**The Conditions Needed for a Buffer to Set the pH in a System**

Norma Rodríguez‐Laguna, Alberto Rojas‐Hernández, María T. Ramírez‐Silva, Rosario Moya‐Hernández, Rodolfo Gómez‐Balderas and Mario A. Romero‐Romo

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69003

#### Abstract

It is a known fact that buffer systems are widely used in industry and diverse laboratories to maintain the pH of a system within desired limits, occasionally narrow. Hence, the aim of the present work is to study the buffer capacity and buffer efficacy in order to determine the useful conditions to impose the pH on a given system. This study is based on the electroneutrality and component balance equations for a mixture of protons polyreceptors. The added volume equations are established, V, for strong acids or bases, as well as the buffer capacity equations with dilution effect, βdil, and the buffer efficacy, ε, considering that the analyte contains a mixture of the species of the same polyacid system or various polyacid systems. The ε index is introduced to define the performance of a buffer solution and find out for certain, whether the buffer is adequate to set the pH of a system, given the proper conditions and characteristics.

Keywords: buffer, buffer capacity, buffer efficacy, polyacid systems, electroneutrality equation.

## 1. Introduction

Currently, there are studies that examine the progress of an acid-base titration for one or various polydonor systems, extending sometimes this study to the theme of buffer capacity [1–16]. In the scientific literature, there are algorithms and simulators to construct acid-base titration curves, even considering a wide range of different mixtures of polydonor systems [17–20].

© 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.

The buffer solutions have a certain buffering capacity that is used to maintain constant the pH of a system, having only a small uncertainty. The buffer capacity, β, has been defined as the quantity of strong acid or strong base (in the buffer solution) that gives rise to a change of one pH unit in 1 L of solution, as an intensive property of the system [15]. This involves using directly the concentration of either a strong base or an acid in the buffer solution, without considering the dilution effect, as King and Kester [2], Segurado [3], Urbansky and Schock [4], De Levie [8] did, among others. Urbansky and Schock also mentioned the use of concentration to simplify the maths. Nevertheless, the dilution effect on buffer capacity was first considered by Michałowski, as Asuero and Michałowski have established in a thorough and holistic review [6].

The buffer capacity considering the effect of dilution, βdil, is defined as the added amount of strong base or strong acid required to change in one unit the pH of an initial V<sup>o</sup> volume of the buffer solution formed by species of only one polydonor system [7]. By their definition, β is an intensive property by considering the concentration, while βdil is an extensive property to include the amount of substance.

A buffer solution is used to impose the pH in a given system; generally speaking, the buffer is added to a working system selected to impose a given pH, thus giving rise to a mixture between both systems. It would be convenient to know the minimum concentration of the buffer components in the mixture as well as the minimum volume that must be added in order for it to fulfill its function. From the existing works in the literature, an evaluation of the buffer performance was attempted, in general a β was provided, although up to now this problem has not been dealt with quantitatively.

Figure 1 shows the scheme, in which a buffer solution is used to buffer the pH of a given system. It is observed that a mixture of both systems exists: the buffer system (BS) and the original system (OS). In general, it is necessary to define which would be the BS and the study system.

Figure 1. Schematization of the manner, in which the pH is generally set in an original system (OS) with a buffer system (BS). C<sup>o</sup> BS is the initial concentration of the buffer system before mixing, VBS is the volume of BS added to OS, C<sup>o</sup> OS and VOS are the initial concentration and initial volume of the original system before mixing, respectively; Vo is the mixture volume formed by the solutions with volumes VBS and VOS, Cmix is the overall concentration of the mixture, Cmix BS is the buffer components' concentration in the mixture; and Cmix OS is the original system's solute concentration in the mixture.

In many applications, it is convenient that the volume of the buffer solution that is added to the system of interest be VBS < VOS, in order not to alter much the composition of the study system.

The present work extends the study of βdil that is contained in a mixture of polydonor systems. Moreover, a new concept is introduced, buffer efficacy (ε), as an index to estimate the performance of a buffer. Finally, it is shown how these indexes allow the determination of the useful conditions (minimum concentration and volume), for a buffer to enable imposing the pH in a system of interest.

## 2. Theoretical background

The buffer solutions have a certain buffering capacity that is used to maintain constant the pH of a system, having only a small uncertainty. The buffer capacity, β, has been defined as the quantity of strong acid or strong base (in the buffer solution) that gives rise to a change of one pH unit in 1 L of solution, as an intensive property of the system [15]. This involves using directly the concentration of either a strong base or an acid in the buffer solution, without considering the dilution effect, as King and Kester [2], Segurado [3], Urbansky and Schock [4], De Levie [8] did, among others. Urbansky and Schock also mentioned the use of concentration to simplify the maths. Nevertheless, the dilution effect on buffer capacity was first considered by Michałowski,

The buffer capacity considering the effect of dilution, βdil, is defined as the added amount of strong base or strong acid required to change in one unit the pH of an initial V<sup>o</sup> volume of the buffer solution formed by species of only one polydonor system [7]. By their definition, β is an intensive property by considering the concentration, while βdil is an extensive property to

A buffer solution is used to impose the pH in a given system; generally speaking, the buffer is added to a working system selected to impose a given pH, thus giving rise to a mixture between both systems. It would be convenient to know the minimum concentration of the buffer components in the mixture as well as the minimum volume that must be added in order for it to fulfill its function. From the existing works in the literature, an evaluation of the buffer performance was attempted, in general a β was provided, although up to now this problem

Figure 1 shows the scheme, in which a buffer solution is used to buffer the pH of a given system. It is observed that a mixture of both systems exists: the buffer system (BS) and the original system (OS). In general, it is necessary to define which would be the BS and the study system.

Figure 1. Schematization of the manner, in which the pH is generally set in an original system (OS) with a buffer system

are the initial concentration and initial volume of the original system before mixing, respectively; Vo is the mixture volume

OS is the original system's solute concentration in the mixture.

OS and VOS

BS is the buffer

BS is the initial concentration of the buffer system before mixing, VBS is the volume of BS added to OS, C<sup>o</sup>

formed by the solutions with volumes VBS and VOS, Cmix is the overall concentration of the mixture, Cmix

as Asuero and Michałowski have established in a thorough and holistic review [6].

include the amount of substance.

4 Advances in Titration Techniques

has not been dealt with quantitatively.

components' concentration in the mixture; and Cmix

(BS). C<sup>o</sup>

## 2.1. Description of the components, species, equilibria, and fractions in a mixture of polydonor systems

In order to evaluate the useful performance of a buffer to impose the pH in a given system, it is necessary to establish the expressions of β or of βdil that consider mixtures of various polydonor systems.

Although Ref. [8] presented equations that describe the behavior of β for polydonor systems, the nomenclature, which has shown in Ref. [7] to study βdil, is considered here to generalize its equations for the case of buffer solutions of mixtures of different polydonor systems.

A polyprotic system [6, 7] can be represented as follows:

$$\mathbf{H}\_n \mathbf{L}^{(n-a)} / \dots / \mathbf{H}\_{\dagger} \mathbf{L}^{(j-a)} / \dots \mathbf{L}^{a-} / \mathbf{H}^+, \quad \text{where } j \in \{0, 1, \dots, a, \dots, n\} \tag{1}$$

HnL(n-a) is the polyprotic acid (weak acid in general), La� is the base of the system, and the neutral species is HaL; H<sup>þ</sup> is the exchanged particle in the reaction, n is the number of protons of the polyprotic acid, a� is the charge of the base (expressed in elementary charge units).

The species that go from Hðn�1ÞLðn�a<sup>Þ</sup> up to HL(a-1) are the system's formal ampholytes.

The global formation equilibria of the species of a polydonor system are represented according to Eq. (2).

$$\mathbf{L}^{a-} + j\mathbf{H}^{+} \rightleftharpoons \mathbf{H}\_{j}\mathbf{L}^{(j+a)} \quad \text{with} \quad \beta\_{j} = \frac{[\mathbf{H}\_{j}\mathbf{L}^{(j+a)}]}{[\mathbf{L}^{a-}][\mathbf{H}^{+}]^{\dagger}}.\tag{2}$$
 
$$\text{where} \begin{split} j \in \{0, 1, ..., a, ..., n\} \end{split} \tag{2}$$

By definition β<sup>o</sup> ¼ 1.

When there is a mixture of c polydonor systems in aqueous solution with (c þ 1) components, a general representation of the set of polyprotic systems is given as:

$$\mathbf{H}\_{n\_k} \text{ (Lk)}^{(n\_k - a\_k)} / \dots / \mathbf{H}\_{j\_k} \text{ (Lk)}^{(j\_k - a\_k)} / \dots / (\mathbf{L}k)^{a\_k -} / \mathbf{H}^+ \tag{3}$$

where k ∈ {1, 2,…, c}, j <sup>k</sup>∈{0, 1,.., ak,…, nk}. Hnk ðLkÞ <sup>ð</sup>nk�ak<sup>Þ</sup> is the polyprotic acid of the <sup>k</sup>th polydonor system, <sup>ð</sup>Lk<sup>Þ</sup> ak� is its polybase, nk is the number of protons of the kth polyprotic acid, and ak is the charge of the kth polybase. The species that go from Hðnk�1<sup>Þ</sup> ðLkÞ <sup>ð</sup>nk�ak�1<sup>Þ</sup> up to H <sup>ð</sup>Lk<sup>Þ</sup> <sup>ð</sup>1�ak<sup>Þ</sup> are the system's ampholytes.

A representation of the kth polydonor system's global formation equilibria in a mixture is given as:

$$\mathbf{H}(\mathbf{L}k)^{a\_k-} + j\_k \mathbf{H}^+ \rightleftharpoons \mathbf{H}\_{\dot{\boldsymbol{\mu}}\_k}(\mathbf{L}k)^{(\dot{\boldsymbol{\mu}}\_k - a\_k)} \quad \text{with} \quad \boldsymbol{\beta}\_{\dot{\boldsymbol{\mu}}\_k} = \frac{[\mathbf{H}\_{\dot{\boldsymbol{\mu}}\_k}(\mathbf{L}k)^{(\dot{\boldsymbol{\mu}}\_k - a\_k)}]}{[(\mathbf{L}k)^{a\_k-}][\mathbf{H}^+]^{\dot{\boldsymbol{\mu}}\_k}} \tag{4}$$

where k ∈ {1, 2,…, c}, j <sup>k</sup> ∈ {0, 1,.., ak,…, nk}.

Also, in this case β<sup>0</sup><sup>k</sup> � 1 needs to be considered.

It can be demonstrated that the molar fraction to describe each of the c distributions of the species of each of the polydonor systems in the mixture with respect to Hþ is given by Eq. (5):

$$f\_{j\_k} = \frac{[\mathbf{H}\_{j\_k}(\mathbf{L}k)^{(j\_k - a\_k)}]}{[\mathbf{L}k]\_\mathbf{T}} = \frac{\beta\_{j\_k}[\mathbf{H}^+]^{j\_k}}{\sum\_{j\_k=0}^{n\_k} \beta\_{j\_k}[\mathbf{H}^+]^{j\_k}} \tag{5}$$

where k ∈ {1, 2,…, c}, j <sup>k</sup> ∈ {0, 1,.., ak,…, nk}.

where [Lk]T is the total concentration of the kth component in the mixture. As can be observed, the molar fractions only depend on pH and on the equilibrium constants β<sup>j</sup> k .

#### 2.2. Description of the mixture to be titrated

There are N solutions, each containing one H<sup>j</sup> <sup>k</sup> ðLkÞ ðj <sup>k</sup>�ak<sup>Þ</sup> species in a Cojk molar concentration, assuming a volume Vojk is taken from each solution to form only one mixture with an overall volume <sup>V</sup>o, then: N <sup>¼</sup> <sup>X</sup><sup>c</sup> k¼1 Xnk j <sup>k</sup>¼0 ð1Þ n o <sup>¼</sup> <sup>X</sup><sup>c</sup> <sup>k</sup>¼<sup>1</sup> f g nk <sup>þ</sup> <sup>1</sup> and Vo <sup>¼</sup> <sup>X</sup><sup>c</sup> k¼1 Xnk j <sup>k</sup>¼<sup>0</sup> Vojk n o. This mixture is titrated with a strong MOH base at Cb concentration or with a strong MX acid at Ca concentration, measuring the pH.

Each species has associated countercations or counteranions (Mzj <sup>k</sup> <sup>þ</sup> or Zzj <sup>k</sup> �) depending on whether (jk-ak) they are negative or positive, which lack the acid-base properties.

#### 2.3. Expressions for the titration plots of polydonor systems mixtures

Although Asuero and Michałowski [6] and De Levie [9] have presented some mathematical representations of added volume as a function of pH for these systems, we have preferred to follow the same procedure and notation used to deduce the added volume equations, proposed by Rojas-Hernández et al. [7]. Then, for the case of the mixtures of species of various polydonor systems, the following expressions can be deduced:

The added volume expression for a strong base, Vb, is

Hnk ðLkÞ

given as:

species that go from Hðnk�1<sup>Þ</sup> ðLkÞ

6 Advances in Titration Techniques

where k ∈ {1, 2,…, c}, j

where k ∈ {1, 2,…, c}, j

volume <sup>V</sup>o, then: N <sup>¼</sup> <sup>X</sup><sup>c</sup>

ðLkÞ

ak� <sup>þ</sup> <sup>j</sup>

Also, in this case β<sup>0</sup><sup>k</sup> � 1 needs to be considered.

2.2. Description of the mixture to be titrated

There are N solutions, each containing one H<sup>j</sup>

at Ca concentration, measuring the pH.

k¼1

Xnk j <sup>k</sup>¼0 ð1Þ n o

Each species has associated countercations or counteranions (Mzj

polydonor systems, the following expressions can be deduced:

2.3. Expressions for the titration plots of polydonor systems mixtures

whether (jk-ak) they are negative or positive, which lack the acid-base properties.

<sup>k</sup>H<sup>þ</sup> ⇄ H<sup>j</sup>

<sup>k</sup> ∈ {0, 1,.., ak,…, nk}.

f j k <sup>¼</sup> <sup>½</sup>H<sup>j</sup>

<sup>k</sup> ∈ {0, 1,.., ak,…, nk}.

the molar fractions only depend on pH and on the equilibrium constants β<sup>j</sup>

k ðLkÞ ðj

<sup>ð</sup>nk�ak<sup>Þ</sup> is the polyprotic acid of the <sup>k</sup>th polydonor system, <sup>ð</sup>Lk<sup>Þ</sup>

<sup>ð</sup>nk�ak�1<sup>Þ</sup> up to H <sup>ð</sup>Lk<sup>Þ</sup>

the number of protons of the kth polyprotic acid, and ak is the charge of the kth polybase. The

A representation of the kth polydonor system's global formation equilibria in a mixture is

It can be demonstrated that the molar fraction to describe each of the c distributions of the species of each of the polydonor systems in the mixture with respect to Hþ is given by Eq. (5):

where [Lk]T is the total concentration of the kth component in the mixture. As can be observed,

<sup>k</sup> ðLkÞ ðj

assuming a volume Vojk is taken from each solution to form only one mixture with an overall

<sup>¼</sup> <sup>X</sup><sup>c</sup>

This mixture is titrated with a strong MOH base at Cb concentration or with a strong MX acid

Although Asuero and Michałowski [6] and De Levie [9] have presented some mathematical representations of added volume as a function of pH for these systems, we have preferred to follow the same procedure and notation used to deduce the added volume equations, proposed by Rojas-Hernández et al. [7]. Then, for the case of the mixtures of species of various

<sup>k</sup> ðLkÞ ðj <sup>k</sup>�ak Þ �

> ½Lk� T

<sup>k</sup>�ak<sup>Þ</sup> with <sup>β</sup><sup>j</sup>

<sup>¼</sup> <sup>β</sup><sup>j</sup> k ½Hþ� j k

Xnk j <sup>k</sup>¼0 βj k ½Hþ� j k

k <sup>¼</sup> <sup>½</sup>H<sup>j</sup> k ðLkÞ ðj <sup>k</sup>�akÞ �

½ðLkÞ

ak� is its polybase, nk is

ð4Þ

ð5Þ

.

<sup>ð</sup>1�ak<sup>Þ</sup> are the system's ampholytes.

ak��½Hþ� j k

> k .

<sup>k</sup>�ak<sup>Þ</sup> species in a Cojk molar concentration,

<sup>k</sup> <sup>þ</sup> or Zzj

k¼1

Xnk j <sup>k</sup>¼<sup>0</sup> Vojk n o

<sup>k</sup> �) depending on

<sup>k</sup>¼<sup>1</sup> f g nk <sup>þ</sup> <sup>1</sup> and Vo <sup>¼</sup> <sup>X</sup><sup>c</sup>

$$V\_{b} = \frac{\sum\_{k=1}^{c} \left\{ \sum\_{j\_{k}=0}^{n\_{k}} \{ (j\_{k} - a\_{k})(V\_{\eta|\_{k}} \mathbb{C}\_{\eta|\_{k}}) \} - \left[ \sum\_{j\_{k}=0}^{n\_{k}} (V\_{\eta|\_{k}} \mathbb{C}\_{\eta|\_{k}}) \right] \left[ \sum\_{j\_{k}=0}^{n\_{k}} \{ (j\_{k} - a\_{k})f\_{j\_{k}} \} \right] \right\} - V\_{o} ([\text{H}^{+}] - [\text{OH}^{-}])} \tag{6}$$

If one considers now that a strong acid is added to the mixture, then the Va expression becomes:

$$V\_d = \frac{-\sum\_{k=1}^{c} \left\{ \sum\_{j\_k=0}^{m\_k} \{ (j\_k - a\_k)(V\_{o\_k^\dagger} \mathbb{C}\_{o\_k^\dagger}) \} + \left[ \sum\_{j\_k=0}^{m\_k} (V\_{o\_k^\dagger} \mathbb{C}\_{o\_k^\dagger}) \right] \left[ \sum\_{j\_k=0}^{n\_k} \{ (j\_k - a\_k)f\_{j\_k} \} \right] \right\} + V\_o \left( [\mathbf{H}^+] - [\mathbf{OH}^-] \right)}{\mathbf{C}\_a - [\mathbf{H}^+] + [\mathbf{OH}^-]} \tag{7}$$

Eqs. (6) and (7), <sup>½</sup>OH�� ¼ Kw <sup>½</sup>Hþ� agree with the water self-protolysis equilibrium.

As can be observed, the added volume equations obtained for a strong base or a strong acid bear the same mathematical form. It is relevant to note which comes from the component's balance of each polyprotic system must be independently added, thus giving rise to the double summations appearing in Eqs. (6) and (7).

When c ¼ 1 hence giving k ¼ 1, the equations are the same as those shown in reference [7], to determine the volume that is added to a strong base and a strong acid (Vb and Va, respectively) in a system formed by species of the same polydonor system.

Eqs. (6) and (7) are exact analytic solutions to obtain titration plots pH ¼ f(V) (estimating the volume from the pH values). These equations also allow obtaining exact equations of dpH/dV, hence the expressions for βdil will be shown in the next section. βdil is the first index used to explore quantitatively the application conditions for a buffer.

## 2.4. General expressions of dpH/dVb and �dpH/dVa

Eqs. (6) and (7) are functions of the pH, thus it becomes possible to obtain analytic expressions for their first derivatives (dV/dpH). With the reciprocals of the first derivatives, exact algebraic expressions of the first derivative of the titration plot are obtained. This is to say dpH/dVb and �dpH/dVa, which are used to detect the volumes at the titration points when the reactions are quantitative.

Extending the expressions for dpH/dVb and �dpH/dVa considering a mixture of the species of various polydonor systems, the expressions obtained are as follows:

$$\frac{d\mathbf{pH}}{dV\_b} = \frac{\mathbf{C}\_b + 10^{-\text{pH}} - 10^{\text{pH}-\text{p}K\_v}}{\sum\_{k=1}^c \left( \left[ \sum\_{j\_k=0}^{n\_k} (V\_{oj\_k} \mathbf{C}\_{oj\_k}) \right] \left[ \sum\_{j\_k=0}^{n\_k} \left\{ j\_k f\_{j\_k} \sum\_{i\_k=0}^{n\_k} \left[ (i\_k - j\_k) f\_{i\_k} \right] \right\} \right] \right)} \tag{8}$$

$$-\frac{d\text{pH}}{dV\_{a}} = \frac{\mathsf{C}\_{a} - 10^{-\text{pH}} + 10^{\text{pH}-pK\_{w}}}{\ \_{-2.303}\sum\_{k=1}^{c} \left( \left[ \sum\_{j\_{k}=0}^{n\_{k}} (V\_{oj\_{k}} \mathsf{C}\_{oj\_{k}}) \right] \left[ \sum\_{j\_{k}=0}^{n\_{k}} \left\{ j\_{k} f\_{j\_{k}} \sum\_{i=0}^{n\_{k}} \left[ (i\_{k} - j\_{k}) f\_{i\_{k}} \right] \right\} \right] \right)} \tag{9}$$

where k ∈ {1, 2, …, c}, ik ∈ {0, 1, …, nk} and jk ∈ {0, 1, …, ak, …, nk}.

Equally, if c ¼ 1 and k ¼ 1, Eqs. (8) and (9) are the same as those shown in Ref. [7] to determine dpH/dVb and �dpH/dVa for a mixture of the species of only one polydonor system.

#### 2.5. General expressions of βdil

In order to determine the buffer capacity considering the dilution, βdil, the derivative is applied to the quantity of strong base or strong acid added as follows:

$$
\beta\_{\rm dil\_b} = \frac{dV\_b \mathbb{C}\_b}{d\mathbf{p} \mathbf{H}} = \mathbb{C}\_b \frac{dV\_b}{d\mathbf{p} \mathbf{H}} \quad \text{or} \quad \beta\_{\rm dil\_s} = -\frac{dV\_a \mathbb{C}\_a}{d\mathbf{p} \mathbf{H}} = -\mathbb{C}\_a \frac{dV\_a}{d\mathbf{p} \mathbf{H}} \tag{10}
$$

where βdil<sup>b</sup> and βdil<sup>a</sup> are units of quantity of substance. The analytic mathematical expressions are shown in Eqs. (11) and (12).

$$2.303\text{C}\_{b}\left\{-\sum\_{k=1}^{c}\left(\left[\sum\_{j=0}^{n\_{k}}(V\_{o|\_{k}}\mathbf{C}\_{o|\_{k}})\right]\left[\sum\_{j\_{c}=0}^{n\_{k}}\left\{j\_{b}f\_{j\_{b}}\sum\_{i=0}^{n\_{k}}[(i\_{k}-j\_{b})f\_{i}]\right\}\right]\right)\right\}\right.\tag{11}$$

$$\beta\_{\text{dil}\_{b}}=\frac{dV\_{b}\mathbf{C}\_{b}}{d\text{pH}}=\frac{(V\_{o}+V\_{b})[10^{-\text{pH}}+10^{\text{pH}-pK\_{v}}]}{\mathbf{C}\_{b}+10^{-\text{pH}}-10^{\text{pH}-pK\_{v}}}\tag{12}$$

$$\beta\_{\rm dil} = -\frac{dV\_a \mathbf{C}\_a}{d\mathbf{pH}} = \frac{\left\{-\sum\_{k=1}^c \left(\left[\sum\_{j=0}^{n\_k} (V\_{\phi\_k^\*} \mathbf{C}\_{\phi\_k^\*})\right] \left[\sum\_{j=0}^{n\_k} \left\{j\mathbf{J}\_j \sum\_{i=0}^{m\_k} [(i\_k - j\_k)\mathbf{f}\_{i\_i}]\right\} \right] \right) \right\}}{+ (V\_o + V\_b)[10^{-\text{pH}} + 10^{\text{pH}-pK\_v}]}\right\}}\tag{12}$$

$$\beta\_{\rm dil} = -\frac{dV\_a \mathbf{C}\_a}{d\mathbf{pH}} = \frac{-\left(\sum\_a - 10^{-\text{pH}} + 10^{\text{pH}-pK\_v}\right)}{\mathbf{C}\_a - 10^{-\text{pH}} + 10^{\text{pH}-pK\_v}}\tag{12}$$

Furthermore, it is worth noting that the different plots presented along this work were constructed from spreadsheets done through Excel 2007 in Microsoft Office using the equations heretofore presented.

#### 3. Some case studies

#### 3.1. Application of expressions for the titration curves fitting experimental data of the Britton-Robinson buffer

Figure 2 shows the pH ¼ f(Vb) and curve of dpH/dVb ¼ f(pH) retaking experimental data from Lange's Handbook of Chemistry [21] (markers • and ▲, respectively). The experimental curve

Figure 2. Titration of 100 mL of equimolar solution of Britton-Robinson 0.04 M ([AcO<sup>0</sup> ] ¼ [PO4 0 ] ¼ [BO3 0 ] ¼ CSB ¼ 0.04 M) with NaOH 0.2M.—represents the calculated curve of pH ¼ f(Vb), • denotes the experimental curve of pH ¼ f(Vb), ---- denotes the calculated curve of dpH/dVb ¼ f(Vb), and ▲ denotes the experimental curve of ΔpH/ΔVb. The pKa values are as follows: pKa ¼ 4.66 for acetic acid [23]; pKa1 ¼ 2.1 [24], pKa2 ¼ 6.75 [25], and pKa3 ¼ 11.71 [26] for phosphoric acid; and pKa ¼ 9.15 for boric acid [27]. pKw ¼ 13.73 [28, 29].

dpH/dVb was calculated as the finite differences quotient of the pH values and the volumes measured during the titration (ΔpH=ΔVb) using the average volumes for each interval. Also shown are the pH ¼ f(Vb) and dpH/dVb ¼ f(pH) curves obtained using Eqs. (6) and (7) (solid and segmented lines, respectively) [22].

## 3.2. Effect of the quantity of a buffer solution on βdil

� <sup>d</sup>pH dVa

8 Advances in Titration Techniques

2.5. General expressions of βdil

are shown in Eqs. (11) and (12).

dpH ¼

dpH ¼

<sup>β</sup>dil<sup>b</sup> <sup>¼</sup> dVbCb

<sup>β</sup>dil<sup>a</sup> ¼ � dVaCa

tions heretofore presented.

3. Some case studies

Britton-Robinson buffer

�2:303 Xc k¼1

<sup>¼</sup> Ca � <sup>10</sup>�pH <sup>þ</sup> 10pH�pKw

dpH/dVb and �dpH/dVa for a mixture of the species of only one polydonor system.

dVb

Xnk j <sup>k</sup>¼0 ðVojk Cojk Þ

> Xnk j <sup>k</sup>¼0 ðVojk Cojk Þ

2 4

0 @

2 4

0 @ 3 <sup>5</sup> <sup>X</sup>nk j <sup>k</sup>¼0

Equally, if c ¼ 1 and k ¼ 1, Eqs. (8) and (9) are the same as those shown in Ref. [7] to determine

In order to determine the buffer capacity considering the dilution, βdil, the derivative is applied

where βdil<sup>b</sup> and βdil<sup>a</sup> are units of quantity of substance. The analytic mathematical expressions

þðVo <sup>þ</sup> VbÞ½10�pH <sup>þ</sup> 10pH�pKw �

þðVo <sup>þ</sup> VbÞ½10�pH <sup>þ</sup> 10pH�pKw �

Furthermore, it is worth noting that the different plots presented along this work were constructed from spreadsheets done through Excel 2007 in Microsoft Office using the equa-

3.1. Application of expressions for the titration curves fitting experimental data of the

Figure 2 shows the pH ¼ f(Vb) and curve of dpH/dVb ¼ f(pH) retaking experimental data from Lange's Handbook of Chemistry [21] (markers • and ▲, respectively). The experimental curve

<sup>d</sup>pH or <sup>β</sup>dil<sup>a</sup> ¼ � dVaCa

3 <sup>5</sup> <sup>X</sup>nk j <sup>k</sup>¼0

4

3 <sup>5</sup> <sup>X</sup>nk j <sup>k</sup>¼0

4

4

j kf j k Xnk ik¼0

ðik � j <sup>k</sup>Þf ik 3 5

1 A ð9Þ

h i <sup>2</sup> ( )

<sup>d</sup>pH ¼ �Ca

j kf j k Xnk ik¼0

> j kf j k Xnk ik¼0

dVa

½ðik � j

Cb <sup>þ</sup> <sup>10</sup>�pH � 10pH�pKw <sup>ð</sup>11<sup>Þ</sup>

2 ( )

Ca � <sup>10</sup>�pH <sup>þ</sup> 10pH�pKw <sup>ð</sup>12<sup>Þ</sup>

½ðik � j

<sup>k</sup>Þf ik �

2 ( )

<sup>k</sup>Þf ik �

<sup>d</sup>pH <sup>ð</sup>10<sup>Þ</sup>

3 5

1 A

9 >>>>=

>>>>;

3 5

1 A

9 >>>>=

>>>>;

Xnk j <sup>k</sup>¼0 ðVojk Cojk Þ

2 4

<sup>þ</sup>2:303ðVoþVbÞ½10�pHþ10pH�pKw �

0 @

where k ∈ {1, 2, …, c}, ik ∈ {0, 1, …, nk} and jk ∈ {0, 1, …, ak, …, nk}.

to the quantity of strong base or strong acid added as follows:

<sup>d</sup>pH <sup>¼</sup> Cb

� Xc k¼1

> � Xc k¼1

8 >>>><

>>>>:

8 >>>><

>>>>:

<sup>β</sup>dil<sup>b</sup> <sup>¼</sup> dVbCb

2:303Cb

2:303Ca

Intuitively, it is known that the performance of a buffer solution is better whenever a larger volume is taken to set the pH. The β shown in the scientific literature [1–14, 20] does not consider this feature, for which it is necessary to have an index that evaluates the effect of the size of the buffer solution to impose the pH. For that purpose, the definition of βdil is used to include in its mathematic expression, the term Vo, as observed in Eqs. (11) and (12). Subsequently, it is shown how βdil takes this effect into account.

Figure 3 shows a series of βdil ¼ f(pH) plots for a 1000 and 10 mL buffer solutions containing the species H3PO4 and H2PO4 � at different concentrations of the PO4' (PO4' ¼ H3PO4/H2PO4 �/ HPO4 <sup>2</sup>�/PO4 <sup>3</sup>�/Hþ) system. In this case, these concentrations can be represented as Co BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix BS because the buffer system is a mixture, in agreement with Figure 1. It is necessary to underline that the axis βdil is log, just as Urbansky and Schok [4] do, in order to compare βdil within an ample PO4' concentration range.

Figure 3a represents a larger buffer system than that represented in Figure 3b, because the initial volumes were 1000 and 10 mL, respectively. As can be observed in Figure 3, βdil increases with

Figure 3. Calculated plots of βdil ¼ f(pH) of systems initially containing the H3PO4 and H2PO4 � at PO4' species at overall concentrations of 10�<sup>1</sup> , 10�<sup>3</sup> , 10�<sup>5</sup> and 10�<sup>6</sup> M. The broken line represents the plot of <sup>β</sup>dil(H2O) <sup>¼</sup> f(pH) for water and its basic and acid particles. Cb ¼ Ca¼0.5 M. pKa1 ¼ 2.1 [24], pKa2 ¼ 6.75 [25], and pKa3 ¼ 11.71 [26]. (a) Vo ¼ 1000 mL. (b) Vo ¼ 10 mL.

increasing quantity of the system (Vo). Hence, βdil indicates well the expected behavior for a buffer system: the pH in a system is better imposed when the buffer amount is larger.

Figure 3 also shows the plot for water, βdil(H2O) ¼ f(pH), titrated with strong base and strong acid. It sets the lower limit given by this solvent and its acid and basic particles (broken line), with respect to all aqueous solutions. Therefore, it is established that any solution, including that of the same solvent, has certain βdil. It can be observed that the concentration diminution of the PO4' system provokes that βdil diminishes and the width of the pH interval also decreases where the PO4' system contributes more to βdil than the solvent.

There is one minimum concentration of the buffer system (CBSmin), small enough, where the PO4' system almost does not contribute to βdil,(CBSmin ≈ 10�<sup>6</sup> M), so that the plot of βdil of the PO4' system can be discerned from the plot of βdil(H2O). Just as a minimum concentration is shown for the PO4' system; whenever there is an acid-base pair (HL/L) with pKa ≈ 7.0, the CBSmin will be the same (10�<sup>6</sup> M). Although in other cases, when pKa < 7.0 or pKa > 7.0, it must be expected that the <sup>C</sup>BSmin be larger; this is to say, for pKa <sup>¼</sup> 5 or pKa <sup>¼</sup> 9, the <sup>C</sup>BSmin <sup>≈</sup> <sup>10</sup>�<sup>5</sup> M and for pKa <sup>¼</sup> 3 or pKa <sup>¼</sup> 11, the <sup>C</sup>BSmin <sup>≈</sup> <sup>10</sup>�<sup>3</sup> M.

#### 3.3. Buffer efficacy (ε) of a buffer system

The previous section showed that the βdil has advantages over β in order to evaluate the buffer performance. However, the shortcomings of this situation refer to βdil, which by definition is the quantity of strong base or strong acid added to change by one unit the system's pH, which is a fairly large change. Therefore, it is necessary to define a new index having a smaller change than βdil.

Following the idea proposed by Christian in his textbook [30], it is possible to approximate the derivative by means of a finite difference quotient. Even when the pH change is acceptable for a buffer system, it depends on the application or on the system to be considered, a ΔpH ≤ 0.1 is sufficiently small to comply with the approximation established through Eq. (13).

The Conditions Needed for a Buffer to Set the pH in a System http://dx.doi.org/10.5772/intechopen.69003 11

$$
\beta\_{\rm dil} = \frac{d(VC)}{d\rm pH} \approx \frac{\Delta(VC)}{\Delta \rm pH} \tag{13}
$$

Then

increasing quantity of the system (Vo). Hence, βdil indicates well the expected behavior for a

basic and acid particles. Cb ¼ Ca¼0.5 M. pKa1 ¼ 2.1 [24], pKa2 ¼ 6.75 [25], and pKa3 ¼ 11.71 [26]. (a) Vo ¼ 1000 mL.

, 10�<sup>5</sup> and 10�<sup>6</sup> M. The broken line represents the plot of <sup>β</sup>dil(H2O) <sup>¼</sup> f(pH) for water and its

� at PO4' species at overall

Figure 3 also shows the plot for water, βdil(H2O) ¼ f(pH), titrated with strong base and strong acid. It sets the lower limit given by this solvent and its acid and basic particles (broken line), with respect to all aqueous solutions. Therefore, it is established that any solution, including that of the same solvent, has certain βdil. It can be observed that the concentration diminution of the PO4' system provokes that βdil diminishes and the width of the pH interval also

There is one minimum concentration of the buffer system (CBSmin), small enough, where the PO4' system almost does not contribute to βdil,(CBSmin ≈ 10�<sup>6</sup> M), so that the plot of βdil of the PO4' system can be discerned from the plot of βdil(H2O). Just as a minimum concentration is shown for the PO4' system; whenever there is an acid-base pair (HL/L) with pKa ≈ 7.0, the CBSmin will be the same (10�<sup>6</sup> M). Although in other cases, when pKa < 7.0 or pKa > 7.0, it must be expected that the <sup>C</sup>BSmin be larger; this is to say, for pKa <sup>¼</sup> 5 or pKa <sup>¼</sup> 9, the <sup>C</sup>BSmin <sup>≈</sup> <sup>10</sup>�<sup>5</sup> M and for pKa

The previous section showed that the βdil has advantages over β in order to evaluate the buffer performance. However, the shortcomings of this situation refer to βdil, which by definition is the quantity of strong base or strong acid added to change by one unit the system's pH, which is a fairly large change. Therefore, it is necessary to define a new index having a smaller change

Following the idea proposed by Christian in his textbook [30], it is possible to approximate the derivative by means of a finite difference quotient. Even when the pH change is acceptable for a buffer system, it depends on the application or on the system to be considered, a ΔpH ≤ 0.1 is

sufficiently small to comply with the approximation established through Eq. (13).

buffer system: the pH in a system is better imposed when the buffer amount is larger.

decreases where the PO4' system contributes more to βdil than the solvent.

Figure 3. Calculated plots of βdil ¼ f(pH) of systems initially containing the H3PO4 and H2PO4

<sup>¼</sup> 3 or pKa <sup>¼</sup> 11, the <sup>C</sup>BSmin <sup>≈</sup> <sup>10</sup>�<sup>3</sup> M.

than βdil.

concentrations of 10�<sup>1</sup>

10 Advances in Titration Techniques

(b) Vo ¼ 10 mL.

, 10�<sup>3</sup>

3.3. Buffer efficacy (ε) of a buffer system

$$
\Delta(VC) \approx \beta\_{\text{dil}} \Delta \text{pH} \tag{14}
$$

In this work, the buffer performance will be assessed considering a ΔpH ¼ 0.1 [30]. The buffer efficacy, ε, is defined as the quantity of strong base or strong acid that provokes a pH change of only one-tenth in a system. The expression of ε is as follows:

$$
\varepsilon \equiv \Delta (VC) \approx \beta\_{\text{dil}} \Delta \text{pH} \;= 0.1 \beta\_{\text{dil}} \tag{15}
$$

#### 3.4. Application of ε: buffer system's concentration threshold

A buffer system is used to set the pH, therefore, it is necessary to know its useful conditions to fulfill its function. Then, for the sake of a deeper understanding it is relevant to establish first a limit to determine the moment in which the buffer system's concentration sets pH conditions over those of the water and of its acid and basic particles, just as those of the system of interest.

#### 3.4.1. Imposing the pH of the buffer system over the water and its acid and basic particles

Because the system's pH needs to be imposed, the efficacy of the system's buffer, the buffer ðεBSÞ should be larger or at least equal to the efficacy of the buffer amplified ten times that of the water, ε<sup>10</sup>ðH2O<sup>Þ</sup> ¼ 10εðH2OÞ, not just the buffer efficacy of the water, εðH2OÞs. In the present work, a factor of 10 is considered sufficiently large to assess a buffer's performance.

Figures 4a and b show the curves of εðH2O<sup>Þ</sup> for Vo ¼ 1000 mL and Vo ¼ 10 mL of water, respectively, titrated with a strong acid and a strong base at 1 M concentration (marker…). Also, it is shown the plot of ε<sup>10</sup>ðH2O<sup>Þ</sup> (marker…), which will be considered as the limit where the buffer concentration is useful to set the pH. It is also observed in Figures 4a and b that both εðH2O<sup>Þ</sup> and ε<sup>10</sup>ðH2O<sup>Þ</sup> depend, as expected, on the quantity of the system.

Figure 4c and d shows, apart from the εðH2O<sup>Þ</sup> and ε<sup>10</sup>ðH2O<sup>Þ</sup> plots, those of εBS of the NH4 <sup>þ</sup>/NH3 buffer solutions at different concentrations for systems with VBS ¼ Vo ¼ 1000 mL and VBS ¼ Vo ¼ 10 mL, respectively, as those shown in Figure 1. In order to establish the limit in which a buffer works to set the pH, it is necessary to compare the εBS curve with that of the ε<sup>10</sup>ðH2OÞ, so that compliance with εBS ≥ ε<sup>10</sup>ðH2O<sup>Þ</sup> can be verified.

In this case, the lowest NH4 <sup>þ</sup>/NH3 buffer concentration falls to a point that almost equals the ε<sup>10</sup>ðH2O<sup>Þ</sup> plot. Therefore, this concentration is termed as threshold buffer concentration (TCBS) with a value of TCBS ≈ 10�<sup>4</sup> M. It can also be seen from Figures 4c and d that TCBS does not depend on the size of the system (Vo).

Then, whenever there is a buffer formed by an acid-base pair HL/L with pKa ≈ 7.0 it must be expected a greater TCBS, of approximately 10�<sup>5</sup> M; when pKa <sup>¼</sup> 5 or pKa <sup>¼</sup> 9, the TCBS will also be approximately 10�<sup>4</sup> M; and for pKa <sup>¼</sup> 3.0 or pKa <sup>¼</sup> 11.0, the TCBS <sup>≈</sup> <sup>10</sup>�<sup>2</sup> M.

Figure 4. Calculated plots of εðH2O<sup>Þ</sup> and ε<sup>10</sup>ðH2O<sup>Þ</sup> for Vo ¼ 1000 mL and Vo ¼ 10 mL of water and its acid and basic particles (marker… and marker… , respectively). The εBS curves of NH4 <sup>þ</sup>/NH3 buffer solutions at different concentrations, [NH3']Tot ¼ Co BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix BS : —10�<sup>1</sup> M, ———10�<sup>3</sup> M, —10�<sup>5</sup> M. Cb <sup>¼</sup> Ca <sup>¼</sup> 1 M. pKa <sup>¼</sup> 9.25 [31]. (a) and (c) 1000 mL. (b) and (d) 10 mL.

Furthermore, in Figures 4c and d it can also be noted that the useful interval for the BS to set the pH, when the Co BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix BS <sup>¼</sup> <sup>10</sup>�<sup>1</sup> M, is pKa – 1 < pH < pKa <sup>þ</sup> 1 [32]. Whereas for Co BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix BS <sup>¼</sup> <sup>10</sup>�<sup>3</sup> M, the interval to set the pH is smaller because the <sup>ε</sup>BS becomes closer to ε<sup>10</sup>ðH2OÞ.

When the εBS < ε<sup>10</sup>ðH2O<sup>Þ</sup> strong acid or strong base must be used also at adequate concentration to set the pH. The strong base and strong acid in these extremes are used because the acid and basic particles of the solvent contribute more to ε than the buffer system components.

Finally, from the analysis of the plots in Figure 4, it can be determined that the pH limits of the buffer performance and the buffer system threshold concentration do not depend on the system's size.

#### 3.4.2. Setting the buffer's system pH over that of the original system

It is necessary to set new limits (of pH and buffer concentration) whenever there is a mixture of the system of interest and the buffer system, because it is not sufficient to consider only the water effect.

The pH of a given system (OS) is set upon adding a buffer system (BS), hence, the buffer component concentration in the mixture (Cmix BS ) must be greater than the concentration of the solutes of the original system (Cmix OS ), because both systems have their own ε, though how large is it? (Figure 1).

An example of the use of ε, to evaluate setting the pH to a 9.0 value at 100 mL (VOS ) of an acetylacetone solution (acac') (OS), is given next, at 10�<sup>3</sup> M concentration (C<sup>o</sup> OS1), at different NH4 <sup>þ</sup>/NH3 buffer concentrations in the mixture (Cmix BS ).

Figure 5a shows the ε of the acetylacetone solution (OS), stated in the previous paragraph (εOS, marker �����), the buffer efficacy amplified 10 times that of the original system (ε10OS, marker ----), also presenting the plot of εðH2O<sup>Þ</sup> (marker…).

Figure 5b shows, apart from the εðH2O<sup>Þ</sup> curve, those corresponding to εBS for 100 mL of the NH4 <sup>þ</sup>/NH3 system solution at different concentrations. The εBS magnitude depends on buffer's concentration and decreases until it reaches a TCBS, which is given when the εBS curve almost becomes equal to that of <sup>ε</sup><sup>10</sup>ðH2OÞ. In this case, TCBS <sup>≈</sup> <sup>10</sup>�<sup>4</sup> M.

Furthermore, in Figures 4c and d it can also be noted that the useful interval for the BS to set

Figure 4. Calculated plots of εðH2O<sup>Þ</sup> and ε<sup>10</sup>ðH2O<sup>Þ</sup> for Vo ¼ 1000 mL and Vo ¼ 10 mL of water and its acid and basic particles

When the εBS < ε<sup>10</sup>ðH2O<sup>Þ</sup> strong acid or strong base must be used also at adequate concentration to set the pH. The strong base and strong acid in these extremes are used because the acid and

Finally, from the analysis of the plots in Figure 4, it can be determined that the pH limits of the buffer performance and the buffer system threshold concentration do not depend on the

It is necessary to set new limits (of pH and buffer concentration) whenever there is a mixture of the system of interest and the buffer system, because it is not sufficient to consider only the

The pH of a given system (OS) is set upon adding a buffer system (BS), hence, the buffer

basic particles of the solvent contribute more to ε than the buffer system components.

BS <sup>¼</sup> <sup>10</sup>�<sup>3</sup> M, the interval to set the pH is smaller because the <sup>ε</sup>BS becomes

BS : —10�<sup>1</sup> M, ———10�<sup>3</sup> M, —10�<sup>5</sup> M. Cb <sup>¼</sup> Ca <sup>¼</sup> 1 M. pKa <sup>¼</sup> 9.25 [31]. (a) and (c) 1000 mL. (b) and (d) 10 mL.

BS <sup>¼</sup> <sup>10</sup>�<sup>1</sup> M, is pKa – 1 < pH < pKa <sup>þ</sup> 1 [32]. Whereas for

<sup>þ</sup>/NH3 buffer solutions at different concentrations, [NH3']Tot ¼

BS ) must be greater than the concentration of the

the pH, when the Co

12 Advances in Titration Techniques

BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix

closer to ε<sup>10</sup>ðH2OÞ.

BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix

system's size.

water effect.

Co

Co

BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix

(marker… and marker… , respectively). The εBS curves of NH4

3.4.2. Setting the buffer's system pH over that of the original system

component concentration in the mixture (Cmix

Figure 5. Curves of ε ¼ f(pH) for 100 mL solutions related to Figure 1. The line marked with ….in all cases represents the buffer efficacy of water, <sup>ε</sup>ðH2OÞ, and its acid and basic particles. (a) Curves <sup>ε</sup>OS and <sup>ε</sup>10OS belong to an acac 10�<sup>3</sup> <sup>M</sup> ([acac']Tot <sup>¼</sup> <sup>C</sup><sup>o</sup> OS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix OS ) solution in the absence of the buffer system (markers ����� and ----, respectively). (b) Curves εBS belong to the NH4 <sup>þ</sup>/NH3 buffer at different concentrations, [NH3']Tot <sup>¼</sup> <sup>C</sup><sup>o</sup> BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix BS : —10�<sup>2</sup> M, ———10�<sup>3</sup> M, —10�<sup>4</sup> M. (c) Curves εmix, εOS, and ε10OS belong to the solutions containing acac 10�<sup>3</sup> M plus NH4 <sup>þ</sup>/NH3 10�<sup>2</sup> M buffer, 10�<sup>3</sup> M, and 10�<sup>4</sup> M in the mixture. 1: εOS, 2:Cmix OS <sup>¼</sup>10-3 <sup>M</sup> <sup>þ</sup>Cmix BS <sup>¼</sup> <sup>10</sup>�<sup>4</sup> M, 3: <sup>C</sup>mix OS <sup>¼</sup> <sup>10</sup>�<sup>3</sup> <sup>M</sup> <sup>þ</sup>Cmix BS <sup>¼</sup> <sup>10</sup>�<sup>3</sup> M, 4:ε10OS, and 5: Cmix OS <sup>¼</sup>10-3 <sup>M</sup> <sup>þ</sup> <sup>C</sup>mix BS <sup>¼</sup> <sup>10</sup>�<sup>2</sup> M. pKa <sup>¼</sup> 9.0 for the system Hacac/acac� and pKa <sup>¼</sup> 9.25 for the NH4 <sup>þ</sup>/NH3 system [31]. Cb ¼ Ca ¼ 0.2 M.

In order to impose the pH of the original system, the buffer efficacy of the mixture (εmix) should be greater or, at least equal to ε10OS. Now, if the buffer is added to the original system to set the pH, the εmix has contributions of the original system and to the buffer system; thus, the buffer does not always set the pH in the system as shown in Figure 5c. The curve 1 that represents a εOS is practically identical to the curve 2 that corresponds to a mixture of the original system, with the buffer solution with concentration 10 times smaller than the solutes in the original system: in these cases, the pH of the system depends only on the original system, because εmix ≈ εOS. The curve 3 shows that the buffer with 10�<sup>3</sup> M concentration does already contribute to εmix apart from the original system, but has not set the pH yet because εOS < εmix < ε10OS.

Finally, Figure 5c depicts the curve 4 as corresponding to ε10OS, whereas the curve 5 represents a εmix corresponding to the buffer threshold concentration for the buffer system component of the mixture, TCmix BS (whenever it is required to set the pH at a value of 9.0), for which εmix ≈ εBS ¼ ε10OS, and therefore, there is an adequate buffer performance to set the pH of the system within the 8.25 < pH < 10.0 interval. It must be noted that if it is required to set pH > 10.0 values, a strong base must be used, apart from the buffer system. To the extent that εmix >> ε10OS the buffer performance to set the system pH becomes better. Approximately TCmix BS must be 10 times larger than Cmix OS . Therefore, it is established that

$$\mathbf{C}\_{\rm BS}^{\rm mix} \ge T \mathbf{C}\_{\rm BS}^{\rm mix} \approx 10 \mathbf{C}\_{\rm OS}^{\rm mix} \tag{16}$$

Eq. (16) becomes specific whenever the original system is set to a pH ¼ 9.0 when this system bears one acid-base pair with pKa ¼ 9.0 (Hacac/acac�) [31]. This example is the most difficult case because this OS acid-base pair competes almost equally with the BS acid-base pair (NH4 þ/ NH3, pKa ¼ 9.25) to set the system's pH.

If the pKa of some species in the original system moves away from the pH value that is desired to impose with the buffer system, the factor of 10 in Eq. (16) becomes smaller.

It must not be forgotten that the buffer system should have a conjugated acid-base pair with a pKa value close to the pH that is desired to impose. As can be observed from Figure 5c, in this case it is difficult that the buffer system imposes the pH to a 9.0 value because both systems (BS and OS) have pKa values similar to that pH value. It is worth clarifying that the NH4 <sup>þ</sup>/NH3 buffer imposes more easily the pH to a 9.0 value to the extent that the pKa of the acid-base pair original system drifts apart from pH <sup>¼</sup> 9.0; of the TCmix BS diminishes till it reaches a limit value given by ε<sup>10</sup>ðH2OÞ.

Consider now the case that a pH 9.0 shall be imposed to the 100 mL (VOS) of the acetylacetone solution (acac') (OS), with a 10�<sup>3</sup> M (Co OS) concentration, using now the Britton-Robinson [12, 21] buffer at different concentrations. For this example, the εOS, ε10OS, and εðH2O<sup>Þ</sup> are the same as those presented in Figure 5a.

Figure 6a shows, apart from the curve of εðH2OÞ, the curves of εBS for 100 mL solution of the Britton-Robinson buffer at different concentrations. In this case, the C<sup>o</sup> BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix BS diminishes till reaching a TCBS, having a value of TCBS ≈ 10�<sup>5</sup> M because at this point the εBS almost equals the ε<sup>10</sup>ðH2O<sup>Þ</sup> curve.

In order to impose the pH of the original system, the buffer efficacy of the mixture (εmix) should be greater or, at least equal to ε10OS. Now, if the buffer is added to the original system to set the pH, the εmix has contributions of the original system and to the buffer system; thus, the buffer does not always set the pH in the system as shown in Figure 5c. The curve 1 that represents a εOS is practically identical to the curve 2 that corresponds to a mixture of the original system, with the buffer solution with concentration 10 times smaller than the solutes in the original system: in these cases, the pH of the system depends only on the original system, because εmix ≈ εOS. The curve 3 shows that the buffer with 10�<sup>3</sup> M concentration does already contribute to εmix apart from the original system, but has not set the pH yet because εOS < εmix < ε10OS.

Finally, Figure 5c depicts the curve 4 as corresponding to ε10OS, whereas the curve 5 represents a εmix corresponding to the buffer threshold concentration for the buffer system component of the

ε10OS, and therefore, there is an adequate buffer performance to set the pH of the system within the 8.25 < pH < 10.0 interval. It must be noted that if it is required to set pH > 10.0 values, a strong base must be used, apart from the buffer system. To the extent that εmix >> ε10OS the

Eq. (16) becomes specific whenever the original system is set to a pH ¼ 9.0 when this system bears one acid-base pair with pKa ¼ 9.0 (Hacac/acac�) [31]. This example is the most difficult case because this OS acid-base pair competes almost equally with the BS acid-base pair (NH4

If the pKa of some species in the original system moves away from the pH value that is desired

It must not be forgotten that the buffer system should have a conjugated acid-base pair with a pKa value close to the pH that is desired to impose. As can be observed from Figure 5c, in this case it is difficult that the buffer system imposes the pH to a 9.0 value because both systems (BS and OS) have pKa values similar to that pH value. It is worth clarifying that the NH4

buffer imposes more easily the pH to a 9.0 value to the extent that the pKa of the acid-base pair

Consider now the case that a pH 9.0 shall be imposed to the 100 mL (VOS) of the acetylacetone

son [12, 21] buffer at different concentrations. For this example, the εOS, ε10OS, and εðH2O<sup>Þ</sup> are

Figure 6a shows, apart from the curve of εðH2OÞ, the curves of εBS for 100 mL solution of the

ishes till reaching a TCBS, having a value of TCBS ≈ 10�<sup>5</sup> M because at this point the εBS almost

Britton-Robinson buffer at different concentrations. In this case, the C<sup>o</sup>

buffer performance to set the system pH becomes better. Approximately TCmix

OS . Therefore, it is established that

Cmix BS ≥ TCmix

to impose with the buffer system, the factor of 10 in Eq. (16) becomes smaller.

BS (whenever it is required to set the pH at a value of 9.0), for which εmix ≈ εBS ¼

BS ≈ 10Cmix

BS must be 10

þ/

<sup>þ</sup>/NH3

BS dimin-

OS ð16Þ

BS diminishes till it reaches a limit value

BS <sup>¼</sup> <sup>C</sup>mix <sup>¼</sup> <sup>C</sup>mix

OS) concentration, using now the Britton-Robin-

mixture, TCmix

times larger than Cmix

14 Advances in Titration Techniques

given by ε<sup>10</sup>ðH2OÞ.

equals the ε<sup>10</sup>ðH2O<sup>Þ</sup> curve.

NH3, pKa ¼ 9.25) to set the system's pH.

original system drifts apart from pH <sup>¼</sup> 9.0; of the TCmix

solution (acac') (OS), with a 10�<sup>3</sup> M (Co

the same as those presented in Figure 5a.

Figure 6. The curves ε¼ f(pH) are for the 100 mL of the solutions related to Figure 1. The broken line in all cases represents the buffer efficacy of the water, εðH2OÞ, and its acid and basic particles. (a) Curves εBS for equimolar Britton-Robinson buffer solutions at different concentrations, [AcO']Tot <sup>¼</sup> [PO4']Tot <sup>¼</sup> [BO3']Tot <sup>¼</sup> <sup>C</sup><sup>o</sup> BS¼Cmix BS : ―10�<sup>1</sup> M, ——— 10�<sup>3</sup> M, ―10�<sup>5</sup> M. (b) Curves εmix, εOS, and ε10OS of solutions that contain 10�<sup>3</sup> M acac plus 10�<sup>2</sup> M, 10�<sup>3</sup> M, and 10�<sup>5</sup> M Britton-Robinson's buffer in the mixture. 1: <sup>ε</sup>OS, 2: [acac'] <sup>¼</sup> <sup>C</sup>mix OS <sup>¼</sup> <sup>10</sup>�<sup>3</sup> <sup>M</sup> <sup>þ</sup> <sup>C</sup>mix BS <sup>¼</sup> <sup>10</sup>�<sup>5</sup> M, 3: [acac'] <sup>¼</sup> <sup>C</sup>mix OS <sup>¼</sup> <sup>10</sup>�<sup>3</sup> <sup>M</sup> <sup>þ</sup> Cmix BS <sup>¼</sup>10�<sup>3</sup> M, 4: <sup>ε</sup>10OS, and 5: [acac'] <sup>¼</sup> <sup>C</sup>mix OS <sup>¼</sup> <sup>10</sup>�<sup>3</sup> <sup>M</sup> <sup>þ</sup> <sup>C</sup>mix BS <sup>¼</sup> <sup>10</sup>�<sup>2</sup> M. The pKa values used in the model are as follows: pKa ¼ 9.0 for acac' [31]; pKa ¼ 4.66 for acetic acid [23]; pKa1 ¼ 2.1 [24], pKa2 ¼ 6.75 [25], and pKa3 ¼ 11.71 [26] for phosphoric acid; and pKa ¼ 9.15 for boric acid [27]. pKw ¼ 13.73 [28, 29]. Cb ¼ Ca ¼ 0.2 M.

As stated, the εmix should be larger or at least equal to ε10OS. Once again, the buffer efficacy of the mixture (εmix) is contributed from both the acac' system and the Britton-Robinson buffer; thus the buffer does not always set the pH in the system as shown in Figure 6b. Curve 1 represents the εOS, that is practically equal to curve 2 corresponding to a mixture of the original system with the buffer solution with a concentration 100 times smaller than the solutes in the original system: in these cases, the system's pH depends only on the original system, because εmix ≈ εOS. Curve 3 shows that in the 4.5 < pH < 7.4 interval, the buffer with 10�<sup>3</sup> M concentration contributes more to the εmix than the original system, consequently, the buffer has the system's capacity to set the pH in this interval but not at pH ¼ 9.0 as required in this example. Curve 3 also has another region where both systems contribute almost the same as εmix (7.4 < pH < 10.7) and because εmix < ε10OS, the buffer is not capable of fulfilling its function in this interval.

Figure 6b also shows curve 4 that corresponds to ε10OS and curve 5 that represents a εmix corresponding to a threshold concentration for the buffer system's components in the mixture, TCmix BS (when the pH to be imposed is 9.0), for which εmix ≈ εBS ¼ ε10OS. At this same concentration, the buffer system's components show a good performance to impose the pH, not only at 9.0 but within the 3.8 < pH < 10.0 interval; which is large because it corresponds to a wide spectrum buffer. Approximately TCmix BS must be 10 times larger than the Cmix OS as established by Eq. (16), because the acid-base pair of the Britton-Robinson buffer is the H3BO3/H2BO3 � with a pKa ¼ 9.15 value. It must be underlined that if it is intended to impose the pH to other value (pH < 8.1), the TCmix BS is smaller.

#### 3.5. Application of ε: threshold volume of the buffer system

The concentration of the buffer components in the mixture (Cmix BS ) must be larger than the solute concentration in the system considered (Cmix OS ) to set the pH of the system. However, it is also necessary to know the buffer minimum volume that must be added to the original system with the aim of fulfilling its function (Figure 1). Next, an example is given to determine this minimum volume on the acetylacetone case using Britton-Robinson buffer to impose pH 9.0 in the system.

If there are 100 mL of an acac (OS1) solution at 10�<sup>3</sup> M concentration (C<sup>o</sup> OS1), and two buffer (BS1 y BS2) solutions with components 10�<sup>1</sup> M (Co BS1) and 1 M (Co OS2) concentration: which is the minimum volume that should be added to each buffer solution to set a pH ¼ 9.0 in the system?

Figure 7a shows that the ε10OS can be attained when 10 mL of the buffer solution with 10�<sup>1</sup> M (SB1) component concentration is added to the original system. Then, this minimum volume is the threshold volume (TVBS1) for this specific buffer solution. If now, the buffer solution of 1 M (SB2) component concentration is added to the system of interest, the threshold volume is different, as shown in Figure 7b, under these conditions the TVBS2 ¼ 1 mL.

The TVBS is related to TCmix BS in the mixtures, with total volumes of 110 mL for the first case and 101 mL for the second case, because in both cases TCmix BS are 10 times greater than the original system's solute concentrations in the mixture (Cmix OS ).

If now, a larger concentration of acac is used in the original system, for example 10�<sup>2</sup> M (Co OS2), it is necessary to use the more concentrated buffer to impose the pH, for example SB2, with 1 M (C<sup>o</sup> BS2) component concentration.

It is clear that if the original system solutes' concentrations in the mixture (Cmix OS ) grow, the TCmix BS also grows, consequently TVBS does it too.

Observing the equations shown in Figure 1, it is possible to demonstrate that the threshold volume, TVBS, can be determined from the initial working conditions using Eq. (15)

$$\mathbf{C}\_{\rm BS}^{\rm mix} \ge T \mathbf{C}\_{\rm BS}^{\rm mix} \approx 10 \mathbf{C}\_{\rm OS}^{\rm mix} \tag{19}$$

which can be rewritten as

$$\frac{\mathcal{C}\_{\rm BS}^{\circ}}{V\_{\vartheta}} \frac{V\_{\rm BS}}{\lambda} \ge \frac{\mathcal{C}\_{\rm BS}^{\circ}}{V\_{\vartheta}} \frac{TV\_{\rm BS}}{V\_{\vartheta}} = 10 \frac{\mathcal{C}\_{\rm OS}^{\circ}}{V\_{\vartheta}} \frac{V\_{\rm OS}}{} \tag{17}$$

From Eq. (17), the following is obtained

$$V\_{\rm BS} \ge TV\_{\rm BS} = 10 \left( \frac{\mathbf{C}\_{\rm OS}^\circ V\_{\rm OS}}{\mathbf{C}\_{\rm BS}^\circ} \right) \tag{18}$$

Figure 7c shows that the TVBS is equal to 10 mL, even if it is a volume added to the buffer solution with 1 M component concentration, because the original system is composed of 100 mL with 10�<sup>2</sup> M solute concentration (C<sup>o</sup> OS2). It can be proved that the TCBS fulfills the condition of being 10 times larger than the original system's solute concentration.

The Conditions Needed for a Buffer to Set the pH in a System http://dx.doi.org/10.5772/intechopen.69003 17

Figure 7. The curves for ε¼ f(pH) for 100 mL of different solutions. The broken line in all cases represents the buffer efficacy of water, εðH2OÞ, and its acid and basic particles. The curves of εOS and ε10OS of acac solutions in the absence of the buffer system (markers ����� and ----, respectively). (a) Curves <sup>ε</sup>OS and <sup>ε</sup>10OS containing 100 mL of the acac solution with a 10�<sup>3</sup> <sup>M</sup> solute concentration, curve εmix containing the same acac solution as εOS and different added volumes of the buffer solution with a concentration CBS ¼ 0.1 M. Volumes added: ―10 mL, ———1 mL and, 0.1 mL. (b) The curves εOS and ε10OS are the same that in (a), curves εmix containing the same acac solution and different added volumes of the buffer solution with CBS¼ 1 M. Added volumes: ―1 mL, ―0.5 mL and ―0.1 mL. (c) Curves εOS and ε10OS containing 100 mL of a acac solution with 10�<sup>2</sup> M solute concentration, curves εmix containing 100 mL of the acac solution with a 10�<sup>2</sup> M solute concentration and different volumes added to the buffer solution with CBS¼ 1 M. Added volumes: ―10 mL, ―5 mL and, ― 1 mL.

Just like in the case of the threshold concentration (TCmix BS ) of Eq. (16), Eq. (18) also corresponds to the most difficult case. The factor of 10 can be smaller to the extent that the pKa values of the original system move away from the pH that is to be set.

## 4. Conclusions

necessary to know the buffer minimum volume that must be added to the original system with the aim of fulfilling its function (Figure 1). Next, an example is given to determine this minimum volume on the acetylacetone case using Britton-Robinson buffer to impose pH 9.0

minimum volume that should be added to each buffer solution to set a pH ¼ 9.0 in the system? Figure 7a shows that the ε10OS can be attained when 10 mL of the buffer solution with 10�<sup>1</sup> M (SB1) component concentration is added to the original system. Then, this minimum volume is the threshold volume (TVBS1) for this specific buffer solution. If now, the buffer solution of 1 M (SB2) component concentration is added to the system of interest, the threshold volume is

OS ).

If now, a larger concentration of acac is used in the original system, for example 10�<sup>2</sup> M (Co

It is clear that if the original system solutes' concentrations in the mixture (Cmix

volume, TVBS, can be determined from the initial working conditions using Eq. (15)

Cmix BS ≥ TCmix

> ≥ Co BS TVBS Vo

<sup>V</sup>BS <sup>≥</sup> TVBS <sup>¼</sup> <sup>10</sup> Co

condition of being 10 times larger than the original system's solute concentration.

Figure 7c shows that the TVBS is equal to 10 mL, even if it is a volume added to the buffer solution with 1 M component concentration, because the original system is composed of 100

Co BS VBS Vo

it is necessary to use the more concentrated buffer to impose the pH, for example SB2, with 1 M

Observing the equations shown in Figure 1, it is possible to demonstrate that the threshold

BS ≈ 10Cmix

<sup>¼</sup> <sup>10</sup> Co

OSVOS Co BS 

OS VOS Vo

BS1) and 1 M (Co

BS in the mixtures, with total volumes of 110 mL for the first case and

OS1), and two buffer (BS1

OS2),

ð17Þ

ð18Þ

OS ) grow, the

OS2) concentration: which is the

BS are 10 times greater than the original

OS ð19Þ

OS2). It can be proved that the TCBS fulfills the

If there are 100 mL of an acac (OS1) solution at 10�<sup>3</sup> M concentration (C<sup>o</sup>

different, as shown in Figure 7b, under these conditions the TVBS2 ¼ 1 mL.

y BS2) solutions with components 10�<sup>1</sup> M (Co

101 mL for the second case, because in both cases TCmix

system's solute concentrations in the mixture (Cmix

BS also grows, consequently TVBS does it too.

in the system.

16 Advances in Titration Techniques

(C<sup>o</sup>

TCmix

The TVBS is related to TCmix

BS2) component concentration.

which can be rewritten as

From Eq. (17), the following is obtained

mL with 10�<sup>2</sup> M solute concentration (C<sup>o</sup>

A model has been proposed to study the buffer capacity with dilution effect, βdil, of mixtures of various polydonor systems in aqueous solutions. From the model, exact algebraic expressions were obtained that describe the buffer capacity with dilution.

Through the study of βdil of solutions containing one or more polydonor systems with different conditions and characteristics, it is concluded that βdil decreases when the total concentration of the polydonor systems (or mixtures of polydonor systems) decreases, and conversely.

When a polydonor system attains a very small concentration, this does not contributes practically to the buffer capacity, only the solvent's acid and basic particles determine this property. Then, it is stated that any solution, even the pure solvent, has a buffer capacity.

It is shown that the βdil depends on the size of the system, which is information that is not considered in the β known and used in the common scientific literature.

A new index has been introduced, ε, to measure the buffer efficacy of a buffer solution such that the quantity added of strong base or strong acid causes a change of only one-tenth of a pH unit instead of one unit as βdil.

From the construction of the different curves of ε ¼ f(pH), it is possible to identify a buffer threshold concentration in the mixture (TCmix BS ), which allows knowing the minimum buffer concentration to set the desired pH of the system of interest. This concentration TCmix BS must be, at least 10 times greater than that of the original system's solutes in the mixture <sup>ð</sup>TCmix BS <sup>¼</sup> <sup>C</sup>mix OS Þ in the most difficult case.

Similarly, the different curves of ε ¼ f(pH) also allow determining the minimum volume (buffer threshold volume, TVBS) that must be added to the system of interest to set its pH

$$\left(TV\_{BS} = 10 \left(\frac{\mathbf{C}\_{\rm OS}^{\circ}V\_{\rm OS}}{\mathbf{C}\_{\rm BS}^{\circ}}\right)\right) \text{ in the most difficult case.}$$

## Acknowledgements

NR-L wants to acknowledge DGAPA-UNAM for a postdoctoral fellowship. The authors also want to acknowledge for partial financial support from PRODEP-SEP through RedNIQAE and CONACyT through 237997 project.

## Author details

Norma Rodríguez-Laguna1,2, Alberto Rojas-Hernández1 \*, María T. Ramírez-Silva1 , Rosario Moya-Hernández<sup>2</sup> , Rodolfo Gómez-Balderas<sup>2</sup> and Mario A. Romero-Romo<sup>3</sup>

\*Address all correspondence to: suemi918@xanum.uam.mx

1 Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Área de Química Analítica, México, DF, Mexico

2 Departamento de Materiales, Universidad Autónoma Metropolitana-Azcapotzalco, Área de Ingeniería de Materiales, México, DF, Mexico

3 Facultad de Estudios Superiores-Cuautitlán, Universidad Nacional Autónoma de México, Estado de México, Mexico

## References

When a polydonor system attains a very small concentration, this does not contributes practically to the buffer capacity, only the solvent's acid and basic particles determine this property.

It is shown that the βdil depends on the size of the system, which is information that is not

A new index has been introduced, ε, to measure the buffer efficacy of a buffer solution such that the quantity added of strong base or strong acid causes a change of only one-tenth of a pH

From the construction of the different curves of ε ¼ f(pH), it is possible to identify a buffer

Similarly, the different curves of ε ¼ f(pH) also allow determining the minimum volume (buffer threshold volume, TVBS) that must be added to the system of interest to set its pH

NR-L wants to acknowledge DGAPA-UNAM for a postdoctoral fellowship. The authors also want to acknowledge for partial financial support from PRODEP-SEP through RedNIQAE and

1 Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Área de

2 Departamento de Materiales, Universidad Autónoma Metropolitana-Azcapotzalco, Área de

3 Facultad de Estudios Superiores-Cuautitlán, Universidad Nacional Autónoma de México,

, Rodolfo Gómez-Balderas<sup>2</sup> and Mario A. Romero-Romo<sup>3</sup>

concentration to set the desired pH of the system of interest. This concentration TCmix

at least 10 times greater than that of the original system's solutes in the mixture <sup>ð</sup>TCmix

in the most difficult case.

BS ), which allows knowing the minimum buffer

\*, María T. Ramírez-Silva1

,

BS must be,

BS <sup>¼</sup> <sup>C</sup>mix OS Þ

Then, it is stated that any solution, even the pure solvent, has a buffer capacity.

considered in the β known and used in the common scientific literature.

unit instead of one unit as βdil.

18 Advances in Titration Techniques

in the most difficult case.

Acknowledgements

OSVOS Co BS

CONACyT through 237997 project.

Química Analítica, México, DF, Mexico

Ingeniería de Materiales, México, DF, Mexico

Norma Rodríguez-Laguna1,2, Alberto Rojas-Hernández1

\*Address all correspondence to: suemi918@xanum.uam.mx

TVBS <sup>¼</sup> <sup>10</sup> <sup>C</sup><sup>o</sup>

Author details

Rosario Moya-Hernández<sup>2</sup>

Estado de México, Mexico

threshold concentration in the mixture (TCmix


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