathbf{x}) \cong \mathbf{x} \tag{40}
$$

to the first term of the related Maclaurin's series [76]

$$\ln(1+\mathbf{x}) = \sum\_{\mathbf{j}=1}^{\circ} (-1)^{\mathbf{j}+1} \cdot \mathbf{x}^{\mathbf{j}}/\mathbf{j}$$

The relation Eq. (40) is valid only at |x| << 1. To extend the x range, Michałowski suggested the approximation [6]

$$\ln(1+\mathbf{x}) = \frac{\mathbf{x}}{1+\mathbf{x}/2} \tag{41}$$

that appeared to be better than expansion of ln(1þx) into the Maclaurin series, up to the 18th term at |x| ≤ 1 [65], see Figure 8.

It is noteworthy that some trials were done by Gran himself [50] to improve G(I), but his proposal based on some empirical formulas was a kind of "prosthesis" applied to the defective model. In further years, the name "Gran method" (in singular) has been factually limited to G(II) i.e., in literature the term "Gran method" is practically perceived as one tantamount with G(II).

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 151

Figure 8. Comparison of the plots for: (1) f1(x) ¼ ln(1 þ x), (2) f2(x) ¼ x/(1 þ x/2), and (3) f3(x) ¼ x at different x-values, 0 < x ≤ 1.

#### 10.2. The original Gran methods: G(I) and G(II)

The principle of the original Gran methods can be illustrated in a modified form [6], starting from titration of V0 mL of C0 mol/L HCl with V mL of C mol/L NaOH, taken as a simplest case. From charge and concentration balances, and C0V0 ¼ CVeq i.e., Φeq ¼ 1 in Eq. (2), we get

$$\mathbf{C} \cdot (\mathbf{H}^{+1} - [\mathbf{O} \mathbf{H}^{-1}]) (\mathbf{V}\_0 + \mathbf{V}) = \mathbf{C} \cdot (\mathbf{V}\_{\text{eq}} - \mathbf{V}) \tag{42}$$

Applying the notations: h <sup>¼</sup> <sup>γ</sup>� [Hþ<sup>1</sup> ], ph ¼ �log h, at [Hþ<sup>1</sup> ] >> [OH�<sup>1</sup> ] (acid branch) i.e., V<Veq, from Eq. (42) we have the relations:

$$(\text{V}\_0 + \text{V}) \cdot 10^{-\text{ph}} = \text{G}\_1 \cdot (\text{V}\_{\text{eq}} - \text{V}) \tag{43}$$

$$\text{ph} \cdot \ln 10 = \ln(\text{V}\_0 + \text{V}) - \ln \text{G}\_1 + \ln \left(\text{V}\_{\text{eq}} - \text{V}\right) \tag{44}$$

#### 10.2.1. G(I) method

Applying Eq. (44) to the pair of points: (Vj , pHj ) and (Vjþ1, pHjþ1), we have, by turns,

$$\ln \text{10 } \cdot (\text{pH}\_{\text{j}+1} - \text{pH}\_{\text{j}}) = \ln \frac{\text{V}\_{0} + \text{V}\_{\text{j}+1}}{\text{V}\_{0} + \text{V}\_{\text{j}}} - \ln \frac{\text{V}\_{\text{eq}} - \text{V}\_{\text{j}+1}}{\text{V}\_{\text{eq}} - \text{V}\_{\text{j}}} \tag{45}$$

$$=\ln(1+\mathbf{x}\_{1\circ}) - \ln(1-\mathbf{x}\_{2\circ})\tag{45a}$$

where:

analysis; the best known ones are the Gran methods considered e.g., in Refs. [3, 6, 65, 77]. The Gran methods of Veq determination can replace the currently used first-derivative method in

In the mathematical model applied for Veq evaluation, it is tacitly assumed that activity coefficients and electrode junction potentials are invariable during the titration. The slope of indicator electrode should be known accurately; the statement that the slope should necessarily be Nernstian [66] is not correct. In reference to acid-base titrations, T and D should not be contaminated by carbonate; it particularly refers to a strong base solution used as

The Gran methods is an eponym of the well–known methods of linearization of the S–shaped curves of potentiometric E or pH titration [69–71]. In principle, there are two original Gran methods, known as Gran I method (abbr. G(I)) [72] and Gran II (abbr. G(II)) method [73, 74]. In current laboratory practice, only G(II) is applied mainly in alkalinity [75] (referred to seawaters, as a rule) and acid–base titrations, in general. The presumable reasons of G(I) factual rejection (this statement was nowhere pronounced in literature) were clearly presented in the chapter [65], where G(I) and G(II) were thoroughly discussed. It was stated that the main reason of rejection was too high error, inherent in the simplified model that can be brought to the approximation

lnð<sup>1</sup> <sup>þ</sup> <sup>x</sup>Þ ¼ <sup>X</sup><sup>∞</sup>

j¼1

The relation Eq. (40) is valid only at |x| << 1. To extend the x range, Michałowski suggested the

lnð<sup>1</sup> <sup>þ</sup> <sup>x</sup>Þ ¼ <sup>x</sup>

that appeared to be better than expansion of ln(1þx) into the Maclaurin series, up to the 18th

It is noteworthy that some trials were done by Gran himself [50] to improve G(I), but his proposal based on some empirical formulas was a kind of "prosthesis" applied to the defective model. In further years, the name "Gran method" (in singular) has been factually limited to G(II) i.e., in literature the term "Gran method" is practically perceived as one tantamount with G(II).

ð � 1Þ

<sup>j</sup>þ<sup>1</sup> � xj =j

ln 1ð Þffi þ x x ð40Þ

<sup>1</sup> <sup>þ</sup> <sup>x</sup>=<sup>2</sup> <sup>ð</sup>41<sup>Þ</sup>

the potentiometric titration procedure.

150 Advances in Titration Techniques

T [67, 68].

10. The Gran methods

10.1. Introductory remarks

approximation [6]

term at |x| ≤ 1 [65], see Figure 8.

to the first term of the related Maclaurin's series [76]

$$\mathbf{x}\_{\text{lj}} = \frac{\mathbf{V}\_{\text{j}+1} - \mathbf{V}\_{\text{j}}}{\mathbf{V}\_{0} + \mathbf{V}\_{\text{j}}} \tag{46a}$$

$$\mathbf{x}\_{2\circ} = \frac{\mathbf{V\_{j+1}} - \mathbf{V\_{j}}}{\mathbf{V\_{eq}} - \mathbf{V\_{j}}} \tag{46b}$$

Applying the approximation Eq. (40), we have:

$$
\ln(1 + \chi\_{\mathbf{i}\mathbf{j}}) \cong \chi\_{\mathbf{i}\mathbf{j}} \colon \ln(1 - \chi\_{\mathbf{2}\mathbf{j}}) \cong -\chi\_{\mathbf{2}\mathbf{j}} \tag{47}
$$

Then we have, by turns,

$$\ln 10 \cdot (\text{pH}\_{\text{j}+1} - \text{pH}\_{\text{j}}) = \text{x}\_{\text{l}\text{j}} + \text{x}\_{\text{2j}} = (\text{V}\_{\text{j}+1} - \text{V}\_{\text{j}}) \cdot \frac{\text{V}\_{0} + \text{V}\_{\text{eq}}}{(\text{V}\_{0} + \text{V}\_{\text{j}})(\text{V}\_{\text{eq}} - \text{V}\_{\text{j}})} \tag{48}$$

$$\mathbf{y}\_{\mathbf{j}} = \mathbf{G}\_{\mathbf{l}} \cdot (\mathbf{V}\_{\mathbf{eq}} - \mathbf{V}\_{\mathbf{j}}) + \varepsilon\_{\mathbf{j}} \tag{49}$$

$$\mathbf{y}\_{\mathbf{j}} = \mathbf{P}\_1 - \mathbf{G}\_1 \cdot \mathbf{V}\_{\mathbf{j}} + \mathbf{e}\_{\mathbf{j}} \tag{50}$$

where P1 ¼ G1Veq, and

$$\mathbf{G}\_1 = \frac{\ln 10}{\mathbf{V}\_0 + \mathbf{V}\_{\text{eq}}} \tag{51}$$

$$\mathbf{y}\_{\circ} = \frac{1}{\mathbf{V}\_0 + \mathbf{V}\_{\circ}} \cdot \frac{\mathbf{V}\_{\circ+1} - \mathbf{V}\_{\circ}}{\mathbf{pH}\_{\circ+1} - \mathbf{pH}\_{\circ}} \tag{52}$$

From Eq. (50) and LSM, we get the formula

$$\mathbf{V\_{eq}} = \frac{\mathbf{P\_1}}{\mathbf{G\_1}} = \frac{\sum \mathbf{y\_j} \mathbf{V\_j} \cdot \sum \mathbf{V\_j} - \sum \mathbf{y\_j} \cdot \sum \mathbf{V\_j^2}}{\mathbf{N} \cdot \sum \mathbf{y\_j} \mathbf{V\_j} - \sum \mathbf{y\_j} \cdot \sum \mathbf{V\_j}} \tag{53}$$

where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>N</sup> j¼1 , and yj is expressed by Eq. (52); it is the essence of G(I).

#### 10.2.2. G(II) method

Eq. (43) can be rewritten into the regression equation

$$\mathbf{y}\_{\mathbf{j}} = \mathbf{P}\_2 - \mathbf{G}\_2 \cdot \mathbf{V}\_{\mathbf{j}} + \mathbf{e}\_{\mathbf{j}} \tag{54}$$

where:

$$\mathbf{G}\_2 = \mathbf{\bar{y}} \cdot \mathbf{C} \tag{55a}$$

$$\mathbf{P\_2 = \gamma \cdot \mathbf{C} \cdot V\_{\text{eq}} = G\_2 \cdot V\_{\text{eq}}} \tag{55b}$$

$$\mathbf{y}\_{\mathbf{j}} = (\mathbf{V}\_0 + \mathbf{V}\_{\mathbf{j}}) \cdot 10^{-\text{ph}} \tag{56}$$

Applying LSM to ph titration data {(Vj , phj )|j¼1,…,N}, from (55b) we get Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 153

$$\mathbf{V\_{eq}} = \frac{\mathbf{P\_2}}{\mathbf{G\_2}} = \frac{\sum \mathbf{y\_j} \mathbf{V\_j} \cdot \sum \mathbf{V\_j} - \sum \mathbf{y\_j} \cdot \sum \mathbf{V\_j}^2}{\mathbf{N} \cdot \sum \mathbf{y\_j} \mathbf{V\_j} - \sum \mathbf{y\_j} \cdot \sum \mathbf{V\_j}} \tag{57}$$

similar to Eq. (53), where yj is expressed by Eq. (56) at this time; it is the essence of G(II).

#### 10.3. The modified Gran methods

#### 10.3.1. MG(I) method

Applying the approximation Eq. (40), we have:

ln10 � ðpHjþ<sup>1</sup> � pHj

From Eq. (50) and LSM, we get the formula

j¼1

Applying LSM to ph titration data {(Vj

Veq <sup>¼</sup> P1 G1 ¼

Eq. (43) can be rewritten into the regression equation

Then we have, by turns,

152 Advances in Titration Techniques

where P1 ¼ G1Veq, and

where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>N</sup>

10.2.2. G(II) method

where:

lnð1 þ x1jÞ ffi x1j; lnð1 � x2jÞffi�x2j ð47Þ

yj ¼ G1 � ðVeq � VjÞ þ ε<sup>j</sup> ð49Þ

yj ¼ P1 � G1 � Vj þ ε<sup>j</sup> ð50Þ

<sup>X</sup>Vj 2

yj ¼ P2 � G2 � Vj þ ε<sup>j</sup> ð54Þ

G2 ¼ γ � C ð55aÞ

P2 ¼ γ � C � Veq ¼ G2 � Veq ð55bÞ

yj ¼ ðV0 <sup>þ</sup> VjÞ � <sup>10</sup>�phj <sup>ð</sup>56<sup>Þ</sup>

)|j¼1,…,N}, from (55b) we get

<sup>X</sup>Vj

<sup>ð</sup>V0 <sup>þ</sup> VjÞðVeq � Vj<sup>Þ</sup> <sup>ð</sup>48<sup>Þ</sup>

ð51Þ

ð52Þ

ð53Þ

Þ ¼ x1j <sup>þ</sup> x2j ¼ ðVjþ<sup>1</sup> � VjÞ � V0 <sup>þ</sup> Veq

� Vjþ<sup>1</sup> � Vj pHjþ<sup>1</sup> � pHj

<sup>X</sup>Vj �Xyj �

Vj �Xyj �

G1 <sup>¼</sup> ln10 V0 þ Veq

yj <sup>¼</sup> <sup>1</sup> V0 þ Vj

> <sup>X</sup>yj Vj �

> > N � <sup>X</sup>yj

, phj

, and yj is expressed by Eq. (52); it is the essence of G(I).

Applying Eq. (41) to Eqs. (45a) and (46), we have

$$\ln(\mathbf{1} + \mathbf{x}\_{\mathrm{Ij}}) \cong \frac{\mathbf{x}\_{\mathrm{Ij}}}{1 + \mathbf{x}\_{\mathrm{Ij}}/2} = \frac{\frac{\mathbf{V}\_{\mathrm{i}+1} - \mathbf{V}\_{\mathrm{i}}}{\mathbf{V}\_{\mathrm{0}} + \mathbf{V}\_{\mathrm{i}}}}{1 + \frac{\mathbf{V}\_{\mathrm{i}+1} - \mathbf{V}\_{\mathrm{i}}}{2(\mathbf{V}\_{\mathrm{0}} + \mathbf{V}\_{\mathrm{i}})}} = \frac{\mathbf{V}\_{\mathrm{j}+1} - \mathbf{V}\_{\mathrm{j}}}{\mathbf{V}\_{\mathrm{0}} + \frac{\mathbf{V}\_{\mathrm{i}} + \mathbf{V}\_{\mathrm{i}+1}}{2}} \tag{58a}$$

$$\ln(\mathbf{1} - \mathbf{x\_{2j}}) \cong \frac{-\mathbf{x\_{2j}}}{\mathbf{1} - \mathbf{x\_{2j}}/2} = \frac{-\frac{\mathbf{V\_{j+1}} - \mathbf{V\_j}}{\mathbf{V\_{eq}} - \mathbf{V\_j}}}{1 - \frac{\mathbf{V\_{j+1}} - \mathbf{V\_j}}{2(\mathbf{V\_{eq}} - \mathbf{V\_j})}} = \frac{-(\mathbf{V\_{j+1}} - \mathbf{V\_j})}{\mathbf{V\_{eq}} - \frac{\mathbf{V\_j} + \mathbf{V\_{j+1}}}{2}} \tag{58b}$$

From Eqs. (58) and (45a) we have, by turns,

$$\mathbf{In10} \cdot (\mathbf{pH\_{j+1}} - \mathbf{pH\_j}) \cong (\mathbf{V\_{j+1}} - \mathbf{V\_j}) \cdot \left(\frac{1}{\mathbf{V\_0} + \mathbf{V\_j^\*}} + \frac{1}{\mathbf{V\_{eq}} - \mathbf{V\_j^\*}}\right) = \frac{(\mathbf{V\_{j+1}} - \mathbf{V\_j}) \cdot (\mathbf{V\_0} + \mathbf{V\_{eq}})}{(\mathbf{V\_0} + \mathbf{V\_j^\*}) \cdot (\mathbf{V\_{eq}} - \mathbf{V\_j^\*})} \quad (59)$$
 
$$\mathbf{y\_j^\*} = \mathbf{G\_1} \cdot (\mathbf{V\_{eq}} - \mathbf{V\_j^\*}) + \mathbf{e\_j}$$

$$\mathbf{y}\_{\mathbf{j}}^{\*} = \mathbf{P}\_{\mathbf{l}} - \mathbf{G}\_{\mathbf{l}} \cdot \mathbf{V}\_{\mathbf{j}}^{\*} \,) + \varepsilon\_{\mathbf{j}} \tag{60}$$

where G1 and Vj \* are as in Eq. (51), and:

$$\mathbf{V}\_{\mathbf{j}}^{\*} = \frac{\mathbf{V}\_{\mathbf{j}} + \mathbf{V}\_{\mathbf{j}+1}}{2} \tag{61}$$

$$\mathbf{y}\_{\mathbf{j}}^{\*} = \frac{1}{\mathbf{V}\_{0} + \mathbf{V}\_{\mathbf{j}}^{\*}} \cdot \frac{\mathbf{V}\_{\mathbf{j}+1} - \mathbf{V}\_{\mathbf{j}}}{\mathbf{pH}\_{\mathbf{j}+1} - \mathbf{pH}\_{\mathbf{j}}} \tag{62}$$

$$\mathbf{V\_{eq}} = \frac{\mathbf{P\_1}}{\mathbf{G\_1}} = \frac{\sum \mathbf{y\_j}^\* \mathbf{V\_j^\*} \cdot \sum \mathbf{V\_j^\*} - \sum \mathbf{y\_j^\*} \cdot \sum \mathbf{V\_j^{\*2}}}{\mathbf{N} \cdot \sum \mathbf{y\_j^\*} \mathbf{V\_j^\*} - \sum \mathbf{y\_j^\*} \cdot \sum \mathbf{V\_j^\*}} \tag{63}$$

Application of Vj \* in Eqs. (59) and (62), suggested in Ref. [6], improves the results of analyses when compared with Eqs. (50) and (52).

#### 10.3.2. New algorithms referred to Feþ<sup>2</sup> <sup>þ</sup> MnO4 –<sup>1</sup> system

The algorithms applied below are referred to the system, where V0 ml of the solution containing FeSO4 (C0) and H2SO4 (C01) as D is titrated with V ml of KMnO4 (C). The simplest form of GEB related to this system has the form [3, 46]

$$\left[\text{Fe}^{+2}\right] + \left[\text{FeOH}^{+1}\right] + \left[\text{FeSO}\_4\right] - \left(5\left[\text{MnO}\_4^{-1}\right] + 4\left[\text{MnO}\_4^{-2}\right] + \left[\text{Mn}^{+3}\right] + \left[\text{MnOOH}^{+2}\right]\right) \tag{64}$$

$$= (\text{C}\_0\text{V}\_0 - 5\text{CV})/(\text{V}\_0 + \text{V}) = (1 - 5\Phi)\text{C}\_0\text{V}\_0/(\text{V}\_0 + \text{V})$$

Concentration balance for Fe has the form

$$\left[\text{Fe}^{+2}\right] + \left[\text{FeOH}^{+1}\right] + \left[\text{FeSO}\_4\right] + \left[\text{Fe}^{3}\right] + \left[\text{FeOH}^{+2}\right] + \left[\text{Fe(OH)}\_2\right]^{+1} + 2\left[\text{Fe}\_2(\text{OH})\_2\right]^{+4}$$

$$+ \left[\text{FeSO}\_4^{+1}\right] + \left[\text{Fe(SO}\_4)\_2^{-1}\right] = \text{C}\_0\text{V}\_0/\text{(V}\_0 + \text{V)}\tag{65}$$

On the basis of Figure 9, at Φ < Φeq ¼ 0.2 and low pH-values, Eqs. (64) and (65) assume simpler forms:

$$\left[\text{Fe}^{+2}\right] + \left[\text{FeSO}\_4\right] = \left(1 - 5\spadesuit\right) \times \text{C}\_0\text{V}\_0/(\text{V}\_0 + \text{V})\tag{66}$$

$$\left[\text{Fe}^{+2}\right] + \left[\text{FeSO}\_4\right] + \left[\text{Fe}^{+3}\right] + \left[\text{FeSO}\_4^{+1}\right] + \left[\text{Fe(SO}\_4\text{)}\_2^{-1}\right] = \text{C}\_0\text{V}\_0/(\text{V}\_0 + \text{V})\tag{67}$$

These simplifications are valid at low pH-values (Figure 6). Eqs. (66) and (67) can be rewritten as follows:

$$\left[\mathbf{F}\mathbf{e}^{+2}\right] \cdot \mathbf{b}\_2 = (\mathbf{1} \text{--} \mathbf{50})\mathbf{C}\_0 \mathbf{V}\_0 / (\mathbf{V}\_0 + \mathbf{V})\tag{68}$$

$$\mathbf{\dot{\color{red}{[Fe}}{[Fe}}^{+2}\mathbf{\dot{\color{red}{0}}}\mathbf{\dot{\color{red}{0}}}\mathbf{\dot{\color{red}{0}}}\mathbf{\dot{\color{red}{0}}}\mathbf{\dot{\color{red}{0}}}\mathbf{\dot{\color{red}{0}}}\mathbf{\dot{\color{red}{0}}}) = \mathbf{\mathbf{C}}\_{0} \cdot \mathbf{V}\_{0} / (\mathbf{V}\_{0} + \mathbf{V})\tag{69}$$

Figure 9. Dynamic speciation curves plotted for (A) Fe-species; (B) Mn-species in D þ T system where V0 ¼ 100 mL of T (FeSO4 (C0 ¼ 0.01) þ H2SO4 (C01 ¼ 1.0) is titrated with V ml of KMnO4 (C ¼ 0.02).

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 155

valid for Φ < Φeq ¼ 0.2, where:

Feþ<sup>2</sup> <sup>þ</sup> FeOHþ<sup>1</sup> <sup>þ</sup> ½ � FeSO4 –ð5 MnO�<sup>1</sup>

þ FeSO4

Feþ<sup>2</sup> <sup>þ</sup> ½ �þ FeSO4 Feþ<sup>3</sup> <sup>þ</sup> FeSO4

<sup>½</sup>Feþ<sup>2</sup>

Concentration balance for Fe has the form

154 Advances in Titration Techniques

forms:

as follows:

4 <sup>þ</sup> 4 MnO�<sup>2</sup>

¼ ð Þ C0V0 � 5CV =ð Þ¼ V0 þ V ð Þ 1 � 5Φ C0V0=ð Þ V0 þ V

On the basis of Figure 9, at Φ < Φeq ¼ 0.2 and low pH-values, Eqs. (64) and (65) assume simpler

These simplifications are valid at low pH-values (Figure 6). Eqs. (66) and (67) can be rewritten

Figure 9. Dynamic speciation curves plotted for (A) Fe-species; (B) Mn-species in D þ T system where V0 ¼ 100 mL of T

(FeSO4 (C0 ¼ 0.01) þ H2SO4 (C01 ¼ 1.0) is titrated with V ml of KMnO4 (C ¼ 0.02).

<sup>þ</sup><sup>1</sup> þ ½Fe SOð Þ<sup>4</sup> �<sup>1</sup>

Feþ<sup>2</sup> <sup>þ</sup> FeOHþ<sup>1</sup> <sup>þ</sup> ½ �þ FeSO4 Feþ<sup>3</sup> <sup>þ</sup> FeOHþ<sup>2</sup> <sup>þ</sup> Fe OH ð Þ<sup>2</sup>

<sup>þ</sup><sup>1</sup> <sup>þ</sup> Fe SOð Þ<sup>4</sup> <sup>2</sup>

4

Feþ<sup>2</sup> <sup>þ</sup> ½ �¼ð FeSO4 <sup>1</sup> � <sup>5</sup>ΦÞ � C0V0=ð Þ V0 <sup>þ</sup> <sup>V</sup> <sup>ð</sup>66<sup>Þ</sup>

Feþ<sup>2</sup> � b2 <sup>¼</sup> ð Þ <sup>1</sup>–5<sup>Φ</sup> C0V0=ð Þ V0 <sup>þ</sup> <sup>V</sup> <sup>ð</sup>68<sup>Þ</sup>

��ðb2 þ f23 � b3Þ ¼ C0 � V0=ð Þ V0 þ V ð69Þ

<sup>þ</sup> Mnþ<sup>3</sup> <sup>þ</sup> MnOHþ<sup>2</sup> <sup>Þ</sup>

<sup>þ</sup><sup>1</sup> <sup>þ</sup> 2 Fe2ð Þ OH <sup>2</sup>


<sup>þ</sup><sup>4</sup>

<sup>2</sup> � ¼ C0V0=ðÞ ð V0 þ V 67Þ

ð64Þ

$$\mathbf{b}\_2 = 1 + \mathbf{K}\_{21} \times \left[ \mathbf{SO}\_4^{-2} \right] \tag{70a}$$

$$\mathbf{b}\_3 = \mathbf{1} + \mathbf{K}\_{31} \times \left[ \mathbf{SO}\_4^{-2} \right] + \mathbf{K}\_{32} \times \left[ \mathbf{SO}\_4^{-2} \right]^2 \tag{70b}$$

$$\text{If}\_{23} = \frac{[\text{Fe}^{+3}]}{[\text{Fe}^{+2}]} = 10^{\text{A}(\text{E} - \text{E}\_0)}\tag{71a}$$

$$\mathbf{A} = \frac{\mathbf{F}}{\mathbf{R} \cdot \mathbf{T} \cdot \ln 10} = \frac{1}{\mathbf{a} \cdot \ln 10} \tag{71b}$$

$$\mathbf{a} = \frac{\mathbf{R}\mathbf{T}}{\mathbf{F}}\tag{71c}$$

and [FeSO4] <sup>¼</sup> K21[Fe<sup>þ</sup><sup>2</sup> ][SO4 –2 ], [FeSO4 þ1 ] <sup>¼</sup> K31[Fe<sup>þ</sup><sup>3</sup> ][SO4 –2 ], [Fe(SO4)2 �1 ] <sup>¼</sup> K32[Fe<sup>þ</sup><sup>3</sup> ][SO4 –2 ] 2 . From Eqs. (68) and (69), we have, by turns,

$$1 + \mathbf{f\_{23}} \cdot \frac{\mathbf{b\_3}}{\mathbf{b\_2}} = \frac{1}{1 - 5\Phi} \tag{72a}$$

$$10^{\text{A}(\text{E} - \text{E}\_0)} \cdot \frac{\mathbf{b}\_3}{\mathbf{b}\_2} = \frac{5\Phi}{1 - 5\Phi} \tag{72\text{b}}$$

$$\mathbf{E} = \mathbf{E}\_0 - \mathbf{a} \cdot \ln\left(\frac{\mathbf{b}\_3}{\mathbf{b}\_2}\right) + \mathbf{a} \cdot \ln(5\Phi) - \mathbf{a} \cdot \ln(1 - 5\Phi) \tag{72c}$$

As results from Figure 10, the term ln(b3/b2) drops monotonically with Φ (and then V) value

$$\ln\left(\frac{\mathbf{b}\_3}{\mathbf{b}\_2}\right) = \alpha - \gamma \cdot \Phi \tag{73a}$$

$$
\ln\left(\frac{\mathbf{b}\_3}{\mathbf{b}\_2}\right) = \alpha - \beta \cdot \mathbf{V} \tag{73b}
$$

The value for β in (73b) is small for higher C01 values, ca. 1 mol/L; in Ref. [77], it was stated that <sup>β</sup> <sup>¼</sup> 1.7 � <sup>10</sup>�<sup>3</sup> at C01 <sup>¼</sup> 1.0 mol/L; this change is small and can be neglected over the V-range covered in the titration. The assumption ln(b3/b2) ¼ const is applied below in the simplified Gran models. For lower C01 values, this assumption provides a kind of drift introduced by the model applied, and then in accurate models, the formula Eq. (72c) is used.

From Eqs. (1) and (2), we have Φ/Φeq ¼ V/Veq; at Φeq ¼ 0.2, we get 5Φ ¼ V/Veq. Then applying Eq. (71b), we have

$$\mathbf{E} = \boldsymbol{\omega} - \mathbf{a} \cdot (\boldsymbol{\alpha} + \boldsymbol{\beta} \cdot \mathbf{V}) + \mathbf{a} \cdot \ln \frac{\mathbf{V}}{\mathbf{V\_{eq}}} - \mathbf{a} \cdot \ln \left( 1 - \frac{\mathbf{V}}{\mathbf{V\_{eq}}} \right) \tag{74}$$

valid for V < Veq, with the parameters: ω, α, β and a assumed constant within the V-range considered.

Figure 10. The ln(b3/b2) versus Φ relationships for the D þ T system where V<sup>0</sup> ¼ 100 mL of T (FeSO4 (C<sup>0</sup> ¼ 0.01) þ H2SO4 (C01) is titrated with V ml of KMnO4 (C ¼ 0.02). The lines are plotted at different concentrations (C01) of H2SO4, indicated at the corresponding curves.

#### 10.3.3. Simplified Gran I method

For jth and jþ1th experimental point, from Eq. (72) we get:

$$\begin{aligned} \mathbf{E}\_{\mathbf{j}} &= \mathbf{E}\_0 - \mathbf{a} \cdot \ln \frac{\mathbf{b}\_3}{\mathbf{b}\_2} + \mathbf{a} \cdot \ln(5\Phi\_{\mathbf{j}}) - \mathbf{a} \cdot \ln(1 - 5\Phi\_{\mathbf{j}}); \\\\ \mathbf{E}\_{\mathbf{j}+1} &= \mathbf{E}\_0 - \mathbf{a} \cdot \ln \frac{\mathbf{b}\_3}{\mathbf{b}\_2} + \mathbf{a} \cdot \ln(5\Phi\_{\mathbf{j}+1}) - \mathbf{a} \cdot \ln(1 - 5\Phi\_{\mathbf{j}+1}) \\\\ \mathbf{E}\_{\mathbf{j}+1} - \mathbf{E}\_{\mathbf{j}} &= \mathbf{a} \cdot \ln \frac{\Phi\_{\mathbf{j}+1}}{\Phi\_{\mathbf{j}}} - \mathbf{a} \cdot \ln \frac{1 - 5\Phi\_{\mathbf{j}+1}}{1 - 5\Phi\_{\mathbf{j}}} \end{aligned} \tag{75}$$

Applying in Eq. (69) the identities: <sup>Φ</sup><sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>Φ</sup><sup>j</sup> <sup>þ</sup> <sup>Φ</sup><sup>j</sup>þ<sup>1</sup> � <sup>Φ</sup><sup>j</sup> and 1 � <sup>5</sup>Φ<sup>j</sup> <sup>¼</sup> <sup>1</sup> � <sup>5</sup>Φ<sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>5</sup>ðΦ<sup>j</sup>þ<sup>1</sup> � <sup>Φ</sup>j<sup>Þ</sup> we have

$$\mathbf{E}\_{\mathbf{j}+1} - \mathbf{E}\_{\mathbf{j}} = \mathbf{a} \cdot \ln(1 + \mathbf{x}\_{\mathbf{i}\mathbf{j}}) - \mathbf{a} \cdot \ln(1 - \mathbf{x}\_{\mathbf{i}\mathbf{j}}) \tag{76}$$

where:

$$\mathbf{x}\_{\mathbf{i}\mathbf{j}} = (\Phi\_{\mathbf{j}+1} - \Phi\_{\mathbf{j}}) / \Phi\_{\mathbf{j}} \quad \text{and} \quad \mathbf{x}\_{\mathbf{2}\mathbf{j}} = \mathbf{S} (\Phi\_{\mathbf{j}+1} - \Phi\_{\mathbf{j}}) / (1 - \mathbf{5}\Phi\_{\mathbf{j}}) \tag{77}$$

Applying the approximation Eq. (41) [6] for x ¼ x1j and x ¼ �x2j in Eq. (69) and putting Φ<sup>j</sup> ¼ C � Vj=ðC0 � V0Þ, Φ<sup>j</sup>þ<sup>1</sup> ¼ C � Vjþ<sup>1</sup>=ðC0 � V0Þ, we get, by turns,

$$\ln(1+\mathbf{x}\_{\text{j}\uparrow}) = \frac{\boldsymbol{\Phi}\_{\text{j}+1}-\boldsymbol{\Phi}\_{\text{j}}}{(\boldsymbol{\Phi}\_{\text{j}}+\boldsymbol{\Phi}\_{\text{j}+1})/2} = \frac{\mathbf{V}\_{\text{j}+1}-\mathbf{V}\_{\text{j}}}{\mathbf{V}\_{\text{j}}^{\*}} \quad \text{and} \quad -\ln(1-\mathbf{x}\_{\text{2}\uparrow}) = \frac{5(\boldsymbol{\Phi}\_{\text{j}+1}-\boldsymbol{\Phi}\_{\text{j}})}{1-5(\boldsymbol{\Phi}\_{\text{j}}+\boldsymbol{\Phi}\_{\text{j}+1})/2} = \frac{\mathbf{V}\_{\text{j}+1}-\mathbf{V}\_{\text{j}}}{\mathbf{V}\_{\text{eq}}-\mathbf{V}\_{\text{j}}^{\*}} \tag{78}$$

$$\frac{1}{\mathbf{V\_j^\*}} \cdot \frac{\mathbf{V\_{j+1}} - \mathbf{V\_j}}{\mathbf{E\_{j+1}} - \mathbf{E\_j}} = \mathbf{G\_1} \cdot (\mathbf{V\_{eq}} - \mathbf{V\_j^\*}) + \varepsilon\_{\mathbf{j}} \tag{79}$$

$$\mathbf{y}\_{\mathbf{j}}^{\*} = \mathbf{P}\_{\mathbf{l}} - \mathbf{G}\_{\mathbf{l}} \cdot \mathbf{V}\_{\mathbf{j}}^{\*} + \varepsilon\_{\mathbf{j}} \tag{80}$$

where Vj \* (Eq. (61)), and

$$\mathbf{y}\_{\mathbf{j}}^{\*} = \frac{1}{\mathbf{V}\_{\mathbf{j}}^{\*}} \cdot \frac{\mathbf{V}\_{\mathbf{j}+1} - \mathbf{V}\_{\mathbf{j}}}{\mathbf{E}\_{\mathbf{j}+1} - \mathbf{E}\_{\mathbf{j}}} \tag{81}$$

$$\mathbf{P}\_1 = \frac{1}{\mathbf{a}}, \mathbf{G}\_1 = \frac{1}{\mathbf{a} \cdot \mathbf{V}\_{\text{eq}}} \tag{82}$$

$$\mathbf{V\_{eq}} = \frac{\mathbf{P\_1}}{\mathbf{G\_1}} \tag{83}$$

P1 and G1 in Eq. (80) are obtained according to LSM, as previously described.

#### 10.3.4. Accurate Gran I method

Applying analogous procedure based on Eqs. (67) and (68), we get, by turns,

$$\mathbf{E}\_{\mathbf{j}+1} - \mathbf{E}\_{\mathbf{j}} = \mathbf{a} \cdot \mathbf{y} \cdot (\Phi\_{\mathbf{j}+1} - \Phi\_{\mathbf{j}}) + \mathbf{a} \cdot \ln(1 + \mathbf{x}\_{\mathbf{l}\mathbf{j}}) - \mathbf{a} \cdot \ln(1 - \mathbf{x}\_{\mathbf{2}\mathbf{j}}) \tag{84}$$

$$\mathbf{E}\_{\mathbf{j}+1} - \mathbf{E}\_{\mathbf{j}} = \mathbf{a} \cdot \mathbf{y} \cdot (\boldsymbol{\Phi}\_{\mathbf{j}+1} - \boldsymbol{\Phi}\_{\mathbf{j}}) + \mathbf{a} \cdot \frac{(\boldsymbol{\Phi}\_{\mathbf{j}+1} - \boldsymbol{\Phi}\_{\mathbf{j}})}{(\boldsymbol{\Phi}\_{\mathbf{j}+1} + \boldsymbol{\Phi}\_{\mathbf{j}})/2} + \mathbf{a} \cdot \frac{\mathbf{5} \cdot (\boldsymbol{\Phi}\_{\mathbf{j}+1} - \boldsymbol{\Phi}\_{\mathbf{j}})}{1 - 5(\boldsymbol{\Phi}\_{\mathbf{j}+1} + \boldsymbol{\Phi}\_{\mathbf{j}})/2} \tag{85}$$

$$\frac{\mathbf{E\_{j+1}} - \mathbf{E\_{j}}}{\mathbf{V\_{j+1}} - \mathbf{V\_{j}}} = \mathbf{B} + \frac{\mathbf{a}}{\mathbf{V\_{j}}^{\*}} + \frac{\mathbf{a}}{\mathbf{V\_{eq}} - \mathbf{V\_{j}}^{\*}} + \varepsilon\_{\mathbf{j}} \tag{86}$$

where

10.3.3. Simplified Gran I method

at the corresponding curves.

156 Advances in Titration Techniques

we have

where:

For jth and jþ1th experimental point, from Eq. (72) we get:

Ej <sup>¼</sup> E0 � <sup>a</sup> � ln b3

Ejþ<sup>1</sup> <sup>¼</sup> E0 � <sup>a</sup> � ln b3

Ejþ<sup>1</sup> � Ej ¼ a � ln

b2

b2

Φ<sup>j</sup>þ<sup>1</sup> Φj

þ a � lnð5Φ<sup>j</sup>

Figure 10. The ln(b3/b2) versus Φ relationships for the D þ T system where V<sup>0</sup> ¼ 100 mL of T (FeSO4 (C<sup>0</sup> ¼ 0.01) þ H2SO4 (C01) is titrated with V ml of KMnO4 (C ¼ 0.02). The lines are plotted at different concentrations (C01) of H2SO4, indicated

� a � ln

Applying in Eq. (69) the identities: <sup>Φ</sup><sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>Φ</sup><sup>j</sup> <sup>þ</sup> <sup>Φ</sup><sup>j</sup>þ<sup>1</sup> � <sup>Φ</sup><sup>j</sup> and 1 � <sup>5</sup>Φ<sup>j</sup> <sup>¼</sup> <sup>1</sup> � <sup>5</sup>Φ<sup>j</sup>þ<sup>1</sup> <sup>þ</sup> <sup>5</sup>ðΦ<sup>j</sup>þ<sup>1</sup> � <sup>Φ</sup>j<sup>Þ</sup>

Þ � a � lnð1 � 5ΦjÞ;

Ejþ<sup>1</sup> � Ej ¼ a � lnð1 þ x1jÞ � a � lnð1 � x2jÞ ð76Þ

x1j ¼ ðΦ<sup>j</sup>þ<sup>1</sup> � <sup>Φ</sup>jÞ=Φ<sup>j</sup> and x2j <sup>¼</sup> <sup>5</sup>ðΦ<sup>j</sup>þ<sup>1</sup> � <sup>Φ</sup>jÞ=ð<sup>1</sup> � <sup>5</sup>ΦjÞ ð77<sup>Þ</sup>

ð75Þ

<sup>þ</sup> <sup>a</sup> � lnð5Φ<sup>j</sup>þ<sup>1</sup>Þ � <sup>a</sup> � lnð<sup>1</sup> � <sup>5</sup>Φ<sup>j</sup>þ<sup>1</sup><sup>Þ</sup>

1 � 5Φ<sup>j</sup>þ<sup>1</sup> 1 � 5Φ<sup>j</sup>

$$\mathbf{B} = \frac{\mathbf{a} \cdot \boldsymbol{\gamma}}{5 \mathbf{V}\_{\text{eq}}} \tag{87}$$

The parameters: B, a and Veq are then found according to iterative procedure; Vj � is defined by Eq. (61).

#### 10.3.5. Simplified Gran II method

From Eqs. (1), (2) and (72a), we have, by turns

$$\mathbf{f}\_{23} \cdot \frac{\mathbf{b}\_3}{\mathbf{b}\_2} = \frac{\boldsymbol{\Phi}}{\boldsymbol{\Phi}\_{\text{eq}} - \boldsymbol{\Phi}} = \frac{\mathbf{V}}{\mathbf{V}\_{\text{eq}} - \mathbf{V}} \tag{88}$$

In this case, the fraction b3/b2 is assumed constant. From Eqs. (88) and (71a), we get, by turns,

$$\mathbf{V} \cdot \mathbf{1}0^{-\mathbf{A} \cdot \mathbf{E}} = \frac{\mathbf{b}\_3}{\mathbf{b}\_2} \cdot \mathbf{1}0^{-\mathbf{A} \cdot \mathbf{E}\_0} \cdot (\mathbf{V}\_{\mathbf{eq}} - \mathbf{V}) \tag{89}$$

If b3/b2 is assumed constant, then G2 <sup>¼</sup> b2/b3�10�A�E0<sup>¼</sup> const, and

$$\mathbf{V\_{j}} \cdot \mathbf{10^{-A \cdot E}} = \mathbf{P\_{2}} - \mathbf{G\_{2}} \cdot \mathbf{V\_{j}} + \varepsilon\_{\mathbf{j}} \tag{90}$$

Then

$$\mathbf{V\_{eq}} = \frac{\mathbf{P\_2}}{\mathbf{G\_2}} \tag{91}$$

where P2 and G2 are calculated according to LSM from the regression equation (90).

#### 10.3.6. MG(II)A method

At β�V << 1, we write

$$\frac{\mathbf{b}\_3}{\mathbf{b}\_2} = \mathbf{e}^{\alpha} \cdot \mathbf{e}^{-\beta \mathbf{V}} \cong \mathbf{e}^{\alpha} \cdot (1 - \beta \cdot \mathbf{V}) \tag{92}$$

From Eqs. (89) and (92), we get

$$
\Omega \Omega = \Omega(\theta, \mathbf{V}) = \mathbf{V} \cdot 10^{-\mathrm{E}/\theta} = \mathbf{G}\_2 \cdot (\mathbf{V}\_{\mathrm{eq}} - \mathbf{V}) \cdot (1 - \beta \cdot \mathbf{V}) \tag{93}
$$

where G2 <sup>¼</sup> <sup>e</sup><sup>α</sup> � <sup>10</sup>�A�E0<sup>¼</sup> const and real slope <sup>ϑ</sup> of an electrode is involved, after putting 1/<sup>ϑ</sup> for A. From Eq. (93), we have

$$
\Omega \Omega = \Omega(\ $, \mathbf{V}) = \mathbf{V} \cdot \mathbf{1} \mathbf{0}^{-\text{E}/\$ } = \mathbf{P} \cdot \mathbf{V}^2 - \mathbf{Q} \cdot \mathbf{V} + \mathbb{R} \tag{94}
$$

where:

$$\mathbf{P} = \mathbf{G}\_2 \cdot \boldsymbol{\beta} \tag{95a}$$

$$\mathbf{Q} = \mathbf{G}\_2 \cdot (\mathbf{\hat{\beta}} \cdot \mathbf{V}\_{\mathbf{eq}} + \mathbf{1}) \tag{95b}$$

$$\mathbf{R} = \mathbf{G}\_2 \times \mathbf{V}\_{\mathbf{eq}} \tag{95c}$$

The P, Q, and R values in Eqs. (95a,b,c) are determined according to LSM, applied to the regression equation

$$\mathbf{Q}\_{\mathbf{j}} = \mathbf{P} \cdot \mathbf{V}\_{\mathbf{j}}^2 - \mathbf{Q} \cdot \mathbf{V}\_{\mathbf{j}} + \mathbf{R} + + \mathbf{e}\_{\mathbf{j}} \tag{96}$$

where

10.3.5. Simplified Gran II method

158 Advances in Titration Techniques

Then

10.3.6. MG(II)A method

At β�V << 1, we write

From Eqs. (89) and (92), we get

for A. From Eq. (93), we have

where:

From Eqs. (1), (2) and (72a), we have, by turns

f23 � b3 b2

If b3/b2 is assumed constant, then G2 <sup>¼</sup> b2/b3�10�A�E0<sup>¼</sup> const, and

b3 b2

<sup>V</sup> � <sup>10</sup>�A�<sup>E</sup> <sup>¼</sup> b3

<sup>¼</sup> <sup>Φ</sup>

b2

<sup>Φ</sup>eq � <sup>Φ</sup> <sup>¼</sup> <sup>V</sup>

In this case, the fraction b3/b2 is assumed constant. From Eqs. (88) and (71a), we get, by turns,

Veq <sup>¼</sup> P2 G2

where G2 <sup>¼</sup> <sup>e</sup><sup>α</sup> � <sup>10</sup>�A�E0<sup>¼</sup> const and real slope <sup>ϑ</sup> of an electrode is involved, after putting 1/<sup>ϑ</sup>

where P2 and G2 are calculated according to LSM from the regression equation (90).

Veq � <sup>V</sup> <sup>ð</sup>88<sup>Þ</sup>

ð91Þ

� <sup>10</sup>�A�E0 � ðVeq � <sup>V</sup>Þ ð89<sup>Þ</sup>

Vj � <sup>10</sup>�A�<sup>E</sup> <sup>¼</sup> P2 � G2 � Vj <sup>þ</sup> <sup>ε</sup><sup>j</sup> <sup>ð</sup>90<sup>Þ</sup>

<sup>¼</sup> <sup>e</sup><sup>α</sup> � <sup>e</sup>�β<sup>V</sup> ffi <sup>e</sup><sup>α</sup> � ð<sup>1</sup> � <sup>β</sup> � <sup>V</sup>Þ ð92<sup>Þ</sup>

<sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup>ðϑ; <sup>V</sup>Þ ¼ <sup>V</sup> � <sup>10</sup>�E=<sup>ϑ</sup> <sup>¼</sup> G2 � ðVeq � <sup>V</sup>Þ�ð<sup>1</sup> � <sup>β</sup> � <sup>V</sup>Þ ð93<sup>Þ</sup>

<sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup>ðϑ; <sup>V</sup>Þ ¼ <sup>V</sup> � <sup>10</sup>�E=<sup>ϑ</sup> <sup>¼</sup> <sup>P</sup> � <sup>V</sup><sup>2</sup> � <sup>Q</sup> � <sup>V</sup> <sup>þ</sup> <sup>R</sup> <sup>ð</sup>94<sup>Þ</sup>

P ¼ G2 � β ð95aÞ

Q ¼ G2 � ðβ � Veq þ 1Þ ð95bÞ

R ¼ G2 � Veq ð95cÞ

$$\mathbf{Q}\_{\mathbf{j}} = \mathbf{V}\_{\mathbf{j}} \cdot \mathbf{1} \mathbf{0}^{-\mathbf{E}\_{\mathbf{j}}/\mathbf{\mathcal{S}}} \tag{97}$$

Then we get, by turns,

$$\frac{\mathbf{R}}{\mathbf{P}} = \frac{\mathbf{V\_{eq}}}{\beta}; \frac{\mathbf{Q}}{\mathbf{R}} = \beta + \frac{1}{\mathbf{V\_{eq}}}; \mathbf{P} \cdot \mathbf{V\_{eq}}^2 - \mathbf{Q} \cdot \mathbf{V\_{eq}} + \mathbf{R} = 0\tag{98}$$

$$\mathbf{V\_{eq}} = \frac{\mathbf{Q} - \sqrt{\mathbf{Q}^2 - 4 \cdot \mathbf{P} \cdot \mathbf{R}}}{2 \cdot \mathbf{P}} \tag{99}$$

Eq. (96) is the basis for the modified G(II) method in its accurate version, denoted as MG(II)A method [77]. This method is especially advantageous in context of the error of analysis resulting from greater discrepancies |ϑ<sup>c</sup> – ϑp| between true (correct, ϑc) and preassumed (ϑp) slope values for RIE has been proved; the error in Veq is significantly decreased even at greater |ϑ<sup>c</sup> – ϑp| values [77].

Numerous modifications of the Gran methods, designed also for calibration of redox indicator electrodes (RIE) purposes, were presented in the Refs. [4–6, 77]. Other calibration methods, related to ISE electrodes, are presented in Ref. [5].

#### 10.4. Modified G(II) methods for carbonate alkalinity (CA) measurements

The G(II) methods were also suggested [28] and applied [78] for determination of carbonate alkalinity (CA) according to the modified CAM method. The CAM is related to the mixtures NaHCO3 þ Na2CO3 (system I) and Na2CO3 þ NaOH (system II), see Table 5. In addition to


Table 5. The modified Gran functions (CAM) related to the systems I and II (see text).

the determination of equivalence volumes, the proposed method gives the possibility of determining the activity coefficient of hydrogen ions (γ). Moreover, CAM can be used to calculate the dissociation constants (K1, K2) for carbonic acid and the ionic product of water (KW) from a single pH titration curve. The parameters of the related functions are calculated according to LSM.

## 11. A brief review of other papers involved with titrimetric methods of analysis

## 11.1. Isohydric systems

Simple acid-acid systems are involved in isohydricity concept, formulated by Michalowski [31, 32, 79]. For the simplest case of acid-acid titration HB (C,V) ! HL (C0, V0), where HB is a strong acid, HL is a weak monoprotic acid (K1), the isohydricity condition, pH ¼ const, occurs at

$$\mathbf{C}\_0 = \mathbf{C} + \mathbf{C}^2 \cdot \mathbf{1}0^{\text{pK}\_1} \tag{100}$$

where pK1 ¼ �logK1.

In such a system, the ionic strength of the D þ T mixture remains constant during the titration, i.e., the isohydricity and isomolarity conditions are fulfilled simultaneously and independently on the volume V of the titrant added. On this basis, a very sensitive method of pK1 determination was suggested [31, 32]. The isohydricity conditions were also formulated for more complex acid-acid, base-base systems, etc.

### 11.2. pH titration in isomolar systems

The method of pH titration in isomolar D þ T systems of concentrated solutions (ionic strength 2–2.5 mol/L) is involved with presence of equal volumes of the sample tested both in D and T. The presence of a strong acid HB in one of the solutions is compensated by a due excess of a salt MB in the second solution [80–90]. In the systems tested, acid-base and complexation equilibria were involved. The method enables to calculate concentrations of components in the sample tested together with equilibrium constants and activity coefficient of hydrogen ions. This method was applied for determination of a complete set of stability constants for mixed complexes [91–94].

#### 11.3. Carbonate alkalinity, total alkalinity, and alkalinity with fulvic acids

Ref. [29] was referred to complex acid-base equilibria related to nonstoichiometric species involved with fulvic acids and their complexes with other metal ions and simpler species present in natural waters. For mathematical description of such systems, the idea of Simms constants was recalled from earlier issues e.g., Refs. [27, 28, 84–88], and the concept of activity/ basicity centers in such systems was introduced.

## 11.4. Binary-solvent systems

the determination of equivalence volumes, the proposed method gives the possibility of determining the activity coefficient of hydrogen ions (γ). Moreover, CAM can be used to calculate the dissociation constants (K1, K2) for carbonic acid and the ionic product of water (KW) from a single pH titration curve. The parameters of the related functions are calculated according to

11. A brief review of other papers involved with titrimetric methods of

Simple acid-acid systems are involved in isohydricity concept, formulated by Michalowski [31, 32, 79]. For the simplest case of acid-acid titration HB (C,V) ! HL (C0, V0), where HB is a strong acid, HL is a weak monoprotic acid (K1), the isohydricity condition, pH ¼ const, occurs at

In such a system, the ionic strength of the D þ T mixture remains constant during the titration, i.e., the isohydricity and isomolarity conditions are fulfilled simultaneously and independently on the volume V of the titrant added. On this basis, a very sensitive method of pK1 determination was suggested [31, 32]. The isohydricity conditions were also formulated for more com-

The method of pH titration in isomolar D þ T systems of concentrated solutions (ionic strength 2–2.5 mol/L) is involved with presence of equal volumes of the sample tested both in D and T. The presence of a strong acid HB in one of the solutions is compensated by a due excess of a salt MB in the second solution [80–90]. In the systems tested, acid-base and complexation equilibria were involved. The method enables to calculate concentrations of components in the sample tested together with equilibrium constants and activity coefficient of hydrogen ions. This method was applied for determination of a complete set of stability constants for

Ref. [29] was referred to complex acid-base equilibria related to nonstoichiometric species involved with fulvic acids and their complexes with other metal ions and simpler species present in natural waters. For mathematical description of such systems, the idea of Simms constants was recalled from earlier issues e.g., Refs. [27, 28, 84–88], and the concept of activity/

11.3. Carbonate alkalinity, total alkalinity, and alkalinity with fulvic acids

C0 <sup>¼</sup> <sup>C</sup> <sup>þ</sup> C2 � <sup>10</sup>pK1 <sup>ð</sup>100<sup>Þ</sup>

LSM.

analysis

11.1. Isohydric systems

160 Advances in Titration Techniques

where pK1 ¼ �logK1.

plex acid-acid, base-base systems, etc.

11.2. pH titration in isomolar systems

basicity centers in such systems was introduced.

mixed complexes [91–94].

Mutual pH titrations of weak acid solutions of the same concentration C in D and T formed in different solvents were applied [33–35] to formulate the pKi ¼ pKi(x) relationships for the acidity parameters, where x is the mole fraction of a cosolvent with higher molar mass in D þ T mixture. The pKi ¼ pKi(x) relationship was based on the Ostwald's formula [95, 96] for monoprotic acid or the Henderson-Hasselbalch functions for diprotic and triprotic acids. The systems were modeled with the use of different nonlinear functions, namely Redlich-Kister and orthogonal (normal, shifted) Legendre polynomials. Asymmetric functions by Myers-Scott and the function suggested by Michałowski were also used for this purpose.

## 11.5. pH-static titration

Two kinds of reactions are necessary in Veq registration according to pH–static titration; one of them has to be an acid–base reaction. The proton consumption or generation occurs in redox, complexation, or precipitation reactions [47], for example in titration of arsenite(þ3) solution with I2 þ KI solution [18]; zinc salt solution with EDTA [97]; cyanide according to a (modified) Liebig-Denigès method [65, 102, 103].

## 11.6. Titration to a preset pH value

A cumulative effect of different factors on precision of Veq determination was considered in [98] for pH titration of a weak monoprotic acids HL with a strong base, MOH. The results of calculations were presented graphically.

## 11.7. Dynamic buffer capacity

The dynamic buffer capacity concept, βV, involving the dilution effect in acid-base D þ T system, has been introduced [99] and extended in further papers [27, 28, 30, 100].

## 11.8. Other examples

The errors involved with more complex titrimetric analyses of chloride (mercurimetric method) [101], and cyanide (modified) Liebig-Denigès method) [97, 102, 103]. A modified, spectro-pH-metric method of dissociation constant determination was presented in Ref. [104]. An overview of potentiometric methods of titrimetric analyses was presented in Ref. [64]. The titration of ammonia in the final step of the Kjeldahl method of nitrogen determination [105, 106] was discussed in Ref. [107].

The proton consumption or generation occurs in redox, complexation, or precipitation reactions [47], for example in titration of arsenite(þ3) solution with I2 þ KI solution [18]; zinc salt solution with EDTA [97]; cyanide according to a (modified) Liebig-Denigès method [65, 102, 103].

Three (complexation, acid-base, precipitation) kinds of reactions occur in the Liebig-Denigès method mentioned above. Four elementary (redox, complexation, acid-base, precipitation of I2) types of reactions occur in the D þ T system described in the legend for Figure 2 and in less complex HCl ! NaIO system presented in Ref. [21]. Other examples of high degree of complexity are shown in the works [9, 11, 12, 14–16]. One of the examples in Ref. [12] concerns a four-step analytical process with the four kinds of reactions, involving three electroactive elements.

## 12. Final comments

The Generalized Approach To Electrolytic Systems (GATES) provides the possibility of thermodynamic description of equilibrium and metastable, redox and non-redox, mono- and twophase systems of any degree of complexity. It gives the possibility of all attainable/pre-selected physicochemical knowledge to be involved, with none simplifying assumptions done for calculation purposes. It can be applied for different types of reactions occurring in batch or dynamic systems, of any degree of complexity. The generalized electron balance (GEB) concept, discovered (1992, 2006) by Michałowski [11, 13] and obligatory for description of redox systems, is fully compatible with charge and concentration balance(s), and relations for the corresponding equilibrium constants.

The chapter provides some examples of dynamic electrolytic systems of different degree of complexity, realized in titrimetric procedure that may be considered from physicochemical and/or analytical viewpoints. In all instances, one can follow measurable quantities (potential E, pH) in dynamic and static processes, and gain the information about details not measurable in real experiments; it particularly refers to dynamic speciation. In the calculations made according to iterative computer programs, all physicochemical knowledge can be involved.

This chapter aims to demonstrate the huge/versatile possibilities inherent in GATES, as a relatively new quality of physicochemical knowledge gaining from electrolytic systems of different degrees of complexity, realizable with use of iterative computer programs.

## Appendix


Expressions for <sup>Φ</sup> related to some D <sup>þ</sup> T acid-base systems [6]; Mþ<sup>1</sup> <sup>¼</sup> Naþ<sup>1</sup> , Kþ<sup>1</sup> ; B�<sup>1</sup> <sup>¼</sup> Cl�<sup>1</sup> , NO3 �1 ; k ¼ 0,…,n (nos. 1–10), k ¼ 0,…,q � n (no. 11); l ¼ 0,…,m.

Principles of Titrimetric Analyses According to Generalized Approach to Electrolytic Systems (GATES) http://dx.doi.org/10.5772/intechopen.69248 163


The symbols:

complex HCl ! NaIO system presented in Ref. [21]. Other examples of high degree of complexity are shown in the works [9, 11, 12, 14–16]. One of the examples in Ref. [12] concerns a four-step

The Generalized Approach To Electrolytic Systems (GATES) provides the possibility of thermodynamic description of equilibrium and metastable, redox and non-redox, mono- and twophase systems of any degree of complexity. It gives the possibility of all attainable/pre-selected physicochemical knowledge to be involved, with none simplifying assumptions done for calculation purposes. It can be applied for different types of reactions occurring in batch or dynamic systems, of any degree of complexity. The generalized electron balance (GEB) concept, discovered (1992, 2006) by Michałowski [11, 13] and obligatory for description of redox systems, is fully compatible with charge and concentration balance(s), and relations for the

The chapter provides some examples of dynamic electrolytic systems of different degree of complexity, realized in titrimetric procedure that may be considered from physicochemical and/or analytical viewpoints. In all instances, one can follow measurable quantities (potential E, pH) in dynamic and static processes, and gain the information about details not measurable in real experiments; it particularly refers to dynamic speciation. In the calculations made according to iterative computer programs, all physicochemical knowledge can be involved.

This chapter aims to demonstrate the huge/versatile possibilities inherent in GATES, as a relatively new quality of physicochemical knowledge gaining from electrolytic systems of

, Kþ<sup>1</sup>

ðn � k � nÞ � C0 � α C þ α

ðn þ k � nÞ � C0 þ α C � α

C0 � C0 � α C þ α

C0 � C0 þ α C � α

C0 �

C0 � ; B�<sup>1</sup> <sup>¼</sup> Cl�<sup>1</sup>

,

different degrees of complexity, realizable with use of iterative computer programs.

Expressions for <sup>Φ</sup> related to some D <sup>þ</sup> T acid-base systems [6]; Mþ<sup>1</sup> <sup>¼</sup> Naþ<sup>1</sup>

No. A B Φ ¼ 1 HCl MOH C

2 MOH HB C

3 MkHn-kL MOH C

4 MkHn-kL HB C

; k ¼ 0,…,n (nos. 1–10), k ¼ 0,…,q � n (no. 11); l ¼ 0,…,m.

analytical process with the four kinds of reactions, involving three electroactive elements.

12. Final comments

162 Advances in Titration Techniques

Appendix

NO3 �1

corresponding equilibrium constants.

n ¼ X<sup>q</sup> <sup>i</sup>¼<sup>1</sup> <sup>i</sup> � ½Hi <sup>L</sup>þi�<sup>n</sup>� X<sup>q</sup> <sup>i</sup>¼<sup>0</sup> <sup>½</sup>Hi <sup>L</sup>þi�<sup>n</sup>� <sup>¼</sup> X<sup>q</sup> <sup>i</sup>¼<sup>1</sup> <sup>i</sup> � 10logK<sup>H</sup> Li�i�pH X<sup>q</sup> <sup>i</sup>¼<sup>0</sup> 10logK<sup>H</sup> Li�i�pH m ¼ X<sup>p</sup> <sup>i</sup>¼<sup>1</sup> <sup>i</sup> � ½Hi <sup>Ł</sup>þi�<sup>m</sup>� X<sup>p</sup> <sup>i</sup>¼<sup>0</sup> <sup>½</sup>Hi <sup>Ł</sup>þi�<sup>n</sup>� <sup>¼</sup> X<sup>p</sup> <sup>i</sup>¼<sup>1</sup> <sup>i</sup> � 10logK<sup>H</sup> Łi �i�pH X<sup>p</sup> <sup>i</sup>¼<sup>0</sup> <sup>10</sup>logKH Łi �i�pH nN <sup>¼</sup> <sup>½</sup>NH4 þ1� ½NH4 <sup>þ</sup><sup>1</sup>�þ½NH3� <sup>¼</sup> 10logK<sup>H</sup> 1N�pH 10logK<sup>H</sup> 1N�pH <sup>þ</sup> <sup>1</sup>

enable to get a compact form of the functions, where:

½Hi <sup>L</sup>þi�<sup>n</sup>� ¼ <sup>K</sup><sup>H</sup> Li � ½Hþ� i ½L�<sup>n</sup> � (i ¼ 0,…,q); ½Hi <sup>Ł</sup>þi�<sup>m</sup>� ¼ KH <sup>Ł</sup><sup>i</sup> � ½Hþ� i <sup>½</sup>Ł�<sup>m</sup>�(i <sup>¼</sup> 0,…,p) ; [NH4 þ1 ] ¼ K<sup>H</sup> 1N[H<sup>þ</sup>][NH3] (logK<sup>H</sup> 1N <sup>¼</sup> <sup>9</sup>:35); KH L0 <sup>¼</sup> <sup>K</sup><sup>H</sup> <sup>Ł</sup><sup>0</sup> <sup>¼</sup> 1; Mþ<sup>1</sup> <sup>¼</sup> <sup>K</sup>þ<sup>1</sup> , Naþ<sup>1</sup> ; [Hþ<sup>1</sup> ] <sup>¼</sup> <sup>10</sup>�pH

and the ubiquitous symbol

$$\alpha = [\text{H}^{+1}] - [\text{OH}^{-1}] = 10^{-\text{pH}} - 10^{\text{pH}-\text{pk}\_w}$$

termed as "proton excess" is used; pKW ¼ 14.0 is assumed here.

#### Notations

D, titrand; T, titrant; V0, volume of D; V, volume of T; all volumes are expressed in mL; all concentrations are expressed in mol/L.

## Author details

Anna Maria Michałowska-Kaczmarczyk<sup>1</sup> , Aneta Spórna-Kucab<sup>2</sup> and Tadeusz Michałowski<sup>2</sup> \*

\*Address all correspondence to: michalot@o2.pl

1 Department of Oncology, The University Hospital in Cracow, Cracow, Poland

2 Faculty of Chemical Engineering and Technology, Cracow University of Technology, Cracow, Poland

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Author details

164 Advances in Titration Techniques

Cracow, Poland

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2 Faculty of Chemical Engineering and Technology, Cracow University of Technology,

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#### **A Distinguishing Feature of the Balance 2∙***f***(O)−***f***(H) in Electrolytic Systems: The Reference to Titrimetric Methods of Analysis** A Distinguishing Feature of the Balance 2∙f(O)�f(H) in Electrolytic Systems: The Reference to Titrimetric Methods of Analysis

DOI: 10.5772/intechopen.69249

Anna Maria Michałowska-Kaczmarczyk, Aneta Spórna-Kucab and Tadeusz Michałowski Anna Maria Michałowska-Kaczmarczyk, Aneta Spórna-Kucab and

Additional information is available at the end of the chapter Tadeusz Michałowski

http://dx.doi.org/10.5772/intechopen.69249 Additional information is available at the end of the chapter

> Motto: 'Nothing is too wonderful to be true if it be consistent with the laws of Nature' (M. Faraday)

#### Abstract

The balance 2∙f(O)�f(H) provides a general criterion distinguishing between electrolytic redox and non-redox systems of any degree of complexity, in aqueous, non-aqueous and mixed-solvent media. When referred to redox systems, it is an equation linearly independent on charge (ChB) and elemental/core balances f(Yg) for elements/cores Yg 6¼ H and O, whereas for non-redox systems, 2∙f(O)�f(H) is linearly dependent on these balances. The balance 2∙f(O)�f(H) formulated for redox systems is the primary form (pr-GEB) of the generalized electron balance (GEB) as the fundamental equation needed for resolution of these systems. Formulation of GEB for redox systems needs no prior knowledge of oxidation numbers for all elements of the system. Any prior knowledge of oxidation numbers for all elements in components forming a redox system and in the species of the system thus formed is not necessary within the Approach II to GEB. Oxidants and reductants are not indicated. Stoichiometry and equivalent mass are redundant concepts only. The GEB, together with charge balance and concentration balances for elements 6¼ H and O, and the complete set of independent equations for equilibrium constants form an algorithm, resolvable with use of an iterative computer program. All attainable physicochemical knowledge can be included in the algorithm. Some variations involved with tests of possible reaction paths for metastable systems can also be made. The effects of incomplete physicochemical knowledge on the system can be also tested. One of the main purposes of this chapter is to provide the GEB formulation needed for resolution of redox systems and familiarize it to a wider community of chemists.

Keywords: electrolytes, redox systems, non-redox systems, generalized electron balance (GEB), titration

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

## 1. Introduction

Scientific theories describe particular units and rules governing the relationships between them. This description is considered as interpretation of reality. In particular, the thermodynamic description of any electrolytic system according to generalized approach to electrolytic systems (GATESs) [1] is based on fundamental and physical rules of conservation, expressed by charge balance (ChB) and elemental balances, for particular elements and/or cores in a closed system, separated from the environment by diathermal (freely permeable by heat) walls. The term 'core' is related to a cluster of elements with the same formula, structure and external charge. For example, HSO4 1 ∙n5H2O, SO4 2 ∙n6H2O and FeSO4∙n27H2O, in Eq. (3) (below), have the common core (SO4 2 ). In this context, the pairs of species: (i) C2O4 <sup>2</sup> and CO3 2 ; (ii) C2O4 <sup>2</sup> (from maleic acid) and C2O4 <sup>2</sup> (from fumaric acid) and (iii) NO2 <sup>1</sup> and NO2 have no common cores.

Chemical interactions in electrolytic systems, for example protonation, neutralization, hydration, hydrolysis or dilution phenomena, are usually accompanied by exothermic or endothermic effects. However, the mass change, Δm, resulting from these thermal phenomena, estimated according to the Einstein's formula ΔE = Δm.c2 and put in context with their enthalpies ΔH (ΔE = ΔH), that is Δm = ΔH/c<sup>2</sup> , is negligibly small (not measurable). Therefore, the mass of a chemical system remains practically unchanged, regardless of whether the chemical reactions take place in it or not.

The heat exchange between the system and its environment through diathermal walls enables the temperature T of the system to be kept constant during the appropriate dynamic processes, such as titration, performed in a quasistatic manner. Stability of temperature T within the titrant (titrating solution, T), titrand (titrated solution, D) and D + T mixture, together with constancy of ionic strength (I) in D+T, is the preliminary condition ensuring stability of the corresponding equilibrium constants, Ki = Ki(T, I), related to the system in question. The diathermal walls separate condensed (liquid or liquid + solid) phases from their environment.

An open chemical system is an approximation of the closed system—provided that the matter (e.g., H2O, CO2 and O2) exchange between the system and its surroundings can be neglected, within a relatively short period of time needed to carry out the process considered, for example, titration.

On the initial stage of ChB and elemental/core balance formulation, it is advisable to start the quantitative considerations from the numbers of particular entities (components, species):


For example, H2O and gaseous HCl, as components, form an aqueous solution of HCl, with H2O and hydrates of H+1,OH<sup>1</sup> and Cl<sup>1</sup> as the species. Generally, when solid, liquid and/or gaseous solutes are introduced into water, a mono- or two-phase system is obtained. The resulting mixture is limited to the condensed (liquid or liquid + solid) phases. We refer mainly to aqueous media (W = H2O), where the physicochemical knowledge is relatively extensive, incomparably better than that for the system with non-aqueous, or mixed-solvent media [2–6], with amphiprotic co-solvents involved, which is also considered in this chapter. For such media, the elemental f(Eg)) or core f(coreg)) balances written in terms of numbers of individual entities containing the elements (Eg) or cores (coreg), are formulated.

1. Introduction

174 Advances in Titration Techniques

example, HSO4

acid) and C2O4

ple, titration.

2

core (SO4

1

∙n5H2O, SO4

enthalpies ΔH (ΔE = ΔH), that is Δm = ΔH/c<sup>2</sup>

• N0j for jth component constituting the system • Ni for ith species in the system thus formed

chemical reactions take place in it or not.

2

). In this context, the pairs of species: (i) C2O4

<sup>2</sup> (from fumaric acid) and (iii) NO2

Scientific theories describe particular units and rules governing the relationships between them. This description is considered as interpretation of reality. In particular, the thermodynamic description of any electrolytic system according to generalized approach to electrolytic systems (GATESs) [1] is based on fundamental and physical rules of conservation, expressed by charge balance (ChB) and elemental balances, for particular elements and/or cores in a closed system, separated from the environment by diathermal (freely permeable by heat) walls. The term 'core' is related to a cluster of elements with the same formula, structure and external charge. For

Chemical interactions in electrolytic systems, for example protonation, neutralization, hydration, hydrolysis or dilution phenomena, are usually accompanied by exothermic or endothermic effects. However, the mass change, Δm, resulting from these thermal phenomena, estimated according to the Einstein's formula ΔE = Δm.c2 and put in context with their

the mass of a chemical system remains practically unchanged, regardless of whether the

The heat exchange between the system and its environment through diathermal walls enables the temperature T of the system to be kept constant during the appropriate dynamic processes, such as titration, performed in a quasistatic manner. Stability of temperature T within the titrant (titrating solution, T), titrand (titrated solution, D) and D + T mixture, together with constancy of ionic strength (I) in D+T, is the preliminary condition ensuring stability of the corresponding equilibrium constants, Ki = Ki(T, I), related to the system in question. The diathermal walls separate condensed (liquid or liquid + solid) phases from their environment.

An open chemical system is an approximation of the closed system—provided that the matter (e.g., H2O, CO2 and O2) exchange between the system and its surroundings can be neglected, within a relatively short period of time needed to carry out the process considered, for exam-

On the initial stage of ChB and elemental/core balance formulation, it is advisable to start the quantitative considerations from the numbers of particular entities (components, species):

For example, H2O and gaseous HCl, as components, form an aqueous solution of HCl, with H2O and hydrates of H+1,OH<sup>1</sup> and Cl<sup>1</sup> as the species. Generally, when solid, liquid and/or gaseous solutes are introduced into water, a mono- or two-phase system is obtained. The resulting mixture is limited to the condensed (liquid or liquid + solid) phases. We refer mainly to aqueous media (W = H2O), where the physicochemical knowledge is relatively extensive, incomparably better than that for the system with non-aqueous, or mixed-solvent media [2–6],

∙n6H2O and FeSO4∙n27H2O, in Eq. (3) (below), have the common

2

, is negligibly small (not measurable). Therefore,

<sup>1</sup> and NO2 have no common cores.

; (ii) C2O4

<sup>2</sup> (from maleic

<sup>2</sup> and CO3

In aqueous electrolytic systems, different entities Xi zi exist as hydrated species, Xi zi � niH2O; ni = niW = niH2O is the mean number of water (W = H2O) molecules attached to Xi zi ðni ≥ 0Þ, zi is a charge of this species, expressed in elementary charge units and e = F/NA (F = Faraday constant, NA = Avogadro number). For these species present in static or dynamic systems, we apply the notation

$$\mathcal{X}\_{\mathbf{i}}^{\,\,\,\mathbf{z}\_{\mathbf{i}}}(\mathcal{N}\_{\mathbf{i}},\mathbf{n}\_{\mathbf{i}}) \tag{1}$$

where Ni is the number of these entities (individual species). On this basis, the numbers of particular elements in these species are calculated; for example, in the solution II (see below), N04 molecules of FeSO4∙7H2O contain 14 N04 atoms of H, 11 N04 atoms of O and N11 atoms of Fe; N5 ions of HSO4 �1 ∙n5H2O (N5,n5) in the set (2) of species specified below contain N5(1 + 2n5) atoms of H, N5(4 + n5) atoms of O and N5 atoms of S.

In further parts of this chapter, the terms linear combination and linear dependency/independency of equations are introduced. These terms, well known from the elementary algebra course, will be applied to elemental/core balances, as a system of algebraic equations. The elemental balances f(H) for hydrogen (H) and f(O) for oxygen (O) and the linear combination 2∙f(O)�f(H) are formulated and then combined with charge balance (ChB) and other elemental/ core balances f(Yg) for other elements (Yg = Eg) or cores (Yg = coreg), Yg 6¼ H and O. This way, the general properties of 2∙f(O)�f(H) in non-redox and redox systems are distinguished, see Refs. [7–18] and earlier references cited therein.

The 2f(O)�f(H), charge balance and elemental/core balances will be expressed first in terms of the numbers of particular entities. Next, the related balances will be presented in terms of molar concentrations, to be fully compatible with expressions for equilibrium constants that are also presented in terms of molar concentrations of the related species.

Static and dynamic systems are distinguished. A static system is obtained after disposable mixing with the respective components. For illustrative purposes, we consider first four solutions, as static non-redox systems, formed from the following components:


The CO2 in the respective solutions is primarily considered as one originated from ambient air, on the step of preparation of these solutions.

From these static systems, we prepare later different dynamic systems: (I) ) (II), (I) ) (III) and (I) ) (IV), where (I) as titrant T is added into (II), (III) or (IV) as titrand D (T ) D), and the D + T mixtures containing different species are formed.

To avoid a redundancy resulting from application of different subscripts within (Ni, ni) ascribed to the same species Xi zi � niH2O in different solutions (I)–(IV), we apply the common basis of the species from which the components will be selected to the respective balances. The set of the species is as follows:

H2OðN1Þ, <sup>H</sup>þ<sup>1</sup> <sup>ð</sup>N2, n2Þ, OH�<sup>1</sup> <sup>ð</sup>N3, n3Þ, <sup>K</sup>þ<sup>1</sup> ðN4, n4Þ, HSO4 �<sup>1</sup>ðN5, n5Þ, SO4 �<sup>2</sup>ðN6, n6Þ, H2C2O4ðN7, n7Þ, and HC2O4 �<sup>1</sup>ðN8, n8Þ, C2O4 �<sup>2</sup>ðN9, n9Þ, H2CO3ðN10, n10Þ, HCO3 �<sup>1</sup>ðN11, n11Þ, CO3 �<sup>2</sup>ðN12, n12Þ, MnO4 �<sup>1</sup>ðN13, n13Þ, MnO4 �<sup>2</sup>ðN14, n14Þ, Mnþ<sup>3</sup> ðN15, n15Þ, MnOH<sup>þ</sup><sup>2</sup> ðN16, n16Þ, MnC2O4 <sup>þ</sup><sup>1</sup>ðN17, n17Þ, MnðC2O4Þ<sup>2</sup> �<sup>1</sup>ðN18, n18Þ, MnðC2O4Þ<sup>3</sup> �<sup>3</sup>ðN19, n19Þ, Mnþ<sup>2</sup> <sup>ð</sup>N20, n20Þ, MnOH<sup>þ</sup><sup>1</sup> ðN21, n21Þ, MnSO4ðN22, n22Þ, MnC2O4ðN23, n23Þ, MnðC2O4Þ<sup>2</sup> �2 <sup>ð</sup>N24, n24Þ, Feþ<sup>2</sup> <sup>ð</sup>N25, n25Þ, FeOH<sup>þ</sup><sup>1</sup> ðN26, n26Þ and FeSO4ðN27, n27ÞFeðC2O4Þ<sup>2</sup> �<sup>2</sup>ðN28, n28Þ, FeðC2O4Þ<sup>3</sup> �<sup>4</sup>ðN29, n29Þ, Feþ<sup>3</sup> <sup>ð</sup>N30, n30Þ, FeOH<sup>þ</sup><sup>2</sup> ðN31, n31Þ, FeðOHÞ<sup>2</sup> <sup>þ</sup><sup>1</sup>ðN32, n32Þ, Fe2ðOHÞ<sup>2</sup> <sup>þ</sup><sup>4</sup>ðN33, n33Þ, FeSO4 <sup>þ</sup><sup>1</sup>ðN34, n34Þ, FeðSO4Þ<sup>2</sup> �<sup>1</sup>ðN35, n35Þ, FeC2O4 <sup>þ</sup><sup>1</sup>ðN36, n36Þ, FeðC2O4Þ<sup>2</sup> �<sup>1</sup>ðN37, n37Þ, FeðC2O4Þ<sup>3</sup> �<sup>3</sup>ðN38, n38Þ, and FeC2O4ðN39, n39Þ, MnC2O4ðN40, n40<sup>Þ</sup> ð2Þ

## 2. A short note

Referring to pure algebra, let us consider the set of G + 1 algebraic equations: fg(x) = ϕg(x)�bg = 0, where g = 0,1,…,G, x<sup>T</sup> = (x1,…,xI), transposed (T ) vector x, composed of independent (scalar) variables xi (i e <1, I>); agi, bg e R are independent (explicitly) on x. After multiplying the equations by the numbers ω<sup>g</sup> e R, and addition of the resulting equations, we get the linear combination <sup>X</sup><sup>G</sup> <sup>g</sup>¼<sup>0</sup> <sup>ω</sup><sup>g</sup> � fgðxÞ ¼ <sup>0</sup> <sup>⇔</sup>X<sup>G</sup> <sup>g</sup>¼<sup>0</sup> <sup>ω</sup><sup>g</sup> � <sup>ϕ</sup>gðxÞ ¼ <sup>X</sup><sup>G</sup> <sup>g</sup>¼<sup>0</sup> <sup>ω</sup><sup>g</sup> � bg of the basic equations.

Formation of linear combinations is applicable to check the linear dependence or independence of the balances. A very useful/effective manner for checking/stating the linear dependence of the balances is the transformation of an appropriate system of equations to the identity 0 = 0 [2, 23]. For this purpose, we will try, in all instances, to obtain the simplest form of the linear combination. To facilitate these operations, carried out by cancellation of the terms on the left and right sides of equations after multiplying and changing sides of these equations, we apply the equivalent forms of the starting equations fg(x) = 0:

$$\mathbf{f}\_{\mathbf{g}}(\mathbf{x}) : \boldsymbol{\varrho}\_{\mathbf{g}}(\mathbf{x}) - \mathbf{b}\_{\mathbf{g}} = \mathbf{0} \Leftrightarrow \boldsymbol{\varrho}\_{\mathbf{g}}(\mathbf{x}) = \mathbf{b}\_{\mathbf{g}} \Leftrightarrow -\mathbf{f}\_{\mathbf{g}}(\mathbf{x}) : -\boldsymbol{\varrho}\_{\mathbf{g}}(\mathbf{x}) = -\mathbf{b}\_{\mathbf{g}} \Leftrightarrow \mathbf{b}\_{\mathbf{g}} = \boldsymbol{\varrho}\_{\mathbf{g}}(\mathbf{x}).\tag{3}$$

In this notation, fg(x) will be essentially treated not as the algebraic expression on the left side of the equation fg(x) = 0 but as an equation that can be expressed in alternative forms presented above.

## 3. Combination of elemental/core balances for non-redox systems

For the solution (I), we have the balances:

The CO2 in the respective solutions is primarily considered as one originated from ambient air,

From these static systems, we prepare later different dynamic systems: (I) ) (II), (I) ) (III) and (I) ) (IV), where (I) as titrant T is added into (II), (III) or (IV) as titrand D (T ) D), and the D + T

To avoid a redundancy resulting from application of different subscripts within (Ni, ni)

basis of the species from which the components will be selected to the respective balances. The

ðN4, n4Þ, HSO4

�<sup>1</sup>ðN13, n13Þ, MnO4

ðN21, n21Þ, MnSO4ðN22, n22Þ, MnC2O4ðN23, n23Þ, MnðC2O4Þ<sup>2</sup>

ðN26, n26Þ and FeSO4ðN27, n27ÞFeðC2O4Þ<sup>2</sup>

ðN31, n31Þ, FeðOHÞ<sup>2</sup>

�<sup>1</sup>ðN35, n35Þ, FeC2O4

�<sup>3</sup>ðN38, n38Þ, and FeC2O4ðN39, n39Þ, MnC2O4ðN40, n40<sup>Þ</sup>

<sup>ð</sup>N3, n3Þ, <sup>K</sup>þ<sup>1</sup>

�<sup>1</sup>ðN8, n8Þ, C2O4

<sup>ð</sup>N30, n30Þ, FeOH<sup>þ</sup><sup>2</sup>

<sup>þ</sup><sup>1</sup>ðN34, n34Þ, FeðSO4Þ<sup>2</sup>

Referring to pure algebra, let us consider the set of G + 1 algebraic equations: fg(x) = ϕg(x)�bg = 0,

variables xi (i e <1, I>); agi, bg e R are independent (explicitly) on x. After multiplying the equations by the numbers ω<sup>g</sup> e R, and addition of the resulting equations, we get the linear

Formation of linear combinations is applicable to check the linear dependence or independence of the balances. A very useful/effective manner for checking/stating the linear dependence of the balances is the transformation of an appropriate system of equations to the identity 0 = 0 [2, 23]. For this purpose, we will try, in all instances, to obtain the simplest form of the linear combination. To facilitate these operations, carried out by cancellation of the terms on the left and right sides of equations after multiplying and changing sides of these equations,

In this notation, fg(x) will be essentially treated not as the algebraic expression on the left side of the equation fg(x) = 0 but as an equation that can be expressed in alternative forms presented

<sup>g</sup>¼<sup>0</sup> <sup>ω</sup><sup>g</sup> � <sup>ϕ</sup>gðxÞ ¼ <sup>X</sup><sup>G</sup>

fgðxÞ : ϕgðxÞ � bg ¼ 0 ⇔ ϕgðxÞ ¼ bg ⇔ –fgðxÞ : �ϕgðxÞ ¼ –bg ⇔ bg ¼ ϕgðxÞ: ð3Þ

<sup>þ</sup><sup>1</sup>ðN17, n17Þ, MnðC2O4Þ<sup>2</sup>

�<sup>2</sup>ðN12, n12Þ, MnO4

zi � niH2O in different solutions (I)–(IV), we apply the common

�<sup>2</sup>ðN9, n9Þ, H2CO3ðN10, n10Þ,

�<sup>1</sup>ðN5, n5Þ, SO4

�<sup>1</sup>ðN18, n18Þ, MnðC2O4Þ<sup>3</sup>

�<sup>2</sup>ðN14, n14Þ, Mnþ<sup>3</sup>

<sup>þ</sup><sup>1</sup>ðN32, n32Þ,

) vector x, composed of independent (scalar)

<sup>g</sup>¼<sup>0</sup> <sup>ω</sup><sup>g</sup> � bg of the basic equations.

<sup>þ</sup><sup>1</sup>ðN36, n36Þ,

�<sup>2</sup>ðN6, n6Þ,

ðN15, n15Þ,

�2

ð2Þ

�<sup>3</sup>ðN19, n19Þ,

�<sup>2</sup>ðN28, n28Þ,

on the step of preparation of these solutions.

mixtures containing different species are formed.

<sup>ð</sup>N2, n2Þ, OH�<sup>1</sup>

<sup>ð</sup>N25, n25Þ, FeOH<sup>þ</sup><sup>1</sup>

ascribed to the same species Xi

176 Advances in Titration Techniques

set of the species is as follows:

H2C2O4ðN7, n7Þ, and HC2O4

�<sup>1</sup>ðN11, n11Þ, CO3

<sup>ð</sup>N20, n20Þ, MnOH<sup>þ</sup><sup>1</sup>

ðN16, n16Þ, MnC2O4

�<sup>4</sup>ðN29, n29Þ, Feþ<sup>3</sup>

<sup>þ</sup><sup>4</sup>ðN33, n33Þ, FeSO4

�<sup>1</sup>ðN37, n37Þ, FeðC2O4Þ<sup>3</sup>

where g = 0,1,…,G, x<sup>T</sup> = (x1,…,xI), transposed (T

<sup>g</sup>¼<sup>0</sup> <sup>ω</sup><sup>g</sup> � fgðxÞ ¼ <sup>0</sup> <sup>⇔</sup>X<sup>G</sup>

we apply the equivalent forms of the starting equations fg(x) = 0:

H2OðN1Þ, <sup>H</sup>þ<sup>1</sup>

<sup>ð</sup>N24, n24Þ, Feþ<sup>2</sup>

2. A short note

combination <sup>X</sup><sup>G</sup>

above.

HCO3

Mnþ<sup>2</sup>

MnOH<sup>þ</sup><sup>2</sup>

FeðC2O4Þ<sup>3</sup>

Fe2ðOHÞ<sup>2</sup>

FeðC2O4Þ<sup>2</sup>

f <sup>1</sup> ¼ fðHÞ : 2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ 2N4n4 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 þ 2N13n13 ¼ 2N03 f <sup>2</sup> ¼ fðOÞ : N1 þ N2n2 þ N3ð1 þ n3Þ þ N4n4 þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ þ N13ð4 þ n13Þ ¼ 4N01 þ 2N02 þ N03 f <sup>12</sup> ¼ 2 � fðOÞ � fðHÞ : �N2 þ N3 þ 4N10 þ 5N11 þ 6N12 þ 8N13 ¼ 8N01 þ 4N02 f <sup>0</sup> ¼ ChB : N2–N3 þ N4 � N11 � 2N12–N13 ¼ 0 �f <sup>3</sup> ¼ –fðKÞ : N01 ¼ N4 –7f <sup>4</sup> ¼ �7fðMnÞ : 7N01 ¼ 7N13 –4f <sup>5</sup> ¼ �4fðCO3Þ : 4N02 ¼ 4N10 þ 4N11 þ 4N12 f <sup>12</sup> þ f <sup>0</sup> � f <sup>3</sup> � 7f <sup>4</sup> � 4f <sup>5</sup> : 0 ¼ 0

For the solution (II), we have the balances:

f <sup>1</sup> ¼ fðHÞ : 2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ N5ð1 þ 2n5Þ þ 2N6n6 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 þ 2N25n25 þ N26ð1 þ 2n26Þ þ 2N27n27 ¼ 14N04 þ 2N05 þ 2N07 f <sup>2</sup> ¼ fðOÞ : N1 þ N2n2 þ N3ð1 þ n3Þ þ N5ð4 þ n5Þ þ N6ð4 þ n6Þ þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ þ N25n25 þ N26ð1 þ n26Þ þ N27ð4 þ n27Þ ¼ 11N04 þ 4N05 þ 2N06 þ N07 f <sup>12</sup> ¼ 2 � fðOÞ � fðHÞ : –N2 þ N3 þ 7N5 þ 8N6 þ 4N10 þ 5N11 þ 6N12 þ N26 þ 8N27 ¼ 8N04 þ 6N05 þ 4N06 f <sup>0</sup> ¼ ChB : N2 � N3 � N5 � 2N6–N11 � 2N12 þ 2N25 þ N26 ¼ 0 �2f <sup>3</sup> ¼ �2fðFeÞ : 2N04 ¼ 2N25 þ 2N26 þ 2N27 �6f <sup>4</sup> ¼ �6fðSO4Þ : 6N04 þ 6N05 ¼ 6N5 þ 6N6 þ 6N27 –4f <sup>5</sup> ¼ �4fðCO3Þ : 4N06 ¼ 4N10 þ 4N11 þ 4N12 f <sup>12</sup> þ f <sup>0</sup> � 2f <sup>3</sup> � 6f <sup>4</sup> � 4f <sup>5</sup> : 0 ¼ 0

For the solution (III), we have the balances:

f <sup>1</sup> ¼ fðHÞ : 2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ N5ð1 þ 2n5Þ þ 2N6n6 þ N7ð2 þ 2n7Þ þ N8ð1 þ 2n8Þ þ 2N9n9 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 ¼ 6N08 þ 2N09 þ 2N011 f <sup>2</sup> ¼ fðOÞ : N1 þ N2n2 þ N3ð1 þ n3Þ þ N5ð4 þ n5Þ þ N6ð4 þ n6Þ þ N7ð4 þ n7Þ þ N8ð4 þ n8Þ þ N9ð4 þ n9Þ þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ ¼ 6N08 þ 4N09 þ 2N010 þ N011 f <sup>12</sup> ¼ 2 � fðOÞ � fðHÞ : �N2 þ N3 þ 7N5 þ 8N6 þ 6N7 þ 7N8 þ 8N9 þ 4N10 þ 5N11 þ 6N12 ¼ 6N08 þ 6N09 þ 4N010 f <sup>0</sup> ¼ ChB : N2 � N3 � N5 � 2N6 � N8 � 2N9 � N11 � 2N12 ¼ 0 �6f <sup>3</sup> ¼ �6fðSO4Þ : 6N09 ¼ 6N5 þ 6N6 �4f <sup>4</sup> ¼ �4fðCO3Þ : 4N010 ¼ 4N10 þ 4N11 þ 4N12 �6f <sup>5</sup> ¼ �6fðC2O4Þ : 6N08 ¼ 6N7 þ 6N8 þ 6N9 f <sup>12</sup> þ f <sup>0</sup> � 6f <sup>3</sup> � 4f <sup>4</sup> � 6f <sup>5</sup> : 0 ¼ 0

For the solution (IV), we have the balances:

f <sup>1</sup> ¼ fðHÞ : 2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ N5ð1 þ 2n5Þ þ 2N6n6 þ N7ð2 þ 2n7Þ þ N8ð1 þ 2n8Þ þ 2N9n9 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 þ 2N25n25 þ N26ð1 þ 2n26Þ þ 2N27n27 þ 2N28n28 þ 2N29n29 þ 2N39n39 ¼ 14N012 þ 6N013 þ 2N014 þ 2N016 f <sup>2</sup> ¼ fðOÞ : N1 þ N2n2 þ N3ð1 þ n3Þ þ N5ð4 þ n5Þ þ N6ð4 þ n6Þ þ N7ð4 þ n7Þ þ N8ð4 þ n8Þ þ N9ð4 þ n9Þ þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ þ N25n25 þ N26ð1 þ n26Þ þ N27ð4 þ n27Þ þ N28ð8 þ n28Þ þ N29ð12 þ n29Þ þ N39ð4 þ n39Þ ¼ 11N012 þ 6N013 þ 4N014 þ 2N015 þ N016 f <sup>12</sup> ¼ 2 � fðOÞ � fðHÞ : �N2 þ N3 þ 7N5 þ 8N6 þ 6N7 þ 7N8 þ 8N9 þ 4N10 þ 5N11 þ 6N12 þ N26 þ 8N27 þ 16N28 þ24N29 þ 8N39 ¼ 8N012 þ 6N013 þ 6N014 þ 4N015 f <sup>0</sup> ¼ ChB : N2–N3–N5–2N6–N8–2N9–N11–2N12 þ 2N26 þ N27–2N29–4N30 ¼ 0 –6f <sup>3</sup> ¼ –6fðSO4Þ : 6N012 þ 6N014 ¼ 6N5 þ 6N6 þ 6N28 –4f <sup>4</sup> ¼ �4fðCO3Þ : 4N015 ¼ 4N10 þ 4N11 þ 4N12 –6f <sup>5</sup> ¼ –6fðC2O4Þ : 6N013 ¼ 6N7 þ 6N8 þ 6N9 þ 12N28 þ 18N29 þ 6N39 –2f <sup>6</sup> ¼ –2fðFeÞ : 2N012 ¼ 2N25 þ 2N26 þ 2N27 þ 2N28 þ 2N29 þ 2N39 f <sup>12</sup> þ f <sup>0</sup>–6f <sup>3</sup> � 4f <sup>4</sup>–6f <sup>5</sup> � 2f <sup>6</sup> : 0 ¼ 0

Summarizing, for all the solutions (I)–(IV), we obtain the identities 0 = 0:

f <sup>1</sup> ¼ fðHÞ :

178 Advances in Titration Techniques

f <sup>2</sup> ¼ fðOÞ :

f <sup>1</sup> ¼ fðHÞ :

f <sup>2</sup> ¼ fðOÞ :

f <sup>12</sup> ¼ 2 � fðOÞ � fðHÞ :

f <sup>0</sup> ¼ ChB :

f <sup>12</sup> ¼ 2 � fðOÞ � fðHÞ :

�6f <sup>3</sup> ¼ �6fðSO4Þ : 6N09 ¼ 6N5 þ 6N6

f <sup>12</sup> þ f <sup>0</sup> � 6f <sup>3</sup> � 4f <sup>4</sup> � 6f <sup>5</sup> : 0 ¼ 0

For the solution (IV), we have the balances:

þ24N29 þ 8N39 ¼ 8N012 þ 6N013 þ 6N014 þ 4N015

f <sup>12</sup> þ f <sup>0</sup>–6f <sup>3</sup> � 4f <sup>4</sup>–6f <sup>5</sup> � 2f <sup>6</sup> : 0 ¼ 0

�4f <sup>4</sup> ¼ �4fðCO3Þ : 4N010 ¼ 4N10 þ 4N11 þ 4N12 �6f <sup>5</sup> ¼ �6fðC2O4Þ : 6N08 ¼ 6N7 þ 6N8 þ 6N9

þ 2N28n28 þ 2N29n29 þ 2N39n39 ¼ 14N012 þ 6N013 þ 2N014 þ 2N016

N2–N3–N5–2N6–N8–2N9–N11–2N12 þ 2N26 þ N27–2N29–4N30 ¼ 0 –6f <sup>3</sup> ¼ –6fðSO4Þ : 6N012 þ 6N014 ¼ 6N5 þ 6N6 þ 6N28

–4f <sup>4</sup> ¼ �4fðCO3Þ : 4N015 ¼ 4N10 þ 4N11 þ 4N12

2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ N5ð1 þ 2n5Þ þ 2N6n6 þ N7ð2 þ 2n7Þ þ N8ð1 þ 2n8Þ

þ N9ð4 þ n9Þ þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ ¼ 6N08 þ 4N09 þ 2N010 þ N011

�N2 þ N3 þ 7N5 þ 8N6 þ 6N7 þ 7N8 þ 8N9 þ 4N10 þ 5N11 þ 6N12 ¼ 6N08 þ 6N09 þ 4N010

þ 2N9n9 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 ¼ 6N08 þ 2N09 þ 2N011

N1 þ N2n2 þ N3ð1 þ n3Þ þ N5ð4 þ n5Þ þ N6ð4 þ n6Þ þ N7ð4 þ n7Þ þ N8ð4 þ n8Þ

f <sup>0</sup> ¼ ChB : N2 � N3 � N5 � 2N6 � N8 � 2N9 � N11 � 2N12 ¼ 0

2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ N5ð1 þ 2n5Þ þ 2N6n6 þ N7ð2 þ 2n7Þ þ N8ð1 þ 2n8Þ þ 2N9n9 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 þ 2N25n25 þ N26ð1 þ 2n26Þ þ 2N27n27

N1 þ N2n2 þ N3ð1 þ n3Þ þ N5ð4 þ n5Þ þ N6ð4 þ n6Þ þ N7ð4 þ n7Þ þ N8ð4 þ n8Þ þ N9ð4 þ n9Þ

þ N28ð8 þ n28Þ þ N29ð12 þ n29Þ þ N39ð4 þ n39Þ ¼ 11N012 þ 6N013 þ 4N014 þ 2N015 þ N016

�N2 þ N3 þ 7N5 þ 8N6 þ 6N7 þ 7N8 þ 8N9 þ 4N10 þ 5N11 þ 6N12 þ N26 þ 8N27 þ 16N28

–6f <sup>5</sup> ¼ –6fðC2O4Þ : 6N013 ¼ 6N7 þ 6N8 þ 6N9 þ 12N28 þ 18N29 þ 6N39 –2f <sup>6</sup> ¼ –2fðFeÞ : 2N012 ¼ 2N25 þ 2N26 þ 2N27 þ 2N28 þ 2N29 þ 2N39

þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ þ N25n25 þ N26ð1 þ n26Þ þ N27ð4 þ n27Þ

$$\begin{aligned} \text{(I)} f\_{12} + f\_0 - f\_3 - 7f\_4 - 4f\_5 &= 0; \text{ (II)} f\_{12} + f\_0 - 2f\_3 - 6f\_4 - 4f\_5 &= 0; \text{ (III)} f\_{12} + f\_0 - 6f\_3 - 4f\_4 - 6f\_5 &= 0; \\ \text{(IV)} f\_{12} + f\_0 - 6f\_3 - 4f\_4 - 6f\_5 - 2f\_6 &= 0 \end{aligned} \tag{1V}$$

All the solutions are non-redox systems. Except protonation/hydrolytic effects in (I)–(IV), the complexation and precipitation occur in (II) and (IV); the precipitation of FeC2O4 does not occur there at sufficiently high concentrations of H2SO4.

The solutions can be mixed according to titrimetric mode. In particular, we refer to the D + T systems obtained in the titrations T ) D indicated above, namely (I) ) (II), (I) ) (III) and (I) ) (IV). According to the notation applied elsewhere, for example, in Refs. [19–23], V0 mL of D is titrated with volume V mL of T, added up to a given point of the titration, and the D + T mixture with volume V0 + V mL is formed at this point if the assumption of the volume additivity is valid.

We assume V1 = V, CV = 103 ∙N01/NA (NA: Avogadro's number) and V2 = V0 and C0V0 = 103 ∙N04/ NA for (I) ) (II); V3 = V0 and C0V0 = 103 <sup>∙</sup>N08/NA for (I) ) (III); V4 = V0 and C01V0 = 103 ∙N013/NA and C02V0 = 103 ∙N012/NA for (I) ) (IV). Concentrations of the species Xi zi � niH2O in the related systems are defined by relation ½Xi zi �ðV0 þ VÞ ¼ Ni=NA, where ½Xi zi � is the molar concentration of Xi zi � niH2O for i <sup>≥</sup> 2. The progress of the titration in (I) ) (II) and (I) ) (III) can be defined by the fraction titrated [24–29] value

$$
\Phi = \frac{\mathbf{C} \cdot \mathbf{V}}{\mathbf{C}\_0 \cdot \mathbf{V}\_0} \tag{5}
$$

whereas V will be taken as a parameter varied on abscissa in the graphical presentation of the system (I) ) (IV).

## 4. Formulation of dynamic redox systems

The D and T, formed out of particular components, are considered as subsystems of the D + T system thus obtained. The titration is considered as a quasistatic process carried out under isothermal conditions and perceived both from physicochemical and analytical viewpoints.

Let us consider four starting solutions composed of:


• N012 molecules of FeSO4∙7H2O, N013 molecules of H2C2O4 �2H2O, N014 molecules of H2SO4, N015 molecules of CO2 and N016 molecules of H2O in V4 mL of the resulting solution

We start our considerations from the most complex dynamic system (I) ) (IV), where V mL KMnO4 (C) + CO2 (C1) is added into V0 mL FeSO4 (C01)+H2C2O4 (C02)+H2SO4 (C03) + CO2 (C04) at the defined point of the titration. The less complex dynamic systems (I) ) (II) and (I) ) (III) will be considered later as a particular case of the system (I) ) (IV).

## 4.1. Formulation of GEB for the system (I) ) (IV)

Referring to the set of species in Eq. (2), we apply a1 = 1 if pr1 = FeC2O4 is the equilibrium solid phase (precipitate) in the system, and a2 = 1 if pr2 = MnC2O4 is the equilibrium solid phase in the system; otherwise, we have a1 = 0 and/or a2 = 0. The elemental/core balances and ChB, formulated on the basis of the set of the species Eq. (2), are as follows:

f <sup>1</sup> ¼ fðHÞ : 2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ 2N4n4 þ N5ð1 þ 2n5Þ þ 2N6n6 þ N7ð2 þ 2n7Þ þ N8ð1 þ 2n8Þ þ 2N9n9 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 þ 2N13n13 þ 2N14n14 þ 2N15n15 þ N16ð1 þ 2n16Þ þ 2N17n17 þ 2N18n18 þ 2N19n19 þ 2N20n20 þ N21ð1 þ 2n21Þ þ 2N22n22 þ 2N23n23 þ 2N24n24 þ 2N25n25 þ N26ð1 þ 2n26Þ þ 2N27n27 þ 2N28n28 þ 2N29n29 þ 2N30n30 þ N31ð1 þ 2n31Þ þ N32ð2 þ 2n32Þ þ N33ð2 þ 2n33Þ þ 2N34n34 þ 2N35n35 þ 2N36n36 þ 2N37n37 þ 2N38n38 þ 2a1N39n39 þ 2a2N40n40 ¼ 2N03 þ 14N012 þ 6N013 þ 2N014 þ 2N016 f <sup>2</sup> ¼ fðOÞ : N1 þ N2n2 þ N3ð1 þ n3Þ þ N4n4 þ N5ð4 þ n5Þ þ N6ð4 þ n6Þ þ N7ð4 þ n7Þ þ N8ð4 þ n8Þ þ N9ð4 þ n9Þ þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ þ N13ð4 þ n13Þ þ N14ð4 þ n14Þ þ N15n15 þ N16ð1 þ n16Þ þ N17ð4 þ n17Þ þ N18ð8 þ n18Þ þ N19ð12 þ n19Þ þ N20n20 þ N21ð1 þ n21Þ þ N22ð4 þ n22Þ þ N23ð4 þ n23Þ þ N24ð8 þ n24Þ þ N25n25 þ N26ð1 þ n26Þ þ N27ð4 þ n27Þ þ N28ð8 þ n28Þ þ N29ð12 þ n29Þ þ N30n30 þ N31ð1 þ n31Þ þ N32ð2 þ n32Þ þ N33ð2 þ n33Þ þ N34ð4 þ n34Þ þ N35ð8 þ n35Þ þ N36ð4 þ n36Þ þ N37ð8 þ n37Þ þ N38ð12 þ n38Þ þ a1N39ð4 þ n39Þ þ a2N40ð4 þ n40Þ ¼ 4N01 þ 2N02 þ N03 þ 11N012 þ 6N013 þ 4N014 þ 2N015 þ N016 f <sup>12</sup> ¼ 2 � fðOÞ–fðHÞ �N2 þ N3 þ 7N5 þ 8N6 þ 6N7 þ 7N8 þ 8N9 þ 4N10 þ 5N11 þ 6N12 þ 8N13 þ 8N14 þ N16 þ 8N17 þ 16N18 þ 24N19N21 þ 8N22 þ 8N23 þ 16N24 þ N26 þ 8N27 þ 16N28 þ 24N29 þ N31 þ 2N32 þ 2N33 þ 8N34 þ 16N35 þ 8N36 þ 16N37 þ 24N38 þ 8a1N39 þ 8a2N40 ¼ 8N01 þ 4N02 þ 8N012 þ 6N013 þ 6N014 þ 4N015

ð6Þ

ð7Þ

ð10Þ

$$\begin{aligned} f\_0 &= \text{ChB} \\ \mathbf{N\_2} - \mathbf{N\_3} + \mathbf{N\_4} - \mathbf{N\_5} - 2\mathbf{N\_6} - \mathbf{N\_8} - 2\mathbf{N\_{7}} - \mathbf{N\_{11}} - 2\mathbf{N\_{12}} - \mathbf{N\_{13}} - 2\mathbf{N\_{14}} + 3\mathbf{N\_{15}} + 2\mathbf{N\_{16}} + \mathbf{N\_{17}} - \mathbf{N\_{18}} - 3\mathbf{N\_{19}} \\ &+ 2\mathbf{N\_{20}} + \mathbf{N\_{21}} - 2\mathbf{N\_{24}} + 2\mathbf{N\_{25}} + \mathbf{N\_{26}} - 2\mathbf{N\_{28}} - 4\mathbf{N\_{29}} + 3\mathbf{N\_{30}} + 2\mathbf{N\_{31}} + \mathbf{N\_{32}} + 4\mathbf{N\_{33}} + \mathbf{N\_{34}} \\ &- \mathbf{N\_{35}} + \mathbf{N\_{36}} - \mathbf{N\_{37}} - 3\mathbf{N\_{38}} = 0 \end{aligned}$$

• N012 molecules of FeSO4∙7H2O, N013 molecules of H2C2O4

4.1. Formulation of GEB for the system (I) ) (IV)

¼ 2N03 þ 14N012 þ 6N013 þ 2N014 þ 2N016

þ 11N012 þ 6N013 þ 4N014 þ 2N015 þ N016

þ 8N012 þ 6N013 þ 6N014 þ 4N015

f <sup>1</sup> ¼ fðHÞ :

180 Advances in Titration Techniques

f <sup>2</sup> ¼ fðOÞ :

f <sup>12</sup> ¼ 2 � fðOÞ–fðHÞ

(III) will be considered later as a particular case of the system (I) ) (IV).

formulated on the basis of the set of the species Eq. (2), are as follows:

N015 molecules of CO2 and N016 molecules of H2O in V4 mL of the resulting solution

We start our considerations from the most complex dynamic system (I) ) (IV), where V mL KMnO4 (C) + CO2 (C1) is added into V0 mL FeSO4 (C01)+H2C2O4 (C02)+H2SO4 (C03) + CO2 (C04) at the defined point of the titration. The less complex dynamic systems (I) ) (II) and (I) )

Referring to the set of species in Eq. (2), we apply a1 = 1 if pr1 = FeC2O4 is the equilibrium solid phase (precipitate) in the system, and a2 = 1 if pr2 = MnC2O4 is the equilibrium solid phase in the system; otherwise, we have a1 = 0 and/or a2 = 0. The elemental/core balances and ChB,

2N1 þ N2ð1 þ 2n2Þ þ N3ð1 þ 2n3Þ þ 2N4n4 þ N5ð1 þ 2n5Þ þ 2N6n6 þ N7ð2 þ 2n7Þ þ N8ð1 þ 2n8Þ þ 2N9n9 þ N10ð2 þ 2n10Þ þ N11ð1 þ 2n11Þ þ 2N12n12 þ 2N13n13 þ 2N14n14 þ 2N15n15 þ N16ð1 þ 2n16Þ þ 2N17n17 þ 2N18n18 þ 2N19n19 þ 2N20n20

þ N21ð1 þ 2n21Þ þ 2N22n22 þ 2N23n23 þ 2N24n24 þ 2N25n25 þ N26ð1 þ 2n26Þ þ 2N27n27 þ 2N28n28 þ 2N29n29 þ 2N30n30 þ N31ð1 þ 2n31Þ þ N32ð2 þ 2n32Þ þ N33ð2 þ 2n33Þ þ 2N34n34 þ 2N35n35 þ 2N36n36 þ 2N37n37 þ 2N38n38 þ 2a1N39n39 þ 2a2N40n40

N1 þ N2n2 þ N3ð1 þ n3Þ þ N4n4 þ N5ð4 þ n5Þ þ N6ð4 þ n6Þ þ N7ð4 þ n7Þ þ N8ð4 þ n8Þ þ N9ð4 þ n9Þ þ N10ð3 þ n10Þ þ N11ð3 þ n11Þ þ N12ð3 þ n12Þ þ N13ð4 þ n13Þ þ N14ð4 þ n14Þ

þ N21ð1 þ n21Þ þ N22ð4 þ n22Þ þ N23ð4 þ n23Þ þ N24ð8 þ n24Þ þ N25n25 þ N26ð1 þ n26Þ þ N27ð4 þ n27Þ þ N28ð8 þ n28Þ þ N29ð12 þ n29Þ þ N30n30 þ N31ð1 þ n31Þ þ N32ð2 þ n32Þ

�N2 þ N3 þ 7N5 þ 8N6 þ 6N7 þ 7N8 þ 8N9 þ 4N10 þ 5N11 þ 6N12 þ 8N13 þ 8N14 þ N16 þ 8N17 þ 16N18 þ 24N19N21 þ 8N22 þ 8N23 þ 16N24 þ N26 þ 8N27 þ 16N28 þ 24N29 þ N31 þ 2N32 þ 2N33 þ 8N34 þ 16N35 þ 8N36 þ 16N37 þ 24N38 þ 8a1N39 þ 8a2N40 ¼ 8N01 þ 4N02

þ N15n15 þ N16ð1 þ n16Þ þ N17ð4 þ n17Þ þ N18ð8 þ n18Þ þ N19ð12 þ n19Þ þ N20n20

þ N33ð2 þ n33Þ þ N34ð4 þ n34Þ þ N35ð8 þ n35Þ þ N36ð4 þ n36Þ þ N37ð8 þ n37Þ þ N38ð12 þ n38Þ þ a1N39ð4 þ n39Þ þ a2N40ð4 þ n40Þ ¼ 4N01 þ 2N02 þ N03

�2H2O, N014 molecules of H2SO4,

ð6Þ

$$-f\_3 = -f(K): \qquad \mathcal{N}\_{01} = \mathcal{N}\_4 \tag{8}$$

$$\begin{aligned} -6f\_4 &= -6f(\mathbf{S}) = -6f(\mathbf{S} \mathbf{O}\_4) : 6\mathbf{N}\_{012} + 6\mathbf{N}\_{014} = 6\mathbf{N}\_5 + 6\mathbf{N}\_6 + 6\mathbf{N}\_{22} + 6\mathbf{N}\_{27} + 6\mathbf{N}\_{34} + 12\mathbf{N}\_{35} \end{aligned} \quad (9)$$

$$\begin{aligned} -4f\_5 &= -4f(\mathbf{C}):\\ 4\mathbf{N}\_{02} + 8\mathbf{N}\_{013} + 4\mathbf{N}\_{015} &= 8\mathbf{N}\_7 + 8\mathbf{N}\_8 + 8\mathbf{N}\_9 + 4\mathbf{N}\_{10} + 4\mathbf{N}\_{11} + 4\mathbf{N}\_{12} + 8\mathbf{N}\_{17} + 16\mathbf{N}\_{18} + 24\mathbf{N}\_{19} \\ &+ 8\mathbf{N}\_{23} + 16\mathbf{N}\_{24} + 16\mathbf{N}\_{28} + 24\mathbf{N}\_{29} + 8\mathbf{N}\_{36} + 16\mathbf{N}\_{37} + 24\mathbf{N}\_{38} + 8\mathbf{a}\_1\mathbf{N}\_{39} + 8\mathbf{a}\_2\mathbf{N}\_{40} \end{aligned}$$

$$\begin{aligned} \text{-3\%}\_6 &= -\text{3f(Fe)} \quad : \\ \text{-3\%}\_{012} &= \text{3N}\_{25} + \text{3N}\_{26} + \text{3N}\_{27} + \text{3N}\_{28} + \text{3N}\_{29} + \text{3N}\_{30} + \text{3N}\_{31} + \text{3N}\_{32} + 6\text{N}\_{33} + \text{3N}\_{34} \\ &+ \text{3N}\_{35} + \text{3N}\_{36} + \text{3N}\_{37} + \text{3N}\_{38} + \text{3a}\_{1}\text{N}\_{39} \end{aligned} \tag{11}$$

$$\begin{aligned} 2 - 2f\_7 &= -2f(\text{Mn}):\\ 2\text{N}\_{01} &= 2\text{N}\_{13} + 2\text{N}\_{14} + 2\text{N}\_{15} + 2\text{N}\_{16} + 2\text{N}\_{17} + 2\text{N}\_{18} + 2\text{N}\_{19} + 2\text{N}\_{20} + 2\text{N}\_{21} + 2\text{N}\_{22} \\ &+ 2\text{N}\_{23} + 2\text{N}\_{24} + 2\text{a}\_{2}\text{N}\_{40} \end{aligned} \tag{12}$$

$$\begin{aligned} &f\_{12} + f\_6 - f\_3 - 6f\_4 - 4f\_5 - 3f\_6 - 2f\_7 : \\ &5 \mathbf{N}\_{13} + 4 \mathbf{N}\_{14} + \mathbf{N}\_{15} + \mathbf{N}\_{16} + \mathbf{N}\_{012} + 2 \mathbf{N}\_{013} = \mathbf{5N}\_{01} + 2\mathbf{N}\_7 + 2\mathbf{N}\_8 + 2\mathbf{N}\_9 + \mathbf{N}\_{17} + 3\mathbf{N}\_{18} \\ &+ 5\mathbf{N}\_{19} + 2\mathbf{N}\_{23} + 4\mathbf{N}\_{24} + \mathbf{N}\_{25} + \mathbf{N}\_{26} + \mathbf{N}\_{27} + 5\mathbf{N}\_{28} + 7\mathbf{N}\_{29} + 2\mathbf{N}\_{36} + 4\mathbf{N}\_{37} \\ &+ 6\mathbf{N}\_{38} + 3\mathbf{a}\_1 \mathbf{N}\_{39} + 2\mathbf{a}\_2 \mathbf{N}\_{40} \end{aligned} \tag{13}$$

<sup>5</sup>½MnO�<sup>1</sup> <sup>4</sup> � þ <sup>4</sup>½MnO�<sup>2</sup> <sup>4</sup> �þ½Mnþ<sup>3</sup> �þ½MnOH<sup>þ</sup><sup>2</sup> ��ð2ð½H2C2O�<sup>1</sup> <sup>4</sup> �þ½HC2O�<sup>1</sup> <sup>4</sup> �þ½C2O�<sup>2</sup> <sup>4</sup> �Þ þ ½MnC2O<sup>þ</sup><sup>1</sup> <sup>4</sup> � þ 3½MnðC2O4Þ �1 <sup>2</sup> � þ 5½MnðC2O4Þ �3 <sup>3</sup> � þ 2½MnC2O4� þ 4½MnðC2O4Þ �2 <sup>2</sup> �þ½Feþ<sup>2</sup> � þ ½FeOH<sup>þ</sup><sup>1</sup> �þ½FeSO4� þ 5½FeðC2O4Þ �2 <sup>2</sup> � þ 7½FeðC2O4Þ �4 <sup>3</sup> � þ <sup>2</sup>½FeC2O<sup>þ</sup><sup>1</sup> <sup>4</sup> � þ 4½FeðC2O4Þ �1 <sup>2</sup> � þ 6½FeðC2O4Þ �3 <sup>3</sup> � þ 3a1½FeC2O4� þ 2a2½MnC2O4�Þ ¼ 5CV=ðV0 þ VÞ – ðC01 þ 2C02ÞV0=ðV0 þ VÞ ) <sup>2</sup>ð½H2C2O4�þ½HC2O�<sup>1</sup> <sup>4</sup> �þ½C2O�<sup>2</sup> <sup>4</sup> �Þ þ ½MnC2O<sup>þ</sup><sup>1</sup> <sup>4</sup> � þ 3½MnðC2O4Þ �1 <sup>2</sup> � þ 5½MnðC2O4Þ �3 <sup>3</sup> � þ 2½MnC2O4� þ 4½MnðC2O4Þ �2 <sup>2</sup> �þ½Fe<sup>þ</sup><sup>2</sup> �þ½FeOH<sup>þ</sup><sup>1</sup> �þ½FeSO4� þ 5½FeðC2O4Þ �2 <sup>2</sup> � þ 7½FeðC2O4Þ �4 <sup>3</sup> � þ <sup>2</sup>½FeC2O<sup>þ</sup><sup>1</sup> <sup>4</sup> � þ 4½FeðC2O4Þ �1 <sup>2</sup> � þ 6½FeðC2O4Þ �3 <sup>3</sup> � þ 3a1½FeC2O4� <sup>þ</sup> 2a2½MnC2O4�–ð5½MnO�<sup>1</sup> <sup>4</sup> � þ <sup>4</sup>½MnO�<sup>2</sup> <sup>4</sup> �þ½Mnþ<sup>3</sup> �þ½MnOH<sup>þ</sup><sup>2</sup> �Þ ¼ ððC01 þ 2C02ÞV0–5CVÞ=ðV0 þ VÞ ð14Þ

Eq. (14) is the shortest/simplest form of GEB for the related system; it is, of course, different from the identity 0 = 0. On the basis of Eq. (14), one can also formulate the GEB for the system (I) ) (II):

$$(\text{KMnO}\_4(\text{C}) + \text{CO}\_2(\text{C}\_1), \text{V}) \Rightarrow (\text{FeSO}\_4(\text{C}\_{01}) + \text{H}\_2\text{SO}\_4(\text{C}\_{03}) + \text{CO}\_2(\text{C}\_{04}), \text{V}\_0) \qquad (15)$$

and for the system (I) ) (III):

$$\text{(KMnO}\_4(\text{C}) + \text{CO}\_2(\text{C}\_1), V) \Rightarrow \text{(H}\_2\text{C}\_2\text{O}\_4(\text{C}\_{02}) + \text{H}\_2\text{SO}\_4(\text{C}\_{03}) + \text{CO}\_2(\text{C}\_{04}), V\_0) \qquad (16)$$

Assuming C02 = 0, after omission of the related species involved with oxalates, from Eq. (14), we have the GEB valid for the system (I) ) (II):

$$\begin{aligned} \left[\text{Fe}^{+2}\right] &+ \left[\text{FeOH}^{+1}\right] + \left[\text{FeSO}\_4\right] \left(5[\text{MnO}\_4^{-1}] + 4[\text{MnO}\_4^{-2}] + [\text{Mn}^{+3}] + [\text{MnOOH}^{+2}]\right) \\ &= (\text{C}\_{01}\text{V}\_0\text{-5CV})(\text{V}\_0 + \text{V}) \end{aligned} \tag{17}$$

Assuming C01 = 0, after omission of the related Fe-species, from Eq. (14), we have the GEB valid for the system (I) ) (III)

$$\begin{aligned} &2(\text{[H}\_{2}\text{O}\_{2}\text{O}\_{4}] + [\text{HC}\_{2}\text{O}\_{4}^{-1}] + [\text{C}\_{2}\text{O}\_{4}^{-2}]) + [\text{MnC}\_{2}\text{O}\_{4}^{+1}] + 3[\text{Mn}(\text{C}\_{2}\text{O}\_{4})\_{2}^{-1}] + 5[\text{Mn}(\text{C}\_{2}\text{O}\_{4})\_{3}^{-3}] \\ &+ 2[\text{MnC}\_{2}\text{O}\_{4}] + 4[\text{Mn}(\text{C}\_{2}\text{O}\_{4})\_{2}^{-2}] + 2\text{a}\_{2}[\text{MnO}\_{2}\text{O}\_{4}] - (5[\text{MnO}\_{4}^{-1}] + 4[\text{MnO}\_{4}^{-2}] + [\text{Mn}^{+3}] \\ &+ [\text{MnO}\text{H}^{+2}]) = (2\text{C}\_{02}\text{V}\_{0} - 5\text{CV})(\text{V}\_{0} + \text{V}) \end{aligned} \tag{18}$$

On the other hand, Eqs. (17) and (18) are the simplest/shortest linear combinations for the related subsystems of the system (I) ) (IV); both are also different from the identity, of course. For comparison, the linear combination f<sup>12</sup> + f<sup>0</sup> – f<sup>3</sup> – 6f<sup>4</sup> – 4f<sup>5</sup> – 2f<sup>6</sup> – 2f<sup>7</sup> = 0, that is where �2f<sup>6</sup> is put for �3f6, gives a more extended equation, where more components are involved. Anyway, we get here the equation, not the identity 0 = 0. It must be stressed that none of the linear combinations of these equations gives the identity. This is the general property of all redox systems, of any degree of complexity.

## 5. Confirmation of linear dependency of balances for non-redox systems

It can be stated that 2f(O)�f(H) is a linear combination of charge and elemental/core balances for non-redox systems of any degree of complexity; this means that 2f(O)�f(H) is not a new, independent balance in non-redox systems. From Eq. (4), we see that for non-redox systems, f<sup>12</sup> can be expressed as the linear combination of other balances of the system considered:

$$\begin{aligned} \text{(I)} f\_{12} &= f\_3 + 7f\_4 + 4f\_5 \text{-f}\_0; & \quad \text{(II)} f\_{12} &= 2f\_3 + 6f\_4 + 4f\_5 \text{-f}\_0; \\ \text{(III)} f\_{12} &= 6f\_3 + 4f\_4 + 6f\_5 - f\_0; & \quad \text{(IV)} f\_{12} &= 6f\_3 + 4f\_4 + 6f\_5 + 2f\_6 \text{-f}\_0 \end{aligned} \tag{19}$$

Eq. (4) can be also rewritten into equivalent forms:

Eq. (14) is the shortest/simplest form of GEB for the related system; it is, of course, different from the identity 0 = 0. On the basis of Eq. (14), one can also formulate the GEB for the system

Assuming C02 = 0, after omission of the related species involved with oxalates, from Eq. (14),

Assuming C01 = 0, after omission of the related Fe-species, from Eq. (14), we have the GEB

<sup>2</sup> � þ 2a2½MnC2O4�–ð5½MnO�<sup>1</sup>

On the other hand, Eqs. (17) and (18) are the simplest/shortest linear combinations for the related subsystems of the system (I) ) (IV); both are also different from the identity, of course. For comparison, the linear combination f<sup>12</sup> + f<sup>0</sup> – f<sup>3</sup> – 6f<sup>4</sup> – 4f<sup>5</sup> – 2f<sup>6</sup> – 2f<sup>7</sup> = 0, that is where �2f<sup>6</sup> is put for �3f6, gives a more extended equation, where more components are involved. Anyway, we get here the equation, not the identity 0 = 0. It must be stressed that none of the linear combinations of these equations gives the identity. This is the general property of all redox

5. Confirmation of linear dependency of balances for non-redox systems

It can be stated that 2f(O)�f(H) is a linear combination of charge and elemental/core balances for non-redox systems of any degree of complexity; this means that 2f(O)�f(H) is not a new, independent balance in non-redox systems. From Eq. (4), we see that for non-redox systems, f<sup>12</sup>

can be expressed as the linear combination of other balances of the system considered:

ðIÞf <sup>12</sup> ¼ f <sup>3</sup> þ 7f <sup>4</sup> þ 4f <sup>5</sup>–f <sup>0</sup>; ðIIÞf <sup>12</sup> ¼ 2f <sup>3</sup> þ 6f <sup>4</sup> þ 4f <sup>5</sup>–f <sup>0</sup>;

ðIIIÞf <sup>12</sup> ¼ 6f <sup>3</sup> þ 4f <sup>4</sup> þ 6f <sup>5</sup> � f <sup>0</sup>; ðIVÞf <sup>12</sup> ¼ 6f <sup>3</sup> þ 4f <sup>4</sup> þ 6f <sup>5</sup> þ 2f <sup>6</sup>–f <sup>0</sup>

<sup>4</sup> �Þ þ ½MnC2O<sup>þ</sup><sup>1</sup>

�þ½FeSO4�–ð5½MnO�<sup>1</sup>

ðKMnO4ðCÞ þ CO2ðC1Þ, VÞ)ðFeSO4ðC01Þ þ H2SO4ðC03Þ þ CO2ðC04Þ, V0Þ ð15Þ

ðKMnO4ðCÞ þ CO2ðC1Þ, VÞ)ðH2C2O4ðC02Þ þ H2SO4ðC03Þ þ CO2ðC04Þ, V0Þ ð16Þ

<sup>4</sup> � þ <sup>4</sup>½MnO�<sup>2</sup>

¼ ðC01V0–5CVÞðV0 <sup>þ</sup> <sup>V</sup><sup>Þ</sup> <sup>ð</sup>17<sup>Þ</sup>

<sup>4</sup> � þ 3½MnðC2O4Þ

<sup>4</sup> �þ½Mnþ<sup>3</sup>

�1

<sup>4</sup> � þ <sup>4</sup>½MnO�<sup>2</sup>

�þ½MnOH<sup>þ</sup><sup>2</sup>

<sup>2</sup> � þ 5½MnðC2O4Þ

<sup>4</sup> �þ½Mnþ<sup>3</sup>

�Þ

�3 <sup>3</sup> �

ð18Þ

ð19Þ

�

(I) ) (II):

182 Advances in Titration Techniques

and for the system (I) ) (III):

½Fe<sup>þ</sup><sup>2</sup>

valid for the system (I) ) (III)

<sup>2</sup>ð½H202O4�þ½HC2O�<sup>1</sup>

þ ½MnOH<sup>þ</sup><sup>2</sup>

þ 2½MnC2O4� þ 4½MnðC2O4Þ

systems, of any degree of complexity.

we have the GEB valid for the system (I) ) (II):

<sup>4</sup> �þ½C2O�<sup>2</sup>

�Þ ¼ ð2C02V0–5CVÞðV0 þ VÞ

�2

�þ½FeOH<sup>þ</sup><sup>1</sup>

$$\begin{aligned} \text{(I)}\dots &(+1)f\_1 + (-2)\cdot f\_2 + (+1)f\_3 + (+7)f\_4 + (+4)\cdot f\_5 - f\_0 = 0; \\ \text{(II)} &(+1)\cdot f\_1 + (-2)\cdot f\_2 + (+2)\cdot f\_3 + (+6)\cdot f\_4 + (+4)\cdot f\_5 - f\_0 = 0; \\ \text{(III)} &(+1)\cdot f\_1 + (-2)\cdot f\_2 + 2(+3)\cdot f\_3 + (+4)\cdot f\_4 + (+6)\cdot f\_5 - f\_0 = 0; \\ \text{(IV)} &(+1)\cdot f\_1 + (-2)\cdot f\_2 + (+2)\cdot f\_6 + 2(+3)\cdot f\_3 + (+4)\cdot f\_4 + (+6)\cdot f\_5 - f\_0 = 0 \end{aligned} \tag{20}$$

As we see, the coefficient at the corresponding elemental/core balance in the related sum is equal to the oxidation number of the corresponding element. The linear dependence will be thus ascertained by multiplying the elemental/core balances by the appropriate oxidation numbers. After consecutive addition of the resulting balances to the sum of 2f(O)�f(H) and charge balance, followed by simplifications, the resulting sum is reduced to the identity 0 = 0. It is the simplest way of checking the linear dependency of the equations related to non-redox systems.

For redox systems, the appropriate linear combination of 2f(O)�f(H) with charge balance and elemental/core balances related to electron-non-active elements in the system in question leads to the simplest form of GEB named as generalized electron balance (GEB). It means that the GEB is a new balance, complementary/compatible with other (charge and elemental/core) balances related to the system in question.

## 6. Confirmation of linear independency of balances for redox systems

Applying a similar procedure, one can also state that 2f(O)�f(H) is not a linear combination of charge and elemental/core balances for redox systems of any degree of complexity; it means that 2f(O)�f(H) is a new/independent balance in redox systems.

The independency/dependency property of the balance 2f(O)�f(H) is the basis for the division of electrolytic systems into redox and non-redox systems [8, 9]. This rule is illustrated by the following examples, related to static and dynamic systems.

## 7. Confirmation of equivalency of approaches I and II to GEB for the system (I) ) (IV)

We apply now the linear combination (algebraic sum) of Eqs. (6–9) for ChB and elemental/core balances, involving electron-non-active elements: H, O, K and S, perceived in terms of the Approach I to GEB as 'fans', we have:

f <sup>12</sup> þ f <sup>0</sup>–f <sup>3</sup>–6f <sup>4</sup> : 6N7 þ 6N8 þ 6N9 þ 4N10 þ 4N11 þ 4N12 þ 7N13 þ 6N14 þ 3N15 þ 3N16 þ 9N17 þ 15N18 þ 21N19 þ 2N20 þ 2N21 þ 2N22 þ 8N23 þ 14N24 þ 2N25 þ 2N26 þ 2N27 þ 14N28 þ 20N29 þ 3N30 þ 3N31 þ 3N32 þ 6N33 þ 3N34 þ 3N35 þ 9N36 þ 15N37 þ 21N38 þ 8a1N39 þ 8a2N40 ¼ 7N01 þ 6N013 þ 4ðN02 þ N015Þ ) 6ðN7 þ N8 þ N9Þ þ 4ðN10 þ N11 þ N12Þ þ 7N13 þ 6N14 þ 3N15 þ 3N16 þ 9N17 þ 15N18 þ 21N19 þ 2ðN20 þ N21 þ N22Þ þ 8N23 þ 14N24 þ 2ðN25 þ N26 þ N27Þ þ 14N28 þ 20N29 þ 3ðN30 þ N31 þ N32 þ 2N33 þ N34 þ N35Þ þ 9N36 þ 15N37 þ 21N38 þ 8a1N39 þ 8a2N40 ¼ 7N01 þ 6N013 þ 4ðN02 þ N015Þ ð21Þ

Denoting by ZC (= 6), ZFe (= 26) and ZMn (= 25), the atomic numbers for electron-active elements ('players') C, Fe and Mn, from Eqs. (10)–(12) and (21), we have, by turns,

ZC � f <sup>5</sup> þ ZFe � f <sup>6</sup> þ ZMn � f <sup>7</sup>–ðf <sup>12</sup> þ f <sup>0</sup>–f <sup>3</sup>–6f <sup>4</sup>Þ ð2ZC–6ÞðN7 þ N8 þ N9ÞþðZC–4ÞðN10 þ N11 þ N12ÞþðZMn–7ÞN13 þ ðZMn–6ÞN14 þ ðZMn–3ÞðN15 þ N16Þ þ N17ðZMn þ 2ZC–9Þ þ N18ðZMn þ 4ZC � 15Þ þ N19ðZMn þ 6ZC–21Þ þ ðZMn � 2ÞðN20 þ N21 þ N22Þ þ N23ðZMn þ 2ZC � 8Þ þ N24ðZMn þ 4ZC � 14ÞþðZFe � 2ÞðN25 þ N26 þ N27ÞðZFe þ 4ZC � 12ÞN28 þ ðZFe þ 6ZC � 20ÞN29 þ ðZFe–3ÞðN30 þ N31 þ N32Þ þ 2ðZFe � 3ÞN33 þ ðZFe � 3ÞðN34 þ N35ÞðZFe þ 2ZC � 9ÞN36 þ ðZFe þ 4ZC � 15ÞN37 þ ðZFe þ 6ZC � 21ÞN38 þ a1ðZFe þ 2ZC � 8ÞN39 þ a2ðZMn þ 2ZC–8ÞN40 ¼ ðZMn � 7ÞN01 þ ðZFe � 2ÞN012 þ 2ðZC � 3ÞN013 þ ðZC � 4ÞðN02 þ N015Þ 2ðZC � 3ÞðN7 þ N8 þ N9Þþ ðZC � 4ÞðN10 þ N11 þ N12ÞþðZMn � 7ÞN13 þ ðZMn � 6ÞN14 þðZMn � 3ÞðN15 þ N16Þ þ N17ðZMn þ 2ZC � 9Þ þ N18ðZMn þ 4ZC � 15Þ þ N19ðZMn þ 6ZC � 21Þ þðZMn � 2ÞðN20 þ N21 þ N22Þ þ N23ðZMn þ 2ZC � 8Þ þ N24ðZMn þ 4ZC � 14ÞþðZFe � 2ÞðN25 þN26 þ N27ÞðZFe þ 4ZC � 12ÞN28 þ ðZFe þ 6ZC � 20ÞN29 þ ðZFe–3ÞðN30 þ N31 þ N32Þ þ 2ðZFe � 3ÞN33 þ ðZFe � 3ÞðN34 þ N35ÞðZFe þ 2ZC � 9ÞN36 þ ðZFe þ 4ZC � 15ÞN37 þ ðZFe þ 6ZC � 21ÞN38 þ a1ðZFe þ 2ZC � 8ÞN39 þ a2ðZMn þ 2ZC–8ÞN40 ¼ ðZMn � 7ÞN01 þ ðZFe � 2ÞN012 þ 2ðZC–3ÞN013 þ ðZC � 4ÞðN02 þ N015Þ <sup>2</sup>ðZC � <sup>3</sup>Þð½H2C2O4�þ½HC2O�<sup>1</sup> <sup>4</sup> �þ½C2O�<sup>2</sup> <sup>4</sup> �Þ þ ðZC � <sup>4</sup>Þð½H2CO3�þ½HCO�<sup>1</sup> <sup>3</sup> �þ½CO�<sup>2</sup> <sup>3</sup> �Þ þ ðZFe–2Þð½Feþ<sup>2</sup> �þ½FeOH<sup>þ</sup><sup>1</sup> �þ½FeSO4�Þ þ ðZFe � <sup>3</sup>Þð½Feþ<sup>3</sup> �þ½FeOH<sup>þ</sup><sup>2</sup> �þ½FeðOHÞ þ1 <sup>2</sup> � þ 2½Fe2ðOHÞ þ1 <sup>2</sup> �þ½FeSO<sup>þ</sup><sup>1</sup> <sup>4</sup> �þ½FeðSO4Þ �1 <sup>2</sup> �Þ þ ðZMn � <sup>7</sup>Þð½MnO�<sup>1</sup> <sup>4</sup> �þðZMn � <sup>6</sup>Þ½MnO�<sup>2</sup> <sup>4</sup> � þ ðZMn–3Þð½Mnþ<sup>3</sup> �þ½MnOH<sup>þ</sup><sup>2</sup> �Þ þ ðZMn � <sup>2</sup>Þð½Mnþ<sup>2</sup> �þ½MnOH<sup>þ</sup><sup>1</sup> �þ½MnSO4�Þ þ ðZFe � 2 þ 4ðZC � 3ÞÞ½FeðC2O4Þ �2 <sup>2</sup> �þðZFe � 2 þ 6ðZC � 3ÞÞ½FeðC2O4Þ �4 <sup>3</sup> � <sup>þ</sup> a1ðZFe � <sup>2</sup> <sup>þ</sup> <sup>2</sup>ðZC � <sup>3</sup>ÞÞ½FeC2O4�þðZFe � <sup>3</sup> <sup>þ</sup> <sup>2</sup>ðZC � <sup>3</sup>ÞÞ½FeC2O<sup>þ</sup><sup>1</sup> <sup>4</sup> � þ ðZFe � 3 þ 4ðZC � 3ÞÞ½FeðC2O4Þ<sup>2</sup> � 1�þðZFe � 3 þ 6ðZC � 3ÞÞ½FeðC2O4Þ �3 <sup>3</sup> � þ ðZMn � 2 þ 2ðZC � 3ÞÞ½MnC2O4�þðZMn � 2 þ 4ðZC � 3ÞÞ½MnðC2O4Þ �2 <sup>2</sup> � þ ðZMn � <sup>3</sup> <sup>þ</sup> <sup>2</sup>ðZC � <sup>3</sup>ÞÞ½MnC2O<sup>þ</sup><sup>1</sup> <sup>4</sup> �þðZMn � 3 þ 4ðZC � 3ÞÞ½MnðC2O4Þ �1 <sup>2</sup> � þ ðZMn � 3 þ 6ðZC � 3ÞÞ½MnðC2O4Þ �3 <sup>3</sup> � þ a2ðZMn–2 þ 2ðZC � 3ÞÞ½MnC2O4� ¼ 2ðZC � 3ÞC01V0=ðV0 þ VÞþðZFe � 2ÞC02V0=ðV0 þ VÞþðZMn � 7ÞCV=ðV0 þ VÞ ð22Þ

Eq. (22) is obtainable immediately according to the Approach I to GEB [19–22]. Note, for example, that:

$$\begin{pmatrix} \mathbf{N\_{17}}(\mathbf{Z\_{Mn}} + 2\mathbf{Z\_{C}} - 9) = \mathbf{N\_{17}}(\mathbf{Z\_{Mn}} - 3 + 1^{\circ}\mathbf{2} \cdot (\mathbf{Z\_{C}} - 3)) \\ \mathbf{N\_{18}}(\mathbf{Z\_{Mn}} + 4\mathbf{Z\_{C}} - 15) = \mathbf{N\_{18}}(\mathbf{Z\_{Mn}} - 3 + 2^{\circ}\mathbf{2} \cdot (\mathbf{Z\_{C}} - 3)) \\ \mathbf{N\_{19}}(\mathbf{Z\_{Mn}} + 6\mathbf{Z\_{C}} - 21) = \mathbf{N\_{19}}(\mathbf{Z\_{Mn}} - 3 + 3^{\circ}\mathbf{2} \cdot (\mathbf{Z\_{C}} - 3)) \end{pmatrix}$$

The equation for GEB thus obtained (according to the Approach II to GEB [1, 4, 7–18, 24, 25]) is then equivalent to GEB, obtained according to the Approach I to GEB, based on the principle of the common pool of electrons introduced by electron-active elements ('players') of the system in question. For redox systems, the GEB is the inherent part of the generalized approach to electrolytic systems (GATES) [1], denoted as GATES/GEB.

## 8. Some generalizing remarks on GEB

f <sup>12</sup> þ f <sup>0</sup>–f <sup>3</sup>–6f <sup>4</sup> :

184 Advances in Titration Techniques

6N7 þ 6N8 þ 6N9 þ 4N10 þ 4N11 þ 4N12 þ 7N13 þ 6N14 þ 3N15 þ 3N16 þ 9N17 þ 15N18 þ 21N19 þ 2N20 þ 2N21 þ 2N22 þ 8N23 þ 14N24 þ 2N25 þ 2N26 þ 2N27 þ 14N28 þ 20N29 þ 3N30 þ 3N31 þ 3N32 þ 6N33 þ 3N34 þ 3N35 þ 9N36 þ 15N37 þ 21N38 þ 8a1N39 þ 8a2N40 ¼ 7N01 þ 6N013 þ 4ðN02 þ N015Þ ) 6ðN7 þ N8 þ N9Þ þ 4ðN10 þ N11 þ N12Þ þ 7N13 þ 6N14

Denoting by ZC (= 6), ZFe (= 26) and ZMn (= 25), the atomic numbers for electron-active

ð21Þ

þ 3N15 þ 3N16 þ 9N17 þ 15N18 þ 21N19 þ 2ðN20 þ N21 þ N22Þ þ 8N23 þ 14N24 þ 2ðN25 þ N26 þ N27Þ þ 14N28 þ 20N29 þ 3ðN30 þ N31 þ N32 þ 2N33 þ N34 þ N35Þ

þ 9N36 þ 15N37 þ 21N38 þ 8a1N39 þ 8a2N40 ¼ 7N01 þ 6N013 þ 4ðN02 þ N015Þ

elements ('players') C, Fe and Mn, from Eqs. (10)–(12) and (21), we have, by turns,

ð2ZC–6ÞðN7 þ N8 þ N9ÞþðZC–4ÞðN10 þ N11 þ N12ÞþðZMn–7ÞN13 þ ðZMn–6ÞN14

þ ðZMn–3ÞðN15 þ N16Þ þ N17ðZMn þ 2ZC–9Þ þ N18ðZMn þ 4ZC � 15Þ þ N19ðZMn þ 6ZC–21Þ þ ðZMn � 2ÞðN20 þ N21 þ N22Þ þ N23ðZMn þ 2ZC � 8Þ þ N24ðZMn þ 4ZC � 14ÞþðZFe � 2ÞðN25

þ N26 þ N27ÞðZFe þ 4ZC � 12ÞN28 þ ðZFe þ 6ZC � 20ÞN29 þ ðZFe–3ÞðN30 þ N31 þ N32Þ þ 2ðZFe � 3ÞN33 þ ðZFe � 3ÞðN34 þ N35ÞðZFe þ 2ZC � 9ÞN36 þ ðZFe þ 4ZC � 15ÞN37 þ ðZFe þ 6ZC � 21ÞN38 þ a1ðZFe þ 2ZC � 8ÞN39 þ a2ðZMn þ 2ZC–8ÞN40 ¼ ðZMn � 7ÞN01

2ðZC � 3ÞðN7 þ N8 þ N9Þþ ðZC � 4ÞðN10 þ N11 þ N12ÞþðZMn � 7ÞN13 þ ðZMn � 6ÞN14 þðZMn � 3ÞðN15 þ N16Þ þ N17ðZMn þ 2ZC � 9Þ þ N18ðZMn þ 4ZC � 15Þ þ N19ðZMn þ 6ZC � 21Þ þðZMn � 2ÞðN20 þ N21 þ N22Þ þ N23ðZMn þ 2ZC � 8Þ þ N24ðZMn þ 4ZC � 14ÞþðZFe � 2ÞðN25

þN26 þ N27ÞðZFe þ 4ZC � 12ÞN28 þ ðZFe þ 6ZC � 20ÞN29 þ ðZFe–3ÞðN30 þ N31 þ N32Þ þ 2ðZFe � 3ÞN33 þ ðZFe � 3ÞðN34 þ N35ÞðZFe þ 2ZC � 9ÞN36 þ ðZFe þ 4ZC � 15ÞN37

�þ½FeSO4�Þ þ ðZFe � <sup>3</sup>Þð½Feþ<sup>3</sup>

�Þ þ ðZMn � <sup>2</sup>Þð½Mnþ<sup>2</sup>

�1

<sup>4</sup> �Þ þ ðZC � <sup>4</sup>Þð½H2CO3�þ½HCO�<sup>1</sup>

�þ½MnOH<sup>þ</sup><sup>1</sup>

<sup>2</sup> �Þ þ ðZMn � <sup>7</sup>Þð½MnO�<sup>1</sup>

<sup>2</sup> �þðZFe � 2 þ 6ðZC � 3ÞÞ½FeðC2O4Þ

<sup>4</sup> �þðZMn � 3 þ 4ðZC � 3ÞÞ½MnðC2O4Þ

<sup>3</sup> � þ a2ðZMn–2 þ 2ðZC � 3ÞÞ½MnC2O4�

�þ½FeOH<sup>þ</sup><sup>2</sup>

<sup>3</sup> �þ½CO�<sup>2</sup>

�þ½FeðOHÞ

<sup>4</sup> �þðZMn � <sup>6</sup>Þ½MnO�<sup>2</sup>

�3 <sup>3</sup> �

�þ½MnSO4�Þ

�4 <sup>3</sup> �

�2 <sup>2</sup> �

> �1 <sup>2</sup> �

<sup>4</sup> �

<sup>3</sup> �Þ

<sup>4</sup> �

ð22Þ

þ1 <sup>2</sup> �

þ ðZFe þ 6ZC � 21ÞN38 þ a1ðZFe þ 2ZC � 8ÞN39 þ a2ðZMn þ 2ZC–8ÞN40 ¼ ðZMn � 7ÞN01 þ ðZFe � 2ÞN012 þ 2ðZC–3ÞN013 þ ðZC � 4ÞðN02 þ N015Þ

<sup>4</sup> �þ½C2O�<sup>2</sup>

�2

<sup>þ</sup> a1ðZFe � <sup>2</sup> <sup>þ</sup> <sup>2</sup>ðZC � <sup>3</sup>ÞÞ½FeC2O4�þðZFe � <sup>3</sup> <sup>þ</sup> <sup>2</sup>ðZC � <sup>3</sup>ÞÞ½FeC2O<sup>þ</sup><sup>1</sup>

þ ðZFe � 3 þ 4ðZC � 3ÞÞ½FeðC2O4Þ<sup>2</sup> � 1�þðZFe � 3 þ 6ðZC � 3ÞÞ½FeðC2O4Þ

�3

¼ 2ðZC � 3ÞC01V0=ðV0 þ VÞþðZFe � 2ÞC02V0=ðV0 þ VÞþðZMn � 7ÞCV=ðV0 þ VÞ

þ ðZMn � 2 þ 2ðZC � 3ÞÞ½MnC2O4�þðZMn � 2 þ 4ðZC � 3ÞÞ½MnðC2O4Þ

<sup>4</sup> �þ½FeðSO4Þ

ZC � f <sup>5</sup> þ ZFe � f <sup>6</sup> þ ZMn � f <sup>7</sup>–ðf <sup>12</sup> þ f <sup>0</sup>–f <sup>3</sup>–6f <sup>4</sup>Þ

<sup>2</sup>ðZC � <sup>3</sup>Þð½H2C2O4�þ½HC2O�<sup>1</sup>

þ ðZFe � 2 þ 4ðZC � 3ÞÞ½FeðC2O4Þ

þ ðZMn � <sup>3</sup> <sup>þ</sup> <sup>2</sup>ðZC � <sup>3</sup>ÞÞ½MnC2O<sup>þ</sup><sup>1</sup>

þ ðZMn � 3 þ 6ðZC � 3ÞÞ½MnðC2O4Þ

þ1

�þ½FeOH<sup>þ</sup><sup>1</sup>

�þ½MnOH<sup>þ</sup><sup>2</sup>

<sup>2</sup> �þ½FeSO<sup>þ</sup><sup>1</sup>

þ ðZFe–2Þð½Feþ<sup>2</sup>

þ ðZMn–3Þð½Mnþ<sup>3</sup>

þ 2½Fe2ðOHÞ

þ ðZFe � 2ÞN012 þ 2ðZC � 3ÞN013 þ ðZC � 4ÞðN02 þ N015Þ

The linear combination 2f(O)�f(H) of elemental balances, f(H) for H and f(O) for O, is a keystone of the overall thermodynamic knowledge on electrolytic systems. The 2f(O) �f(H) can be formulated both for non-redox and redox systems, with amphiprotic (co)solvent(s) involved. It is the basis for the Generalized Electron Balance (GEB) formulated according to the Approach II to GEB.

The principle of GEB formulation was presented for the first time in Refs. [30, 31] and then in Refs. [19–22, 31–36] as the Approach I to GEB. The GEB formulation according to the Approach I is based on the 'card game' principle, with a common pool of electrons as money, electron-active elements as players and electron-non-active elements as fans—not changing their oxidation degree, that is the fans' accounts are intact in this convention [13, 23], see an illustration below. Electrons are considered as money, transferred between players; the knowledge of oxidation numbers of all elements in the system in question is needed there.

The Approach I to GEB, named also as the 'short' version of GEB, needs a knowledge of oxidation numbers for all elements in the species participating in the system that is considered. The equivalency of the Approaches I and II means that the equation obtained by a suitable linear combination of pr-GEB with charge balance and other elemental/core balances becomes identical with the one obtained directly from the Approach I to GEB.

Although derivation of GEB according to the Approach II is more extensive/laborious, it enables to formulate this balance without prior knowledge of oxidation numbers for the elements involved in the system. It is the paramount advantage of the Approach II to GEB, particularly when applied to more complex organic species, with radicals and ion-radicals involved. Moreover, within the Approaches I and II, the roles of oxidants and reducers are not ascribed a priori to particular components forming the redox system and the species formed in this system.

Ultimately, GEB, charge and elemental/core balances are expressed in terms of molar concentrations—to be fully compatible with expressions for equilibrium constants, interrelating molar concentrations of defined species on the basis of the mass action law applied to the correctly written reaction equation. The law of mass action is the one and only chemical law applied in GATES.

GEB is perceived as the law of matter conservation, as the general law of nature related to electrolytic (aqueous, non-aqueous or mixed-solvent media) redox systems and as a synthesis of physical and chemical laws [1, 14, 15, 23, 24, 27]. This law can also be extended on the systems with mixed (e.g., binary) solvents with amphiprotic (protophilic and protogenic) and aprotic properties. GEB is a rather unexpected consequence of the concentration balances for H and O, and therefore the formulation of GEB, especially as the Approach II to GEB, is regarded as the scientific discovery and not as a confirmation of the obvious fact arising from other, fundamental laws of nature. This fact is emphasized in this chapter in the context of philosophical understanding of the scientific discoveries in the aspect of the laws of nature.

The GEB, together with charge and concentration/core balances and a set of independent equilibrium constants, provides a complete set of equations used for a thermodynamic description of a redox system taken for quantitative considerations within GATES/GEB ∈ GATES.

The roles of oxidants and reductants are not ascribed a priori to particular components forming the redox system and to the species formed in this system. In other words, full 'democracy' is established a priori within GATES/GEB.

The Approach II to GEB shows that the equivalent equations for GEB are derived from the common root of the elements conservation and then GEB is fully compatible with charge and concentration balances like 'the lotus flower, lotus leaf and lotus seed come from the same root' [13]. This compatibility is directly visible from the viewpoint of the Approach II to GEB. The GEB, based on a reliable law of the matter conservation, is equally robust as equations for charge and concentration balances. The complementarity of the GEB (Approaches I and II) to other balances is regarded as the expression of harmony of nature, and GATES/GEB is an example of excellent epistemological paradigm [27].

The number of electron-active elements (considered as players, in terms of Approach I to GEB) in a redox system, considered according to GATES/GEB principles, is practically unlimited; among others, the systems with three [24] or four [1] players were considered.

In the modeling of real systems, it is assumed that an effect of the matter (such as H2O, CO2 and O2) exchange with the environment is negligibly small within the period designed for certain chemical operations made on the system.

## 9. Completion of balances

The set of balances for the system (I) ) (IV) is composed of GEB (e.g., 14 or 22) and equations obtained from the balances (7)–(12) are expressed in terms of molar concentrations, namely:

A Distinguishing Feature of the Balance 2∙*f*(O)−*f*(H)… http://dx.doi.org/10.5772/intechopen.69249 187

$$\begin{aligned} &[\text{H}^{+}]-[\text{OH}^{-}]+[\text{K}^{+}]-[\text{HSO}\_{4}^{-}]-2[\text{SO}\_{4}^{-}]-[\text{HCO}\_{2}\text{O}\_{4}^{-}]-2[\text{C}\_{2}\text{O}\_{4}^{-}]-[\text{HCO}\_{3}^{-}]-[\text{MnO}\_{4}^{-}] \\ &-2[\text{MnO}\_{4}^{-}]+3[\text{Mn}^{+}]+2[\text{MnO}\text{H}^{+}^{2}]+[\text{MnO}\text{C}\_{2}\text{O}\_{4}^{+}]-[\text{Mn}(\text{C}\_{2}\text{O}\_{4})^{2}]-3[\text{Mn}(\text{C}\_{2}\text{O}\_{4})^{3}]+2[\text{Mn}^{+}] \\ &+[\text{MnO}\text{H}^{+}]-2[\text{Mn}(\text{C}\_{2}\text{O}\_{4})^{2}]+2[\text{Fe}^{+}]+[\text{FeOH}^{+}]-2[\text{Fe}(\text{C}\_{2}\text{O}\_{4})^{2}]-4[\text{Fe}(\text{C}\_{2}\text{O}\_{4})^{3}] +3[\text{Fe}^{+}] \\ &+2[\text{FeOH}^{+}]+[\text{Fe(OH)}\_{2}^{+}]+4[\text{Fe}(\text{OH})\_{2}^{+}]+[\text{Fe}(\text{O})\_{2}^{+}] + [\text{FeSO}\_{4}^{+}]-[\text{Fe(SO}\_{4})\_{2}^{-}]+[\text{Fe}\_{2}\text{O}\_{4}^{+}] \\ &-[\text{Fe(C}\_{2}\text{O}\_{4})^{2}]-3[\text{Fe(C}\_{2}\text{O}\_{4})^{3}] = 0 \end{aligned}$$

molar concentrations of defined species on the basis of the mass action law applied to the correctly written reaction equation. The law of mass action is the one and only chemical law

GEB is perceived as the law of matter conservation, as the general law of nature related to electrolytic (aqueous, non-aqueous or mixed-solvent media) redox systems and as a synthesis of physical and chemical laws [1, 14, 15, 23, 24, 27]. This law can also be extended on the systems with mixed (e.g., binary) solvents with amphiprotic (protophilic and protogenic) and aprotic properties. GEB is a rather unexpected consequence of the concentration balances for H and O, and therefore the formulation of GEB, especially as the Approach II to GEB, is regarded as the scientific discovery and not as a confirmation of the obvious fact arising from other, fundamental laws of nature. This fact is emphasized in this chapter in the context of philosophical understanding of the scientific discoveries in the aspect of the laws

The GEB, together with charge and concentration/core balances and a set of independent equilibrium constants, provides a complete set of equations used for a thermodynamic description of a redox system taken for quantitative considerations within GATES/GEB ∈

The roles of oxidants and reductants are not ascribed a priori to particular components forming the redox system and to the species formed in this system. In other words, full

The Approach II to GEB shows that the equivalent equations for GEB are derived from the common root of the elements conservation and then GEB is fully compatible with charge and concentration balances like 'the lotus flower, lotus leaf and lotus seed come from the same root' [13]. This compatibility is directly visible from the viewpoint of the Approach II to GEB. The GEB, based on a reliable law of the matter conservation, is equally robust as equations for charge and concentration balances. The complementarity of the GEB (Approaches I and II) to other balances is regarded as the expression of harmony of nature, and GATES/GEB is an

The number of electron-active elements (considered as players, in terms of Approach I to GEB) in a redox system, considered according to GATES/GEB principles, is practically unlimited;

In the modeling of real systems, it is assumed that an effect of the matter (such as H2O, CO2 and O2) exchange with the environment is negligibly small within the period designed for

The set of balances for the system (I) ) (IV) is composed of GEB (e.g., 14 or 22) and equations obtained from the balances (7)–(12) are expressed in terms of molar concentrations, namely:

among others, the systems with three [24] or four [1] players were considered.

'democracy' is established a priori within GATES/GEB.

example of excellent epistemological paradigm [27].

certain chemical operations made on the system.

9. Completion of balances

applied in GATES.

186 Advances in Titration Techniques

of nature.

GATES.

$$\left(\mathbf{23}\right)$$

$$\mathbf{[K}^{+1}] = \mathbf{C}\mathbf{V}/(\mathbf{V}\_0 + \mathbf{V})\tag{24}$$

$$\begin{aligned} \text{CB(S)}:\\ \text{[HSO}\_4^{-1}] + [\text{SO}\_4^{-2}] + [\text{MnSO}\_4] + [\text{FeSO}\_4] + [\text{FeSO}\_4^{+1}] + 2[\text{Fe(SO}\_4)\_2^{-1}] &= (\text{C}\_{01} + \text{C}\_{03})\text{V}\_0/(\text{V}\_0 + \text{V}) \end{aligned} \tag{25}$$

CBðCÞ : 2ð½H2C2O4�þ½HC2O4 �<sup>1</sup>�þ½C2O4 �<sup>2</sup>�Þ þ ½H2CO3�þ½HCO3 �<sup>1</sup>�þ½CO3 �2� <sup>þ</sup>2½MnC2O4� þ <sup>4</sup>½MnðC2O4Þ<sup>2</sup> �<sup>2</sup>� þ <sup>4</sup>½FeðC2O4Þ2�<sup>2</sup>� þ <sup>6</sup>½FeðC2O4Þ<sup>3</sup> �<sup>4</sup>� þ <sup>2</sup>½FeC2O4 þ1� þ4½FeðC2O4Þ<sup>2</sup> �<sup>1</sup>� þ <sup>6</sup>½FeðC2O4Þ<sup>3</sup> �<sup>3</sup>� þ 2a1½FeC2O4� þ 2a2½MnC2O4� ¼ ðð2C02 þ C04ÞV0 þ C1VÞ=ðV0 þ VÞ ð26Þ CBðFeÞ : <sup>½</sup>Feþ<sup>2</sup> �þ½FeOH<sup>þ</sup><sup>1</sup> �þ½FeSO4�þ½FeðC2O4Þ<sup>2</sup> �<sup>2</sup>�þ½FeðC2O4Þ<sup>3</sup> �<sup>4</sup>�þ½Feþ<sup>3</sup> �þ½FeOH<sup>þ</sup><sup>2</sup> � þ½FeðOHÞ<sup>2</sup> <sup>þ</sup><sup>1</sup>� þ <sup>2</sup>½Fe2ðOHÞ<sup>2</sup> <sup>þ</sup><sup>4</sup>�þ½FeSO4 <sup>þ</sup><sup>1</sup>�þ½FeðSO4Þ<sup>2</sup> �<sup>1</sup>�þ½FeC2O4 <sup>þ</sup><sup>1</sup>�þ½FeðC2O4Þ2�<sup>1</sup>� þ½FeðC2O4Þ3�<sup>3</sup>� þ a1½FeC2O4� ¼ C01V0=ðV0 <sup>þ</sup> <sup>V</sup><sup>Þ</sup> ð27Þ CBðMnÞ : ½MnO4 �<sup>1</sup>�þ½MnO4 �<sup>2</sup>�þ½Mnþ<sup>3</sup> �þ½MnOH<sup>þ</sup><sup>2</sup> �þ½MnC2O4 <sup>þ</sup><sup>1</sup>�þ½MnðC2O4Þ2�<sup>1</sup>� þ½MnðC2O4Þ3�<sup>3</sup>� þ½Mn<sup>þ</sup><sup>2</sup> �þ½MnOH<sup>þ</sup><sup>1</sup> �þ½MnSO4�þ½MnC2O4�þ½MnðC2O4Þ2�<sup>2</sup>� þa2½MnC2O4� ¼ CV=ðV0 þ VÞ ð28Þ

The balances can be specified as equations or equalities. The equality is represented here by relation (24), where only one species is involved. In Eqs. (23) and (25)–(28), we have concentrations of more species, interrelated in expressions for equilibrium constants, formulated on the basis of the proper stoichiometric reaction notations. As with above results, we have seven balances: six equations and one equality for the system (I) ) (IV); the equality (24) can enter immediately the charge balance (23). The current volume V of titrant T added is a parameter (steering variable) in these balances.

### 10. The arrangement of relations for equilibrium constants

The balances written in terms of molar concentrations are congruent with a complete set of independent (non-contradictory [9, 16]) relations for the equilibrium constants, interrelating concentrations of some species in the balances.

We refer first to non-redox systems discussed in Section 3, and we have:


Referring now to the redox systems, we have:

The complete set of equilibrium constants, namely ionic product of water, dissociation constants, standard potentials, stability constants of complexes and interrelated concentrations of

<sup>½</sup>Hþ<sup>1</sup>

�1

�½SO4 �2

� ¼ 1010:<sup>1</sup>

�2

� ¼ 104:<sup>5</sup>

<sup>½</sup>Hþ<sup>1</sup>

�¼½Mnþ<sup>2</sup>

; {15}½FeðC2O4Þ3�<sup>4</sup>� ¼ <sup>10</sup><sup>5</sup>:<sup>22</sup>½Feþ<sup>2</sup>

½ Feþ3�<sup>2</sup> ½ OH�<sup>1</sup> �2

� ¼ 107:<sup>53</sup>½Feþ<sup>3</sup>

� ¼ 103:<sup>4</sup>

� ¼ 105:<sup>25</sup>½Mnþ<sup>2</sup>

<sup>½</sup>Feþ<sup>2</sup>

<sup>þ</sup><sup>4</sup>� ¼ 1025:<sup>1</sup>

þ1

; {24}½MnOH<sup>þ</sup><sup>1</sup>

� ; {30}½MnðC2O4Þ2�<sup>1</sup>

�; {27}½MnðC2O4Þ2�<sup>2</sup>

for pr1 <sup>¼</sup> FeC2O4 and }7B33 }7D½Mn<sup>þ</sup><sup>2</sup>

11. Relation between the numbers of species, balances and equilibrium

For any electrolytic system, one can define the relationship between the numbers of (i) kinds of species (P) (with free H2O molecules included), (ii) independent balances (Q) and (iii) indepen-

�½CO3 �2

�½OH�<sup>1</sup>

�;{3}½H2C2O4� ¼ 105:<sup>2</sup>

<sup>½</sup>Hþ<sup>1</sup>

�; {7}½H2CO3�

� ; {13}½FeSO4�

�½C2O4 �2

<sup>½</sup>Mnþ<sup>2</sup>

� ¼ <sup>10</sup><sup>16</sup>:<sup>57</sup>½Mnþ<sup>3</sup>

�½C2O4 �2 �

� � 104AðE�1:743Þþ8pH; {10}½Mnþ<sup>3</sup>

�½C2O4 �2� 3 ;

; {19}½FeSO4

�½OH�<sup>1</sup>

�½C2O4 �2 � 2 ;

> �½C2O4 �2 � 2 ;

þ1�

�; {22}½FeðC2O4Þ2�<sup>1</sup>

�; {25}½MnSO4�

�½C2O4 �2 � ;

�

�

ð29Þ

the species (except K+1) is presented in Eq. (2).

� ¼ <sup>10</sup>�<sup>14</sup>; {2}½HSO4

<sup>½</sup>Hþ<sup>1</sup>

<sup>½</sup>Hþ<sup>1</sup> � 2 ½CO3 �2

10AðEþ0:396ÞþpH;

;

� ¼ 1011:<sup>0</sup>

� � 10AðE�0:771<sup>Þ</sup>

<sup>½</sup>Feþ<sup>3</sup>

<sup>½</sup>Feþ<sup>3</sup>

<sup>½</sup>Feþ<sup>3</sup>

� ¼ 1018:<sup>46</sup>½Feþ<sup>3</sup>

<sup>½</sup>Mnþ<sup>3</sup>

� ¼ 109:<sup>98</sup>½Mnþ<sup>3</sup>

� ¼ <sup>10</sup>�6:<sup>7</sup>

�½C2O4 �2 � ;

�1

� ¼ <sup>10</sup><sup>1</sup>:<sup>8</sup>

�; {6}½HCO3

� � 105AðE�1:507Þþ8pH; {9}½MnO4

; {12}½FeOH<sup>þ</sup><sup>1</sup>

; {18}½Fe2ðOHÞ<sup>2</sup>

; {21}½FeC2O4

�½C2O4 �2� 2

�½C2O4 �2 � 3

�½OH<sup>1</sup> � ;

�½OH�<sup>1</sup> �2

�½SO4 �2 � 2

> �½C2O4 �2

�½OH�<sup>1</sup> � ;

> �½C2O4 �2

{1}½Hþ<sup>1</sup>

f4g½HC2O4

¼ ½H2C2O4

{8}½MnO4

¼ ½Mnþ<sup>2</sup>

<sup>½</sup>11�½Feþ<sup>3</sup>

<sup>¼</sup> 102:<sup>3</sup>

�½OH�<sup>1</sup>

188 Advances in Titration Techniques

{5}½H2CO3� ¼ 1016:<sup>4</sup>

�1

<sup>½</sup>Feþ<sup>2</sup>

{16}½FeOH<sup>þ</sup><sup>2</sup>

{17}½FeðOHÞ<sup>2</sup>

<sup>¼</sup> <sup>10</sup><sup>4</sup>:<sup>18</sup>½Feþ<sup>3</sup>

{20}½FeðSO4Þ<sup>2</sup>

<sup>¼</sup> 1013:<sup>64</sup>½Feþ<sup>3</sup>

{28}½MnOH<sup>þ</sup><sup>2</sup>

{29}½MnC2O4

{32}½Feþ<sup>2</sup>

<sup>¼</sup> <sup>10</sup>�5:<sup>3</sup>

constants

{31}½MnðC2O4Þ3�<sup>3</sup>

{23}½FeðC2O4Þ3�<sup>3</sup>

�1

0:5�

� ¼ 103:<sup>8</sup>

�¼½Mnþ<sup>2</sup>

� � 10Að1:509<sup>Þ</sup>

�½SO4 �2 � ;

{14}½FeðC2O4Þ2�<sup>2</sup>� ¼ 104:<sup>52</sup>½Feþ<sup>2</sup>

<sup>þ</sup><sup>1</sup>� ¼ 1021:<sup>7</sup>

� ¼ 107:<sup>4</sup>

� ¼ 1014:<sup>2</sup>

�

for pr2 ¼ MnC2O4

dent equilibrium constant (R) values.

�½C2O4 �2 � 2 ;

�½SO4 �<sup>2</sup>� ;

�1

{26}½MnC2O4� ¼ 103:<sup>82</sup>½Mnþ<sup>2</sup>

þ1

�½C2O4 �2

�¼½Feþ<sup>2</sup>


On this basis, one can state the relationships:


To standardize this relationship, it is (informally) assumed that the electron is one of the species in the redox systems. In this way, the number of species is increased by 1, and we can suggest the relationship

$$\mathbf{P} = \mathbf{Q} + \mathbf{R} + \mathbf{1} \tag{30}$$

as common for both redox and non-redox electrolytic systems, regardless of their degree of complexity. The relation (30), applicable in resolution of electrolytic systems, was first presented in Ref. [37]; it can be perceived as a useful counterpart of the Gibbs' phase rule (of a similar 'degree of complexity') in this area of the knowledge.

It should be noted that the total number, P = 40, of kinds of species involved in the system (I) ) (IV) is relatively high.

## 12. The steps realized within GATES/GEB

Modeling the electrolytic systems according to GATES/GEB consists of several interacting steps [1]: (1) collection of preliminary data; (2) preparation of computer programs; (3) calculations and data handling and (4) knowledge gaining; all the steps are indicated in Figure 1.

#### 12.1. Collection of preliminary data

The necessary physicochemical knowledge is mainly attainable in tables of physicochemical data, exemplified by monographs [38–40]. It should be noted that the period of interest in this

Figure 1. The steps of modeling any electrolytic system [1].

field of research is currently the past. On the other hand, the physicochemical constants originate from works released over several decades, which are clearly seen in Ref. [40], where the relevant information is included. The point is that these physicochemical constants were determined using models that had been adapted to current computing capabilities, especially in the precomputer era; these calculations were based (exclusively, in principle) on the reaction stoichiometry. For example, the solubility products were determined on the basis of molar solubility, see for example, the remark in Refs. [41, 42], without checking whether the precipitate is the equilibrium solid phase in the system [43–48]. Acid dissociation constants were mainly determined on the basis of the Ostwald's formula (see e.g., [49, 50]). Conditional ('formal', not normal) potentials were determined for many redox pairs [51]. In the computer era, some new models resolved with use of iterative methods were elaborated. The assumptions and implementation of these models in relevant experimental studies aroused a number of concerns expressed, among others, in Refs. [52, 53]. Despite these circumstances, GATES and GATES/GEB, in particular, provides a new and reliable tool, applicable for physicochemical knowledge gaining. Thanks to this tool, it will be possible, in the immediate future, for a renaissance of interest in this—so important, after all!—field of fundamental research, which cannot be creatively developed on the basis of the previous 'paradigm' [27] based on the stoichiometry concept.

#### 12.2. Preparation of computer program

Modeling of electrolytic systems can be realized with the use of iterative computer programs, for example MATLAB, perceived as a universal and high-level programming language [54, 55]. From the viewpoint of the form of mathematical models, MATLAB is focused on the matrix algebra procedure. MATLAB allows to make a quick and accurate computation and visualization of numerical data.

The iterative computer programs, written in MATLAB language, are exemplified in Refs. [11, 12, 16].

## 12.3. Calculations and data handling

field of research is currently the past. On the other hand, the physicochemical constants originate from works released over several decades, which are clearly seen in Ref. [40], where the relevant information is included. The point is that these physicochemical constants were determined using models that had been adapted to current computing capabilities, especially in the precomputer era; these calculations were based (exclusively, in principle) on the reaction stoichiometry. For example, the solubility products were determined on the basis of molar solubility, see for example, the remark in Refs. [41, 42], without checking whether the precipitate is the equilibrium solid phase in the system [43–48]. Acid dissociation constants were mainly determined on the basis of the Ostwald's formula (see e.g., [49, 50]). Conditional ('formal', not normal) potentials were determined for many redox pairs [51]. In the computer era, some new models resolved with use of iterative methods were elaborated. The assumptions and implementation of these models in relevant experimental studies aroused a number of concerns expressed, among others, in Refs. [52, 53]. Despite these circumstances, GATES and GATES/GEB, in particular, provides a new and reliable tool, applicable for physicochemical knowledge gaining. Thanks to this tool, it will be possible, in the immediate future, for a renaissance of interest in this—so important, after all!—field of fundamental research, which cannot be creatively developed on the

Modeling of electrolytic systems can be realized with the use of iterative computer programs, for example MATLAB, perceived as a universal and high-level programming language [54, 55]. From the viewpoint of the form of mathematical models, MATLAB is focused on the matrix algebra procedure. MATLAB allows to make a quick and accurate computation and visualiza-

basis of the previous 'paradigm' [27] based on the stoichiometry concept.

12.2. Preparation of computer program

Figure 1. The steps of modeling any electrolytic system [1].

190 Advances in Titration Techniques

tion of numerical data.

The calculations can be made at different levels of the preliminary, physicochemical knowledge about the system in question. What is more, some 'variations on the subject' can also be done for this purpose; it particularly refers to metastable and non-equilibrium systems. A special emphasis will be put on complex redox systems, where all types of elementary chemical reactions proceed simultaneously and/or sequentially. In all instances, one can follow measurable quantities (potential E, pH) in dynamic and static processes and gain the information about many details not measurable in real experiments; it particularly refers to dynamic speciation.

We refer here to dynamic (titration) redox systems, represented by the system of 2 + k nonlinear equations composed of GEB, charge balance and k (≥ 1) concentration balances. The results of calculations, made with the use of an iterative computer program, are presented graphically. Thus, the plots E = E(Φ) and pH = pH(Φ) for potential E and pH of the solution and log[Xzi <sup>i</sup> ] versus Φ (Eq. (5)) relationships (speciation curves) will be drawn.

The Φ concept is used for simpler systems, providing a kind of normalization (independence on V0) in the systems considered. Φ plays a key role in the formulation of the generalized equivalence mass (GEM) concept, introduced also by Michałowski [25]. In more complex systems, the volume V is put on the abscissa.

The numerical data can be visualized in the form of two- or three-dimensional graphs (2D, 3D), see for example Ref. [12].

## 12.4. Computer program for the system (I) ) (III)

The set of independent equilibrium constants is involved in the algorithm needed for calculation purposes [14], realized in the system (I) ) (IV) as specified above. An algorithm is a welldefined procedure, expressed by a sequence of unambiguous instructions, which allows a computer to solve a problem according to a computer program implemented for this purpose. The term 'unambiguous' indicates that there is no room for subjective interpretation.

The system (I) ) (IV) and its subsystems (I) ) (II) and (I) ) (III) were simulated using an iterative computer program (MATLAB software, included in the optimization toolbox™). In particular, the computer program for the system (I) ) (III) is as follows.

```
function F = Function_KMnO4_Na2C2O4(x)
```
global V Vmin Vstep Vmax V0 C C1 C0 C01 C02 fi H OH pH E Kw pKw A aa

global H2C2O4 HC2O4 C2O4 H2CO3 HCO3 CO3 K

global logH2C2O4 logHC2O4 logC2O4 logH2CO3 logHCO3 logCO3 logK

global Mn7O4 Mn6O4 HSO4 SO4 Na global logMn7O4 logMn6O4 logHSO4 logSO4 logNa global Mn3 Mn3OH Mn3C2O4 Mn3C2O42 Mn3C2O43 global logMn3 logMn3OH logMn3C2O4 logMn3C2O42 logMn3C2O43 global Mn2 Mn2OH Mn2SO4 Mn2C2O4 Mn2C2O42 global logMn2 logMn2OH logMn2SO4 logMn2C2O4 logMn2C2O42 global pr logpr q logq

```
pH=x(1);
```
E=x(2);

if aa==0

Mn2=10.^-x(3);

pr=0;

#### else

```
pr=10.^-x(3);
```
end;

```
H2C2O4=10.^-x(4);
```

```
SO4=10.^-x(5);
```
H=10.^-pH;

pKw=14;

Kw=10.^-14;

```
OH=Kw./H;
```

```
A=16.92;
```
ZMn=25;

ZC=6;

Ksp=10.^-5.3;

HC2O4=10.^(pH-1.25).\*H2C2O4;

C2O4=10.^(pH-4.27).\*HC2O4;

H2CO3=10.^(A.\*(E+0.386)).\*H2C2O4.^0.5;

HCO3=10.^(pH-6.3).\*H2CO3;

CO3=10.^(pH-10.1).\*HCO3;

if aa==1

global Mn7O4 Mn6O4 HSO4 SO4 Na

global pr logpr q logq

192 Advances in Titration Techniques

Mn2=10.^-x(3);

pr=10.^-x(3);

H2C2O4=10.^-x(4);

SO4=10.^-x(5);

H=10.^-pH;

Kw=10.^-14; OH=Kw./H;

pKw=14;

A=16.92; ZMn=25;

ZC=6;

Ksp=10.^-5.3;

HC2O4=10.^(pH-1.25).\*H2C2O4;

H2CO3=10.^(A.\*(E+0.386)).\*H2C2O4.^0.5;

C2O4=10.^(pH-4.27).\*HC2O4;

HCO3=10.^(pH-6.3).\*H2CO3;

pH=x(1); E=x(2); if aa==0

pr=0;

else

end;

global logMn7O4 logMn6O4 logHSO4 logSO4 logNa global Mn3 Mn3OH Mn3C2O4 Mn3C2O42 Mn3C2O43

global Mn2 Mn2OH Mn2SO4 Mn2C2O4 Mn2C2O42

global logMn3 logMn3OH logMn3C2O4 logMn3C2O42 logMn3C2O43

global logMn2 logMn2OH logMn2SO4 logMn2C2O4 logMn2C2O42

Mn2=Ksp./C2O4;

end;

HSO4=10.^(1.8-pH).\*SO4;

Mn7O4=Mn2.\*10.^(5.\*A.\*(E-1.507)+8.\*pH);

Mn6O4=10.^(A.\*(0.56-E)).\*Mn7O4;

Mn2OH=10.^3.4.\*Mn2.\*OH;

Mn2SO4=10.^2.28.\*Mn2.\*SO4;

Mn2C2O4=10.^3.82.\*Mn2.\*C2O4;

Mn2C2O42=10.^5.25.\*Mn2.\*C2O4.^2;

Mn3=Mn2.\*10.^(A.\*(E-1.509));

Mn3OH=10.^(pH-0.2).\*Mn3;

Mn3C2O4=10.^9.98.\*Mn3.\*C2O4;

Mn3C2O42=10.^16.57.\*Mn3.\*C2O4.^2;

Mn3C2O43=10.^19.42.\*Mn3.\*C2O4.^3;

K=C.\*V./(V0+V);

Na=C0.\*V0./(V0+V);

%Charge balance

```
F=[(H-OH+K+Na-HSO4-2.*SO4-HC2O4-2.*C2O4-HCO3-2.*CO3-Mn7O4-2.*Mn6O4…
```
+3.\*Mn3+2.\*Mn3OH+Mn3C2O4-Mn3C2O42-3.\*Mn3C2O43+2.\*Mn2+Mn2OH…


%Concentration balance of Mn

(Mn7O4+Mn6O4+Mn3+Mn3OH+Mn3C2O4+Mn3C2O42+Mn3C2O43…

+Mn2+Mn2OH+Mn2SO4+Mn2C2O4+Mn2C2O42+aa.\*pr-C.\*V./(V0+V));

%Concentration balance of C

(2.\*H2C2O4+2.\*HC2O4+2.\*C2O4+H2CO3+HCO3+CO3+2.\*Mn2C2O4…

+4.\*Mn2C2O42+2.\*Mn3C2O4+4.\*Mn3C2O42+6.\*Mn3C2O43…

```
+2.*aa.*pr-(2.*C0.*V0+C02.*V0+C1.*V)./(V0+V));
```
%Concentration balance of S

(HSO4+SO4+Mn2SO4-C01.\*V0./(V0+V));

%Electron balance

((ZMn-7).\*Mn7O4+(ZMn-6).\*Mn6O4+(ZMn-3).\*(Mn3+Mn3OH)…

+(ZMn-2).\*(Mn2+Mn2OH+Mn2SO4)+(ZC-4).\*(H2CO3+HCO3+CO3)…

+2.\*(ZC-3).\*(H2C2O4+HC2O4+C2O4)+(ZMn-3+2.\*ZC-6).\*Mn3C2O4…

+(ZMn-3+4.\*ZC-12).\*Mn3C2O42+(ZMn-3+6.\*ZC-18).\*Mn3C2O43…

+(ZMn-2+2.\*ZC-6).\*Mn2C2O4+(ZMn-2+4.\*ZC-12).\*Mn2C2O42+…

+(ZMn-2+2.\*ZC-6).\*aa.\*pr…


+(ZMn-7).\*C.\*V)./(V0+V))];

q=Mn2.\*C2O4./Ksp;

logMn2=log10(Mn2);

logMn2OH=log10(Mn2OH);

logMn2SO4=log10(Mn2SO4);

logq=log10(q);

logpr=log10(pr);

logMn2C2O4=log10(Mn2C2O4);

logMn2C2O42=log10(Mn2C2O42);

logMn3=log10(Mn3);

logMn3OH=log10(Mn3OH);

logMn3C2O4=log10(Mn3C2O4);

logMn3C2O42=log10(Mn3C2O42);

logMn3C2O43=log10(Mn3C2O43);

logMn6O4=log10(Mn6O4);

logMn7O4=log10(Mn7O4);

logH2CO3=log10(H2CO3);

logHCO3=log10(HCO3); logCO3=log10(CO3); logH2C2O4=log10(H2C2O4); logHC2O4=log10(HC2O4); logC2O4=log10(C2O4); logHSO4=log10(HSO4); logSO4=log10(SO4); logNa=log10(Na); logK=log10(K); %The end of program

+2.\*aa.\*pr-(2.\*C0.\*V0+C02.\*V0+C1.\*V)./(V0+V));

((ZMn-7).\*Mn7O4+(ZMn-6).\*Mn6O4+(ZMn-3).\*(Mn3+Mn3OH)…

+(ZMn-2).\*(Mn2+Mn2OH+Mn2SO4)+(ZC-4).\*(H2CO3+HCO3+CO3)… +2.\*(ZC-3).\*(H2C2O4+HC2O4+C2O4)+(ZMn-3+2.\*ZC-6).\*Mn3C2O4…

+(ZMn-3+4.\*ZC-12).\*Mn3C2O42+(ZMn-3+6.\*ZC-18).\*Mn3C2O43… +(ZMn-2+2.\*ZC-6).\*Mn2C2O4+(ZMn-2+4.\*ZC-12).\*Mn2C2O42+…


(HSO4+SO4+Mn2SO4-C01.\*V0./(V0+V));

%Concentration balance of S

+(ZMn-2+2.\*ZC-6).\*aa.\*pr…

+(ZMn-7).\*C.\*V)./(V0+V))];

q=Mn2.\*C2O4./Ksp;

logMn2=log10(Mn2);

logMn3=log10(Mn3);

logMn3OH=log10(Mn3OH);

logMn6O4=log10(Mn6O4); logMn7O4=log10(Mn7O4); logH2CO3=log10(H2CO3);

logMn3C2O4=log10(Mn3C2O4);

logMn3C2O42=log10(Mn3C2O42); logMn3C2O43=log10(Mn3C2O43);

logq=log10(q); logpr=log10(pr);

logMn2OH=log10(Mn2OH); logMn2SO4=log10(Mn2SO4);

logMn2C2O4=log10(Mn2C2O4); logMn2C2O42=log10(Mn2C2O42);

%Electron balance

194 Advances in Titration Techniques

## 13. Graphical presentation of the data

The results of calculations in the system (I) ) (IV) are presented in Figures 2–4. More detailed, numerical data are specified in Ref. [14].

Figure 2. The (a) E versus V and (b) pH versus V relationships plotted at V0 = 100, C = 0.02, C03 = 0.5, C1 = C04 = 0.001 and indicated pairs of C01 and C02 values.

Figure 3. The speciation curves plotted for (a) Mn and (b) Fe species at V0 = 100, C = 0.02, C03 = 0.5, C1 = C04 = 0.001 and C01 = C02 = 0.002.

Figure 4. The log(qi) versus V relationships (see Eq. (31)) plotted for (a) Fe (i = 1) and (b) Mn (i = 2) oxalates.

There are valid relationships: CVeq1 = 0.2∙C01V0 for iron and CVeq2 = 0.4∙C02V0 for oxalate. For V0 = 100, C = 0.02 we have, in particular, Veq1 = 10 and Veq2 = 20 at C01 = 0.01, C02 = 0.01 and Veq1 = 20 and Veq2 = 40 at C01 = 0.02 and C02 = 0.02. This agrees exactly with the position of the points

Figure 5. The relationships: (a) E = E(Φ), (b) pH = pH(Φ) and the speciation curves log[Xi zi] versus Φ for (c) manganese and (d) iron species plotted for titration of V0 = 100 mL of C0 = 0.01 mol/L FeSO4 + C01 mol/L H2SO4 as D with V mL of C = 0.02 mol/L KMnO4 as T.

where jumps of potential E occur and should be compared with the plots of titration curves for individual analytes: FeSO4 (Figure 5a) and H2C2O4 (Figure 6a), where abscissas are expressed in terms of the fraction titrated Φ (Eq. (5)). The related pH versus Φ relationships are presented in Figures 5b and 6b. To explain/formulate the reactions occurred in the systems together with

There are valid relationships: CVeq1 = 0.2∙C01V0 for iron and CVeq2 = 0.4∙C02V0 for oxalate. For V0 = 100, C = 0.02 we have, in particular, Veq1 = 10 and Veq2 = 20 at C01 = 0.01, C02 = 0.01 and Veq1 = 20 and Veq2 = 40 at C01 = 0.02 and C02 = 0.02. This agrees exactly with the position of the points

Figure 4. The log(qi) versus V relationships (see Eq. (31)) plotted for (a) Fe (i = 1) and (b) Mn (i = 2) oxalates.

(a) (b)

(a) (b)

C01 = C02 = 0.002.

196 Advances in Titration Techniques

Figure 3. The speciation curves plotted for (a) Mn and (b) Fe species at V0 = 100, C = 0.02, C03 = 0.5, C1 = C04 = 0.001 and

Figure 6. The relationships: (a) E = E(Φ), (b) pH = pH(Φ) and the speciation curves log[Xi zi] for (c) Mn-, (d) C-species for titration of V0 = 100 mL of C0 = 0.01 mol/L H2C2O4 + C01 mol/L H2SO4 as D with V mL of C = 0.02 mol/L KMnO4 as T.

their relative efficiencies, the speciation diagrams depicted in Figures 3a, b, and 5c, d are used. From Figure 4a and b, we see that

$$\mathbf{q}\_1 = [\mathbf{F}\mathbf{e}^{+2}][\mathbf{C}\_2\mathbf{O}\_4^{-2}]/\mathbf{K}\_{\mathrm{sp1}} < 1 \quad \text{and} \quad \mathbf{q}\_2 = [\mathbf{M}\mathbf{n}^{+2}][\mathbf{C}\_2\mathbf{O}\_4^{-2}]/\mathbf{K}\_{\mathrm{sp2}} < 1 \tag{31}$$

i.e., the precipitates FeC2O4 (Ksp1) and MnC2O4 (Ksp2) do not exist in this system as the equilibrium solid phases at the pre-assumed sufficiently high concentration C03 of H2SO4; MnO2 is not formed there as well, i.e., a1 = a2 = 0 in Eqs. (26)–(28).

## 14. Deficiency and veracity of equilibrium data

In some 'variations on the subject', we try to know what would happen if some constraints put on the metastable system are removed and the reaction is conducted in a thermodynamic manner, in accordance with the conditions imposed by the equilibrium constants [1, 36]. One can also analyze the data resulting from (intentional) omission or (factual, presupposed) incomplete physicochemical knowledge on the system studied.

Some computer simulations can be used to check some effects involved with complexation phenomena. For example, we intend to check the effect involved with formation of sulfate complexes FeSO4, MnSO4 and (particularly) FeSO4 +1, Fe(SO4)2 �<sup>1</sup> in the system (I) ) (II). The shapes of the titration curves E = E(Φ) are compared in Figure 7.

Figure 7. The E versus Φ relationships plotted for the system (I) ) (II): (1) at pre-assumed physicochemical knowledge and (2) after intentional omission of all sulfate complexes; C0 = 0.01, C01 = 1.0 and C = 0.02.

their relative efficiencies, the speciation diagrams depicted in Figures 3a, b, and 5c, d are used.

titration of V0 = 100 mL of C0 = 0.01 mol/L H2C2O4 + C01 mol/L H2SO4 as D with V mL of C = 0.02 mol/L KMnO4 as T.

(a) (b)

(c) (d)

Figure 6. The relationships: (a) E = E(Φ), (b) pH = pH(Φ) and the speciation curves log[Xi

�=Ksp1 <sup>&</sup>lt; 1 and q2 ¼ ½Mn<sup>þ</sup><sup>2</sup>

�½C2O4 �2

�=Ksp2 < 1 ð31Þ

zi] for (c) Mn-, (d) C-species for

From Figure 4a and b, we see that

198 Advances in Titration Techniques

q1 ¼ ½Fe<sup>þ</sup><sup>2</sup>

�½C2O4 �2

Figure 8. Fragments of hypothetical titration curves for V0 = 100 mL of FeSO4 (C0 = 0.01 mol/L) + H2SO4 (Ca0 = 0.1 mol/L) titrated with C = 0.02 mol/L KMnO4, plotted at different pairs of stability constants (K31, K32) of the sulfate complexes Mn (SO4)i +3–2i: (1) (104 , 107 ), (2) (10<sup>3</sup> , 106 ), (3) (102.5, 105 ), (4) (10<sup>2</sup> , 104 ), (5) (10<sup>4</sup> , 0), (6) (10<sup>3</sup> , 0), (7) (10<sup>2</sup> , 0) and (8) (0, 0).

Some equilibrium constants used in calculations may be unknown/doubtful on the stage of collection of equilibrium data. In such instances, the pre-assumed/virtual data can be introduced for comparative purposes, and the effects involved with omission/inclusion of some types of complexes can be checked.

The possible a priori complexes of Mn(SO4)i +3�2i are unknown in literature. To check the effect of formation of these complexes on the shape of the titration curve E = E(Φ) in the system (I) ) (II), the pre-assumed stability constants K3i of the complexes, [Mn(SO4)i +3�2i]=K3i[Mn3+][SO4 <sup>2</sup>�] i , specified in legend for Figure 8, were applied in the related algorithm, where concentrations of MnSO4 +1 and Mn(SO4)2 �<sup>1</sup> with the corresponding multipliers were inserted in electron (GEB) and charge balances and in concentration balances for Mn and sulfate. As we see, at higher K3i values (comparable to ones related to Fe(SO4)i +3�2i (i=1,2) complexes [39]), the new inflection point appears at Φ = 0.25 and disappears at lower K3i values assumed in the simulating procedure. Comparing the simulated curves with the one obtained experimentally [25], one can conclude that the complexes Mn(SO4)i +3�2i do not exist at all or the K3i values are small, when compared with those for Fe(SO4)i +3�2i [33].

Other interesting examples involved with 'variations on the subject' are presented in Ref. [1], and other references cited therein.

## 15. Advantages of GATES and usefulness of chemical processes simulation

Some equilibrium constants used in calculations may be unknown/doubtful on the stage of collection of equilibrium data. In such instances, the pre-assumed/virtual data can be introduced for comparative purposes, and the effects involved with omission/inclusion of some

Figure 8. Fragments of hypothetical titration curves for V0 = 100 mL of FeSO4 (C0 = 0.01 mol/L) + H2SO4 (Ca0 = 0.1 mol/L) titrated with C = 0.02 mol/L KMnO4, plotted at different pairs of stability constants (K31, K32) of the sulfate complexes Mn

, 104

), (5) (10<sup>4</sup>

, 0), (6) (10<sup>3</sup>

), (4) (10<sup>2</sup>

formation of these complexes on the shape of the titration curve E = E(Φ) in the system (I) ) (II),

specified in legend for Figure 8, were applied in the related algorithm, where concentrations of

and charge balances and in concentration balances for Mn and sulfate. As we see, at higher K3i

point appears at Φ = 0.25 and disappears at lower K3i values assumed in the simulating procedure. Comparing the simulated curves with the one obtained experimentally [25], one can

the pre-assumed stability constants K3i of the complexes, [Mn(SO4)i

), (3) (102.5, 105

+3�2i [33].

+3�2i are unknown in literature. To check the effect of

, 0), (7) (10<sup>2</sup>

+3�2i (i=1,2) complexes [39]), the new inflection

+3�2i do not exist at all or the K3i values are small, when

�<sup>1</sup> with the corresponding multipliers were inserted in electron (GEB)

+3�2i]=K3i[Mn3+][SO4

, 0) and (8) (0, 0).

<sup>2</sup>�] i ,

types of complexes can be checked.

, 107

), (2) (10<sup>3</sup>

, 106

+1 and Mn(SO4)2

MnSO4

(SO4)i

+3–2i: (1) (104

200 Advances in Titration Techniques

The possible a priori complexes of Mn(SO4)i

values (comparable to ones related to Fe(SO4)i

conclude that the complexes Mn(SO4)i

compared with those for Fe(SO4)i

Mathematical formalism of electrolytic systems tested and resolved according to GATES principles formulated by Michałowski (1992) arises from synthesis of the three laws: (1o ) law of charge conservation, (2<sup>o</sup> ) law of elements conservation and (3<sup>o</sup> ) law of mass action. All other chemical laws result from conjunction of those laws; it particularly refers to the stoichiometry and equivalent mass concepts.

GATES, based on physical, physicochemical and chemical laws, is considered as the best thermodynamic approach to equilibrium, non-equilibrium and metastable, mono- and polyphase, static and dynamic, and redox and non-redox systems, of any degree of complexity, with liquid-liquid extraction systems included [31].

GATES related to redox systems is denoted as GATES/GEB. All these systems are resolvable with use of iterative computer programs, for example, MATLAB. The complexity of chemical systems is here of a secondary importance from the point of view of the computational capabilities inherent in iterative computer programs.

GATES is a confirmation of the thesis that 'everything brilliant is simple'. GATES/GEB is the unique tool to obtain information about the thermodynamics of redox systems on the basis of balances and equilibrium constants values.

GATES enables to simulate all possible (from a thermodynamic point of view) processes obtained after pre-assumed crossing of one or more reaction paths in metastable systems.

GATES enables to simulate the processes impossible to track experimentally; for example, dissolving a solid phase in the electrolytic system of a pre-established composition.

GATES relies on the assumption that the chemistry involved with such systems is predictable on the basis of knowledge of physicochemical properties of the species involved in the system in question. A complete set of non-contradictory relations for the equilibrium constants must be used in calculations; this 'iron rule' of mathematics is then obligatory also in calculations related to electrolytic systems.

GATES is the intrinsically consistent theory, joining fundamental laws of physics and chemistry [10, 28]. The knowledge gaining from redox systems is the most comprehensive way for studying such systems. Note that equations-based simulations are most commonly used in physics and related sciences.

GATES joins, on the thermodynamic basis, four kinds of chemical interactions, named as acidbase, redox, precipitation and complexation reactions, extended on a liquid-liquid extraction in mono- and poly-phase systems. To a certain degree—one can perceive GATES as a spitting image of theory of everything (ToE), as the main, still unresolved issue in physics, aiming to elaborate the consistent theory, that links together four: strong, weak, electromagnetic and gravitational interactions.

GATES referred to electrolytic non-redox and redox systems and is considered as the best thermodynamic approach to such systems. GATES, based on physical (charge conservation), physicochemical (conservation of elements) and chemical (mass action) laws, is the best tool applicable for computer simulation of equilibrium, non-equilibrium and metastable, and mono- and poly-phase electrolytic systems. GATES is the basis for the Generalized Equivalence Mass (GEM) concept, with no relevance to the chemical reaction notation.

One can also express a conviction that the discovery of the Approach II to GEB in context with GATES will lead to gradual elimination of the stoichiometry concept from the consciousness of chemists.

## 16. Final comments

This chapter provides comprehensive, compatible and consistent knowledge on modeling electrolytic redox and non-redox systems and further steps applied to gain the thermodynamic knowledge on the systems, referred mainly to aqueous media.

The Generalized Electron Balance (GEB) concept, related to electrolytic redox systems, is put in context with the principle of conservation of all elements in electrolytic redox systems, in aqueous, non-aqueous or mixed-solvent media. The GEB is fully compatible with charge and concentration balances, and completes the set of 2 + k equations needed for quantitative description of a redox system, with 2 + k independent/scalar variables <sup>x</sup><sup>T</sup> ¼ ðE; pH; pX1;…; pXkÞ. Two equivalent approaches (I and II) to GEB were proposed (1992, 2006) by Michałowski. The Approach I to GEB is based on a card-game principle, with electron-active elements as gamblers, electron-non-active elements as fans and common pool of electrons introduced by electron-active elements as money. The Approach II to GEB is based on the linear combination 2f(O)�f(H) of elemental balances: f(H) for H and f(O) for O. The linear independency/dependency of 2f(O)�f(H) from charge and other elemental/core balances referred to the system in question provides the general criterion distinguishing between redox and non-redox systems. For non-redox systems, 2f(O)�f(H) is the linear combination of those balances, that is, it is not a new, independent equation in such systems. In redox systems, 2f(O)�f(H) is the independent equation, considered as the primary form of GEB and denoted as pr-GEB. The balances for elements/cores 6¼ H, O are the basis for k concentration balances, forming—with GEB and charge balance—the set of 2+k independent balances, expressed in terms of concentrations. The Approach I to GEB, considered as the 'short' version of GEB, can be applied if the oxidation numbers for all elements in components forming a system and in the species of the system are known beforehand. The Approach II to GEB needs none prior information on oxidation numbers of all elements in the components and species in the system. Within the Approaches I and II to GEB, the roles of oxidants and reducers are not ascribed to the components and particular species. The GEB is put in context with the Generalized Approach to Electrolytic Systems (GATES) as GATES/GEB, where all quantitative thermodynamic knowledge on the redox system is involved in the complete set of independent equilibrium constants, where standard potentials E0i are involved. The GATES/GEB provides the best thermodynamic formulation of electrolytic redox systems of any degree of complexity, namely: equilibrium, non-equilibrium and metastable, mono- and poly-phase and static and dynamic electrolytic systems, resolvable with the use of iterative computer programs, applied to the set of nonlinear equations, with no simplifying assumptions needed. The GATES/GEB can also be referred to as redox systems in mixed-solvent media, provided that the related thermodynamic knowledge is attainable. This chapter is referred to dynamic systems, realized according to the titrimetric mode. The results obtained from the calculations can be presented graphically on 2D or 3D diagrams. The speciation diagrams obtained according to GATES/GEB have indisputable advantage over Pourbaix predominance diagrams. The GEB concept, unknown before 1992, is perceived as an emanation of the matter/elements conservation, as the general law of nature. The redox systems are formulated on simple principles, unknown in earlier literature. Earlier approaches to electrolytic redox systems, based on stoichiometric principles, are thus invalidated.

## Author details

mono- and poly-phase systems. To a certain degree—one can perceive GATES as a spitting image of theory of everything (ToE), as the main, still unresolved issue in physics, aiming to elaborate the consistent theory, that links together four: strong, weak, electromagnetic and

GATES referred to electrolytic non-redox and redox systems and is considered as the best thermodynamic approach to such systems. GATES, based on physical (charge conservation), physicochemical (conservation of elements) and chemical (mass action) laws, is the best tool applicable for computer simulation of equilibrium, non-equilibrium and metastable, and mono- and poly-phase electrolytic systems. GATES is the basis for the Generalized Equiva-

One can also express a conviction that the discovery of the Approach II to GEB in context with GATES will lead to gradual elimination of the stoichiometry concept from the consciousness of

This chapter provides comprehensive, compatible and consistent knowledge on modeling electrolytic redox and non-redox systems and further steps applied to gain the thermodynamic

The Generalized Electron Balance (GEB) concept, related to electrolytic redox systems, is put in context with the principle of conservation of all elements in electrolytic redox systems, in aqueous, non-aqueous or mixed-solvent media. The GEB is fully compatible with charge and concentration balances, and completes the set of 2 + k equations needed for quantitative description of a redox system, with 2 + k independent/scalar variables <sup>x</sup><sup>T</sup> ¼ ðE; pH; pX1;…; pXkÞ. Two equivalent approaches (I and II) to GEB were proposed (1992, 2006) by Michałowski. The Approach I to GEB is based on a card-game principle, with electron-active elements as gamblers, electron-non-active elements as fans and common pool of electrons introduced by electron-active elements as money. The Approach II to GEB is based on the linear combination 2f(O)�f(H) of elemental balances: f(H) for H and f(O) for O. The linear independency/dependency of 2f(O)�f(H) from charge and other elemental/core balances referred to the system in question provides the general criterion distinguishing between redox and non-redox systems. For non-redox systems, 2f(O)�f(H) is the linear combination of those balances, that is, it is not a new, independent equation in such systems. In redox systems, 2f(O)�f(H) is the independent equation, considered as the primary form of GEB and denoted as pr-GEB. The balances for elements/cores 6¼ H, O are the basis for k concentration balances, forming—with GEB and charge balance—the set of 2+k independent balances, expressed in terms of concentrations. The Approach I to GEB, considered as the 'short' version of GEB, can be applied if the oxidation numbers for all elements in components forming a system and in the species of the system are known beforehand. The Approach II to GEB needs none prior information on oxidation numbers of all elements in the components and species in the system.

lence Mass (GEM) concept, with no relevance to the chemical reaction notation.

knowledge on the systems, referred mainly to aqueous media.

gravitational interactions.

202 Advances in Titration Techniques

16. Final comments

chemists.

Anna Maria Michałowska-Kaczmarczyk<sup>1</sup> , Aneta Spórna-Kucab<sup>2</sup> and Tadeusz Michałowski<sup>2</sup> \*

\*Address all correspondence to: michanku@poczta.onet.pl

1 Department of Oncology, The University Hospital in Cracow, Cracow, Poland

2 Faculty of Chemical Engineering and Technology, Cracow University of Technology, Warszawska, Cracow, Poland

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## *Edited by Vu Dang Hoang*

In chemistry, titration (a.k.a. titrimetry) is a common laboratory technique used for the determination of the unknown concentration of an analyte. Because of its versatility, the application of various forms of titration can affect nearly all aspects of society. This book is specifically aimed at broadening and deepening the theory and applications of titration. It contains six chapters being organized into three main sections: Volumetric Titration, Isothermal Titration Calorimetry, and Titrimetric Principles in Electrolytic Systems. Each chapter has been well written by internationally renowned experts in the field of chemistry, with mathematical expressions and illustrative examples selectively and logically presented. It is highly recommended for postgraduate students and scientists alike.

Advances in Titration Techniques

Advances in

Titration Techniques

*Edited by Vu Dang Hoang*

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