**Distribution Diagrams and Graphical Methods to Determine or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions**

Alberto Rojas-Hernández1, Norma Rodríguez-Laguna1, María Teresa Ramírez-Silva1 and Rosario Moya-Hernández2 *1Depto. de Química, Área de Química Analítica, Universidad Autónoma Metropolitana-Iztapalapa, 2Facultad de Estudios Superiores-Cuautitlán, Lab. 10, UIM, Universidad Nacional Autónoma de México, Mexico* 

### **1. Introduction**

286 Stoichiometry and Research – The Importance of Quantity in Biomedicine

Wells, J.E. & Russel, J.B. (1996). The effect of growth and starvation on the lysis of the

*Microbiology*, Vol. 62, No.4, pp.1342-1346.

ruminal cellulolytic bacterium *Fibrobacter succinogenes*. *Applied and Environmental* 

Graphical methods to study the behaviour of systems showing different chemical equilibrium are known and very used in several fields of Chemistry (particularly in Bioinorganic, Medicinal and Pharmaceutical Chemistry) in order to establish the chemical species that are related with drugs behaviour in different systems. Among these methods, the most commonly used are the distribution diagrams (Högfeldt, 1979), the titration curves (Asuero & Michałowski, 2011) and the molar ratio and continuous variations methods (Hartley et al., 1980).

In the present work we have selected two molecules used extensively like drugs. In order to exemplify some novelties related with distribution diagrams and titration curves for acidbase systems we have selected the case of oxine (HOX, 8-hydroxiquinoline) that has been used as antiseptic and disinfectant. On the other hand, we have selected the complexation interaction between Fe(III) and tenoxicam (Tenox) to show other novelties related with more complicated distribution diagrams and molar ratio and continuous variations methods, because tenoxicam has been extensively used as non-steroidal anti-inflammatory drug that may be complexed with several metal ions. The chemical developed formulae of these compounds are presented in Scheme 1.

### **2. Distribution diagrams for acid-base and complexation systems**

Graphic representations of chemical systems have found wide application because a simple look at them allows for solve specific problems and have a panorama, qualitative and quantitative, for different problems and phenomena. Moreover, some of these representations also permit to graphically solve the stated problem with a predetermined error (Vicente-Pérez, 1985). Distribution diagrams of species are some of the most used

Distribution Diagrams and Graphical Methods to Determine

*n =*

aqueous solutions.

considerations.

section 2.2.

**0.00** 

**2.1.2 Complexation systems** 

**0 5 10** 

ν

where *i*∈ {1, 2, ..., *m*} and *j*∈ {(*0*, *1*, ..., *n[i]*}

HOX H2OX+

**pH**

**0.25** 

**0.50** 

**Molar fractions**

function of pH. *j* (or

p*Ka1* = 5.0, p*Ka2* = 9.7.

**0.75** 

**1.00** 

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 289

factorization of *n* as a function of pH (taking into account the sets of Eq. 1 and 2) for

1

<sup>−</sup> = = +

*<sup>j</sup> <sup>n</sup> <sup>T</sup> <sup>j</sup>*

*jβ*

[ ] 1 []

[*H+*]*T* is the protons total concentration in the system, which requires the balance of proton equation. In aqueous solutions, this equation needs the autoprotolysis constant and special

The chemical information given in the curves of Fig. 1 is the same, as expected from Eq. 2 and 3. In fact, the Average Proton Number is the set of statistical means of the subjacent distributions resumed by the distribution diagram of Fig. 1, as it will be explained in the

> **0 1 2**

**Average Proton Number (***n***)**

(a) (b)

Fig. 1. Typical graphic representations of the oxine hydrochloride (H2OXCl) in water as a

a) Distribution diagram. b) Average proton number. Data were taken from Ringbom (1963):

Even though the knowledge of Brφnsted acid-base behaviour is important, the interaction of substances with metal ions is remarkable as well (e.g., the interaction of drugs with metal

The formation of several coordination compounds, or complexes, between a metal ion (*M*) and a ligand (*L*) can be described by the global formation equilibrium represented in Eq. 4.

*iM + jL MLi <sup>j</sup>* ←⎯⎯⎯⎯→ with *<sup>i</sup> [M L ]j <sup>β</sup> <sup>=</sup> ij <sup>i</sup> <sup>j</sup> [M] [L]*

ions may potentiate or suppress its pharmacological activity or toxicity).

**0.0**

) represents the proton stoichiometric coefficient for each species.

**0 5 10** 

**pH**

**0.5**

**1.0**

**1.5**

**2.0**

*j or* ν

OX-

*<sup>L</sup> <sup>H</sup>*

[][]

+ + <sup>=</sup>

*H H*

1

*β*

*<sup>T</sup> <sup>j</sup> <sup>j</sup> <sup>j</sup> j*

=

[ ]

+

*H*

*n j*

+ =

*<sup>n</sup> <sup>j</sup>*

1

*jf*

(3)

(4)

graphic representations since the second half of 20th century; nevertheless, there have been some novelties on the field in the last decade (e.g. Moya-Hernández et al., 2002a, 2002b).

Scheme 1. Developed formulae of a) oxine (HOX) and b) tenoxicam (Tenox).

### **2.1 Typical representations of distribution diagrams**

In this section some typical representations will be quickly reviewed to introduce the newest representations given in section 2.2.

### **2.1.1 Acid-base systems**

In Chemistry the study of polyprotic systems, whose global formation equilibrium are represented in Eq. 1, is of crucial importance, because many substances, drugs among them, follow this Brφnsted acid-base behaviour.

$$\text{L}^{a-} + j\text{H}^{+} \xleftarrow{\text{g}} \text{L} \text{H}\_{\text{j}}^{j-a} \quad \text{with} \ \beta\_{\text{j}} = \frac{[\text{L} \text{H}\_{\text{j}}^{j-a}]}{[\text{L}^{a-}][\text{H}^{+} \text{J}^{j}]} \tag{1}$$

The typical way to define the molar fractions to describe the species distribution of the component *L* in this system with respect to the proton (H+) is given in Eq. 2, as well as its factorization in the substance amount balance equation of this component in the system – using the set of Eq. 1 (Rojas-Hernández, 1995).

$$f\_j = \frac{[L H\_j^{j-a}]}{[L]\_r} = \frac{\beta\_j [H^+]^j}{1 + \sum\_{j=1}^n \beta\_j [H^+]^j} \tag{2}$$

where [*L*]*T* is L total concentration in the system.

An example of distribution diagram is given in Fig. 1a, showing the case of oxine hydrochloride (H2OXCl, 8-hydroxiquinolinol = HOX).

Fig. 1b represents the function known as Average Proton Number ( *n* ), introduced and developed by Niels and Jannik Bjerrum (Hartley et al., 1980). Eq. 3 shows the definition and factorization of *n* as a function of pH (taking into account the sets of Eq. 1 and 2) for aqueous solutions.

$$\overline{m} = \frac{[H^+]\_{\mathbb{T}} - [H^+]}{[L]\_{\mathbb{T}}} = \frac{\sum\_{j=1}^n j \beta\_j [H^+]^j}{1 + \sum\_{j=1}^n \beta\_j [H^+]^j} = \sum\_{j=1}^n j f\_j \tag{3}$$

[*H+*]*T* is the protons total concentration in the system, which requires the balance of proton equation. In aqueous solutions, this equation needs the autoprotolysis constant and special considerations.

The chemical information given in the curves of Fig. 1 is the same, as expected from Eq. 2 and 3. In fact, the Average Proton Number is the set of statistical means of the subjacent distributions resumed by the distribution diagram of Fig. 1, as it will be explained in the section 2.2.

Fig. 1. Typical graphic representations of the oxine hydrochloride (H2OXCl) in water as a function of pH. *j* (or ν) represents the proton stoichiometric coefficient for each species. a) Distribution diagram. b) Average proton number. Data were taken from Ringbom (1963): p*Ka1* = 5.0, p*Ka2* = 9.7.

### **2.1.2 Complexation systems**

288 Stoichiometry and Research – The Importance of Quantity in Biomedicine

graphic representations since the second half of 20th century; nevertheless, there have been some novelties on the field in the last decade (e.g. Moya-Hernández et al., 2002a, 2002b).

N

CH3

*j-a [LH ] <sup>j</sup> a- + <sup>j</sup> [L ][H ]*

*β* = (1)

(2)

O

N

N

H

S O O

In this section some typical representations will be quickly reviewed to introduce the newest

In Chemistry the study of polyprotic systems, whose global formation equilibrium are represented in Eq. 1, is of crucial importance, because many substances, drugs among them,

The typical way to define the molar fractions to describe the species distribution of the component *L* in this system with respect to the proton (H+) is given in Eq. 2, as well as its factorization in the substance amount balance equation of this component in the system –

[ ] [ ]

An example of distribution diagram is given in Fig. 1a, showing the case of oxine

Fig. 1b represents the function known as Average Proton Number ( *n* ), introduced and developed by Niels and Jannik Bjerrum (Hartley et al., 1980). Eq. 3 shows the definition and

*LH H f = <sup>L</sup> <sup>H</sup>*

=

[ ] 1 []

*j a j j j <sup>j</sup> <sup>n</sup> <sup>T</sup> <sup>j</sup>*

− +

1

=

<sup>+</sup>

*β*

*j j*

*β*

+

OH

S

Scheme 1. Developed formulae of a) oxine (HOX) and b) tenoxicam (Tenox).

*a- + j-a L + jH LH <sup>j</sup>* ←⎯⎯⎯⎯→ with *<sup>j</sup>*

N

(a) (b)

**2.1 Typical representations of distribution diagrams** 

OH

representations given in section 2.2.

follow this Brφnsted acid-base behaviour.

using the set of Eq. 1 (Rojas-Hernández, 1995).

where [*L*]*T* is L total concentration in the system.

hydrochloride (H2OXCl, 8-hydroxiquinolinol = HOX).

**2.1.1 Acid-base systems** 

Even though the knowledge of Brφnsted acid-base behaviour is important, the interaction of substances with metal ions is remarkable as well (e.g., the interaction of drugs with metal ions may potentiate or suppress its pharmacological activity or toxicity).

The formation of several coordination compounds, or complexes, between a metal ion (*M*) and a ligand (*L*) can be described by the global formation equilibrium represented in Eq. 4.

$$\text{LiM} + \text{jL} \xrightarrow{\text{g}} \begin{array}{c} M\_{\text{i}} \text{L}\_{\text{j}} \text{ with } \rho\_{\text{ij}} = \frac{[M\_{\text{i}} \text{L}\_{\text{j}}]}{[M]^{\text{i}} [\text{L}]^{\text{j}}} \end{array} \tag{4}$$

where *i*∈ {1, 2, ..., *m*} and *j*∈ {(*0*, *1*, ..., *n[i]*}

Distribution Diagrams and Graphical Methods to Determine

*M/L*

*M/L*

ϕ

because the following inequality is always confirmed.

*Distribution diagrams for the Fe(III)-tenoxicam system in acetone* 

where *i*∈ {0, 1, ..., *m[j]*} and *j*∈ {*1*, 2, ..., *n*}

where *i*∈ {0, 1, ..., *m[j]*} and *j*∈ {*1*, 2, ..., *n*}

*<sup>L</sup> f = <sup>L</sup>*

*L = L*

ϕ

*Distributions of M in L* 

described in Eq. 10.

Eq. 11.

collected in Table 1.

Hernández et al., 2009).

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 291

Following the same approach, several distributions of *L* species may be defined. Then, the L substance amount fractions are defined in Eq. 9, while the L concentration fractions are

[ ]

*M/L i j M/L i (j ) ij 01 ij T*

*jML f = f ML <sup>L</sup>*

[ ]

*M/L i j M/L i (j ) ij 01 ij T*

*j ML <sup>=</sup> M L*

<sup>=</sup> + 

> ϕ

*ij*

=1 =0

*j i*

[ ] and ( [ ][ ] ) [ ]

=

The substance amount and concentration fractions of *L* species are related by means of

*M/L M/L* [ ]*<sup>T</sup> ij ij*

[ ] [ ]

Σ [ ] [ ] []

*L i j i j T*

In order to exemplify the typical distributions diagrams of *M* and *L* species, Fig. 2 is reported for the Fe(III) - tenoxicam (Tenox) system in acetone from previously reported data by Moya-Hernández et al. (2009). The equilibrium constants of this reference have been

**Species** *i j* **log**

Table 1. Global formation constants of Fe(III)-tenoxicam species in acetone (Moya-

*Fe*2*Tenox* 2 1 9.04 ± 0.03 *Fe*2*Tenox*2 2 2 14.75 ± 0.06 *Fe*2*Tenox*3 2 3 18.45 ± 0.07

 =< =

=1 =0 =1 =0

*j i j i*

*n n m j m j*

[ ] <sup>1</sup> [ ][ ]

*jβ*

*<sup>01</sup> <sup>n</sup> m j <sup>T</sup> <sup>i</sup> (j )*

[ ] 1

*L*

ϕ

*ij*

<sup>=</sup> + 

*M L*

*jβ*

*M L*

*jβ*

*L L*

*M L j M L L*

=1 =0

*j i*

[ ] and ( [ ][ ] ) [ ]

=

[ ] <sup>1</sup> [ ][ ]

*jβ*

*<sup>01</sup> <sup>n</sup> m j <sup>T</sup> <sup>i</sup> (j )*

[ ] 1





*= f* Σ (11)

(12)

β*ij* (9)

(10)

In Eq. 4, the charge of the species has been omitted for notation simplicity. When *i* = 1 the complexes are called mononuclear, but when *i* ≥ 2 the corresponding complexes are called polynuclear. The formation of polynuclear complexes on a given system is thermodynamically favoured when the total concentration of *M* ([*M*]*T*) is high and placed over the mononuclear wall (Ringbom, 1963).

In complexation chemistry, several distributions have to be considered. In general, it is preferred to study the way the component *L* is distributing on *M* species, but studying how the component *M* is distributing on *L* species may be interesting as well. In both cases there are two possible descriptions, depending on what is considered between the amount or the concentrations of the species. When polynuclear compexes are formed in the system all the distribution diagrams representing the formation of the species depends on [*M*]*T* and [*L*]*T*.

### *Distributions of L in M*

The M substance amount fractions are defined by Eq. 5.

$$\begin{aligned} \, \_{L/M} f\_{10} &= \frac{[M]}{[M]\_{\Gamma}} = \frac{1}{1 + \sum\_{i=1}^{m} \left( \sum\_{j=0}^{n[i]} i \beta\_{ij} [M]^{(i-1)} [L]^{j} \right)} \\ \, \_{\text{and}} \quad ^{L/M} f\_{ij} &= \frac{i [M\_i L\_j]}{[M]\_{\Gamma}} = ^{L/M} f\_{10} (i \theta\_{ij} [M]^{(i-1)} [L]^{j}) \end{aligned} \tag{5}$$

where *i*∈ {1, 2, ..., *m*} and *j*∈ {0, *1*, ..., *n[i]*}

When polynuclear M species are not forming in a given system ( *i* = 1), this distribution only depends on [*L*]*T*, but when polynuclear species appear, the substance amount fractions depend on [*L*]*T* and [*M*]*T*; furthermore in this last case the simple sum of the concentrations of M species (Σ*M*) is lower than [*M*]*T*, in agreement with Eq. 6.

$$\Sigma\_M = \sum\_{i=1}^m \left( \sum\_{j=0}^{n[i]} [M\_i L\_j] \right) < \sum\_{i=1}^m \left( \sum\_{j=0}^{n[i]} i [M\_i L\_j] \right) = [M]\_\Gamma \tag{6}$$

In this particular case, M concentration fractions can be defined by Eq. 7.

$$\mathcal{I}^{L/M} \varphi\_{10} = \frac{[M]}{\Sigma\_M} = \frac{1}{1 + \sum\_{i=1}^m \left(\sum\_{j=0}^{n[i]} \mathcal{J}\_{\vec{\eta}}[M]^{\vec{\eta}-1}[L]^j\right)} \quad \text{and} \quad {}^{L/M} \varphi\_{\vec{\eta}} = \frac{[M\_i L\_j]}{\Sigma\_M} = {}^{L/M} \varphi\_{10} (\mathcal{J}\_{\vec{\eta}}[M]^{\vec{\eta}-1}[L]^j) \tag{7}$$

where *i*∈ {1, 2, ..., *m*} and *j*∈ {0, *1*, ..., *n[i]*}

The substance amount and concentration fractions of *M* species are related by means of Eq. 8.

$$\rho\_{i\uparrow}^{L\uparrow M} \varphi\_{i\downarrow} = {}^{L\uparrow M} f\_{i\uparrow} \frac{\{M\}\_{\Gamma}}{\Sigma\_M} \tag{8}$$

### *Distributions of M in L*

290 Stoichiometry and Research – The Importance of Quantity in Biomedicine

In Eq. 4, the charge of the species has been omitted for notation simplicity. When *i* = 1 the complexes are called mononuclear, but when *i* ≥ 2 the corresponding complexes are called polynuclear. The formation of polynuclear complexes on a given system is thermodynamically favoured when the total concentration of *M* ([*M*]*T*) is high and placed

In complexation chemistry, several distributions have to be considered. In general, it is preferred to study the way the component *L* is distributing on *M* species, but studying how the component *M* is distributing on *L* species may be interesting as well. In both cases there are two possible descriptions, depending on what is considered between the amount or the concentrations of the species. When polynuclear compexes are formed in the system all the distribution diagrams representing the formation of the species depends on [*M*]*T* and [*L*]*T*.

[ ]

*i j L/M L/M i j ij 10 ij T*

*f = f M L*

*ij*

<sup>=</sup> + 

=1 =0

[ ] and ( [ ] [ ]) [ ]

When polynuclear M species are not forming in a given system ( *i* = 1), this distribution only depends on [*L*]*T*, but when polynuclear species appear, the substance amount fractions depend on [*L*]*T* and [*M*]*T*; furthermore in this last case the simple sum of the concentrations

[ ] [ ]

Σ [ ] [ ] []

*i j L/M L/M L/M i j <sup>10</sup> ij 10 ij <sup>m</sup> n i M M (i ) <sup>j</sup>*

The substance amount and concentration fractions of *M* species are related by means of

[ ] *L/M L/M <sup>T</sup> ij ij*

ϕ

*M i j i j T*

 =< = 

*ML iML M*

[ ] [ ] <sup>1</sup> and ( [ ] [ ])

 ϕ

*M M*

*<sup>M</sup> M L <sup>=</sup> <sup>=</sup> M L*

=1 =0 =1 =0

= =

*i j i j*

In this particular case, M concentration fractions can be defined by Eq. 7.


Σ Σ

*M L*

*m m n i n i*

=

[ ] 1 [ ] []

*iβ*

*<sup>10</sup> <sup>m</sup> n i <sup>T</sup> (i ) <sup>j</sup>*

*i j*

[ ] 1

*iML*

*M*


*M L*

*iβ*

( -1)

(6)

(5)

( -1)

(7)

ϕ

*= f* Σ (8)

*β*

over the mononuclear wall (Ringbom, 1963).

The M substance amount fractions are defined by Eq. 5.

*<sup>M</sup> f = <sup>M</sup>*

of M species (Σ*M*) is lower than [*M*]*T*, in agreement with Eq. 6.

[ ]

1 [ ] []

*β*

*ij*

+ 

=1 =0

*i j*

where *i*∈ {1, 2, ..., *m*} and *j*∈ {0, *1*, ..., *n[i]*}

*L/M*

where *i*∈ {1, 2, ..., *m*} and *j*∈ {0, *1*, ..., *n[i]*}

*Distributions of L in M* 

ϕ

Eq. 8.

Following the same approach, several distributions of *L* species may be defined. Then, the L substance amount fractions are defined in Eq. 9, while the L concentration fractions are described in Eq. 10.

$$\begin{aligned} \,^{M/L}f\_{01} &= \frac{[L]}{[L]\_{\mathrm{T}}} = \frac{1}{1 + \sum\_{j=1}^{n} \left( \sum\_{i=0}^{\mathrm{m}[j]} j \beta\_{ij} [M]^{\mathrm{i}} [L]^{\mathrm{j}-1} \right)} \\ \,^{\mathrm{and}}\mathbf{1} & \quad ^{M/L}f\_{ij} = \frac{j [M\_i L\_j]}{[L]\_{\mathrm{T}}} = \,^{M/L}f\_{01} (j \beta\_{ij} [M]^{\mathrm{i}} [L]^{\mathrm{j}-1}) \end{aligned} \tag{9}$$

where *i*∈ {0, 1, ..., *m[j]*} and *j*∈ {*1*, 2, ..., *n*}

$$\begin{aligned} \boldsymbol{\varphi}\_{01}^{M/L} \boldsymbol{\varphi}\_{01} &= \frac{[L]}{[L]\_{\mathrm{T}}} = \frac{1}{1 + \sum\_{j=1}^{n} \left( \sum\_{i=0}^{\mathrm{mL}[j]} j \boldsymbol{\beta}\_{ij} [\boldsymbol{\mathsf{M}}]^{\mathrm{i}} [\boldsymbol{L}]^{\circ(\cdot 1)} \right)} \\ \text{and} \quad {}^{M/L} \boldsymbol{\varphi}\_{ij} &= \frac{j [\boldsymbol{\mathsf{M}}\_{i} \boldsymbol{L}\_{j}]}{[\boldsymbol{L}]\_{\mathrm{T}}} = {}^{M/L} \boldsymbol{\varphi}\_{01} (j \boldsymbol{\beta}\_{ij} [\boldsymbol{\mathsf{M}}]^{\mathrm{i}} [\boldsymbol{L}]^{\circ(\cdot 1)}) \end{aligned} \tag{10}$$

where *i*∈ {0, 1, ..., *m[j]*} and *j*∈ {*1*, 2, ..., *n*}

The substance amount and concentration fractions of *L* species are related by means of Eq. 11.

$$\rho^{M/L} \varphi\_{ij} = {}^{M/L} f\_{ij} \frac{\{L\}\_{\Gamma}}{\Sigma\_L} \tag{11}$$

because the following inequality is always confirmed.

$$\Sigma\_L = \sum\_{j=1}^{n} \left( \sum\_{i=0}^{m[j]} [M\_i L\_j] \right) < \sum\_{j=1}^{n} \left( \sum\_{i=0}^{m[j]} j [M\_i L\_j] \right) = [L]\_{\Gamma} \tag{12}$$

*Distribution diagrams for the Fe(III)-tenoxicam system in acetone* 

In order to exemplify the typical distributions diagrams of *M* and *L* species, Fig. 2 is reported for the Fe(III) - tenoxicam (Tenox) system in acetone from previously reported data by Moya-Hernández et al. (2009). The equilibrium constants of this reference have been collected in Table 1.


Table 1. Global formation constants of Fe(III)-tenoxicam species in acetone (Moya-Hernández et al., 2009).

Distribution Diagrams and Graphical Methods to Determine

**coefficient for an acid-base system species )** 

each distribution of discrete variable is defined by Eq. 13.

distribution of discrete variable is defined by Eq. 14.

meaning of each distribution.

the system.

are forming.

could represent another variable (like pΣ*M*, pΣ*L*, p*L*, p*M*, etc.)

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 293

The independent variable of these figures, in all cases, has been selected to be the p*TenoxT*, in order to compare directly the shape of each fraction in Fig. 2 and 3. Nevertheless this axis

It is noteworthy than distribution of substance amount and concentration are similar for the same component, even though slightly differences can be shown comparing Fig. 2a and 2b, or Fig. 3a or 3b; some of their differences are crucial and related with the physical-chemical

A distributions of Fe(III) and tenoxicam species comparison allows for the conclusion that they are very different, obviously representing distributions of two different components in

The sentence "distribution diagram" could have an implicit idea concerning possible statistical distributions subjacent to the graphic representations given in Fig. 1, 2 and 3. This idea was first explored by Moya-Hernández et al. (2002a) for the case of Brφnsted acid-base systems. In the next subsection (2.2.1) the conclusions of this work will be applied to the case of oxine species as a function of pH while in the other subsection (2.2.2) this treatment will be generalized to the case of complex systems of the M-L kind, where polynuclear species

**2.2.1 Distribution diagrams of one discrete variable (the proton stoichiometric** 

If an aqueous solution of oxine at a certain concentration has a known pH value, the distribution of the oxine species will be fixed, as it is shown in Fig. 1a. Then a 3D graph could represent all the distributions of oxine species for each pH value. The set of distributions of oxine species is represented in Fig. 4 as well as one of them, at pH = 5.0.

In agreement with Mathematical Statistics (Kreyszig, 1970; Reichl, 1980), the meaning of

*n*

0

*j* ν *=* =

Furthermore, each of these statistical distributions has a variance. The variance of each

2 2 0

The equality of Eq. 3 and 13 demonstrate that the set of means of the oxine species distributions is the one given in Fig. 1b. In this way, the mean of the distribution offers the

The set of variances of these distributions is related with an intrinsic buffer capacity (Moya-Hernández et al., 2002a). A graphic representation of the oxine set of variances as a function

ν

*n*

*j s =* ν

value in which the proton number is centered, as an average, at each pH value.

=

*j*

*j*

*jf* (13)

*(j - ) f* (14)

**2.2 Distribution diagrams as a function of stoichiometric coefficients** 

The distribution diagrams that represent the substance amount fractions, calculated from global formation constants given in Table 1, have been constructed with the aid of program MEDUSA (Puigdomenech, 2010). The distribution diagrams that represent the concentration fractions have been obtained by means of Excel (*Microsoft*®) worksheets applying Eq. 8 and 11. Some of the typical distribution diagrams are shown in Fig. 2 and 3.

Fig. 2. Typical graphic representations of the Fe(III)-tenoxicam system in acetone as a function of p*TenoxT*. [*Fe(III)*]*T* = 1×10-3 M and p*TenoxT* = -log[*Tenox*]*T*. a) Distribution diagram of substance amount of Fe(III) species taking into account the quantity of Fe(III) in each species. b) Distribution diagram of concentration of Fe(III) species taking in account which Fe(III) species is more concentrated in the system.

Fig. 3. Typical graphic representations of the Fe(III)-tenoxicam system in acetone as a function of p*TenoxT*. [*Fe(III)*]*T* = 1×10-3 M and p*TenoxT* = -log[*Tenox*]*T*. a) Distribution diagram of substance amount of Tenoxicam species, taking in account the quantity of tenoxicam in each species. b) Distribution diagram of concentration of tenoxicam species, taking in account which tenoxicam species is more concentrated in the system.

The distribution diagrams that represent the substance amount fractions, calculated from global formation constants given in Table 1, have been constructed with the aid of program MEDUSA (Puigdomenech, 2010). The distribution diagrams that represent the concentration fractions have been obtained by means of Excel (*Microsoft*®) worksheets applying Eq. 8 and

**0.0**

**0.0**

**0.3**

**0.5**

**0.8**

**1.0**

**1.0 2.0 3.0 4.0 5.0** *pTenoxT*

**1.0 2.0 3.0 4.0 5.0**

*pTenoxT*

*Fe2Tenox2*

*Fe2Tenox Tenox*

*Fe2Tenox3*

*Fe2Tenox2*

*Fe2Tenox*

*Fe*

**0.3**

**0.5**

**0.8**

**1.0**

*Fe2Tenox3*

*Fe*

Fig. 2. Typical graphic representations of the Fe(III)-tenoxicam system in acetone as a function of p*TenoxT*. [*Fe(III)*]*T* = 1×10-3 M and p*TenoxT* = -log[*Tenox*]*T*. a) Distribution diagram of substance amount of Fe(III) species taking into account the quantity of Fe(III) in each species. b) Distribution diagram of concentration of Fe(III) species taking in account which

11. Some of the typical distribution diagrams are shown in Fig. 2 and 3.

**1.0 2.0 3.0 4.0 5.0** *pTenoxT*

Fe(III) species is more concentrated in the system.

*Fe2Tenox2*

*Fe2Tenox Tenox*

*Fe2Tenox3*

**1.0 2.0 3.0 4.0 5.0**

*pTenoxT*

(a) (b)

account which tenoxicam species is more concentrated in the system.

Fig. 3. Typical graphic representations of the Fe(III)-tenoxicam system in acetone as a function of p*TenoxT*. [*Fe(III)*]*T* = 1×10-3 M and p*TenoxT* = -log[*Tenox*]*T*. a) Distribution diagram of substance amount of Tenoxicam species, taking in account the quantity of tenoxicam in each species. b) Distribution diagram of concentration of tenoxicam species, taking in

*Fe2Tenox2*

*Fe2Tenox*

(a) (b)

**0.0**

**0.0**

**0.3**

**0.5**

*Fe/Tenoxfij*

**0.8**

**1.0**

**0.3**

**0.5**

*Tenox/Fefij*

**0.8**

**1.0**

*Fe2Tenox3*

The independent variable of these figures, in all cases, has been selected to be the p*TenoxT*, in order to compare directly the shape of each fraction in Fig. 2 and 3. Nevertheless this axis could represent another variable (like pΣ*M*, pΣ*L*, p*L*, p*M*, etc.)

It is noteworthy than distribution of substance amount and concentration are similar for the same component, even though slightly differences can be shown comparing Fig. 2a and 2b, or Fig. 3a or 3b; some of their differences are crucial and related with the physical-chemical meaning of each distribution.

A distributions of Fe(III) and tenoxicam species comparison allows for the conclusion that they are very different, obviously representing distributions of two different components in the system.

### **2.2 Distribution diagrams as a function of stoichiometric coefficients**

The sentence "distribution diagram" could have an implicit idea concerning possible statistical distributions subjacent to the graphic representations given in Fig. 1, 2 and 3. This idea was first explored by Moya-Hernández et al. (2002a) for the case of Brφnsted acid-base systems. In the next subsection (2.2.1) the conclusions of this work will be applied to the case of oxine species as a function of pH while in the other subsection (2.2.2) this treatment will be generalized to the case of complex systems of the M-L kind, where polynuclear species are forming.

### **2.2.1 Distribution diagrams of one discrete variable (the proton stoichiometric coefficient for an acid-base system species )**

If an aqueous solution of oxine at a certain concentration has a known pH value, the distribution of the oxine species will be fixed, as it is shown in Fig. 1a. Then a 3D graph could represent all the distributions of oxine species for each pH value. The set of distributions of oxine species is represented in Fig. 4 as well as one of them, at pH = 5.0.

In agreement with Mathematical Statistics (Kreyszig, 1970; Reichl, 1980), the meaning of each distribution of discrete variable is defined by Eq. 13.

$$\overline{\nabla} = \sum\_{j=0}^{n} j f\_j \tag{13}$$

Furthermore, each of these statistical distributions has a variance. The variance of each distribution of discrete variable is defined by Eq. 14.

$$\left\|s\_{\nu}^{2} = \sum\_{j=0}^{n} (j \cdot \nabla)^{2} f\_{j}\right\|\tag{14}$$

The equality of Eq. 3 and 13 demonstrate that the set of means of the oxine species distributions is the one given in Fig. 1b. In this way, the mean of the distribution offers the value in which the proton number is centered, as an average, at each pH value.

The set of variances of these distributions is related with an intrinsic buffer capacity (Moya-Hernández et al., 2002a). A graphic representation of the oxine set of variances as a function

Distribution Diagrams and Graphical Methods to Determine

covariance for the same distribution is given in Eq. 16.

variances and one covariance (Reichl, 1980).

fractions of Fe(III) species.

following sections.

coefficient of the corresponding component.

only one of the systems represented in Fig. 2 are shown in Fig. 6.

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 295

Examples of substance amount and concentration distributions of tenoxicam in Fe(III), for

In this case, each one of the statistical distributions that can be defined has two means, two

Just as a mere example, the definition of the two means of the substance amount distribution of Tenoxicam in Fe(III) is given in Eq. 15, while the definition of the two variances and the

[ ]

[ )

*i f*

](

](

(15)

[ )

*j f*

=1 =0 [ ]

*m n i L/M L/M f M ij i j n m j L/M L/M f L ij j i*

*ν*

=

=

*ν*

=0 =1

(a) (b)

Fig. 6. Statistical distributions of tenoxicam in Fe(III) species in acetone. [*Fe(III)*]*T* = 1×10-3M and [*Tenox*]*T* = 6×10-4M. a) Substance amount fractions of Fe(III) species. b) Concentration

The set of means and variances for the distributions of two discrete variables of Tenoxicam in Fe(III), represented typically in Fig. 2a like a "distribution diagram", are shown in Fig. 7. The interpretation of each of the two means is to be the stoichiometric coefficient average for the corresponding component. The two variances grow when two or more species are forming: higher variance increases the number of species with different stoichiometric

The possible consequences of this statistical view should be studied exhaustively, but this kind of study would be beyond the objectives of the present work. The distribution diagrams applications to determine or to use stoichiometric coefficients are developed in the

of pH is given in Fig. 5. When the variance has a value near to zero one species is present in the system with a fraction near 1; when the variance reaches a maximum, two or more species are present in the system with comparable fraction values. In the case of oxine system the maxima of the variance function are reached for pH values equal to p*Ka1* and p*Ka2*, because p*Ka2* >> p*Ka1*.

Fig. 4. Statistical distributions of oxine species. a) Set of distributions of the stoichiometric coefficient of protons as the discrete variable. b) Specific distribution diagram at pH = 5.0.

Fig. 5. Set of variances of the distributions of discrete variable of oxine as a function of pH. The maxima of these function are placed in the p*Ka* values.

### **2.2.2 Distribution diagrams of two discrete variables (the M and L components stoichiometric coefficients for complexation systems)**

It can be demonstrated that the distribution diagrams as those of Fig. 2 and 3 for *M*-*L* complexation systems represent sets of distributions of two discrete variables,where these variables are the M and L stoichiometric coefficients.

of pH is given in Fig. 5. When the variance has a value near to zero one species is present in the system with a fraction near 1; when the variance reaches a maximum, two or more species are present in the system with comparable fraction values. In the case of oxine system the maxima of the variance function are reached for pH values equal to p*Ka1* and

**0 1 2** 

(a) (b)

**0.0**

The maxima of these function are placed in the p*Ka* values.

**stoichiometric coefficients for complexation systems)** 

variables are the M and L stoichiometric coefficients.

**0.1**

**0.2**

**Set of variances**

**0.3**

**OX-**

**0.0** 

**0.2** 

**0.4** 

**0.6** 

**Molar fractions**

Fig. 4. Statistical distributions of oxine species. a) Set of distributions of the stoichiometric coefficient of protons as the discrete variable. b) Specific distribution diagram at pH = 5.0.

**5.0 9.7** 

**0 2 4 6 8 10 12 14** 

**pH**

Fig. 5. Set of variances of the distributions of discrete variable of oxine as a function of pH.

It can be demonstrated that the distribution diagrams as those of Fig. 2 and 3 for *M*-*L* complexation systems represent sets of distributions of two discrete variables,where these

**2.2.2 Distribution diagrams of two discrete variables (the M and L components** 

**Molar fractions**

**0.0**

**OX-**

**2 1 0** 

**H2OX HOX <sup>+</sup>**

*j =* ν

pH = p*Ka1* =5.0

**0.2**

**0.4**

**0.6**

**0.8**

**1.0**

**0.8** 

**1.0** 

p*Ka2*, because p*Ka2* >> p*Ka1*.

**H2OX+ HOX**

**0 2 4 6 8 10 12 14**  Examples of substance amount and concentration distributions of tenoxicam in Fe(III), for only one of the systems represented in Fig. 2 are shown in Fig. 6.

In this case, each one of the statistical distributions that can be defined has two means, two variances and one covariance (Reichl, 1980).

Just as a mere example, the definition of the two means of the substance amount distribution of Tenoxicam in Fe(III) is given in Eq. 15, while the definition of the two variances and the covariance for the same distribution is given in Eq. 16.

$$\begin{aligned} \, ^{L/M}\_{\!\!\!f} \overline{\nu}\_{\mathcal{M}} &= \sum\_{i=1}^{m} \sum\_{j=0}^{n[i]} [i(^{L/M} f\_{ij})] \\ ^{L/M}\_{\!\!\!f} \overline{\nu}\_{\mathcal{L}} &= \sum\_{j=0}^{n} \sum\_{i=1}^{m[j]} [j(^{L/M} f\_{ij})] \end{aligned} \tag{15}$$

Fig. 6. Statistical distributions of tenoxicam in Fe(III) species in acetone. [*Fe(III)*]*T* = 1×10-3M and [*Tenox*]*T* = 6×10-4M. a) Substance amount fractions of Fe(III) species. b) Concentration fractions of Fe(III) species.

The set of means and variances for the distributions of two discrete variables of Tenoxicam in Fe(III), represented typically in Fig. 2a like a "distribution diagram", are shown in Fig. 7.

The interpretation of each of the two means is to be the stoichiometric coefficient average for the corresponding component. The two variances grow when two or more species are forming: higher variance increases the number of species with different stoichiometric coefficient of the corresponding component.

The possible consequences of this statistical view should be studied exhaustively, but this kind of study would be beyond the objectives of the present work. The distribution diagrams applications to determine or to use stoichiometric coefficients are developed in the following sections.

Distribution Diagrams and Graphical Methods to Determine

1 0

−

the cations of the polyprotic system for *j* ∈ {*a*+1, *a*+2, ..., *n*}.

+

*a*

−

*n*

<sup>1</sup> <sup>1</sup>

+

*o b j*

( ){ }

0 0 0

*j j j <sup>b</sup> <sup>w</sup> <sup>b</sup>*

*<sup>V</sup> <sup>K</sup> C H*

**3.1.2 Titrations with a strong acid:** *HX*

1 0

*n*

*a*

−

( ){ }

−

algebraically rearranging it, it is possible to arrive to Eq. 20.

+

*a j VC*

+

−

volume of *HX* (*Va*), in the form:

{( )( )} ( ) {( ) }

−

*j a VC*

( ){ }

*a j VC*

**3.1.1 Titrations with a strong base:** *MOH*

of *MOH* (*Vb*), in the form:

Eq. 18.

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 297

If the mixture described at the beginning of section 3.1 is titrated with a strong base *MOH,* of *Cb* concentration, it is possible to write the electroneutrality equation, for each added volume

1

+ + − +=

( )[ ] [ ]

*j*

[ ]

+

+

*H*

( )[ ] [ ]

+ − +=

*j*

*j*

( )[ ] [ ] [ ]

*j*

(17)

(18)

(19)

0

<sup>−</sup> = + <sup>−</sup> <sup>−</sup> =

*a j LH OH V V*

where [*M*+] = *VbCb*/(*Vo* + *Vb*), the second term in the first member of Eq. 17, represents the contra-cations charge associated to the anions of the polyprotic system for *j* ∈ {0, 1, ..., *a*-1}, while the first term in the second member represents the contra-anions charge associated to

Introducing Eq. 1 and 2 in Eq. 17 and algebraically rearranging it, it is possible to arrive to

*n n <sup>n</sup> <sup>w</sup> oj oj oj oj j o*

<sup>+</sup> <sup>=</sup> = = +

+ −

If the mixture described at the beginning of the section 3.1 is titrated with a strong acid *HX,* of *Ca* concentration, it is possible to write the electroneutrality equation, for each added

1

= − + = +

*j a LH H V V*

0

= + − + +

=

*a j LH OH X V V*

<sup>−</sup> = + <sup>−</sup> − −

where [*X*<sup>−</sup>] = *VaCa*/(*Vo* + *Va*). Introducing Eq. 1 and 2 conveniently in Eq. 19 and

<sup>1</sup> <sup>1</sup>

*o b j*

*oj oj <sup>n</sup> <sup>j</sup> j a*

*oj oj <sup>a</sup> j a j a*

( ){ }

*j a VC*

*o b j a*

− − −− − <sup>=</sup>

*<sup>K</sup> j a VC VC j af V H <sup>H</sup>*

[ ] ( )[ ] [ ]

+ + = − = +

*<sup>M</sup> j a LH H V V*

*o b j a*

*oj oj <sup>a</sup> j a j a*

= + − +

*oj oj <sup>n</sup> <sup>j</sup> j a*

Fig. 7. Statistical parameters for the substance amount distributions of Tenoxicam in Fe(III) in acetone corresponding to Fig. 2a. a) Set of the two means of the distribution. b) Set of the two variances of the distribution.

### **3. Acid-base titration curves for polyprotic systems**

This section deals with titration curves, pH = f(volume of strong acid or base), of polyacid and polybase systems by means of a description given by a thermodynamic model. This allows for the theoretical building of this kind of curves as well as their first derivative and the buffer capacity curves as a function of pH. The model is deducted from electroneutrality equation relating the global formation equilibrium and molar fractions of the distribution diagram of the system. Some previous works concerning this mater can be found in literature, by Fleck, 1967; Högfeldt, 1979; King & Kester, 1990; Efstathiou, 2000; Rojas-Hernández & Ramírez-Silva, 2002; Tarapčík & Beinrohr, 2003; Asuero, 2007; Gutz, 2010; Asuero & Michałowski, 2011.

### **3.1 Titration curves, pH = f(added volume of strong acid or base), of a mixture of species in the same polyprotic system**

A system formed by a mixture of *Voj* volumes of solutions of *HjLj-a* species, with concentrations *Coj*, giving a total initial volume *Vo =* Σ*Voj*, has been considered. The *HjLj-a* species form part of the same polyprotic system (*HnL(n-a)+/H(n-1)L(n-a-1)+/.../HaL/.../HL(a-1)-/La-* /*H*+). Each charged species has associated a contra-cation or a contra-anion (*M*z+ or *X*z-) that do not have acid-base properties, depending if (*j-a*) is negative or positive.

### **3.1.1 Titrations with a strong base:** *MOH*

296 Stoichiometry and Research – The Importance of Quantity in Biomedicine

[ )

[ )

*L/M L/M L/M L/M f f M LM f L*

*S - i - ν j ν f*

2

**0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7**

Fig. 7. Statistical parameters for the substance amount distributions of Tenoxicam in Fe(III) in acetone corresponding to Fig. 2a. a) Set of the two means of the distribution. b) Set of the

This section deals with titration curves, pH = f(volume of strong acid or base), of polyacid and polybase systems by means of a description given by a thermodynamic model. This allows for the theoretical building of this kind of curves as well as their first derivative and the buffer capacity curves as a function of pH. The model is deducted from electroneutrality equation relating the global formation equilibrium and molar fractions of the distribution diagram of the system. Some previous works concerning this mater can be found in literature, by Fleck, 1967; Högfeldt, 1979; King & Kester, 1990; Efstathiou, 2000; Rojas-Hernández & Ramírez-Silva, 2002; Tarapčík & Beinrohr, 2003; Asuero, 2007; Gutz, 2010;

**3.1 Titration curves, pH = f(added volume of strong acid or base), of a mixture of** 

do not have acid-base properties, depending if (*j-a*) is negative or positive.

A system formed by a mixture of *Voj* volumes of solutions of *HjLj-a* species, with concentrations *Coj*, giving a total initial volume *Vo =* Σ*Voj*, has been considered. The *HjLj-a* species form part of the same polyprotic system (*HnL(n-a)+/H(n-1)L(n-a-1)+/.../HaL/.../HL(a-1)-/La-* /*H*+). Each charged species has associated a contra-cation or a contra-anion (*M*z+ or *X*z-) that

*Tenox/Fe Sf* **2**

[( ( ( ]

)))

*i j*

*Tenox/Fef S2 Tenox*

**1.0 2.0 3.0 4.0 5.0 p***TenoxT*

*Tenox/Fef S2 Fe*

(16)

*i j*

{ ( ] ( )}

*j - ν f*

<sup>2</sup> { ( ] ( )}

*i - ν f*

[ ]

=1 =0 [ ]

*i j n m j L/M L/M L/M f f L ij*

*M,L*

=

=

=

*S*

*S*

**0.0 0.5 1.0 1.5 2.0 2.5 3.0**

*Tenox/Fef* ν *Tenox* **—**

> *Tenox/Fef* ν *Fe* **—**

two variances of the distribution.

Asuero & Michałowski, 2011.

**species in the same polyprotic system** 

*M,L*

*M,L*

**1.0 2.0 3.0 4.0 5.0 p***TenoxT*

(a) (b)

**3. Acid-base titration curves for polyprotic systems** 

*m n i L/M L/M L/M f f M M L*

=0 =1 [ ]

*j i m n i*

=1 =0

*j*

*i*

If the mixture described at the beginning of section 3.1 is titrated with a strong base *MOH,* of *Cb* concentration, it is possible to write the electroneutrality equation, for each added volume of *MOH* (*Vb*), in the form:

$$\begin{aligned} \sum\_{j=0}^{a-1} & (a-j) \{ V\_{\eta} \mathbb{C}\_{\eta \bar{\eta}} \} \\ \{ M^+ \} + \frac{j=0}{V\_o + V\_b} + \sum\_{j=a+1}^n & (j-a) \{ L H\_j^{j-a} \} + \{ H^+ \} = \\ \sum\_{j=a+1}^n & (j-a) \{ V\_{\eta} \mathbb{C}\_{\eta \bar{\eta}} \} \\ = & \frac{\sum\_{j=a+1}^n (j-a) \{ V\_{\eta} \mathbb{C}\_{\eta j} \}}{V\_o + V\_b} + \sum\_{j=0}^{a-1} (a-j) \{ L H\_j^{j-a} \} + \{ OH^- \} \end{aligned} \tag{17}$$

where [*M*+] = *VbCb*/(*Vo* + *Vb*), the second term in the first member of Eq. 17, represents the contra-cations charge associated to the anions of the polyprotic system for *j* ∈ {0, 1, ..., *a*-1}, while the first term in the second member represents the contra-anions charge associated to the cations of the polyprotic system for *j* ∈ {*a*+1, *a*+2, ..., *n*}.

Introducing Eq. 1 and 2 in Eq. 17 and algebraically rearranging it, it is possible to arrive to Eq. 18.

$$V\_b = \frac{\sum\_{j=0}^{n} \{(j-a)(V\_{\neq} \mathbb{C}\_{\neq j})\} - \left[\sum\_{j=0}^{n} (V\_{\neq} \mathbb{C}\_{\neq j})\right] \left[\sum\_{j=0}^{n} \{(j-a)f\_j\}\right] - V\_o \left(\left[H^+\right] - \frac{K\_w}{\left[H^+\right]}\right)}{\mathbb{C}\_b + \left[H^+\right] - \frac{K\_w}{\left[H^+\right]}}\tag{18}$$

### **3.1.2 Titrations with a strong acid:** *HX*

If the mixture described at the beginning of the section 3.1 is titrated with a strong acid *HX,* of *Ca* concentration, it is possible to write the electroneutrality equation, for each added volume of *HX* (*Va*), in the form:

$$\begin{aligned} \sum\_{j=0}^{a-1} & (a-j)\{V\_{\eta}\mathbb{C}\_{\eta j}\} \\ \frac{\sum\_{j=0}^{a} & (j-a)\{LH\_j^{j-a}\} + \{H^+\} = \\ & \sum\_{j=a+1}^{n} \{j-a\} \{V\_{\eta}\mathbb{C}\_{\eta j}\} \\ = & \frac{\sum\_{j=a+1}^{n} \{j-a\} \{V\_{\eta}\mathbb{C}\_{\eta j}\}}{V\_o + V\_b} + \sum\_{j=0}^{a-1} \{a-j\} \{LH\_j^{j-a}\} + \{OH^-\} + \{X^-\} \end{aligned} \tag{19}$$

where [*X*<sup>−</sup>] = *VaCa*/(*Vo* + *Va*). Introducing Eq. 1 and 2 conveniently in Eq. 19 and algebraically rearranging it, it is possible to arrive to Eq. 20.

Distribution Diagrams and Graphical Methods to Determine

0 0 0

*j j j <sup>a</sup> <sup>w</sup> <sup>b</sup>*

*<sup>V</sup> <sup>K</sup> C H*

{( )( )} ( ) {( ) }

**3.1.3 Titration of aqueous solution of oxine hydrochloride (***H2OXCl***) with** *NaOH*

solution 0.001M *H2OXCl* where titrated with a solution of NaOH 0.1315 M at 25°C.

quantitative reaction consumes 1 mol of *H2OX+* for each added mol of *OH*

applications of these equations.

given by Eq. 18 is shown in Fig. 8a.

quantification of oxine with the hydroxide ion:

expressions are given in Eq. 22 and 23.

*dV*

*dV*

*dpH C*

*dpH C*

**3.2 The first derivative of the titration curve: dpH/dV** 

0 00

0 00

*j ji*

= ==

*j ji*

= ==

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 299

Practically all the simulators, available nowadays, to predict acid-base titration curves, using strong base or acid as titrand agent, are based on Eq. 18 and 20. They are analytical expressions to calculate exactly the added volume of strong base or acid, given a set of pH values. In the following subsections this feature will be used to obtain important

*n n <sup>n</sup> <sup>w</sup> oj oj oj oj j o*

<sup>+</sup> <sup>=</sup> = = +

−− + −+ − <sup>=</sup>

− +

In order to compare the predicted results with the experimental ones, 50 mL of an aqueous

The comparison of the experimental titration curve with the fitted curve through the model

As it is shown in curves 1 of Fig. 8, the observed fitting for the titration curve is good. In the case of Fig. 8b, the molar ratio (*r*) is defined as the ratio of the added titrand (NaOH), with respect to the analyte (*OXT*). The solution pH has a great change near 1, meaning that the

the stoichiometric coefficients of these reagents are 1 and 1 for the reaction that permits the

The first derivative method of an acid-base titration curve is well known to determine the

Nevertheless, being Eq. 18 and 20 functions of only one variable (the pH), it is possible to obtain the analytical expressions of their first derivatives and their reciprocal functions, to arrive to the exact algebraic expressions of the titration first derivatives curves. These

[ ]

*V C jf i j f V V*

2.303 ( ) ( ) 2.303( ) 10 10

[ ]

− + − = <sup>−</sup> −+ + +

*V C jf i j f V V*

2.303 ( ) ( ) 2.303( ) 10 10

*n nn <sup>a</sup> pH pH pK oj oj j i o a*

*n nn <sup>b</sup> pH pH pK oj oj j i o b*

10 10

+ − <sup>=</sup> <sup>−</sup> −+ + +

10 10

*pH pH pK*

*pH pH pK*

− −

− −

equivalence point volumes in order to accomplish quantitative chemical analysis.

*b*

*a*

*<sup>K</sup> j a VC VC j af V H <sup>H</sup>*

[ ]

+

*+ - H OX + OH HOX + H O 2 2* ←⎯⎯⎯⎯→ (21)

*w*

*w*

+

−

. In other words,

*w*

*w*

(22)

(23)

− −

− −

(20)

*H*

Fig. 8. Titration curve of 50 mL of aqueous solution of *H2OXCl* with an aqueous solution of NaOH 0.1315 M. The values used to obtain the fitting shown are the following: p*Ka1* = 5.0, p*Ka2* = 9.7, p*Kw* = 13.7, [*H2OXCl*]initial = 0.00101 M.

1) The circles represent the experimental points of the pH = f(*Vb*) curve and the solid line is the fitted curve obtained with Eq. 18. 2) The rhombuses represent the experimental points of the (dpH/d*Vb*) = g(*Vb*) curve and the dashed line is the fitted curve obtained with Eq. 22. a) Curves as a function of added volume of the titrand. b) Curves as a function of molar ratio (*r*) of the titrand.

**(1)**

**(2)**

**0 0.2 0.4 0.6 0.8 1 1.2 1.4**

*Vb* **=** *VNaOH* **/mL**

(a)

**(2)**

**(1)**

**0.0 0.5 1.0 1.5 2.0 2.5 3.0**

*r* **=** *nNaOH* **/***nOXT*

(b) Fig. 8. Titration curve of 50 mL of aqueous solution of *H2OXCl* with an aqueous solution of NaOH 0.1315 M. The values used to obtain the fitting shown are the following: p*Ka1* = 5.0,

1) The circles represent the experimental points of the pH = f(*Vb*) curve and the solid line is the fitted curve obtained with Eq. 18. 2) The rhombuses represent the experimental points of the (dpH/d*Vb*) = g(*Vb*) curve and the dashed line is the fitted curve obtained with Eq. 22. a) Curves as a function of added volume of the titrand. b) Curves as a function of molar

**4**

**4**

ratio (*r*) of the titrand.

p*Ka2* = 9.7, p*Kw* = 13.7, [*H2OXCl*]initial = 0.00101 M.

**5**

**6**

**7**

**8**

**pH**

**9**

**10**

**11**

**12**

**5**

**6**

**7**

**8**

**pH**

**9**

**10**

**11**

**12**

**0**

**0**

**20**

**40**

**dpH/d**

*Vb*

**/mL**

**-1**

**60**

**80**

**100**

**120**

**140**

**20**

**40**

**dpH/d**

*Vb*

**/mL**

**-1**

**60**

**80**

**100**

**120**

**140**

Practically all the simulators, available nowadays, to predict acid-base titration curves, using strong base or acid as titrand agent, are based on Eq. 18 and 20. They are analytical expressions to calculate exactly the added volume of strong base or acid, given a set of pH values. In the following subsections this feature will be used to obtain important applications of these equations.

$$V\_a = \frac{-\sum\_{j=0}^{n} \{(j-a)(V\_{\neq} \mathbb{C}\_{\neq j})\} + \left[\sum\_{j=0}^{n} (V\_{\neq} \mathbb{C}\_{\neq j})\right] \left[\sum\_{j=0}^{n} \{(j-a)f\_j\}\right] + V\_o \left(\left[H^+\right] - \frac{K\_w}{\left[H^+\right]}\right)}{\mathbf{C}\_b - \left[H^+\right] + \frac{K\_w}{\left[H^+\right]}}\tag{20}$$

### **3.1.3 Titration of aqueous solution of oxine hydrochloride (***H2OXCl***) with** *NaOH*

In order to compare the predicted results with the experimental ones, 50 mL of an aqueous solution 0.001M *H2OXCl* where titrated with a solution of NaOH 0.1315 M at 25°C.

The comparison of the experimental titration curve with the fitted curve through the model given by Eq. 18 is shown in Fig. 8a.

As it is shown in curves 1 of Fig. 8, the observed fitting for the titration curve is good. In the case of Fig. 8b, the molar ratio (*r*) is defined as the ratio of the added titrand (NaOH), with respect to the analyte (*OXT*). The solution pH has a great change near 1, meaning that the quantitative reaction consumes 1 mol of *H2OX+* for each added mol of *OH*− . In other words, the stoichiometric coefficients of these reagents are 1 and 1 for the reaction that permits the quantification of oxine with the hydroxide ion:

$$H\_2OX^\* + OH^- \xleftarrow{\colon-} HOX + H\_2O \tag{21}$$

### **3.2 The first derivative of the titration curve: dpH/dV**

The first derivative method of an acid-base titration curve is well known to determine the equivalence point volumes in order to accomplish quantitative chemical analysis.

Nevertheless, being Eq. 18 and 20 functions of only one variable (the pH), it is possible to obtain the analytical expressions of their first derivatives and their reciprocal functions, to arrive to the exact algebraic expressions of the titration first derivatives curves. These expressions are given in Eq. 22 and 23.

$$\frac{dpH}{dV\_b} = \frac{C\_b + 10^{-pH} - 10^{pH - pK\_w}}{-2.303 \left[ \sum\_{j=0}^{n} (V\_{oj}C\_{oj}) \right] \left[ \sum\_{j=0}^{n} \left\{ j f\_j \sum\_{i=0}^{n} [(i-j)f\_i] \right\} \right] + 2.303 (V\_o + V\_b) \left[ 10^{-pH} + 10^{pH - pK\_w} \right]} \quad \text{(22)}$$

$$-\frac{dpH}{dV\_a} = \frac{C\_a - 10^{-pH} + 10^{pH - pK\_w}}{-2.303 \left[ \sum\_{j=0}^{n} (V\_{oj}C\_{oj}) \right] \left[ \sum\_{j=0}^{n} \left\{ j f\_j \sum\_{i=0}^{n} [(i-j)f\_i] \right\} \right] + 2.303 (V\_o + V\_a) \left[ 10^{-pH} + 10^{pH - pK\_w} \right]} \quad \text{(23)}$$

Distribution Diagrams and Graphical Methods to Determine

0 00

0 00

*dpH C*

*j ji*

intrinsic buffer capacity of a polyprotic system.

ν

**0 0.02 0.04 0.06 0.08 0.1 0.12 0.14**

ν

amphiprotic solvent (H+ and *OH*

Schock (2000).)

**(***H2OXCl***) with** *NaOH*

subsection 3.1.3.

effect of dilution.

= ==

*j ji*

= ==

*dpH C*

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 301

*w*

*w*

**<sup>β</sup>dil** *<sup>b</sup>* (24)

**<sup>β</sup>dil** *<sup>a</sup>* (25)

*(j - ) f* (26)

*w*

*w*

− −

− −

in the case of water). (See Segurado (2003), Urbansky &

[ ]

2.303 (10 10 )

[ ]

*oj oj j i oa*

*j ji*

*V C jf i j f V V*

( ) ( ) ( ) 10 10

− −

( ) ( ) ( ) 10 10

*n nn pH pH pK*

 − − ++ + 

*pH pH pK*

*n nn pH pH pK*

 − − ++ + 

As Moya-Hernández et al. (2002b) have demonstrated, Eq. 14 is equal to the double sum between brackets in Eq. 24 and 25. For this reason, the set of variances is related with the

( )

It is noteworthy that the intrinsic buffer capacity, i.e. the set of variances of the proton stoichiometric coefficient distributions for this kind of systems, is a function that only

The second term between keys in Eq. 24 and 25 is due to the acid and basic particles of the

Fig. 9 shows the comparison of experimental buffer capacity, with effect of dilution, and the curve obtained by Eq. 24 and 25 for the titrations of the system defined at the beginning of

> **0 2 4 6 8 10 12 14 pH**

Fig. 9. Comparison of experimental (markers) and calculated (solid line) buffer capacity with

**3.3.2 Buffer capacity with dilution of aqueous solution of oxine hydrochloride** 

*s = jf i j f intrinsic buffer capacity*

=− − =

*pH pH pK*

− −

2.303 (10 10 )

*oj oj j i ob*

*V C jf i j f V V*

*a a a*

*a*

=− = − −

[ ] 2 2 0 0 0

depends on the molar fractions and the stoichiometric coefficients.

−

*n n n*

*j j i*

= = =

*dV C C*

*b b b*

*b*

= = + −

*dV C C*

### **3.2.1 Titration first derivative curve of an oxine hydrochloride (***H2OXCl***) aqueous solution with** *NaOH*

The curves 2 in Fig. 8 compare the titration experimental first derivative, described at the beginning of subsection 3.1.3, with the curves obtained by Eq. 22.

The experimental first derivative was obtained approximately as the ratio of finite differences of measured pH values and volumes during titration, using the average of volumes, or molar ratios, for each interval.

As it can be seen, the fitting attained is quite good and the maximum observed in Fig. 8a and 8b is sharp. In the first case, this maximum signals the volume position of the first equivalence point, while in the second case it indicates the ratio of the titrand stoichiometric coefficients with respect to the analyte for the quantitative reaction.

The first derivative is usually better than the pH curve to experimentally determine the volumes of the equivalence points (Fig. 8a) and the titration reactions (titrand/analyte) ratio of stoichometric coefficients (Fig. 8b) when these are quantitative.

In the case of the *HOX* reaction with *OH*− , the second titration reaction, is not quantitative because its corresponding equilibrium constant is not high enough for a 0.001 M analyte initial concentration. For this reason there are no visible changes in the second equivalence point volume in curves 1 or 2 in Fig. 8a, nor in Fig. 8b for *r* = 2.

### **3.3 The buffer capacity (**β**) of a polyprotic system**

In many chemical and biological processes it is essential that the medium pH be kept within certain limits, which is possible through the use of buffer solutions. They possess a specific buffer capacity and are used to maintain constant the pH with a very small uncertainty.

In the chemical literature, there are two ways to define a buffer capacity (β): one is defined in terms of concentration of strong acid or base added to the system, in order to simplify the mathematical treatment, as firstly proposed by Van Slyke (1922), and then used by others, as Urbansky & Schock (2000) or Segurado (2003). The other way to define the buffer capacity is in terms of the amount of strong acid or base added to the system, as King and Kester did (1990), as well as Skoog et al. (2005); they also derive equations with the concentration, but considering explicitly 1L of solution.

### **3.3.1 A buffer capacity with dilution effect (**β**dil)**

According to the definition given by King & Kester (1990) it is possible to apply the derivative of the added amount of strong base or acid with respect to the pH, and then obtaining mathematical expressions for the buffer capacity as function of pH, by considering the dilution effect, i.e. β**dil** = f(pH). In the present work, this implies to take Eq. 22 and 23 reciprocals and multiply them by *Cb* or *Ca*, respectively.

The typical way to represent a buffer capacity curve, consists in plotting it as a function of pH, even though it may be represented as a function of the titrand volume or the molar ratio titrand/analyte.

The curves 2 in Fig. 8 compare the titration experimental first derivative, described at the

The experimental first derivative was obtained approximately as the ratio of finite differences of measured pH values and volumes during titration, using the average of

As it can be seen, the fitting attained is quite good and the maximum observed in Fig. 8a and 8b is sharp. In the first case, this maximum signals the volume position of the first equivalence point, while in the second case it indicates the ratio of the titrand stoichiometric

The first derivative is usually better than the pH curve to experimentally determine the volumes of the equivalence points (Fig. 8a) and the titration reactions (titrand/analyte) ratio

because its corresponding equilibrium constant is not high enough for a 0.001 M analyte initial concentration. For this reason there are no visible changes in the second equivalence

In many chemical and biological processes it is essential that the medium pH be kept within certain limits, which is possible through the use of buffer solutions. They possess a specific buffer capacity and are used to maintain constant the pH with a very small uncertainty.

In the chemical literature, there are two ways to define a buffer capacity (β): one is defined in terms of concentration of strong acid or base added to the system, in order to simplify the mathematical treatment, as firstly proposed by Van Slyke (1922), and then used by others, as Urbansky & Schock (2000) or Segurado (2003). The other way to define the buffer capacity is in terms of the amount of strong acid or base added to the system, as King and Kester did (1990), as well as Skoog et al. (2005); they also derive equations with the concentration, but

According to the definition given by King & Kester (1990) it is possible to apply the derivative of the added amount of strong base or acid with respect to the pH, and then obtaining mathematical expressions for the buffer capacity as function of pH, by considering the dilution effect, i.e. β**dil** = f(pH). In the present work, this implies to take Eq. 22 and 23

The typical way to represent a buffer capacity curve, consists in plotting it as a function of pH, even though it may be represented as a function of the titrand volume or the molar ratio

, the second titration reaction, is not quantitative

−

**3.2.1 Titration first derivative curve of an oxine hydrochloride (***H2OXCl***) aqueous** 

beginning of subsection 3.1.3, with the curves obtained by Eq. 22.

coefficients with respect to the analyte for the quantitative reaction.

of stoichometric coefficients (Fig. 8b) when these are quantitative.

point volume in curves 1 or 2 in Fig. 8a, nor in Fig. 8b for *r* = 2.

**3.3 The buffer capacity (**β**) of a polyprotic system** 

volumes, or molar ratios, for each interval.

In the case of the *HOX* reaction with *OH*

considering explicitly 1L of solution.

titrand/analyte.

**3.3.1 A buffer capacity with dilution effect (**β**dil)** 

reciprocals and multiply them by *Cb* or *Ca*, respectively.

**solution with** *NaOH*

$$\begin{split} \mathfrak{g}\_{\text{dil}} \stackrel{d}{=} & \frac{d \, V\_b \mathbb{C}\_b}{dpH} = \frac{2.303 \mathbb{C}\_b}{\mathbb{C}\_b + (10^{-pH} - 10^{pH - pK\_w})} \\ \mathbb{E}\left\{ \left[ \sum\_{j=0}^n (V\_{oj} \mathbb{C}\_{oj}) \right] \left[ - \sum\_{j=0}^n \left( \dot{y} f\_j \sum\_{i=0}^n (i - j) f\_i \right) \right] + (V\_o + V\_b) \left[ 10^{-pH} + 10^{pH - pK\_w} \right] \right\} \\ \mathfrak{g}\_{\text{dil}} = & -\frac{dV\_a \mathbb{C}\_a}{\mathbb{P}\_{\text{initial}} + \frac{\text{vol}(W)}{\mathbb{P}\_{\text{initial}} + \text{vol}(W) + \text{vol}(W)}}. \end{split} \tag{24}$$

$$\begin{aligned} \mathsf{P}\_{\text{dil}\_{H}} &= -\frac{\mathsf{u} \cdot \mathsf{u}}{d\mathfrak{p}H} = \frac{\mathsf{u}}{\mathsf{C}\_{a} - (\mathbbm{1}0^{-\mathsf{p}H} - \mathbbm{1}0^{\mathsf{p}H - \mathsf{p}K\_{w}})} \\ &+ \left\{ \left[ \sum\_{j=0}^{n} \langle \mathsf{V}\_{o} \mathsf{C}\_{oj} \rangle \right] \left[ -\sum\_{j=0}^{n} \langle \dot{\mathsf{r}} \dot{f}\_{j} \sum\_{i=0}^{n} (\dot{\mathsf{i}} - \dot{\mathsf{j}}) f\_{i} \rangle \right] + \langle \mathsf{V}\_{o} + \mathsf{V}\_{o} \rangle \Big[ \mathbbm{1}0^{-\mathsf{p}H} + \mathbbm{1}0^{\mathsf{p}H - \mathsf{p}K\_{w}} \Big] \right\} \end{aligned} \tag{25}$$

As Moya-Hernández et al. (2002b) have demonstrated, Eq. 14 is equal to the double sum between brackets in Eq. 24 and 25. For this reason, the set of variances is related with the intrinsic buffer capacity of a polyprotic system.

$$s\_{\nu}^{2} = \sum\_{j=0}^{n} (j \cdot \nabla)^{2} f\_{j} = \left[ -\sum\_{j=0}^{n} \left( jf\_{j} \sum\_{i=0}^{n} [(i-j)f\_{i}] \right) \right] = \text{intrinsic buffer capacity} \tag{26}$$

It is noteworthy that the intrinsic buffer capacity, i.e. the set of variances of the proton stoichiometric coefficient distributions for this kind of systems, is a function that only depends on the molar fractions and the stoichiometric coefficients.

The second term between keys in Eq. 24 and 25 is due to the acid and basic particles of the amphiprotic solvent (H+ and *OH*− in the case of water). (See Segurado (2003), Urbansky & Schock (2000).)

### **3.3.2 Buffer capacity with dilution of aqueous solution of oxine hydrochloride (***H2OXCl***) with** *NaOH*

Fig. 9 shows the comparison of experimental buffer capacity, with effect of dilution, and the curve obtained by Eq. 24 and 25 for the titrations of the system defined at the beginning of subsection 3.1.3.

Fig. 9. Comparison of experimental (markers) and calculated (solid line) buffer capacity with effect of dilution.

Distribution Diagrams and Graphical Methods to Determine

*L M b n n a*

*b*

*b n <n a*

arbitrary units.

*L b r a*

*SC L*

*b r*

*L b < r a*

coefficient : *SCrL* = *b/a*.

*R*

*<sup>a</sup>* <sup>=</sup> *<sup>a</sup>*

*M SC L*

*M L*

*<sup>a</sup>* <sup>=</sup> *<sup>a</sup>*

*n n*

*Initial nM nL*

< *nM - (a/b)nL b*

*Initial nM nL*

< *kM(nM - (a/b)nL)/VT b*

∈ ≈

*a*∈ ≈

For this study it has been assumed that *kM* = *kL* = 0.

**012345**

*rL*

(a) (b)

∈ ≈

*a*∈ ≈

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 303

∈ ≈

∈ ≈

∈ ≈

∈ ≈

 *0 kL(nL - (b/a)nM)/VT kMaLb(1/a)nM/VT*

 *0 nL - (b/a)nM (1/a)nM*

 *0 (1/b)nL*

 *0 (1/a)nM*

*rL* **012345**

*b*

*MaLb*

 *0 kMaLb(1/b)rL(nM)/VT*

 *0 kMaLb(1/a)nM/VT*

*MaLb*

*a M + b L*

 *0 b*

equal to *VT*. *SCnL* represents the amount of *L* for the stoichiometric condition.

*a M + b L*

 *0 b*

Table 3. TVSA for molar ratio method expressed in terms of molar ratio of *L* (*rL*) and response (*R*) of the system. *SCrL* represents the amount of *L* for the stoichiometric condition.

It should be noticed in Fig. 10 that the intersection between the two formed straight lines indicates the molar ratio corresponding to stoichiometric conditions; in other words, this value is equal to the L stoichiometric coefficient ratio divided by the M stoichiometric

Fig. 10. Curves *Response* = *R*= f(*rL*) for experiments of molar ratio method. a) *a* = 1 for the formation of species of the *MLb* kind. b) *a* = 2 for the formation of species of the *M2Lb* kind.

*R*

*b*

Table 2. TVSA for molar ratio method. *nM* is a constant for all the systems, *nL* is variable, all systems are in solution in thermodynamic equilibrium and the total volume is the same and

The functions deducted for molar ratio method and presented in Table 3, with the given restrictions, lead to represent their behaviour in Fig. 10 assuming Eq. 28 and giving *R* in

The experimental β**dil** has been determined approximately by taking the ratio of the titrand concentration and the first derivative experimentally obtained as explained in subsection 3.2.1.

As it can be seen in Fig. 9, there is a good agreement between the theoretical and the experimental data. The expected maximum of the curve at pH = p*Ka2* = 9.7 (see Fig. 5) is lost due to the effect of the hydroxide ion over the pair *HOX*/*OX*− by the low concentration of oxine in the system (0.001 M).

### **4. Molar ratio and continuous variations methods for complexation systems**

For complexation reactions the two traditional methods to determine the stoichiometry of formation reactions and formation equilibrium constants are the molar ratio and the continuous variations methods (this last also known as Job's method). (See Hartley et al., 1980.)

Nevertheless, the explanation of these methods has not always been given in a clear way, and this is particularly evident for equilibrium considering the formation of polynuclear species. Furthermore, these treatments are disappearing from modern text books. For this reason, the aim of this part of the chapter is to describe a way to deduct the equations and the curves that appear in some Inorganic and Analytical Chemistry books.

### **4.1 Tables of Variation of Substance Amount (TVSA)**

Professor Gaston Charlot (1969), in France, has used a teaching tool to explain the advance degree in the formation of acid-base, complexation and redox reactions. It has been called by him the Table of Variation of Substance Amount. In the following subsections this tool will be applied to the systems equilibrium states using the molar ratio and continuous variations methods.

For a system where only one reaction takes place, quantitatively:

$$aM + b\mathbf{L} \xleftarrow{\text{ $\bf b$ }} \mathbf{M}\_a \mathbf{L}\_b \quad \text{with} \quad \beta\_{ab} = \frac{\text{ $\bf $ M\_a \mathbf{L}\_b $}{\text{$ \bf  $\bf \mathbf{L}$ }^a} \tag{27}$$

### **4.1.1 Table of Variation of Substance Amount for the molar ratio method**

In the molar ratio method several amounts of one of the reagents, i. e. *L*, is added to another reagent, i. e. *M*, and typically the dilution of the solutions is controlled in volumetric flasks of the same capacity *VT* in order to measure a response *R* that is directly proportional to the reaction product, *MaLb*, through the equation:

$$R = k\_{M\_a L\_b} \left[ M\_a L\_b \right] = k\_{M\_a L\_b} \left( n\_{M\_a L\_b} \ne V\_T \right) \tag{28}$$

This experiment is described in Table 2 for a quantitative reaction.

If all the amounts in the first column of Table 2 are divided by the *M* quantity (*nM*), and the amounts in the other columns are divided by *VT* and multiplied by the corresponding response factor, Table 3 will be obtained.

The experimental β**dil** has been determined approximately by taking the ratio of the titrand concentration and the first derivative experimentally obtained as explained in subsection

As it can be seen in Fig. 9, there is a good agreement between the theoretical and the experimental data. The expected maximum of the curve at pH = p*Ka2* = 9.7 (see Fig. 5) is lost

**4. Molar ratio and continuous variations methods for complexation systems**  For complexation reactions the two traditional methods to determine the stoichiometry of formation reactions and formation equilibrium constants are the molar ratio and the continuous variations methods (this last also known as Job's method). (See Hartley et al.,

Nevertheless, the explanation of these methods has not always been given in a clear way, and this is particularly evident for equilibrium considering the formation of polynuclear species. Furthermore, these treatments are disappearing from modern text books. For this reason, the aim of this part of the chapter is to describe a way to deduct the equations and

Professor Gaston Charlot (1969), in France, has used a teaching tool to explain the advance degree in the formation of acid-base, complexation and redox reactions. It has been called by him the Table of Variation of Substance Amount. In the following subsections this tool will be applied to the systems equilibrium states using the molar ratio and continuous variations

In the molar ratio method several amounts of one of the reagents, i. e. *L*, is added to another reagent, i. e. *M*, and typically the dilution of the solutions is controlled in volumetric flasks of the same capacity *VT* in order to measure a response *R* that is directly proportional to the

If all the amounts in the first column of Table 2 are divided by the *M* quantity (*nM*), and the amounts in the other columns are divided by *VT* and multiplied by the corresponding

with >> 1 *a b*

*[M L ] aM + bL M L <sup>β</sup> [M] [L]* ←⎯⎯⎯⎯→ <sup>=</sup> (27)

[ ] /) *Ma b L a b ML ML T ab ab R = k M L k (n V* = (28)

*a b ab a b*

the curves that appear in some Inorganic and Analytical Chemistry books.

**4.1 Tables of Variation of Substance Amount (TVSA)** 

reaction product, *MaLb*, through the equation:

response factor, Table 3 will be obtained.

For a system where only one reaction takes place, quantitatively:

This experiment is described in Table 2 for a quantitative reaction.

**4.1.1 Table of Variation of Substance Amount for the molar ratio method** 

−

by the low concentration of

due to the effect of the hydroxide ion over the pair *HOX*/*OX*

3.2.1.

1980.)

methods.

oxine in the system (0.001 M).


Table 2. TVSA for molar ratio method. *nM* is a constant for all the systems, *nL* is variable, all systems are in solution in thermodynamic equilibrium and the total volume is the same and equal to *VT*. *SCnL* represents the amount of *L* for the stoichiometric condition.

The functions deducted for molar ratio method and presented in Table 3, with the given restrictions, lead to represent their behaviour in Fig. 10 assuming Eq. 28 and giving *R* in arbitrary units.


Table 3. TVSA for molar ratio method expressed in terms of molar ratio of *L* (*rL*) and response (*R*) of the system. *SCrL* represents the amount of *L* for the stoichiometric condition. For this study it has been assumed that *kM* = *kL* = 0.

It should be noticed in Fig. 10 that the intersection between the two formed straight lines indicates the molar ratio corresponding to stoichiometric conditions; in other words, this value is equal to the L stoichiometric coefficient ratio divided by the M stoichiometric coefficient : *SCrL* = *b/a*.

Fig. 10. Curves *Response* = *R*= f(*rL*) for experiments of molar ratio method. a) *a* = 1 for the formation of species of the *MLb* kind. b) *a* = 2 for the formation of species of the *M2Lb* kind.

Distribution Diagrams and Graphical Methods to Determine

*a*∈ ≈

*a*∈ ≈

*L <sup>b</sup> 0 x*

≤ ≤

*b*

*b*

*a b*

+

*R*

kind.

*a b* <sup>=</sup> <sup>+</sup>

*a b*

+

*SC L*

*x*

1 *<sup>L</sup>*

factors are different enough).

**0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0**

*xL*

(a) (b)

*x*

≤ ≤

*xM+xL=1 a M + b L*

*Initial nT - nL nL* 

*kM(nT-((b-a)/b)nL)/VT b*

molar fraction is equal to the following relationship: *SCxL* = *b/(a+b)*.

 *0 b*

and response (*R*) of the system. *SCxL* represents the amount of L substance for the stoichiometric condition. For this study it has been assumed that *kM* = *kL* = 0.

Table 5. TVSA for continuous variations method expressed in terms of L molar fraction (*xL*)

It should be remarked in Fig. 11 that the intersection between the two formed straight lines indicates the molar fraction corresponding to stoichiometric conditions; in other words, this

Moreover, in this method, as it has been discussed for the molar ratio method, if some of the assumptions formulated in this work are avoided there will be some deviations of the curves shown in Fig. 11. As an example, if more species contribute to the response, the parameters of the straight line change, but their intersection still indicates the molar fraction of the stiochiometric conditions. If the formation reaction in Eq. 27 is not quantitative, there will be some deviations of the linearity near the stoichiometric conditions. Finally, if there are more species forming in the system at the same time, in principle, there will be more straight lines in the molar ratio curve, one for each species appearing in the system (if all the reactions are quantitative, if all the contributions to the response are linear and the response

*R*

Fig. 11. Curves *Response* = *R*= f(*xL*) for experiments of continuous variations method. a) *a* = 1 for the formation of species of the *MLb* kind. b) *a* = 2 for the formation of species of the *M2Lb*

*b*

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 305

∈ ≈

∈ ≈

 *0 kL(nL-(b/a)(nT-nL))/VT (kMaLb/a)(1-xL)nT/VT*

**0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0**

*xL*

*b*

*MaLb*

 *0 (kMaLb/b)xLnT/VT*

 *0 (kMaLb/(a+b))nT/VT*

If some of the assumptions formulated in this work are avoided there will be some deviations of the curves shown in Fig. 10. As an example, if more species linearly contribute to the response, the parameters of the straight lines change, but their intersection still indicates the molar ratio of the stiochiometric conditions. If the formation reaction in Eq. 27 is not quantitative there will be some deviations of the linearity near the stoichiometric conditions. Finally, if there are more species forming in the system at the same time, in principle, there will be more straight lines in the molar ratio curve, one for each species appearing in the system (if all the reactions are quantitative, if all the contributions to the response are linear and the response factors are different enough).

### **4.1.2 Tables of Variation of Substance Amount for the continuous variations method**

In the continuous variations method several amounts of the reagents, *L* and *M*, are mixed in such a way that the total amount of both reagents is constant: *nM* + *nL* = *nT* = constant. Typically, the solutions dilution is controlled in volumetric flasks of the same capacity *VT* in order to measure a response *R* that is directly proportional to the reaction product, *MaLb*, through Eq. 28. This experiment is described in Table 4 for a quantitative reaction.

Generally, in this method the equilibrium conditions are expressed in terms of the molar fraction of one of the components, *xL* or *xM*. So the *SCxL* (the component *L* molar fraction at stoichiometric conditions) can be deducted, as it is shown in Eq. 29.

$$\frac{b}{a} = \frac{\_{\text{SC}} n\_L}{n\_T - \_{\text{SC}} n\_L} \Longrightarrow \frac{b}{a} (n\_T - \_{\text{SC}} n\_L) = \_{\text{SC}} n\_L$$

$$\Rightarrow \, \_{\text{SC}} \mathbf{x}\_L = \frac{b}{a} (1 - \_{\text{SC}} \mathbf{x}\_L) \Rightarrow \, \_{\text{SC}} \mathbf{x}\_L = \frac{\frac{b}{a}}{1 + \frac{b}{a}} \tag{29}$$

$$\therefore \, \_{\text{SC}} \mathbf{x}\_L = \frac{b}{a + b}$$


*a b*

+

Table 4. TVSA for continuous variations method. *nT* is a constant for all the systems, *nM* and *nL* are variable, all systems are in solution in thermodynamic equilibrium and the total volume is the same and equal to *VT*. *SCnL* represents the amount of substance of *L* for the stoichiometric condition.

If all the amounts in the first column of Table 4 are divided by *nT*, while the amounts in the other columns are divided by *VT* and multiplied by the corresponding response factor, Table 5 will be obtained.

The functions deducted for continuous variations method and presented in Table 5, with the given restrictions, lead to represent their behaviour in Fig. 11 assuming Eq. 28 and giving *R* in arbitrary units.

If some of the assumptions formulated in this work are avoided there will be some deviations of the curves shown in Fig. 10. As an example, if more species linearly contribute to the response, the parameters of the straight lines change, but their intersection still indicates the molar ratio of the stiochiometric conditions. If the formation reaction in Eq. 27 is not quantitative there will be some deviations of the linearity near the stoichiometric conditions. Finally, if there are more species forming in the system at the same time, in principle, there will be more straight lines in the molar ratio curve, one for each species appearing in the system (if all the reactions are quantitative, if all the contributions to the

**4.1.2 Tables of Variation of Substance Amount for the continuous variations method**  In the continuous variations method several amounts of the reagents, *L* and *M*, are mixed in such a way that the total amount of both reagents is constant: *nM* + *nL* = *nT* = constant. Typically, the solutions dilution is controlled in volumetric flasks of the same capacity *VT* in order to measure a response *R* that is directly proportional to the reaction product, *MaLb*,

Generally, in this method the equilibrium conditions are expressed in terms of the molar fraction of one of the components, *xL* or *xM*. So the *SCxL* (the component *L* molar fraction at

(1 )

<sup>=</sup> − = <sup>−</sup>

*<sup>b</sup> <sup>a</sup> x xx a b*

*b*

*a b*

∈ ≈

∈ ≈

Table 4. TVSA for continuous variations method. *nT* is a constant for all the systems, *nM* and *nL* are variable, all systems are in solution in thermodynamic equilibrium and the total volume is the same and equal to *VT*. *SCnL* represents the amount of substance of *L* for the

If all the amounts in the first column of Table 4 are divided by *nT*, while the amounts in the other columns are divided by *VT* and multiplied by the corresponding response factor, Table

The functions deducted for continuous variations method and presented in Table 5, with the given restrictions, lead to represent their behaviour in Fig. 11 assuming Eq. 28 and giving *R*

+

*SC L SC L SC L*

= − =

*SC L*

∴ =

 *0 b*

*x*

( )

*T SC L SC L*

*nn n*

1

 *0 nL - (b/a)( nT - nL) (1/a)( nT -nL)* 

*b*

+

*a*

 *0 (1/b)nL*

 *0 (1/b)( SCnL)* 

*MaLb*

(29)

through Eq. 28. This experiment is described in Table 4 for a quantitative reaction.

response are linear and the response factors are different enough).

stoichiometric conditions) can be deducted, as it is shown in Eq. 29.

*nM + nL = nT a M + b L*

∈ ≈

∈ ≈

*n n L SC L* = *a*

*SC L L n n* < *a*

stoichiometric condition.

5 will be obtained.

in arbitrary units.

*Initial nT - nL nL n n L SC L* < *nT - nL - (a/b)nL b*

*SC L*

*b b n*

*T SC L*

*an n a*


Table 5. TVSA for continuous variations method expressed in terms of L molar fraction (*xL*) and response (*R*) of the system. *SCxL* represents the amount of L substance for the stoichiometric condition. For this study it has been assumed that *kM* = *kL* = 0.

It should be remarked in Fig. 11 that the intersection between the two formed straight lines indicates the molar fraction corresponding to stoichiometric conditions; in other words, this molar fraction is equal to the following relationship: *SCxL* = *b/(a+b)*.

Moreover, in this method, as it has been discussed for the molar ratio method, if some of the assumptions formulated in this work are avoided there will be some deviations of the curves shown in Fig. 11. As an example, if more species contribute to the response, the parameters of the straight line change, but their intersection still indicates the molar fraction of the stiochiometric conditions. If the formation reaction in Eq. 27 is not quantitative, there will be some deviations of the linearity near the stoichiometric conditions. Finally, if there are more species forming in the system at the same time, in principle, there will be more straight lines in the molar ratio curve, one for each species appearing in the system (if all the reactions are quantitative, if all the contributions to the response are linear and the response factors are different enough).

Fig. 11. Curves *Response* = *R*= f(*xL*) for experiments of continuous variations method. a) *a* = 1 for the formation of species of the *MLb* kind. b) *a* = 2 for the formation of species of the *M2Lb* kind.

Distribution Diagrams and Graphical Methods to Determine

*rTenox*

(a) (b)

only have been added to help follow the discussion given in the text.

**0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0**

**A b s o r b a n c e**

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9**

Fig. 14.

Hernández, 2003).

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 307

**A b s o r b a n c e**

**0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8**

Fig. 13. Spectral behaviour, at 520 nm, of the Fe(III)-tenoxicam system in acetone where different complexes are forming. [Fe(III)] = 0.001M or [Fe(III)] + [Tenox] = 0.001M. a) Molar ratio method. b) Continuous variations method. The dashed lines and the thin vertical lines

The shape of the curves presented in Fig. 13 demonstrates the formation of polynuclear species in the system, both methods clearly showing an abrupt change of slope before the molar ratio or molar fraction corresponding to 1:1 stoichiometry (*rTenox* < 1 and *xTenox* < 0.5), as it can be concluded form Fig. 10b and 11b. The other changes in slope support the possibility of the formation of several complexes; in other words, two straight lines are not enough to explain the shape of the curves. The thin vertical lines indicate the position of the molar ratio or the molar fraction that fulfil the stoichiometric conditions of the Fe(III)-tenoxicam complexes (*Fe2Tenox*, *Fe2Tenox2* and *Fe2Tenox3*) obtained in the

In order to better describe the physical-chemical behaviour observed in the molar ratio method, the distribution diagrams of discrete variable of tenoxicam in Fe(III) (for the concentration of Fe(III) species) for the thin vertical lines shown in Fig. 13a, are presented in

As it can be seen in Fig. 14, the predominant complex in the system corresponds to the molar ratio of each system, with other Fe(III) species been important as well. This is the reason why the experimental points are placed outside straight (dashed) lines in Fig. 13a.

Although some of these distribution diagrams of discrete variable could be represented for the continuous variations method, Fig. 13b may be the best graphic representation could be a distribution that consider Fe(III) and tenoxicam species at the same time, being the sum of substance amount of Fe(III) and tenoxicam (*nFe* +*nTenox* = *nT* = constant) the biggest restriction of this method. A look of this kind of distributions have been previously developed (Moya-

treatment of spectrophotometric data (Moya-Hernández et al., 2009 and Table 1).

**0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0**

*xTenox*

### **4.1.3 Some remarks for the use of Tables of Variation of Substance Amount**

The TVSA are a very powerful tool to understand the composition of thermodynamic equilibrium states when chemical reactions are involved. In the subsections 4.1.1 and 4.1.2 the dependent and independent variables are those typically selected. Nevertheless other selections could be used if the description and interpretation of chemical behaviour require it. Then, in the molar ratio method the representation of the curves could be done as a function of metal ion ratio (*rM*), while for the continuous variations method the molar fraction of metal ion (*xM*) could be used.

Furthermore, the TVSA could be used for the treatment of several complexes forming in the system at the same time.

In this work we have presented only the curves that may help interpreting the system that will be treated in the following section.

### **4.2 Application of the methods to determine the stoichiometry and equilibrium constants: The case of Fe(III)-tenoxicam system in acetone**

A spectroscopic study in the visible region of electromagnetic spectrum was undertaken to determine the complexes formed as well as their formation constants for the Fe(III) tenoxicam system in acetone, due to the red colour observed when both reagents are mixed. The solutions preparation and the details of the spectra acquisition have been given elsewhere (Moya-Hernández et al., 2009).

The absorption spectra obtained in this study are presented in Fig. 12, for molar ratio and continuous variations methods.

In order to be near the fulfilment of the conditions selected in section 4.1, (e.g. absorption only due to the complexes formed) a wavelength of 520 nm was selected to represent the typical curves of molar ratio and continuous variations, which are shown in Figure 13.

Fig. 12. Absorption spectra in the visible region for the Fe(III)-tenoxicam system in acetone. [Fe(III)] = 0.001M or [Fe(III)] + [Tenox] = 0.001M. a) Molar ratio method. b) Continuous variations method.

The TVSA are a very powerful tool to understand the composition of thermodynamic equilibrium states when chemical reactions are involved. In the subsections 4.1.1 and 4.1.2 the dependent and independent variables are those typically selected. Nevertheless other selections could be used if the description and interpretation of chemical behaviour require it. Then, in the molar ratio method the representation of the curves could be done as a function of metal ion ratio (*rM*), while for the continuous variations method the molar

Furthermore, the TVSA could be used for the treatment of several complexes forming in the

In this work we have presented only the curves that may help interpreting the system that

A spectroscopic study in the visible region of electromagnetic spectrum was undertaken to determine the complexes formed as well as their formation constants for the Fe(III) tenoxicam system in acetone, due to the red colour observed when both reagents are mixed. The solutions preparation and the details of the spectra acquisition have been given

The absorption spectra obtained in this study are presented in Fig. 12, for molar ratio and

In order to be near the fulfilment of the conditions selected in section 4.1, (e.g. absorption only due to the complexes formed) a wavelength of 520 nm was selected to represent the typical curves of molar ratio and continuous variations, which are shown in Figure 13.

> **0.00 0.13 0.25 0.50 1.00 1.50 2.00 3.00 4.00**

**A b s o r b a n c e**

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Fig. 12. Absorption spectra in the visible region for the Fe(III)-tenoxicam system in acetone. [Fe(III)] = 0.001M or [Fe(III)] + [Tenox] = 0.001M. a) Molar ratio method. b) Continuous

465 515 565 615 665

λ**/nm** **( )**

0.00 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

*xTenox*

*rTenox*

**4.2 Application of the methods to determine the stoichiometry and equilibrium** 

**constants: The case of Fe(III)-tenoxicam system in acetone** 

**4.1.3 Some remarks for the use of Tables of Variation of Substance Amount** 

fraction of metal ion (*xM*) could be used.

will be treated in the following section.

elsewhere (Moya-Hernández et al., 2009).

**485 505 525 545 565 585 605 625 645 665 685**

λ**/nm**

(a) (b)

continuous variations methods.

**0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9**

variations method.

**A b s o r b a n c e**

system at the same time.

Fig. 13. Spectral behaviour, at 520 nm, of the Fe(III)-tenoxicam system in acetone where different complexes are forming. [Fe(III)] = 0.001M or [Fe(III)] + [Tenox] = 0.001M. a) Molar ratio method. b) Continuous variations method. The dashed lines and the thin vertical lines only have been added to help follow the discussion given in the text.

The shape of the curves presented in Fig. 13 demonstrates the formation of polynuclear species in the system, both methods clearly showing an abrupt change of slope before the molar ratio or molar fraction corresponding to 1:1 stoichiometry (*rTenox* < 1 and *xTenox* < 0.5), as it can be concluded form Fig. 10b and 11b. The other changes in slope support the possibility of the formation of several complexes; in other words, two straight lines are not enough to explain the shape of the curves. The thin vertical lines indicate the position of the molar ratio or the molar fraction that fulfil the stoichiometric conditions of the Fe(III)-tenoxicam complexes (*Fe2Tenox*, *Fe2Tenox2* and *Fe2Tenox3*) obtained in the treatment of spectrophotometric data (Moya-Hernández et al., 2009 and Table 1).

In order to better describe the physical-chemical behaviour observed in the molar ratio method, the distribution diagrams of discrete variable of tenoxicam in Fe(III) (for the concentration of Fe(III) species) for the thin vertical lines shown in Fig. 13a, are presented in Fig. 14.

As it can be seen in Fig. 14, the predominant complex in the system corresponds to the molar ratio of each system, with other Fe(III) species been important as well. This is the reason why the experimental points are placed outside straight (dashed) lines in Fig. 13a.

Although some of these distribution diagrams of discrete variable could be represented for the continuous variations method, Fig. 13b may be the best graphic representation could be a distribution that consider Fe(III) and tenoxicam species at the same time, being the sum of substance amount of Fe(III) and tenoxicam (*nFe* +*nTenox* = *nT* = constant) the biggest restriction of this method. A look of this kind of distributions have been previously developed (Moya-Hernández, 2003).

Distribution Diagrams and Graphical Methods to Determine

*Chemistry*, Vol.37, (July 2007), pp. 269-301.

University of Athens, Greece, Available from

http://www2.iq.usp.br/docente/gutz/Curtipot\_.html

Aqueuses, Masson, Paris, France

Chichester, UK

932-933.

York, USA

pp389.html

*Analytical Chemistry*, Vol.41, (May 2011), pp. 151-187.

Fleck, G. M. (1967). *Equilibrios en Disolución,* Alhambra, Madrid, Spain

Chemistry, University of Sao Paulo, Brazil. Available from

comments.

**7. References** 

or to Use the Stoichiometric Coefficients of Acid-Base and Complexation Reactions 309

MTR-S and AR-H acknowledge PROMEP for partial financial support, through Cuerpo

RM-H and AR-H acknowledge αlfa Program of the EC and UNAM (through the 201855(B1),

AR-H acknowledge Dr. Julio Arturo Soto-Guerrero, from 3M company, for helpful

Asuero, A. G. Buffer Capacity of a Polyprotic Acid: First Derivative of the Buffer Capacity

Asuero, A. G. & Michałowski, T. Comprehensive Formulation of Titration Curves for

Charlot, G. (1969). Cours de Chimie Analytique Générale, Vol.1. Solutions Aqueuses et Non

Efstathiou, C. E. (2000). Acid-base titration curves, Applet, Department of Chemistry,

Gutz, I. G. R. (September 2011). Simulator: CurTiPot. Version 3.5.4, MS Excel. Institute of

Hartley, F. R., Burgess, C. & Alcock R. M. (1980). *Solution Equilibria*, Ellis Horwood,

Högfeldt, E. (1979). Chapter 15, In: *Treatise on Analytical Chemistry*, Kolthoff, I. M., Elving, P. J. Part 1, Vol.2, Section D, Interscience, ISBN: 978-047-1055-10-5, New York, USA King, D. W.; Kester, D. R. (1990). A General Approach for Calculating Polyprotic Acid

Kreyszig, E. (1970). *Introductory Mathematical Statistics: Principles and Methods*, Wiley, New

Moya-Hernández, R.; Rueda-Jackson, J. C.; Ramírez-Silva, M. T.; Vázquez-Coutiño, G. A.;

*Journal of Chemical Education*, Vol.79, No.3, (March 2002), pp. 389-392. Moya-Hernández, R.; Rueda-Jackson, J. C.; Ramírez-Silva, M. T.; Vázquez-Coutiño, G. A.;

Material for Journal of Chemical Education On Line, Available form http://jchemed.chem.wisc.edu/Journal/Issues/2002/Mar/PlusSub/JCESupp/su

Speciation and Buffer Capacity. *Journal of Chemical Education*, Vol.67, No. 11, pp.

Havel, J. & Rojas-Hernández, A. (2002a). Statistical Study of Distribution Diagrams for Two-component Systems: Relationships of Means and Variances of the Discrete Variable Distributions with Average Ligand Number and Intrinsic Buffer Capacity.

Havel, J. & Rojas-Hernández, A. (2002b). Statistical Study of Distribution Diagrams for Two-component Systems: Relationships of Means and Variances of the Discrete Variable Distributions with Average Ligand Number and Intrinsic Buffer Capacity. *Journal of Chemical Education*, Vol.79, No.3, (March 2002), pp. 389-392. Supplemental

http://www.chem.uoa.gr/Applets/AppletTitration/Appl\_Titration2.html

and p*Ka* Values of Single and Overlapping Equilibria. *Critical Reviews in Analytical* 

Complex Acid-Base Systems and Its Analytical Implications. *Critical Reviews in* 

Académico de Química Analítica (CA-UAMI-33), to develop this study.

PAPIIT-IN222011and PACIVE GC-11 projects) for partial financial support.

(c)

Fig. 14. Distribution diagrams of concentration of tenoxicam in Fe(III), for Fe(III) species, of some of the systems related with Fig. 13a. a) *rTenox* = 0.5. b) *rTenox* = 1.0. c) *rTenox* = 1.5.

### **5. Conclusion**

In this work, some novelties related with the distribution diagrams, understood as statistical distributions of discrete variables, have been presented. We have extended this vision for two component systems. Moreover, the fractions of the distribution diagrams have been used to develop analytical equations to calculate exactly pH-metric titration curves as well as their first derivatives and a buffer capacity with effect of dilution. Finally the molar ratio and continuous variations methods have been reviewed, according to Charlot's methods, and explained with the discrete variables statistical distributions for complexation systems.

### **6. Acknowledgments**

NR-L wants to acknowledge CONACyT for the stipend to follow PhD studies. MTR-S, RM-H and AR-H are in debt with SNI by the stipend and recognition like National Researchers. MTR-S and AR-H acknowledge PROMEP for partial financial support, through Cuerpo Académico de Química Analítica (CA-UAMI-33), to develop this study.

RM-H and AR-H acknowledge αlfa Program of the EC and UNAM (through the 201855(B1), PAPIIT-IN222011and PACIVE GC-11 projects) for partial financial support.

AR-H acknowledge Dr. Julio Arturo Soto-Guerrero, from 3M company, for helpful comments.

### **7. References**

308 Stoichiometry and Research – The Importance of Quantity in Biomedicine

0 0.2 0.4 0.6

**Fe**

**Fe2pirox3**

*Fe2Tenox3*

pirox/Fe

φFeipiroxj

0.8

i

i

**Fe2pirox3**

*Fe2Tenox3*

2

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>1</sup>

**Fe2pirox**

**Fe2pirox2**

*Fe2Tenox2*

j

*Fe Fe 2Tenox*

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>1</sup> <sup>2</sup>

**Fe2pirox**

*Fe2Tenox*

i j

(a) (b)

0 0.1 0.2 0.3 0.4 0.5 0.6

pirox/Fe

φFeipiroxj

**Fe2pirox2**

*Fe2Tenox2*

**Fe2pirox3** *Fe2Tenox3*

<sup>0</sup> <sup>1</sup>

some of the systems related with Fig. 13a. a) *rTenox* = 0.5. b) *rTenox* = 1.0. c) *rTenox* = 1.5.

**Fe**

2

(c) Fig. 14. Distribution diagrams of concentration of tenoxicam in Fe(III), for Fe(III) species, of

In this work, some novelties related with the distribution diagrams, understood as statistical distributions of discrete variables, have been presented. We have extended this vision for two component systems. Moreover, the fractions of the distribution diagrams have been used to develop analytical equations to calculate exactly pH-metric titration curves as well as their first derivatives and a buffer capacity with effect of dilution. Finally the molar ratio and continuous variations methods have been reviewed, according to Charlot's methods, and explained with the discrete variables statistical distributions for complexation systems.

NR-L wants to acknowledge CONACyT for the stipend to follow PhD studies. MTR-S, RM-H and AR-H are in debt with SNI by the stipend and recognition like National Researchers.

**Fe2pirox**

*Fe2Tenox Fe*

3 1

i j

**Fe2pirox2**

*Fe2Tenox2*

2

0 0.1 0.2 0.3 0.4 0.5

**5. Conclusion** 

**6. Acknowledgments** 

**Fe**

*Fe*

pirox/Fe

φFeipiroxj


 http://jchemed.chem.wisc.edu/Journal/Issues/2002/Mar/PlusSub/JCESupp/su pp389.html

**14** 

*USA* 

**Limiting Reactants in Chemical Analysis:** 

**Constants for Selected Iron-Ligand Chelates** 

The concept of the limiting reactant in chemical reactions is one of the fundamental concepts learned by all students enrolled in introductory courses in chemistry at the high school and early undergraduate levels. Practicing chemists, and just about every technical professional involved in at least some aspect of chemistry, use the concepts of stoichiometry, and particularly the limiting reactant concept, in the workplace daily. In the practice of analytical chemistry, one makes use of an analytical method that relies upon a chemical reaction of an *analyte* (i.e., a chemical species present in a sample to be analyzed, for which concentration and other information is to be sought) with a suitable analytical reagent to generate a desired product, which yields a signal due to a change in a physical property (e.g., color, temperature, current, conductivity, etc.) of the product that is quantifiable and can be related to analyte concentration. The balanced chemical equation for the reaction must also be known. In a chemical analysis, the analyte is deliberately made the limiting reactant, with the analytical reagent added in excess, to ensure that **all** of the analyte reacts to yield the product and its signal, which in turn are proportional to the analyte concentration. The stoichiometric ratios among the analyte, analytical reagent, and product must be established

Analytical chemistry makes use of chemical reactions that reach equilibrium. The Law of Mass Action (Baird, 1999) describes the relatively fixed ratio of product to reactant concentrations at the equilibrium state. For a generic chemical reaction described by a

In Equation 1, the uppercase and lowercase letters represent the reactants and products and their stoichiometric coefficients, respectively, the square brackets represent mole/L

**1. Introduction** 

**1.1 The limiting reactant concept in chemical analysis** 

for the analytical determination to yield reliable results.

balanced equation, the Law of Mass Action is described in Equation 1:

aA �aq� + bB �aq� ⇌ cC �aq� + dD �aq� K= [C]<sup>c</sup>

**Influences of Metals and Ligands** 

Mark T. Stauffer, William E. Weller, Kimberly R. Kubas and Kelly A. Casoni

> [D]<sup>d</sup> [A]<sup>a</sup>

[B]b (1)

**on Calibration Curves and Formation** 

*University of Pittsburgh at Greensburg, Greensburg, Pennsylvania* 


## **Limiting Reactants in Chemical Analysis: Influences of Metals and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates**

Mark T. Stauffer, William E. Weller, Kimberly R. Kubas and Kelly A. Casoni *University of Pittsburgh at Greensburg, Greensburg, Pennsylvania USA* 

### **1. Introduction**

310 Stoichiometry and Research – The Importance of Quantity in Biomedicine

Moya-Hernández, R. (2003). Estudio de Especiación Química de los Fármacos

Moya-Hernández, R., Gómez-Balderas R., Mederos A., Domínguez S., Ramírez-Silva M. T. &

Puigdomenech, I. (May 2011). Make Equilibrium Diagrams Using Sophisticated Algorithms

Reichl, L. E. (1980). A *Modern Course in Statistical Physics*, University of Texas Press, Austin,

Rojas-Hernández, A.; Ramírez, M. T.; González, I. & Ibanez, J. G. Predominance-Zone

Rojas-Hernández, A. & Ramírez-Silva M. T. (2002). Modelo Termodinámico General para

Segurado, M. A. P. (2003). Extreme Values in Chemistry: Buffer capacity. *Chemical Educator*,

Skoog, D. A. West, D. M., Holler, F. J. & Crouch, S. R. (2004). Fundamentals of Analytical Chemistry, 8th Ed., Brooks/Cole, Thompson Learning, Belmont, USA Tarapčík, P., & Beinrohr, E. (2003). Implementation of a Universal Algorithm for pH

http://jchemed.chem.wisc.edu/JCEDLib/WebWare/collection/open/JCEWWOR

Urbansky, E. & Schock, M. R. (2000). Understanding, Deriving, and Computing Buffer

Van Slyke, D. D. (1922). On the Measurement of Buffer Values and on the Relationship of

Capacity. *Journal of Chemical Education*, Vol.77, pp. 1640-1644

(MEDUSA), Available from http//www.kemi.kth.se/medusa

Ringbom, A. (1963). *Complexation in Analytical Chemistry*, Wiley, New York, USA

Iztapalapa, México, D. F., Mexico.

ISBN: 970-31-0149-6, México, D. F., Mexico

Bratislava, Eslovaquia, Available from

51.

USA

1099-1105

Vol.8, pp. 22-27

Madrid, Spain

012/

Antiinflamatorios Tenoxicam y Piroxicam con Cationes Metálicos de Interés Biológico, Ph. D. Thesis, Universidad Autónoma Metropolitana, Unidad

Rojas-Hernández A. (2009). *Journal of Coordination Chemistry*, Vol.62, No.1, pp. 40-

Diagrams in Solution Chemistry. Dismutation Processes in Two-Component Systems (M-L). *Journal of Chemical Education*, Vol.72, No.12, (December 1995), pp.

Curvas de Valoración Ácido-Base de Mezclas de Sistemas Poliácidos o Polibásicos (Sin Polinucleación) Con Ácido o Base Fuertes, In: *Química Inorgánica en la UAM-Iztapalapa 2002*, López-Goerne, T. M. & Rojas-Hernández, A. Vol. 1. pp. 133-158,

Calculation into Spreadsheet and its Use in Teaching in Analytical Chemistry,

Buffer Values to the Dissociation Constant of the Buffer and the Concentration and Reaction of the Buffer Solution. *Journal of Biological Chemistry*, Vol. 52, pp. 525-570 Vicente-Pérez, S. (1985). *Química de las Disoluciones. Diagramas y Cálculos Gráficos.* Alhambra.

### **1.1 The limiting reactant concept in chemical analysis**

The concept of the limiting reactant in chemical reactions is one of the fundamental concepts learned by all students enrolled in introductory courses in chemistry at the high school and early undergraduate levels. Practicing chemists, and just about every technical professional involved in at least some aspect of chemistry, use the concepts of stoichiometry, and particularly the limiting reactant concept, in the workplace daily. In the practice of analytical chemistry, one makes use of an analytical method that relies upon a chemical reaction of an *analyte* (i.e., a chemical species present in a sample to be analyzed, for which concentration and other information is to be sought) with a suitable analytical reagent to generate a desired product, which yields a signal due to a change in a physical property (e.g., color, temperature, current, conductivity, etc.) of the product that is quantifiable and can be related to analyte concentration. The balanced chemical equation for the reaction must also be known. In a chemical analysis, the analyte is deliberately made the limiting reactant, with the analytical reagent added in excess, to ensure that **all** of the analyte reacts to yield the product and its signal, which in turn are proportional to the analyte concentration. The stoichiometric ratios among the analyte, analytical reagent, and product must be established for the analytical determination to yield reliable results.

Analytical chemistry makes use of chemical reactions that reach equilibrium. The Law of Mass Action (Baird, 1999) describes the relatively fixed ratio of product to reactant concentrations at the equilibrium state. For a generic chemical reaction described by a balanced equation, the Law of Mass Action is described in Equation 1:

$$\text{aA (aq)} + \text{bB (aq)} \rightleftharpoons \text{cC (aq)} + \text{dD (aq)} \qquad \text{K} = \frac{[\text{C}]^{\text{d}}[\text{D}]^{\text{d}}}{[\text{A}]^{\text{b}}[\text{B}]^{\text{b}}} \tag{1}$$

In Equation 1, the uppercase and lowercase letters represent the reactants and products and their stoichiometric coefficients, respectively, the square brackets represent mole/L

Limiting Reactants in Chemical Analysis: Influences of Metals

**1.2 Spectrophotometric determinations of metal ions** 

1998; Harris, 2010), which is perhaps most familiar as

**1.3 Stoichiometry of metal ion chelates** 

must follow Beer's Law.

Fe3+) (McBryde, 1964; Stookey, 1970).

at low pH.

analyte.

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 313

The deprotonation steps occur sequentially, with each step possessing its own *acid dissociation* or *acid ionization* constant, Ka. Ka values are less than unity; the smaller the magnitude of Ka, the more difficult the acidic hydrogen is to ionize. Weak bases, on the other hand, can directly chelate a metal ion such as Fe2+, and can also undergo protonation

One of the most established and widely used analytical methods employed by analytical chemists and many other technical professionals, for determination of a plethora of analytes in a wide variety of samples, is ultraviolet-visible spectrophotometry, also known as UVvisible spectrophotometry, UV-vis, spectrophotometry, or colorimetry. As this topic alone is immense in scope, and most readers are perhaps already familiar with spectrophotometry, the reader is referred to any number of references on this subject (Thomas, 1996; Skoog et al., 2007; Kellner, 1998). Most metal ions such as iron are weakly absorbing species that can be made very strongly absorbing via complexation of iron ions with an appropriate complexing agent, which is usually selective for a particular valence of iron (i.e., Fe2+ or

As with any analytical method, UV-visible spectrophotometry requires prior knowledge, e.g,, the stoichiometric ratio of metal ion to ligand, of the chemical reaction that occurs between the metal ion and its chelating ligand, equilibrium information such as formation constants, and the useful concentration range for determination of the metal analyte. Knowledge of the stoichiometry of the metal-ion complexation required for the spectrophotometric determination of a given metal ion guides the analyst in proper selection of the amount of ligand to use to ensure complete coordination of the metal ion

As this chapter will focus on the stoichiometric aspects of the spectrophotometric determination of iron (as Fe2+) using the clinical iron-marking chelators Ferene S and Ferrozine, one needs to know about Beer's Law (Thomas, 1996; Skoog et al., 2007; Kellner,

The *molar* absorptivity (ε, in L mole-1 cm-1), is a function of the wavelength used for absorbance measurements and the nature of the analyte (Harris, 2010); thus, it is actually a measure of the light-absorbing ability of the analyte in its absorbing form. The term b is the path length of the cuvet or sample container used for absorbance measurements. For the most sensitive results, the wavelength of maximum absorption (λmax) should be used in a

Perhaps the best known and most widely used spectrophotometric method for studying metal-ion complex stoichiometry is the *method of continuous variations* (Harvey & Manning, 1950). Also known as the *Job method* (Job, 1928), it is best suited for metal-ion complex systems in which only one complex predominates. Additionally, the chelate in question

spectrophotometric analysis, to yield the maximum absorptivity possible.

A = εbC (10)

concentrations of the species at equilibrium, and K is the *equilibrium constant*, which is the numerical value of the product-to-reactant concentration ratio. The magnitude of the equilibrium constant K indicates the extent of conversion of reactants to products upon attainment of equilibrium. As K values are generally either greater than or less than unity, a K value of 1016 is considered very large and characteristic of a reaction that goes essentially to completion (i.e., the equilibrium favors the products). Conversely, K = 10-5 represents a reaction in which there is partial formation of products and most of the reactants still intact (i.e., the equilibrium favors the reactants) (Brown et al., 2006). Metal-ion chelation reactions have values of K that are generally much greater than unity, indicating that such reactions proceed spontaneously, without energy assistance from outside (Harris, 2010)

Complexation of a metal ion Mn+ with a coordinating ligand L, both in aqueous solution, proceeds sequentially in a manner analogous to that described by Equations 2-4:

$$\text{M}^{n^{+}}\text{ (aq)} + \text{L (aq)} \rightleftharpoons \text{ML}^{n^{+}}\text{ (aq)}\qquad \text{K}\_{\text{fl}} = \frac{[\text{ML}^{n^{+}}]}{[\text{M}^{n^{+}}][\text{L}]}\tag{2}$$

$$\text{ML}^{n+}\text{ (aq)} + \text{L (aq)} \rightleftharpoons \text{ML}\_2^{n+}\text{ (aq)}\qquad \text{K}\_{\text{f2}} = \frac{[\text{ML}\_2^{n+}]}{[\text{ML}^{n+}][\text{L}]}\tag{3}$$

$$\text{ML}\_2^{n+}\text{ (aq)} + \text{L (aq)} \rightleftharpoons \text{ML}\_3^{n+}\text{ (aq)}\qquad \text{K}\_\text{f3} = \frac{[\text{ML}\_3^{n+}]}{[\text{ML}\_2^{n+}][\text{L}]}\tag{4}$$

Each step in the overall complexation process reaches equilibrium and thus has an associated equilibrium constant, Kf, the *formation* or *stability* constant, the magnitude of which denotes the strength or tightness of binding of Mn+ by L (Butler, 1998). The stepwise coordination sequence described by Equations 2-4 may be described as well in terms of summary, or *overall*, formation constants, described by Equations 5–7, and denoted by the lowercase Greek βx, in which x is the number of individual complex-forming reaction steps that contribute to the overall complexation reaction (Butler, 1998):

$$\text{M}^{n+}\text{ (aq)} + \text{L (aq)} \rightleftharpoons \text{ML}^{n+}\text{ (aq)}\qquad \text{ }\beta\_1 = \frac{[\text{ML}^{n+}]}{[\text{M}^{n+}][\text{L}]}\tag{5}$$

$$\text{M}^{\text{n}^+} \text{ (aq)} + 2 \text{ L (aq)} \rightleftharpoons \text{ML}\_2^{\text{n}^+} \text{ (aq)} \qquad \text{ } \beta\_2 = \frac{[\text{ML}\_2^{\text{n}^+}]}{[\text{M}^{\text{n}^+}][\text{L}]^2} \tag{6}$$

$$\text{M}^{\text{n}^+} \text{ (aq)} + 3 \text{ L (aq)} \rightleftharpoons \text{ML}^{\text{n}^+} \text{ (aq)} \qquad \text{\beta}\_3 = \frac{[\text{ML}^{\text{n}^+}]}{[\text{M}^{\text{n}^+}] [\text{L}]^3} \tag{7}$$

Ligands that are weak acids can undergo dissociation of their acidic hydrogens during metal-ion complexation, in a manner similar to that shown in Equations 8 and 9 for the diprotic weak acid Tiron (1,2-dihydroxybenzene-3,5-disulfonic acid disodium salt, Na2H2L) (Cheng et al., 1992):

$$\text{H}\_2\text{L}^{2\*}\text{ (aq)} \rightleftharpoons \text{H}^\*\text{(aq)} + \text{HL}^\*\text{ (aq)}\qquad \text{K}\_{\text{al}} = \frac{[\text{H}^\*][\text{HL}^\*]}{[\text{H}\_2\text{L}^2]}\tag{8}$$

$$\text{HL}^-\text{ (aq)} \rightleftharpoons \text{H}^+\text{(aq)} + \text{L}^{2-}\text{ (aq)}\qquad\qquad \text{K}\_{\text{a2}} = \frac{[\text{H}^+][\text{L}^{2-}]}{[\text{HL}^-]}\tag{9}$$

The deprotonation steps occur sequentially, with each step possessing its own *acid dissociation* or *acid ionization* constant, Ka. Ka values are less than unity; the smaller the magnitude of Ka, the more difficult the acidic hydrogen is to ionize. Weak bases, on the other hand, can directly chelate a metal ion such as Fe2+, and can also undergo protonation at low pH.

### **1.2 Spectrophotometric determinations of metal ions**

312 Stoichiometry and Research – The Importance of Quantity in Biomedicine

concentrations of the species at equilibrium, and K is the *equilibrium constant*, which is the numerical value of the product-to-reactant concentration ratio. The magnitude of the equilibrium constant K indicates the extent of conversion of reactants to products upon attainment of equilibrium. As K values are generally either greater than or less than unity, a K value of 1016 is considered very large and characteristic of a reaction that goes essentially to completion (i.e., the equilibrium favors the products). Conversely, K = 10-5 represents a reaction in which there is partial formation of products and most of the reactants still intact (i.e., the equilibrium favors the reactants) (Brown et al., 2006). Metal-ion chelation reactions have values of K that are generally much greater than unity, indicating that such reactions

Complexation of a metal ion Mn+ with a coordinating ligand L, both in aqueous solution,

Each step in the overall complexation process reaches equilibrium and thus has an associated equilibrium constant, Kf, the *formation* or *stability* constant, the magnitude of which denotes the strength or tightness of binding of Mn+ by L (Butler, 1998). The stepwise coordination sequence described by Equations 2-4 may be described as well in terms of summary, or *overall*, formation constants, described by Equations 5–7, and denoted by the lowercase Greek βx, in which x is the number of individual complex-forming reaction steps

Ligands that are weak acids can undergo dissociation of their acidic hydrogens during metal-ion complexation, in a manner similar to that shown in Equations 8 and 9 for the diprotic weak acid Tiron (1,2-dihydroxybenzene-3,5-disulfonic acid disodium salt, Na2H2L)

HL� �aq� ⇌ H+�aq� + L2- (aq) Ka2= �H+�[L2-]

n+ �aq� Kf2= [ML2

n+ �aq� Kf3= [ML3

n+ �aq� 2= [ML2

n+ �aq� 3= [ML3

(aq) Ka1= �H+�[HL-

�Mn+�[L] (2)

�MLn+�[L] (3)

n+�[L] (4)

�Mn+�[L] (5)

�Mn+�[L]2 (6)

�Mn+�[L]3 (7)

(8)

] (9)

n+]

n+] �ML2

n+]

n+]

] [H2L2-]

[HL-

proceed spontaneously, without energy assistance from outside (Harris, 2010)

proceeds sequentially in a manner analogous to that described by Equations 2-4:

<sup>M</sup>n+ �aq� + L �aq� ⇌ MLn+ �aq� Kf1= [MLn+]

n+ �aq� + L �aq� ⇌ ML3

that contribute to the overall complexation reaction (Butler, 1998):

<sup>M</sup>n+ �aq� + L �aq� ⇌ MLn+ �aq� 1= [MLn+]

MLn+ �aq� <sup>+</sup> <sup>L</sup>�aq� ⇌ ML2

<sup>M</sup>n+ �aq� + 2 L �aq� ⇌ ML2

<sup>M</sup>n+ �aq� + 3 L �aq� ⇌ ML3

H2L2- �aq� ⇌ H+�aq� + HL-

ML2

(Cheng et al., 1992):

One of the most established and widely used analytical methods employed by analytical chemists and many other technical professionals, for determination of a plethora of analytes in a wide variety of samples, is ultraviolet-visible spectrophotometry, also known as UVvisible spectrophotometry, UV-vis, spectrophotometry, or colorimetry. As this topic alone is immense in scope, and most readers are perhaps already familiar with spectrophotometry, the reader is referred to any number of references on this subject (Thomas, 1996; Skoog et al., 2007; Kellner, 1998). Most metal ions such as iron are weakly absorbing species that can be made very strongly absorbing via complexation of iron ions with an appropriate complexing agent, which is usually selective for a particular valence of iron (i.e., Fe2+ or Fe3+) (McBryde, 1964; Stookey, 1970).

As with any analytical method, UV-visible spectrophotometry requires prior knowledge, e.g,, the stoichiometric ratio of metal ion to ligand, of the chemical reaction that occurs between the metal ion and its chelating ligand, equilibrium information such as formation constants, and the useful concentration range for determination of the metal analyte. Knowledge of the stoichiometry of the metal-ion complexation required for the spectrophotometric determination of a given metal ion guides the analyst in proper selection of the amount of ligand to use to ensure complete coordination of the metal ion analyte.

As this chapter will focus on the stoichiometric aspects of the spectrophotometric determination of iron (as Fe2+) using the clinical iron-marking chelators Ferene S and Ferrozine, one needs to know about Beer's Law (Thomas, 1996; Skoog et al., 2007; Kellner, 1998; Harris, 2010), which is perhaps most familiar as

$$\mathbf{A} = \mathbf{e} \mathbf{b} \mathbf{C} \tag{10}$$

The *molar* absorptivity (ε, in L mole-1 cm-1), is a function of the wavelength used for absorbance measurements and the nature of the analyte (Harris, 2010); thus, it is actually a measure of the light-absorbing ability of the analyte in its absorbing form. The term b is the path length of the cuvet or sample container used for absorbance measurements. For the most sensitive results, the wavelength of maximum absorption (λmax) should be used in a spectrophotometric analysis, to yield the maximum absorptivity possible.

### **1.3 Stoichiometry of metal ion chelates**

Perhaps the best known and most widely used spectrophotometric method for studying metal-ion complex stoichiometry is the *method of continuous variations* (Harvey & Manning, 1950). Also known as the *Job method* (Job, 1928), it is best suited for metal-ion complex systems in which only one complex predominates. Additionally, the chelate in question must follow Beer's Law.

Limiting Reactants in Chemical Analysis: Influences of Metals

Ferrozine chelates of iron(II).

solution is calculated from Beer's Law:

Equation 15.

βn, for the complexation reaction

by use of its βn expression:

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 315

e.g., NMR, have been employed for formation constant determinations (Irwin et al., 1996). In this chapter, the focus will be on the pH-based potentiometric and spectrophotometric methods for determination of the overall formation constant β3 for the Ferene S and

The Bjerrum pH-spectrophotometric method (Billo, 2001) utilizes low values of pH to promote only partial complexation and thus permits calculation of equilibrium concentrations of the chelate, free metal ion, and free ligand, and subsequently, the overall formation constant. The procedure involves preparation of a series of solutions, all of the same volume and containing the same concentrations of metal ion and ligand, but at different low values of pH. The absorbance of each solution is measured at the wavelength of maximum absorption (λmax) for the chelate (MLn), and the chelate concentration in each

�MLn� <sup>=</sup>AMLn

From here, the equilibrium concentrations of free M ([M]) and free L in all its forms at the

From the Ka of the ligand and the pH (to yield [H+]), the concentration of free ligand in its unprotonated form ([L]) is calculated from the fraction of free L to free L' in solution (i.e., αL = [L]/[L']). For a monoprotic weak acid or a weak monobase, αL is described by

> <sup>α</sup>L = Ka Ka + [H+]

�L� = αL [L'

Having now obtained [M], [L], and [MLn], one can calculate the overall formation constant,

M �aq� + n L �aq� ⇌ MLn (aq) (17)

<sup>β</sup>n = [MLn]

The Bjerrum pH-potentiometric method makes use of a pH titration of a solution containing known concentrations of metal ion M, ligand L, and an acid. If the ligand is a weak acid, then no additional acid is necessary. If the ligand is a weak base, then either a stoichiometric amount of strong acid (e.g., HCl) is added, or the conjugate acid of the weak base ligand is used. Details on the procedures used in this type of formation constant determination are obtainable from the published literature (Rossotti & Rossotti, 1961; Martell & Motekaitis,

selected pH ([L']) are calculated according to Equations 13 and 14:

The concentration of free L is then calculated by Equation 16:

�L'

εMLnb

(12)

(15)

] (16)

�M�[L]n (18)

�M� = CM - [MLn] (13)

� = CL' - n[MLn] (14)

The Job method involves mixing aliquots of metal-ion (M) and ligand (L) solutions, with dilution to constant volume, to obtain solutions with a constant total concentration of metal ion (CM) plus ligand (CL), or CM + CL = constant. A series of solutions of the chelate are prepared, with the ratios of M and L varied but the total concentration of each solution kept constant, as depicted in Figure 1. The absorbance of each solution in the series is measured at the maximum absorbing wavelength of the chelate (λmax), followed by generation of a plot of corrected absorbance as a function of the mole fraction of L (XL) in each solution. The maximum corrected absorbance in the continuous variations plot corresponds to the stoichiometric ratio M:L, which gives the chelate formula. The corrected absorbance is calculated according to Equation 11 (Harris, 2010):

$$\mathbf{A}\_{\rm corr} = \mathbf{A}\_{\rm meas} \cdot \mathbf{A}\_{\rm M} \{ \mathbf{1} \cdot \mathbf{X}\_{\rm L} \} \cdot \mathbf{A}\_{\rm L} \{ \mathbf{X}\_{\rm L} \} \tag{11}$$

In Equation 11, Ameas is the measured absorbance of the solution, and AM and AL are the absorbances of the all-metal-ion and all-ligand solutions, respectively. XL is simply the ratio of the moles of L to the total moles of (M + L), or XL = moles L / total moles of M + L. The value of XL at the apex of the Job plot represents the stoichiometry of the complex, e.g., when XL = 0.5, the complex has a 1:1 M:L ratio.

If equimolar solutions of M and L are used, calculation of XL is then based on the volumes of stock M and stock L used in each solution. Again, the reader is referred to the published literature regarding the method of continuous variations (Harvey& Manning, 1950; Job, 1928; Harris, 2010; Vosburgh & Cooper, 1941; Foley & Anderson, 1948).

### **1.4 Formation constants of metal ion chelates**

Formation constants for metal-ion complexes provide quantitative information on the extent of coordination of a metal ion M by a ligand L. Bjerrum and coworkers (Bjerrum, 1941; Bjerrum & Nielsen, 1948) have contributed immensely to the development and implementation of potentiometric and spectrophotometric techniques for determination of formation constants of a huge number of metal-ion complexes. Other analytical methods, e.g., NMR, have been employed for formation constant determinations (Irwin et al., 1996). In this chapter, the focus will be on the pH-based potentiometric and spectrophotometric methods for determination of the overall formation constant β3 for the Ferene S and Ferrozine chelates of iron(II).

The Bjerrum pH-spectrophotometric method (Billo, 2001) utilizes low values of pH to promote only partial complexation and thus permits calculation of equilibrium concentrations of the chelate, free metal ion, and free ligand, and subsequently, the overall formation constant. The procedure involves preparation of a series of solutions, all of the same volume and containing the same concentrations of metal ion and ligand, but at different low values of pH. The absorbance of each solution is measured at the wavelength of maximum absorption (λmax) for the chelate (MLn), and the chelate concentration in each solution is calculated from Beer's Law:

$$\left[\mathrm{ML}\_{\mathrm{n}}\right] = \begin{array}{c} \mathrm{A}\_{\mathrm{ML}\_{\mathrm{n}}}\\ \mathrm{e}\_{\mathrm{ML}\_{\mathrm{n}}} \mathrm{b} \end{array} \tag{12}$$

From here, the equilibrium concentrations of free M ([M]) and free L in all its forms at the selected pH ([L']) are calculated according to Equations 13 and 14:

$$\text{[M]} = \text{C}\_{\text{M}} \text{ - [ML}\_{\text{n}}] \tag{13}$$

$$\begin{bmatrix} \mathbf{L} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{\mathbf{L}'} \ - \ \mathbf{n} \begin{bmatrix} \mathbf{M} \mathbf{L}\_{\mathbf{n}} \end{bmatrix} \tag{14}$$

From the Ka of the ligand and the pH (to yield [H+]), the concentration of free ligand in its unprotonated form ([L]) is calculated from the fraction of free L to free L' in solution (i.e., αL = [L]/[L']). For a monoprotic weak acid or a weak monobase, αL is described by Equation 15.

$$\mathbf{q}\_{\rm L} = \begin{array}{c} \mathbf{K}\_{\rm u} \\ \hline \text{K}\_{\rm u} + \begin{bmatrix} \text{H}^{+} \end{bmatrix} \end{array} \tag{15}$$

The concentration of free L is then calculated by Equation 16:

$$\begin{bmatrix} \mathbf{L} \end{bmatrix} = \begin{bmatrix} \mathbf{G}\_{\mathbf{L}} \begin{bmatrix} \mathbf{L} \end{bmatrix} \end{bmatrix} \tag{16}$$

Having now obtained [M], [L], and [MLn], one can calculate the overall formation constant, βn, for the complexation reaction

$$\text{M (aq)} + \text{ n L (aq)} \rightleftharpoons \text{ML}\_{\text{n}} \text{ (aq)}\tag{17}$$

by use of its βn expression:

314 Stoichiometry and Research – The Importance of Quantity in Biomedicine

The Job method involves mixing aliquots of metal-ion (M) and ligand (L) solutions, with dilution to constant volume, to obtain solutions with a constant total concentration of metal ion (CM) plus ligand (CL), or CM + CL = constant. A series of solutions of the chelate are prepared, with the ratios of M and L varied but the total concentration of each solution kept constant, as depicted in Figure 1. The absorbance of each solution in the series is measured at the maximum absorbing wavelength of the chelate (λmax), followed by generation of a plot of corrected absorbance as a function of the mole fraction of L (XL) in each solution. The maximum corrected absorbance in the continuous variations plot corresponds to the stoichiometric ratio M:L, which gives the chelate formula. The corrected absorbance is

 Acorr = Ameas - AM�1 - XL� - AL�XL� (11) In Equation 11, Ameas is the measured absorbance of the solution, and AM and AL are the absorbances of the all-metal-ion and all-ligand solutions, respectively. XL is simply the ratio of the moles of L to the total moles of (M + L), or XL = moles L / total moles of M + L. The value of XL at the apex of the Job plot represents the stoichiometry of the complex, e.g.,

Fig. 1**.** Preparation of solutions of metal ion M and ligand L for a continuous variation

1928; Harris, 2010; Vosburgh & Cooper, 1941; Foley & Anderson, 1948).

**1.4 Formation constants of metal ion chelates** 

If equimolar solutions of M and L are used, calculation of XL is then based on the volumes of stock M and stock L used in each solution. Again, the reader is referred to the published literature regarding the method of continuous variations (Harvey& Manning, 1950; Job,

Formation constants for metal-ion complexes provide quantitative information on the extent of coordination of a metal ion M by a ligand L. Bjerrum and coworkers (Bjerrum, 1941; Bjerrum & Nielsen, 1948) have contributed immensely to the development and implementation of potentiometric and spectrophotometric techniques for determination of formation constants of a huge number of metal-ion complexes. Other analytical methods,

calculated according to Equation 11 (Harris, 2010):

when XL = 0.5, the complex has a 1:1 M:L ratio.

experiment.

$$\beta\_n = \frac{[\text{ML}\_n]}{[\text{M}][\text{L}]^n} \tag{18}$$

The Bjerrum pH-potentiometric method makes use of a pH titration of a solution containing known concentrations of metal ion M, ligand L, and an acid. If the ligand is a weak acid, then no additional acid is necessary. If the ligand is a weak base, then either a stoichiometric amount of strong acid (e.g., HCl) is added, or the conjugate acid of the weak base ligand is used. Details on the procedures used in this type of formation constant determination are obtainable from the published literature (Rossotti & Rossotti, 1961; Martell & Motekaitis,

Limiting Reactants in Chemical Analysis: Influences of Metals

Fig. 3. The UV-visible spectra of (a) Fe(Ferene S)3

2+, 1.00 x 10-4 mol L-1.

(Stauffer, 2007).

(b) Fe(Ferrozine)3

water (Stookey, 1970).

**1.6 Purpose and aims of this chapter** 

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 317

on the analytical utility of Ferene S (Artiss et al., 1981) and its unsulfonated analogue, Ferene Triazine (Smith et al., 1984), for determination of serum iron. More recent papers (Camberlein et al., 2010) indicate that Ferene S is still widely used in biological iron determinations. A paper describing application of flow-injection spectrophotometry to the determination of aluminum and iron in potable and treated waters, using Ferene S for determination of iron, shows an environmental application of Ferene S (Benson & Worsfold, 1993). There is also a paper describing the pedagogical use of a number of ferroin ligands, Ferene S and Ferrozine among them, for teaching Beer's Law in analytical chemistry

Ferrozine has experienced extensive application inside and outside the realm of biological/biomedical analysis. Ferrozine was introduced in 1970 as a potential Fe(II) chelator, and was first used extensively for determinations of iron in serum (Carter, 1971). Other applications over the years have included hemoglobin (Riemer et al., 2004), wine (Argyri et al., 2006), plants (Yamamoto et al., 2010), foods (Alexandropoulou et al., 2006), natural waters (Viollier et al., 2000), and a host of other sample types. Ferrozine is a trade name of the Hach Chemical Company and was originally applied to iron determinations in

The purpose of this chapter is to present and emphasize the role of the metal ion as the limiting reactant in spectrophotometric determinations of metal ion analytes. For our studies of this topic, the ligands Ferene S and Ferrozine were chosen for their ability to chelate divalent iron ions (Fe(II)) strongly and selectively as well as their prevalent use as iron markers in clinical analysis. In this chapter, the effect of the concentration ratio of Fe(II) to ligand on the Beer's Law behavior of calibration curves for the determination of iron will be presented and discussed. Along with considerations of Fe(II):ligand ratio effects on calibration curves, the stoichiometries of the Ferene S and Ferrozine chelates of Fe(II) will be examined in light of the results obtained from the calibration curve study. This chapter will explore our attempts to determine formation constants for the Fe(II)-Ferene S and Fe(II)-

2+, 1.00 x 10-4 mol L-1, and

1992). The calculations involved in obtaining formation constants (usually stepwise formation constants, or Kf values) for the complex MLn are extensive and involved. The reader is referred to appropriate texts on the pH-potentiometric method and its variations (Rossotti & Rossotti, 1961; Martell & Motekaitis, 1992).

### **1.5 The ligands Ferene S and Ferrozine in the spectrophotometric determination of iron, and their role as iron markers in clinical analysis**

Two organic molecules that are known chelating agents for iron in its divalent state, Ferene S and Ferrozine, have been selected by the authors for a study of the effects of the iron-toligand ratio on the linearity of calibration curves produced for the spectrophotometric determination of iron by these two chelators, and on the determination of formation constants for the iron chelates of the two ligands.

Ferene S (3-(2-pyridyl)-5,6-difurylsulfonic acid-1,2,4-triazine disodium salt, Figure 2a) and Ferrozine (3-(2-pyridyl)-5,6-bis(4-phenylsulfonic acid)-1,2,4-triazine disodium salt, Figure 2b) are members of a class of pyridyl- and triazine-containing molecules referred to as "ferroins" due to their renowned capacity to bind with iron in its divalent oxidation state (Fe2+) (Almog et al., 1996). Other members of this class of well-known iron(II) chelators include 1,10-phenanthroline, 2,2'-bipyridine, and 2,4,6-tripyridyl-1,3,5-triazine (TPTZ). Both Ferene S and Ferrozine are structurally analogous except for the different groups attached to the 5- and 6- positions on the triazine ring. The iron(II) chelates of both ligands possess high molar absorptivities – 3.55 x 104 L mole-1 cm-1 (λmax = 593 nm) for Fe(II)-Ferene S (Higgins, 1981), and 2.79 x 104 L mole-1 cm-1 (λmax = 562 nm) for Fe(II)-Ferrozine (Stookey, 1970) – making the two ligands ideal for sub-μg-per-mL iron determinations, particularly Ferene S. The UV-vis spectra of the Fe(II) chelates of Ferene S and Ferrozine are shown in Figure 3. Neither Ferene S nor Ferrozine absorbs in the visible region. Both compounds are sulfonated derivatives of their parent ferroins, making both molecules water-soluble and thus desirable for analytical applications involving quantitation of iron in aqueous media, and they are commercially available from major chemicals suppliers.

Fig. 2. The iron(II)-chelating ferroin-type ligands (a) Ferene S and (b) Ferrozine.

Ferene S and Ferrozine are strong, effective, and highly selective chelators for iron as the divalent cation Fe2+. Both molecules, particularly Ferene S, are well known iron markers in clinical determinations of iron in blood cells, serum, and a host of biological and biomedical sample types. The first account of the use of Ferene S as an iron(II) chelating agent for determination of iron in serum was published in 1981 (Higgins, 1981), followed by papers on the analytical utility of Ferene S (Artiss et al., 1981) and its unsulfonated analogue, Ferene Triazine (Smith et al., 1984), for determination of serum iron. More recent papers (Camberlein et al., 2010) indicate that Ferene S is still widely used in biological iron determinations. A paper describing application of flow-injection spectrophotometry to the determination of aluminum and iron in potable and treated waters, using Ferene S for determination of iron, shows an environmental application of Ferene S (Benson & Worsfold, 1993). There is also a paper describing the pedagogical use of a number of ferroin ligands, Ferene S and Ferrozine among them, for teaching Beer's Law in analytical chemistry (Stauffer, 2007).

Fig. 3. The UV-visible spectra of (a) Fe(Ferene S)3 2+, 1.00 x 10-4 mol L-1, and (b) Fe(Ferrozine)32+, 1.00 x 10-4 mol L-1.

Ferrozine has experienced extensive application inside and outside the realm of biological/biomedical analysis. Ferrozine was introduced in 1970 as a potential Fe(II) chelator, and was first used extensively for determinations of iron in serum (Carter, 1971). Other applications over the years have included hemoglobin (Riemer et al., 2004), wine (Argyri et al., 2006), plants (Yamamoto et al., 2010), foods (Alexandropoulou et al., 2006), natural waters (Viollier et al., 2000), and a host of other sample types. Ferrozine is a trade name of the Hach Chemical Company and was originally applied to iron determinations in water (Stookey, 1970).

### **1.6 Purpose and aims of this chapter**

316 Stoichiometry and Research – The Importance of Quantity in Biomedicine

1992). The calculations involved in obtaining formation constants (usually stepwise formation constants, or Kf values) for the complex MLn are extensive and involved. The reader is referred to appropriate texts on the pH-potentiometric method and its variations

**1.5 The ligands Ferene S and Ferrozine in the spectrophotometric determination of** 

Two organic molecules that are known chelating agents for iron in its divalent state, Ferene S and Ferrozine, have been selected by the authors for a study of the effects of the iron-toligand ratio on the linearity of calibration curves produced for the spectrophotometric determination of iron by these two chelators, and on the determination of formation

Ferene S (3-(2-pyridyl)-5,6-difurylsulfonic acid-1,2,4-triazine disodium salt, Figure 2a) and Ferrozine (3-(2-pyridyl)-5,6-bis(4-phenylsulfonic acid)-1,2,4-triazine disodium salt, Figure 2b) are members of a class of pyridyl- and triazine-containing molecules referred to as "ferroins" due to their renowned capacity to bind with iron in its divalent oxidation state (Fe2+) (Almog et al., 1996). Other members of this class of well-known iron(II) chelators include 1,10-phenanthroline, 2,2'-bipyridine, and 2,4,6-tripyridyl-1,3,5-triazine (TPTZ). Both Ferene S and Ferrozine are structurally analogous except for the different groups attached to the 5- and 6- positions on the triazine ring. The iron(II) chelates of both ligands possess high molar absorptivities – 3.55 x 104 L mole-1 cm-1 (λmax = 593 nm) for Fe(II)-Ferene S (Higgins, 1981), and 2.79 x 104 L mole-1 cm-1 (λmax = 562 nm) for Fe(II)-Ferrozine (Stookey, 1970) – making the two ligands ideal for sub-μg-per-mL iron determinations, particularly Ferene S. The UV-vis spectra of the Fe(II) chelates of Ferene S and Ferrozine are shown in Figure 3. Neither Ferene S nor Ferrozine absorbs in the visible region. Both compounds are sulfonated derivatives of their parent ferroins, making both molecules water-soluble and thus desirable for analytical applications involving quantitation of iron in aqueous media, and they are

(Rossotti & Rossotti, 1961; Martell & Motekaitis, 1992).

constants for the iron chelates of the two ligands.

commercially available from major chemicals suppliers.

Fig. 2. The iron(II)-chelating ferroin-type ligands (a) Ferene S and (b) Ferrozine.

Ferene S and Ferrozine are strong, effective, and highly selective chelators for iron as the divalent cation Fe2+. Both molecules, particularly Ferene S, are well known iron markers in clinical determinations of iron in blood cells, serum, and a host of biological and biomedical sample types. The first account of the use of Ferene S as an iron(II) chelating agent for determination of iron in serum was published in 1981 (Higgins, 1981), followed by papers

**iron, and their role as iron markers in clinical analysis** 

The purpose of this chapter is to present and emphasize the role of the metal ion as the limiting reactant in spectrophotometric determinations of metal ion analytes. For our studies of this topic, the ligands Ferene S and Ferrozine were chosen for their ability to chelate divalent iron ions (Fe(II)) strongly and selectively as well as their prevalent use as iron markers in clinical analysis. In this chapter, the effect of the concentration ratio of Fe(II) to ligand on the Beer's Law behavior of calibration curves for the determination of iron will be presented and discussed. Along with considerations of Fe(II):ligand ratio effects on calibration curves, the stoichiometries of the Ferene S and Ferrozine chelates of Fe(II) will be examined in light of the results obtained from the calibration curve study. This chapter will explore our attempts to determine formation constants for the Fe(II)-Ferene S and Fe(II)-

Limiting Reactants in Chemical Analysis: Influences of Metals

ligand for each calibration set is given in Table 1.

**Highest Fe Concentration in Set (mole Fe L-1)** 

1 1.00 x 10-4 5.00 x 10-5 1 : 0.5 2 1.00 x 10-4 1.00 x 10-4 1 : 1 3 1.00 x 10-4 2.00 x 10-4 1 : 2 4 1.00 x 10-4 5.00 x 10-4 1 : 5 5 1.00 x 10-4 1.00 x 10-3 1 : 10 6 1.00 x 10-4 2.00 x 10-3 1 : 20

Table 1. Ratios of Fe:ligand used in the calibration curve experiments involving the Ferene S and Ferrozine chelates of Fe(II). The ratio is based upon the highest Fe concentration in the set and the ligand concentration (which is kept constant in all the solutions in the set).

For each calibration set, an absorbance-concentration plot was generated, and the Beer's Law region was subjected to linear regression to obtain the calibration slope and the molar absorptivity (ελ). The absorbance data used were those measured at 593 nm (vs. H2O) for the

Fe(II)-Ferene S chelate, and at 562 nm (vs. H2O) for the Fe(II)-Ferrozine chelate.

**Concentration of Ligand Used in Set (mole ligand L-1)** 

**Ratio of Fe to Ligand** 

**Calibration Solution Set Number** 

**2.2 Experimental procedures** 

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 319

solutions (1000 mg Fe L-1 or 1.79 x 10-2 mole Fe L-1, Fisher Scientific, Inc.) were used to prepare aqueous Fe working standard solutions (100 mg Fe L-1 or 1.79 x 10-3 mole Fe L-1), according to established procedures (McBride, 1980). Calibration standards ranging from 0.05 – 10 mg Fe L-1 (or approximately 10-6 – 10-4 mole Fe L-1) were prepared from the working standard. The 1 mole L-1 NaOH titrant solution for the pH titrations of the ligands

A series of six sets of calibration solutions, each containing at least one reagent blank and ten Fe-containing calibration solutions (2.00 x10-6, 4.00 x10-6, 6.00 x10-6, 8.00 x10-6, 1.00 x10- 5, 2.00 x10-5, 4.00 x10-5, 6.00 x10-5, 8.00 x10-5, 1.00 x10-4 mole Fe L-1), were prepared according to a series of steps common to the colorimetric determination of iron by ferroin ligands (McBride, 1980). These steps are: (1) pipetting an appropriate aliquot of Fe working standard into a small (30-60 mL) cup, beaker, or equivalent containing a small (5- 10 mL) amount of distilled or deionized water; (2) pipetting fixed aliquots of ligand (1.00 mL of Ferene S or Ferrozine) solution, Fe(III) reducing agent (hydroquinone, 1.00 mL) solution, and buffering (sodium acetate, 2.00 mL) solution; (3) pH adjustment of the analysis solution to pH 5.0 ± 0.05 for optimum color formation (for both Ferene S and Ferrozine chelates of Fe(II)); (4) quantitative transfer of the analysis solution to a small (e.g., 25-mL) volumetric flask and dilution to volume, followed by spectrophotometric determination of iron, with absorbance measurements at 591, 593, and 595 nm (vs. H2O) for the Fe(II)-Ferene S chelate, and 560, 562, and 564 nm (vs. H2O) for the Fe(II)-ferrozine chelate. For each set of calibration solutions, the ligand concentration was based on the highest Fe concentration in the set (i.e., 1.00 x 10-4 mole Fe L-1), increased from one set to the next, and kept constant throughout each calibration solutions set. The ratio of Fe to

and their Fe(II) chelates was standardized against potassium hydrogen phthalate.

**2.2.1 Calibration curve study for Fe(II)-Ferene S and Fe(II)-Ferrozine chelates** 

Ferrozine chelates by spectrophotometry and potentiometry, and to gain insights into the binding of Fe(II) by each of the two ligands. The chapter will conclude with presentation and discussion of an application of Ferene S to the spectrophotometric determination of iron outside the realm of clinical analysis, e.g., in abandoned mine drainage (AMD) and other natural waters.

## **2. Experimental section**

### **2.1 Equipment, reagents, solutions**

### **2.1.1 Equipment and reagents**

Spectral measurements were made with a Hitachi Model U-3010 scanning double-beam UVvisible spectrophotometer (Hitachi High Technologies America, San Diego, CA, USA) and 1.00-cm quartz cuvets (Fisher Scientific, Inc., Pittsburgh, PA, USA), a battery-operated WPA Colourwave 7500B eight-filter colorimeter and 1.00-cm plastic cuvets (Biochrom, Ltd., Cambridge, UK), and a Vernier SpectroVis diode-array spectrophotometer plus batteryoperated LabQuest microprocessor with software (for operation of the SpectroVis spectrophotometer), and 1.00-cm plastic cuvets (Vernier Software & Technology, Beaverton, OR, USA). A Perkin-Elmer AAnalyst 100 flame atomic absorption spectrophotometer with air-acetylene flame, 10-cm single-slot burner head, and Lumina iron hollow cathode lamp Model No. N3050126 (Perkin-Elmer Corporation, Norwalk, CT, USA), was used for verification of total iron concentration in some field determinations of iron. Adjustments of pH for optimum color formation of the Fe2+-Ferene S and Fe2+-Ferrozine chelate solutions, and for carrying out pH titrations of ligand and chelate solutions, were made with an Oakton pHTestr 30 handheld combination electrode pH meter with built-in temperature probe (Fisher Scientific, Inc.). Temperature control for the spectrophotometric and potentiometric formation constant experiments was achieved using an Isotemp Model 2340 programmable constant temperature water bath (Fisher Scientific, Inc.).

Ferene S (3-(2-pyridyl)-5,6-difurylsulfonic acid-1,2,4-triazine disodium salt) and Ferrozine (3-(2-pyridyl)-5,6-bis(4-phenylsulfonic acid)-1,2,4-triazine disodium salt) were purchased from GFS Chemicals, Inc. (Powell, OH, USA) and Sigma-Aldrich (St. Louis, MO, USA) and used without further purification. Other reagents include hydroquinone (C4H6O2, reducing agent for Fe3+), sodium acetate trihydrate (NaC2H3O2 **.** 3H2O, pH buffering agent), iron(II) ammonium sulfate hexahydrate (Fe(NH4)2(SO4)2 **.** 6H2O), potassium hydrogen phthalate (KHC8H4O4), concentrated hydrochloric acid (12 mole HCl L-1), sodium hydroxide (NaOH), 1000 mg Fe L-1 stock standard solution, and NIST-traceable pH 5.00 buffer solution (Fisher Scientific, Inc.).

### **2.1.2 Solution preparation**

Aqueous solutions of hydroquinone were prepared at 2-3% (w/v) concentration. Aqueous solutions of sodium acetate were prepared at 2.0 mole NaC2H3O2 L-1 concentration. Dilute (1 mole L-1) solutions of HCl and NaOH were prepared from reagent grade concentrated HCl and NaOH pellets, respectively. Solutions of the water-soluble Ferene S and Ferrozine were easily prepared. Aqueous stock solutions of Ferene S and Ferrozine were prepared at concentrations between 5.00 x 10-3 to 5.00 x 10-2 mole ligand L-1. Standard aqueous Fe stock

solutions (1000 mg Fe L-1 or 1.79 x 10-2 mole Fe L-1, Fisher Scientific, Inc.) were used to prepare aqueous Fe working standard solutions (100 mg Fe L-1 or 1.79 x 10-3 mole Fe L-1), according to established procedures (McBride, 1980). Calibration standards ranging from 0.05 – 10 mg Fe L-1 (or approximately 10-6 – 10-4 mole Fe L-1) were prepared from the working standard. The 1 mole L-1 NaOH titrant solution for the pH titrations of the ligands and their Fe(II) chelates was standardized against potassium hydrogen phthalate.

### **2.2 Experimental procedures**

318 Stoichiometry and Research – The Importance of Quantity in Biomedicine

Ferrozine chelates by spectrophotometry and potentiometry, and to gain insights into the binding of Fe(II) by each of the two ligands. The chapter will conclude with presentation and discussion of an application of Ferene S to the spectrophotometric determination of iron outside the realm of clinical analysis, e.g., in abandoned mine drainage (AMD) and other

Spectral measurements were made with a Hitachi Model U-3010 scanning double-beam UVvisible spectrophotometer (Hitachi High Technologies America, San Diego, CA, USA) and 1.00-cm quartz cuvets (Fisher Scientific, Inc., Pittsburgh, PA, USA), a battery-operated WPA Colourwave 7500B eight-filter colorimeter and 1.00-cm plastic cuvets (Biochrom, Ltd., Cambridge, UK), and a Vernier SpectroVis diode-array spectrophotometer plus batteryoperated LabQuest microprocessor with software (for operation of the SpectroVis spectrophotometer), and 1.00-cm plastic cuvets (Vernier Software & Technology, Beaverton, OR, USA). A Perkin-Elmer AAnalyst 100 flame atomic absorption spectrophotometer with air-acetylene flame, 10-cm single-slot burner head, and Lumina iron hollow cathode lamp Model No. N3050126 (Perkin-Elmer Corporation, Norwalk, CT, USA), was used for verification of total iron concentration in some field determinations of iron. Adjustments of pH for optimum color formation of the Fe2+-Ferene S and Fe2+-Ferrozine chelate solutions, and for carrying out pH titrations of ligand and chelate solutions, were made with an Oakton pHTestr 30 handheld combination electrode pH meter with built-in temperature probe (Fisher Scientific, Inc.). Temperature control for the spectrophotometric and potentiometric formation constant experiments was achieved using an Isotemp Model 2340

Ferene S (3-(2-pyridyl)-5,6-difurylsulfonic acid-1,2,4-triazine disodium salt) and Ferrozine (3-(2-pyridyl)-5,6-bis(4-phenylsulfonic acid)-1,2,4-triazine disodium salt) were purchased from GFS Chemicals, Inc. (Powell, OH, USA) and Sigma-Aldrich (St. Louis, MO, USA) and used without further purification. Other reagents include hydroquinone (C4H6O2, reducing agent for Fe3+), sodium acetate trihydrate (NaC2H3O2 **.** 3H2O, pH buffering agent), iron(II) ammonium sulfate hexahydrate (Fe(NH4)2(SO4)2 **.** 6H2O), potassium hydrogen phthalate (KHC8H4O4), concentrated hydrochloric acid (12 mole HCl L-1), sodium hydroxide (NaOH), 1000 mg Fe L-1 stock standard solution, and NIST-traceable pH 5.00 buffer solution (Fisher

Aqueous solutions of hydroquinone were prepared at 2-3% (w/v) concentration. Aqueous solutions of sodium acetate were prepared at 2.0 mole NaC2H3O2 L-1 concentration. Dilute (1 mole L-1) solutions of HCl and NaOH were prepared from reagent grade concentrated HCl and NaOH pellets, respectively. Solutions of the water-soluble Ferene S and Ferrozine were easily prepared. Aqueous stock solutions of Ferene S and Ferrozine were prepared at concentrations between 5.00 x 10-3 to 5.00 x 10-2 mole ligand L-1. Standard aqueous Fe stock

programmable constant temperature water bath (Fisher Scientific, Inc.).

natural waters.

Scientific, Inc.).

**2.1.2 Solution preparation** 

**2. Experimental section** 

**2.1 Equipment, reagents, solutions** 

**2.1.1 Equipment and reagents** 

### **2.2.1 Calibration curve study for Fe(II)-Ferene S and Fe(II)-Ferrozine chelates**

A series of six sets of calibration solutions, each containing at least one reagent blank and ten Fe-containing calibration solutions (2.00 x10-6, 4.00 x10-6, 6.00 x10-6, 8.00 x10-6, 1.00 x10- 5, 2.00 x10-5, 4.00 x10-5, 6.00 x10-5, 8.00 x10-5, 1.00 x10-4 mole Fe L-1), were prepared according to a series of steps common to the colorimetric determination of iron by ferroin ligands (McBride, 1980). These steps are: (1) pipetting an appropriate aliquot of Fe working standard into a small (30-60 mL) cup, beaker, or equivalent containing a small (5- 10 mL) amount of distilled or deionized water; (2) pipetting fixed aliquots of ligand (1.00 mL of Ferene S or Ferrozine) solution, Fe(III) reducing agent (hydroquinone, 1.00 mL) solution, and buffering (sodium acetate, 2.00 mL) solution; (3) pH adjustment of the analysis solution to pH 5.0 ± 0.05 for optimum color formation (for both Ferene S and Ferrozine chelates of Fe(II)); (4) quantitative transfer of the analysis solution to a small (e.g., 25-mL) volumetric flask and dilution to volume, followed by spectrophotometric determination of iron, with absorbance measurements at 591, 593, and 595 nm (vs. H2O) for the Fe(II)-Ferene S chelate, and 560, 562, and 564 nm (vs. H2O) for the Fe(II)-ferrozine chelate. For each set of calibration solutions, the ligand concentration was based on the highest Fe concentration in the set (i.e., 1.00 x 10-4 mole Fe L-1), increased from one set to the next, and kept constant throughout each calibration solutions set. The ratio of Fe to ligand for each calibration set is given in Table 1.


Table 1. Ratios of Fe:ligand used in the calibration curve experiments involving the Ferene S and Ferrozine chelates of Fe(II). The ratio is based upon the highest Fe concentration in the set and the ligand concentration (which is kept constant in all the solutions in the set).

For each calibration set, an absorbance-concentration plot was generated, and the Beer's Law region was subjected to linear regression to obtain the calibration slope and the molar absorptivity (ελ). The absorbance data used were those measured at 593 nm (vs. H2O) for the Fe(II)-Ferene S chelate, and at 562 nm (vs. H2O) for the Fe(II)-Ferrozine chelate.

Limiting Reactants in Chemical Analysis: Influences of Metals

aforementioned manner.

unnecessary.

at 1.88 x 10-4 mol L-1 in both experiments.

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 321

containing 2.00 x 10-5 mole Fe(II) L-1, were prepared and analyzed for Fe in the manner just described. The Fe(II):Ferene S concentration ratios in the four series were 1:1, 1:4, 1:5, and 1:10, respectively. For the Fe(II)-Ferrozine chelate, two series made up of 7-11 solutions with the same Fe(II) concentration in each solution, the same pH ranges, and with 1:1, 1:4, 1:5, and 1:10 Fe(II):Ferrozine ratios, were prepared and analyzed for Fe in the

For the potentiometric experiments, 50.00-mL titration solutions containing 1:1, 1:2, 1:3, and 1:4 ratios of Fe(II):Ferene S, in which the Fe(II) concentration was 5 x 10-3 mol L-1 and which were prepared in a manner analogous to that described in section 2.2.3, were titrated with standard 1 mole NaOH L-1 titrant solution and measured for pH in the same manner as described in section 2.2.3. In the titrations involving Ferene S, one mole equivalent of HCl was added for every mole of Ferene S in the titration mixture. For the titrations involving Ferrozine, Fe(II):ligand ratios of 1:2 and 1:4 were used, and the Fe(II) concentration was kept

Samples of abandoned mine drainage (AMD) and other types of natural waters were analyzed for total Fe (as Fe(II)) by the spectrophotometric Ferene S method, according to the common protocol described earlier in section 2.2.1. Water samples that contained suspended solids or coarser debris were filtered by established procedures (Stauffer, 2007; Stauffer et al., 2007). If necessary, samples were digested by traditional wet ashing or microwave digestion (Eperesi et al., 2010); the digestion step was, in the bulk of determinations,

AMD samples that were investigated for Fe(II)/Fe(III) speciation were analyzed for Fe, as Fe(II) and Fe(III), by the spectrophotometric Ferene S method, in a somewhat different manner. Between 50-200 µL of water sample were used for this determination, along with 80.0 µL of 0.01-0.05 mole Ferene S L-1 solution and 80.0 µL of 2-3% (w/v) hydroquinone solution. An aliquot of pH 5.00 buffer solution (KHP/NaOH), ranging from 1640-1790 µL, was selected to bring the total volume of the analysis solution to 2000 µL (2.000 mL). The color-forming reaction was carried out in a 1.00-cm plastic cuvet, with calibrated adjustable-volume micropipettors used for transfer of solution and sample aliquots to the cuvet. Absorbance measurements were made at 590 nm (vs. H2O) with a WPA Colourwave 7500B colorimeter, or at 593 nm (vs. H2O) with a Vernier SpectroVis diode-array spectrophotometer, operated in kinetics mode. Once the desired sample volume was selected, the appropriate volume of pH 5.00 buffer solution was added first, followed by the Ferene S solution aliquot and then the sample aliquot. At this point, the solution volume in the cuvet was 1.920 mL. Upon addition of the sample to the cuvet, with brief (2-3 seconds) stirring, absorbance measurements were initiated and recorded at a predetermined time interval throughout the course of the determination. When the absorbance began to plateau after an initial increase, hydroquinone solution was added to the cuvet (with brief stirring) to reduce Fe(III) to Fe(II). Depending on the amount of Fe(III) present in the sample, an additional absorbance increase occurred, with subsequent leveling of absorbance to a fairly constant value. All recorded absorbances were corrected to a total volume of 2.000 mL for the Fe determination, with an average corrected absorbance (Acorr,1) calculated from the first plateau (due to Fe(II)) and one (Acorr,2) calculated from the second plateau (due to total Fe).

**2.2.5 Applications of Ferene S to the determination of iron in natural waters** 

Absorption spectra of Ferene S and Ferrozine, and their Fe(II) chelates, were measured from 300-800 nm (vs. H2O) using the Hitachi U-3010 spectrophotometer.

### **2.2.2 Stoichiometries of Fe(II)-Ferene S and Fe(II)-Ferrozine chelates**

The method of continuous variations was employed for determination of the stoichiometries of the two Fe(II) chelates. For each chelate, three series of solutions, of total concentrations (Fe(II) + ligand) of 5.00-6.00 x 10-5, 1.00 x 10-4, and 2.50 x 10-4 mol L-1 and containing from 11 to 15 solutions each, were prepared according to the common steps outlined in section 2.2.1 and analyzed for Fe. In each series, the moles of Fe and ligand were varied in accordance with the method of continuous variations while keeping the total moles of all solutions in the series constant. The measured absorbances (at 593 nm vs. H2O for Fe(II)-Ferene S and 562 nm vs. H2O for Fe(II)-Ferrozine) were corrected as recommended by the Job method, and the mole fraction of ligand (XL) calculated for each solution in the series. The corrected absorbances were plotted as a function of XL, and the Fe:ligand ratio for the chelates determined from the apex of each plot (Stauffer, 2007).

### **2.2.3 Determination of Ka for conjugate acids of Ferene S and Ferrozine**

The determination of acid dissociation constants (Ka) for Ferene S and Ferrozine was performed by manual pH titration. For each ligand, a 50.00-mL titration solution containing 0.01 mole ligand L-1 and 0.10 mole KCl L-1 to maintain constant ionic strength was prepared and transferred to a beaker, which was placed in a constant-temperature water bath at 20oC and allowed to stand for several minutes before the titration began. Titration of Ferrozine was carried out using the monosodium salt; titration of Ferene S was performed in the presence of a mole equivalent of HCl (per mole of Ferene S) added to the titration solution before dilution to volume. The titration solution was stirred manually (glass stirring rod) for at least 5 seconds after addition of each increment of standard NaOH, followed by measurement of the solution pH (no stirring) with a calibrated Oakton pHTestr 30 handheld pH meter, with combination glass electrode. The titration was carried out from the initial solution pH (usually pH 2.0-3.0) to pH 11.5-12.0. Between 60-100 data points were collected for each titration. The pKa of each ligand was determined graphically (Harris, 2010).

### **2.2.4 Determination of formation constants for Fe(II)-Ferene S and Fe(II)-Ferrozine chelates**

Both spectrophotometric and potentiometric methods (Billo, 2001; Martell & Motekaitis, 1992; Rossotti & Rossotti, 1961) were employed for the determination of the overall formation constant (β3) for the Fe(II)-Ferene S and Fe(II)-Ferrozine chelates.

For the spectrophotometric experiments, a series of solutions containing known, fixed concentrations of Fe(II) and ligand were prepared according to the common protocol given in section 2.2.1, except that no sodium acetate solution was added, and the pH of the solutions in the series ranged from 1.20 to 2.80 via adjustment with dilute HCl and NaOH. The solutions were placed in a constant-temperature bath at 20oC for 30-60 minutes prior to measuring the absorbance of each solution (593 nm vs. H2O for Fe(II)-Ferene S and 562 nm vs. H2O for Fe(II)-Ferrozine). For the Fe(II)-Ferene S chelate, four series of 5-11 solutions, ranging in pH from 1.20-2.80, and with each solution

Absorption spectra of Ferene S and Ferrozine, and their Fe(II) chelates, were measured from

The method of continuous variations was employed for determination of the stoichiometries of the two Fe(II) chelates. For each chelate, three series of solutions, of total concentrations (Fe(II) + ligand) of 5.00-6.00 x 10-5, 1.00 x 10-4, and 2.50 x 10-4 mol L-1 and containing from 11 to 15 solutions each, were prepared according to the common steps outlined in section 2.2.1 and analyzed for Fe. In each series, the moles of Fe and ligand were varied in accordance with the method of continuous variations while keeping the total moles of all solutions in the series constant. The measured absorbances (at 593 nm vs. H2O for Fe(II)-Ferene S and 562 nm vs. H2O for Fe(II)-Ferrozine) were corrected as recommended by the Job method, and the mole fraction of ligand (XL) calculated for each solution in the series. The corrected absorbances were plotted as a function of XL, and the Fe:ligand ratio for the chelates

The determination of acid dissociation constants (Ka) for Ferene S and Ferrozine was performed by manual pH titration. For each ligand, a 50.00-mL titration solution containing 0.01 mole ligand L-1 and 0.10 mole KCl L-1 to maintain constant ionic strength was prepared and transferred to a beaker, which was placed in a constant-temperature water bath at 20oC and allowed to stand for several minutes before the titration began. Titration of Ferrozine was carried out using the monosodium salt; titration of Ferene S was performed in the presence of a mole equivalent of HCl (per mole of Ferene S) added to the titration solution before dilution to volume. The titration solution was stirred manually (glass stirring rod) for at least 5 seconds after addition of each increment of standard NaOH, followed by measurement of the solution pH (no stirring) with a calibrated Oakton pHTestr 30 handheld pH meter, with combination glass electrode. The titration was carried out from the initial solution pH (usually pH 2.0-3.0) to pH 11.5-12.0. Between 60-100 data points were collected

for each titration. The pKa of each ligand was determined graphically (Harris, 2010).

formation constant (β3) for the Fe(II)-Ferene S and Fe(II)-Ferrozine chelates.

**2.2.4 Determination of formation constants for Fe(II)-Ferene S and Fe(II)-Ferrozine** 

Both spectrophotometric and potentiometric methods (Billo, 2001; Martell & Motekaitis, 1992; Rossotti & Rossotti, 1961) were employed for the determination of the overall

For the spectrophotometric experiments, a series of solutions containing known, fixed concentrations of Fe(II) and ligand were prepared according to the common protocol given in section 2.2.1, except that no sodium acetate solution was added, and the pH of the solutions in the series ranged from 1.20 to 2.80 via adjustment with dilute HCl and NaOH. The solutions were placed in a constant-temperature bath at 20oC for 30-60 minutes prior to measuring the absorbance of each solution (593 nm vs. H2O for Fe(II)-Ferene S and 562 nm vs. H2O for Fe(II)-Ferrozine). For the Fe(II)-Ferene S chelate, four series of 5-11 solutions, ranging in pH from 1.20-2.80, and with each solution

300-800 nm (vs. H2O) using the Hitachi U-3010 spectrophotometer.

determined from the apex of each plot (Stauffer, 2007).

**chelates** 

**2.2.2 Stoichiometries of Fe(II)-Ferene S and Fe(II)-Ferrozine chelates** 

**2.2.3 Determination of Ka for conjugate acids of Ferene S and Ferrozine** 

containing 2.00 x 10-5 mole Fe(II) L-1, were prepared and analyzed for Fe in the manner just described. The Fe(II):Ferene S concentration ratios in the four series were 1:1, 1:4, 1:5, and 1:10, respectively. For the Fe(II)-Ferrozine chelate, two series made up of 7-11 solutions with the same Fe(II) concentration in each solution, the same pH ranges, and with 1:1, 1:4, 1:5, and 1:10 Fe(II):Ferrozine ratios, were prepared and analyzed for Fe in the aforementioned manner.

For the potentiometric experiments, 50.00-mL titration solutions containing 1:1, 1:2, 1:3, and 1:4 ratios of Fe(II):Ferene S, in which the Fe(II) concentration was 5 x 10-3 mol L-1 and which were prepared in a manner analogous to that described in section 2.2.3, were titrated with standard 1 mole NaOH L-1 titrant solution and measured for pH in the same manner as described in section 2.2.3. In the titrations involving Ferene S, one mole equivalent of HCl was added for every mole of Ferene S in the titration mixture. For the titrations involving Ferrozine, Fe(II):ligand ratios of 1:2 and 1:4 were used, and the Fe(II) concentration was kept at 1.88 x 10-4 mol L-1 in both experiments.

### **2.2.5 Applications of Ferene S to the determination of iron in natural waters**

Samples of abandoned mine drainage (AMD) and other types of natural waters were analyzed for total Fe (as Fe(II)) by the spectrophotometric Ferene S method, according to the common protocol described earlier in section 2.2.1. Water samples that contained suspended solids or coarser debris were filtered by established procedures (Stauffer, 2007; Stauffer et al., 2007). If necessary, samples were digested by traditional wet ashing or microwave digestion (Eperesi et al., 2010); the digestion step was, in the bulk of determinations, unnecessary.

AMD samples that were investigated for Fe(II)/Fe(III) speciation were analyzed for Fe, as Fe(II) and Fe(III), by the spectrophotometric Ferene S method, in a somewhat different manner. Between 50-200 µL of water sample were used for this determination, along with 80.0 µL of 0.01-0.05 mole Ferene S L-1 solution and 80.0 µL of 2-3% (w/v) hydroquinone solution. An aliquot of pH 5.00 buffer solution (KHP/NaOH), ranging from 1640-1790 µL, was selected to bring the total volume of the analysis solution to 2000 µL (2.000 mL). The color-forming reaction was carried out in a 1.00-cm plastic cuvet, with calibrated adjustable-volume micropipettors used for transfer of solution and sample aliquots to the cuvet. Absorbance measurements were made at 590 nm (vs. H2O) with a WPA Colourwave 7500B colorimeter, or at 593 nm (vs. H2O) with a Vernier SpectroVis diode-array spectrophotometer, operated in kinetics mode. Once the desired sample volume was selected, the appropriate volume of pH 5.00 buffer solution was added first, followed by the Ferene S solution aliquot and then the sample aliquot. At this point, the solution volume in the cuvet was 1.920 mL. Upon addition of the sample to the cuvet, with brief (2-3 seconds) stirring, absorbance measurements were initiated and recorded at a predetermined time interval throughout the course of the determination. When the absorbance began to plateau after an initial increase, hydroquinone solution was added to the cuvet (with brief stirring) to reduce Fe(III) to Fe(II). Depending on the amount of Fe(III) present in the sample, an additional absorbance increase occurred, with subsequent leveling of absorbance to a fairly constant value. All recorded absorbances were corrected to a total volume of 2.000 mL for the Fe determination, with an average corrected absorbance (Acorr,1) calculated from the first plateau (due to Fe(II)) and one (Acorr,2) calculated from the second plateau (due to total Fe).

Limiting Reactants in Chemical Analysis: Influences of Metals

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 323

Fig. 4**.** Calibration curves for spectrophotometric determination of Fe(II) using (a) Ferene S

Table 2. Calibration slopes (L (mol Fe)-1) and corresponding Beer's Law concentration ranges (mol Fe L-1) for the calibration plots for Fe(II)-Ferene S (Figure 4a) and Fe(II)-Ferrozine

and (b) Ferrozine at varying ratios of Fe(II) to ligand.

(Figure 4b).

Blanks and calibration standards were analyzed for Fe in the same manner. The calibration factor obtained was expressed in terms of the total Fe concentration in the standard as Fe(II). The concentrations of Fe(II) and total Fe were determined from Acorr,1 and Acorr,2, respectively. The concentration of Fe(III) was determined by the difference between the total Fe and Fe(II) concentrations.

### **2.3 Data manipulations**

Data and results from the calibration curve, stoichiometry, formation constant, and Ferene S application studies were processed using Microsoft® Excel™.

### **3. Results and discussion**

### **3.1 Study of calibration curves for iron(II) chelates of Ferene S and Ferrozine**

The effect of varying Fe:ligand ratios on the Beer's Law behavior of calibration curves for colorimetric determination of Fe using Ferene S and Ferrozine is depicted by Figures 4a and 4b, respectively. For each Fe(II) complex, six sets of calibration solutions, composed of one blank and ten Fe standard solutions (2.00 x 10-6 to 1.00 x 10-4 mol Fe2+ L-1 concentrations) were prepared, and their absorbances measured, according to the common protocols outlined in section 2.2.1. The stated ratio of Fe(II) to ligand is the ratio of the concentration of the highest Fe(II) standard in the series (1.00 x 10-4 mol Fe2+ L-1, for all six calibration sets) to the concentration of ligand used for the particular set (refer to Table 1). From Figures 4a and 4b, the Beer's Law region for each calibration set is readily apparent. For both iron(II) chelates, the trends are almost identical, with the calibration set possessing the lowest ligand concentration exhibiting a shortened Beer's Law range in concentration to 1.00 x 10-5 mol Fe2+ L-1, and the set with the highest ligand concentration exhibiting linearity of absorbance with Fe(II) concentration nearly throughout the entire Fe(II) concentration range used for our studies. It becomes apparent that for equimolar concentrations of Fe(II) and ligand, the ligand, and not the analyte iron, becomes the limiting reactant at higher iron concentrations in the calibration scheme.

The calibration results shown in Figures 4a and 4b for Fe(II)-Ferene S and Fe(II)-Ferrozine, respectively, are based on absorbance measurements at 593 nm for the Ferene S chelate and at 562 nm for the Ferrozine chelate. Calibration slopes (Table 2) are the product of molar absorptivity and cuvet path length (1.00 cm), and ranged from 3.12 x 104 to 3.34 x 104 L (mol Fe2+)-1 for Fe(II)-Ferene S, and from 2.54 x 104 to 2.64 x 104 L (mol Fe2+)-1 for Fe(II)-Ferrozine. The R2 and standard error of the estimate were greater than 0.999 and less than 0.009, respectively, indicating excellent linearity of the Beer's Law regions for each calibration set. The lower limit of the Beer's Law concentration range (Table 2) for each set is an estimated limit of detection (LOD), or minimum Fe concentration detectable by the colorimetric method, and is calculated by Equation 19 (Miller & Miller, 1993):

$$\text{LOD} = \text{3s}\_{\text{bbl}} / \text{m}\_{\text{calib}} \tag{19}$$

where sblk is the standard deviation of the blank (i.e., the standard deviation of the y intercept of the Beer's Law curve), mcalib is the calibration slope, and the factor of 3 corresponds to the 99% confidence level (Miller & Miller, 1993).

Blanks and calibration standards were analyzed for Fe in the same manner. The calibration factor obtained was expressed in terms of the total Fe concentration in the standard as Fe(II). The concentrations of Fe(II) and total Fe were determined from Acorr,1 and Acorr,2, respectively. The concentration of Fe(III) was determined by the difference between the total

Data and results from the calibration curve, stoichiometry, formation constant, and Ferene S

The effect of varying Fe:ligand ratios on the Beer's Law behavior of calibration curves for colorimetric determination of Fe using Ferene S and Ferrozine is depicted by Figures 4a and 4b, respectively. For each Fe(II) complex, six sets of calibration solutions, composed of one blank and ten Fe standard solutions (2.00 x 10-6 to 1.00 x 10-4 mol Fe2+ L-1 concentrations) were prepared, and their absorbances measured, according to the common protocols outlined in section 2.2.1. The stated ratio of Fe(II) to ligand is the ratio of the concentration of the highest Fe(II) standard in the series (1.00 x 10-4 mol Fe2+ L-1, for all six calibration sets) to the concentration of ligand used for the particular set (refer to Table 1). From Figures 4a and 4b, the Beer's Law region for each calibration set is readily apparent. For both iron(II) chelates, the trends are almost identical, with the calibration set possessing the lowest ligand concentration exhibiting a shortened Beer's Law range in concentration to 1.00 x 10-5 mol Fe2+ L-1, and the set with the highest ligand concentration exhibiting linearity of absorbance with Fe(II) concentration nearly throughout the entire Fe(II) concentration range used for our studies. It becomes apparent that for equimolar concentrations of Fe(II) and ligand, the ligand, and not the analyte iron, becomes the limiting reactant at higher iron concentrations

The calibration results shown in Figures 4a and 4b for Fe(II)-Ferene S and Fe(II)-Ferrozine, respectively, are based on absorbance measurements at 593 nm for the Ferene S chelate and at 562 nm for the Ferrozine chelate. Calibration slopes (Table 2) are the product of molar absorptivity and cuvet path length (1.00 cm), and ranged from 3.12 x 104 to 3.34 x 104 L (mol Fe2+)-1 for Fe(II)-Ferene S, and from 2.54 x 104 to 2.64 x 104 L (mol Fe2+)-1 for Fe(II)-Ferrozine. The R2 and standard error of the estimate were greater than 0.999 and less than 0.009, respectively, indicating excellent linearity of the Beer's Law regions for each calibration set. The lower limit of the Beer's Law concentration range (Table 2) for each set is an estimated limit of detection (LOD), or minimum Fe concentration detectable by the colorimetric

where sblk is the standard deviation of the blank (i.e., the standard deviation of the y intercept of the Beer's Law curve), mcalib is the calibration slope, and the factor of 3

LOD = 3sblk/mcalib (19)

method, and is calculated by Equation 19 (Miller & Miller, 1993):

corresponds to the 99% confidence level (Miller & Miller, 1993).

**3.1 Study of calibration curves for iron(II) chelates of Ferene S and Ferrozine** 

application studies were processed using Microsoft® Excel™.

Fe and Fe(II) concentrations.

**3. Results and discussion** 

in the calibration scheme.

**2.3 Data manipulations** 

Fig. 4**.** Calibration curves for spectrophotometric determination of Fe(II) using (a) Ferene S and (b) Ferrozine at varying ratios of Fe(II) to ligand.


Table 2. Calibration slopes (L (mol Fe)-1) and corresponding Beer's Law concentration ranges (mol Fe L-1) for the calibration plots for Fe(II)-Ferene S (Figure 4a) and Fe(II)-Ferrozine (Figure 4b).

Limiting Reactants in Chemical Analysis: Influences of Metals

view of the structural similarities between Ferene S and Ferrozine.

the overall formation constant for Fe(Ferene S)32+ may be reasonable.

**with iron(II)** 

for Fe(Ferene S)3

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 325

Prior to determination of formation constants for the Fe(II) chelates of Ferene S and Ferrozine, determination of the pKa of each ligand was carried out by pH titration, according to the procedure described in section 2.2.3, followed by graphical analysis of the generated titration curves to obtain the pKa of each ligand from the midpoint of the buffer region (Harris, 2010). The pKa values of Ferene S and Ferrozine were found to be 3.13 and 3.21, respectively. Our experimental pKa for Ferrozine is in fairly good agreement with the published value (Thompsen & Mottola, 1984). Attempts to locate a published pKa for Ferene S were unsuccessful; the experimental pKa for Ferene S appears to be a reasonable one, in

Potentiometric determination of formation constants for Fe(Ferene S)32+ and Fe(Ferrozine)32+ proved exceedingly difficult; thus, values of the overall formation constants (β3) of both chelates were determined spectrophotometrically, as described in sections 1.4 and 2.2.4. The log10β3 values ranged from 16.30 to 16.82 (mean = 16.45 ± 0.25, %RSD = 1.5) for Fe(Ferene S)32+, and from 17.24 to 18.01 (mean = 17.46 ± 0.37, %RSD = 2.1) for Fe(Ferrozine)32+. The average log10β3 results indicate that β3 is not affected by changes in ligand concentration, as predicted by LeChatelier's Principle. The results also indicate that Ferrozine chelates Fe(II) more tightly than Ferene S. The experimental log10β3 obtained for Ferrozine differs from a published value (Thompsen & Mottola, 1984) by about 2 units; the difference is most likely due to any number of experimental factors, such as temperature control, instrument problems, or differences in the methods for determination of log10β3. Attempts to find a published value for comparison with our experimental log10β3 for Fe(Ferene S)32+ were unsuccessful. Again, the structural similarities between Ferene S and Ferrozine suggest that

Figures 6a and 6b are the absorbance-pH curves ("formation curves") associated with the spectrophotometric determination of β3 for Fe(Ferene S)32+ and Fe(Ferrozine)32+ and, as such, illustrate the formation of the two chelates at low pH, at which partial complexation is expected to occur due to competition between Fe2+ and H+ for the binding site on the ligand. For Fe(Ferene S)32+ and Fe(Ferrozine)32+, a predicted absorbance for complete chelation was calculated using Beer's Law and a cuvet path length of 1.00 cm and a Fe(II) concentration of 2.00 x 10-5 mol L-1. This absorbance is indicated in Figures 6a and 6b by a dashed line. The results provided by Figures 6a and 6b suggest that while β3 does not appreciably change due to increasing ligand concentration, the extent of complexation does. This trend is observed for both Fe(Ferene S)32+ and Fe(Ferrozine)32+, and even more so for Fe(Ferrozine)32+, as its formation curves at higher Ferrozine concentrations are closer together near the predicted absorbance for complete chelation than are the formation curves

increases beyond a value of 2.00, Fe(II) is essentially completely chelated by Ferrozine. A similar trend is observed for the 1:10 Fe(II):Ferene S formation curve. For both chelates, the 1:1 Fe(II):ligand formation curves are expected to give much lower absorbances, as the ligand now becomes the limiting reactant and only partial complexation occurs. It was noted

magenta color of Fe(Ferrozine)32+ are readily formed and apparent. This suggests tight binding of Fe(II) by the ligands and a tendency to preferentially form Fe(Ferene S)32+ and

that, even for the 1:1 Fe(II):ligand experiments, the indigo color of Fe(Ferene S)3

2+. The 1:10 Fe(II):Ferrozine formation curve in particular shows that, as pH

2+ and the

**3.3 Determination of formation constants for the chelates of Ferene S and Ferrozine** 

### **3.2 Stoichiometries of the iron(II) chelates of Ferene S and Ferrozine by the method of continuous variations**

The stoichiometries of the iron(II) chelates of Ferene S and Ferrozine were investigated by the method of continuous variations (a.k.a. Job method) according to the procedure described in section 2.2.2, and are illustrated by the Job plots of Figures 5a and 5b for Fe(II)- Ferene S and Fe(II)-Ferrozine, respectively. For all six Job plots, the Fe(II):ligand ratio corresponding to the apex of each plot is 1:3 (XL = 0.75) and is readily apparent from the shape of each plot. Thus, the stoichiometry of Fe(II)-Ferene S and Fe(II)-Ferrozine is 1:3 for each chelate, indicating compositional formulae of Fe(Ferene S)3 2+ and Fe(Ferrozine)32+. The 1:3 stoichiometry found in our studies agrees with published findings for the Ferrozine chelate (Gibbs, 1976) and those for the unsulfonated analogue of Ferene S (Smith et al., 1984).

Fig. 5. Continuous variation plots for (a) Fe(Ferene S)32+ and (b) Fe(Ferrozine)3 2+, at three different total concentrations of Fe(II) + ligand (in mol L-1).

The degree of sharpness at the apex of a continuous variations plot gives a qualitative estimate of the tightness of binding of the metal ion by the ligand in a complex, and can also be used for determination of an overall formation constant for the complex if there is sufficient rounding at the apex (Harris, 2010). The lower-concentration Job plots shown in Figures 5a and 5b appear quite rounded at their apices relative to the highest-concentration Job plots. Upon closer inspection of each Job plot individually, it was found that the lowerconcentration plots are much sharper than expected; the roundedness observed in Figures 5a and 5b may simply be a result of plotting together three continuous variations plots of differing total concentrations. Yet, in the Job plots of Figures 5a and 5b, there is slight rounding at the apex evident in all of them, which may be characteristic of the chelates themselves or due to the concentrations of Fe(II) and ligand employed in these experiments. Nonetheless, the sharp apices of the continuous variation plots of Figures 5a and 5b indicate strong coordination of Fe2+ by Ferene S and Ferrozine. The results obtained for Fe(Ferene S)3 2+ and Fe(Ferrozine)3 2+ appear to be typical of stoichiometric Fe(II):ligand ratios for other ferroins, e.g. 1,10-phenanthroline and 2,2'-bipyridine (Stauffer, 2007 and references cited therein).

### **3.3 Determination of formation constants for the chelates of Ferene S and Ferrozine with iron(II)**

324 Stoichiometry and Research – The Importance of Quantity in Biomedicine

**3.2 Stoichiometries of the iron(II) chelates of Ferene S and Ferrozine by the method of** 

The stoichiometries of the iron(II) chelates of Ferene S and Ferrozine were investigated by the method of continuous variations (a.k.a. Job method) according to the procedure described in section 2.2.2, and are illustrated by the Job plots of Figures 5a and 5b for Fe(II)- Ferene S and Fe(II)-Ferrozine, respectively. For all six Job plots, the Fe(II):ligand ratio corresponding to the apex of each plot is 1:3 (XL = 0.75) and is readily apparent from the shape of each plot. Thus, the stoichiometry of Fe(II)-Ferene S and Fe(II)-Ferrozine is 1:3 for each chelate, indicating compositional formulae of Fe(Ferene S)32+ and Fe(Ferrozine)32+. The 1:3 stoichiometry found in our studies agrees with published findings for the Ferrozine chelate (Gibbs, 1976) and those for the unsulfonated analogue of Ferene S (Smith et al.,

Fig. 5. Continuous variation plots for (a) Fe(Ferene S)32+ and (b) Fe(Ferrozine)32+, at three

The degree of sharpness at the apex of a continuous variations plot gives a qualitative estimate of the tightness of binding of the metal ion by the ligand in a complex, and can also be used for determination of an overall formation constant for the complex if there is sufficient rounding at the apex (Harris, 2010). The lower-concentration Job plots shown in Figures 5a and 5b appear quite rounded at their apices relative to the highest-concentration Job plots. Upon closer inspection of each Job plot individually, it was found that the lowerconcentration plots are much sharper than expected; the roundedness observed in Figures 5a and 5b may simply be a result of plotting together three continuous variations plots of differing total concentrations. Yet, in the Job plots of Figures 5a and 5b, there is slight rounding at the apex evident in all of them, which may be characteristic of the chelates themselves or due to the concentrations of Fe(II) and ligand employed in these experiments. Nonetheless, the sharp apices of the continuous variation plots of Figures 5a and 5b indicate strong coordination of Fe2+ by Ferene S and Ferrozine. The results obtained for Fe(Ferene

ferroins, e.g. 1,10-phenanthroline and 2,2'-bipyridine (Stauffer, 2007 and references cited

2+ appear to be typical of stoichiometric Fe(II):ligand ratios for other

different total concentrations of Fe(II) + ligand (in mol L-1).

**continuous variations** 

1984).

S)3

therein).

2+ and Fe(Ferrozine)3

Prior to determination of formation constants for the Fe(II) chelates of Ferene S and Ferrozine, determination of the pKa of each ligand was carried out by pH titration, according to the procedure described in section 2.2.3, followed by graphical analysis of the generated titration curves to obtain the pKa of each ligand from the midpoint of the buffer region (Harris, 2010). The pKa values of Ferene S and Ferrozine were found to be 3.13 and 3.21, respectively. Our experimental pKa for Ferrozine is in fairly good agreement with the published value (Thompsen & Mottola, 1984). Attempts to locate a published pKa for Ferene S were unsuccessful; the experimental pKa for Ferene S appears to be a reasonable one, in view of the structural similarities between Ferene S and Ferrozine.

Potentiometric determination of formation constants for Fe(Ferene S)3 2+ and Fe(Ferrozine)32+ proved exceedingly difficult; thus, values of the overall formation constants (β3) of both chelates were determined spectrophotometrically, as described in sections 1.4 and 2.2.4. The log10β3 values ranged from 16.30 to 16.82 (mean = 16.45 ± 0.25, %RSD = 1.5) for Fe(Ferene S)3 2+, and from 17.24 to 18.01 (mean = 17.46 ± 0.37, %RSD = 2.1) for Fe(Ferrozine)3 2+. The average log10β3 results indicate that β3 is not affected by changes in ligand concentration, as predicted by LeChatelier's Principle. The results also indicate that Ferrozine chelates Fe(II) more tightly than Ferene S. The experimental log10β3 obtained for Ferrozine differs from a published value (Thompsen & Mottola, 1984) by about 2 units; the difference is most likely due to any number of experimental factors, such as temperature control, instrument problems, or differences in the methods for determination of log10β3. Attempts to find a published value for comparison with our experimental log10β3 for Fe(Ferene S)3 2+ were unsuccessful. Again, the structural similarities between Ferene S and Ferrozine suggest that the overall formation constant for Fe(Ferene S)32+ may be reasonable.

Figures 6a and 6b are the absorbance-pH curves ("formation curves") associated with the spectrophotometric determination of β3 for Fe(Ferene S)32+ and Fe(Ferrozine)32+ and, as such, illustrate the formation of the two chelates at low pH, at which partial complexation is expected to occur due to competition between Fe2+ and H+ for the binding site on the ligand. For Fe(Ferene S)32+ and Fe(Ferrozine)32+, a predicted absorbance for complete chelation was calculated using Beer's Law and a cuvet path length of 1.00 cm and a Fe(II) concentration of 2.00 x 10-5 mol L-1. This absorbance is indicated in Figures 6a and 6b by a dashed line. The results provided by Figures 6a and 6b suggest that while β3 does not appreciably change due to increasing ligand concentration, the extent of complexation does. This trend is observed for both Fe(Ferene S)32+ and Fe(Ferrozine)3 2+, and even more so for Fe(Ferrozine)32+, as its formation curves at higher Ferrozine concentrations are closer together near the predicted absorbance for complete chelation than are the formation curves for Fe(Ferene S)32+. The 1:10 Fe(II):Ferrozine formation curve in particular shows that, as pH increases beyond a value of 2.00, Fe(II) is essentially completely chelated by Ferrozine. A similar trend is observed for the 1:10 Fe(II):Ferene S formation curve. For both chelates, the 1:1 Fe(II):ligand formation curves are expected to give much lower absorbances, as the ligand now becomes the limiting reactant and only partial complexation occurs. It was noted that, even for the 1:1 Fe(II):ligand experiments, the indigo color of Fe(Ferene S)3 2+ and the magenta color of Fe(Ferrozine)3 2+ are readily formed and apparent. This suggests tight binding of Fe(II) by the ligands and a tendency to preferentially form Fe(Ferene S)32+ and

Limiting Reactants in Chemical Analysis: Influences of Metals

base neutralizations were obtained for Fe(Ferene S)3

results obtained by the pH-spectrophotometric method.

**3.4 Application of Ferene S to iron determinations in natural waters** 

abandoned mine drainage (AMD) and other types of natural waters.

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 327

For the higher Fe(II):ligand ratios, titration curves resembling those for strong acid-strong

suggest that Fe(II) is coordinated strongly by Ferene S and Ferrozine, and support the

The results presented in this section will focus on the application of Ferene S to the spectrophotometric determination of total iron, and speciation of Fe2+ and Fe3+, in

Table 3 gives results for determination of total iron in water from Pigeon Creek (Washington County, Pennsylvania, USA) during March and October 2007. The iron concentrations (± their standard deviations) given are averages of triplicate and quadruplicate determinations, and are reported as mol Fe L-1 and mg Fe L-1. Relative precision of replicates ranged from 2.2-10.2 percent, which is fairly good for these low part-per-million Fe levels. Calibration slopes given in Table 3 are in the range expected for Fe(Ferene S)32+, with R2 values approaching unity, indicating excellent linearity of the Fe calibration range (7.16 x 10-7 – 6.00 x 10-5 mol Fe L-1, or 0.04 – 3.35 mg Fe L-1) used in these determinations. The estimated limit of detection (LOD) was calculated to be 7.2 x 10-7 mol Fe L-1 (0.04 mg Fe L-1), using the standard deviation of the y intercept obtained from linear regression of the calibration data via the LINEST function in Microsoft® Excel™ (Billo, 2001) and Equation 19 (section 3.1).

Table 3. Results from determination of total iron in water from Pigeon Creek (Washington

A precision study of the spectrophotometric Ferene S method for determination of iron in AMD was conducted by the authors on runoff water collected near an abandoned coal mine in the vicinity of Coal Bluff in Washington County, PA in October 2006. The mean total Fe concentration in this AMD sample was found to be 13.9 ± 0.09 mg Fe L-1 (n = 8 replicates), with a 95% confidence interval of ± 0.07 mg Fe L-1, suggesting good accuracy. A separate accuracy study for the Ferene S method was conducted in January 2007 in our laboratory on a series of water samples prepared in our laboratory and spiked with standard Fe solution to yield concentrations of 5.00 mg Fe L-1 in each sample. The samples were analyzed for total

County, PA, USA) in 2007, using the colorimetric Ferene S method.

2+ and Fe(Ferrozine)3

2+. These results

Fe(Ferrozine)32+, even at low pH. Additionally, a data point each in the 1:1 and 1:10 Fe(II):Ferrozine formation curves in Figure 6b is anomalous and is outside the trend exhibited by the other data points in the curve. This may be due to a pipetting error for either the stock Fe2+ or Ferrozine solutions.

Fig. 6. Absorbance-pH formation curves for the spectrophotometric determination of the overall formation constant β3 for (a) Fe(Ferene S)32+ and (b) Fe(Ferrozine)32+ at varying Fe(II):ligand ratios.

Even though the potentiometric determination of formation constants was unsuccessful, the pH titrations of Fe(Ferene S)3 2+ and Fe(Ferrozine)32+ at different ratios of Fe(II) to ligand, performed as described in section 2.2.4, yielded useful information toward understanding the complexation of Fe(II) by Ferene S and Ferrozine. Figures 7a and 7b depict the pH-vs. titrant volume curves generated by titration of the chelates with standard 1 mol NaOH L-1 titrant, at Fe(II):ligand ratios of 1:1 through 1:4 for Fe(Ferene S)32+, and ratios of 1:2 and 1:4 for Fe(Ferrozine)32+. At the lower ratios of Fe(II) to ligand, a slight stepwise curve was obtained for Fe(Ferene S)32+, and a curve with a large single equivalence point inflection, similar to that for a strong acid-strong base titration, was observed for Fe(Ferrozine)32+.

Fig. 7. pH titration curves for (a) Fe(Ferene S)32+ at Fe(II):Ferene S ratios of 1:1 through 1:4, and (b) Fe(Ferrozine)32+ at Fe(II):Ferrozine ratios of 1:2 and 1:4.

For the higher Fe(II):ligand ratios, titration curves resembling those for strong acid-strong base neutralizations were obtained for Fe(Ferene S)3 2+ and Fe(Ferrozine)3 2+. These results suggest that Fe(II) is coordinated strongly by Ferene S and Ferrozine, and support the results obtained by the pH-spectrophotometric method.

### **3.4 Application of Ferene S to iron determinations in natural waters**

326 Stoichiometry and Research – The Importance of Quantity in Biomedicine

Fe(II):Ferrozine formation curves in Figure 6b is anomalous and is outside the trend exhibited by the other data points in the curve. This may be due to a pipetting error for

Fig. 6. Absorbance-pH formation curves for the spectrophotometric determination of the overall formation constant β3 for (a) Fe(Ferene S)32+ and (b) Fe(Ferrozine)32+ at varying

Even though the potentiometric determination of formation constants was unsuccessful, the

performed as described in section 2.2.4, yielded useful information toward understanding the complexation of Fe(II) by Ferene S and Ferrozine. Figures 7a and 7b depict the pH-vs. titrant volume curves generated by titration of the chelates with standard 1 mol NaOH L-1 titrant, at Fe(II):ligand ratios of 1:1 through 1:4 for Fe(Ferene S)32+, and ratios of 1:2 and 1:4 for Fe(Ferrozine)32+. At the lower ratios of Fe(II) to ligand, a slight stepwise curve was obtained for Fe(Ferene S)32+, and a curve with a large single equivalence point inflection, similar to that for a strong acid-strong base titration, was observed for Fe(Ferrozine)32+.

Fig. 7. pH titration curves for (a) Fe(Ferene S)32+ at Fe(II):Ferene S ratios of 1:1 through 1:4,

and (b) Fe(Ferrozine)32+ at Fe(II):Ferrozine ratios of 1:2 and 1:4.

2+ and Fe(Ferrozine)32+ at different ratios of Fe(II) to ligand,

2+, even at low pH. Additionally, a data point each in the 1:1 and 1:10

Fe(Ferrozine)3

Fe(II):ligand ratios.

pH titrations of Fe(Ferene S)3

either the stock Fe2+ or Ferrozine solutions.

The results presented in this section will focus on the application of Ferene S to the spectrophotometric determination of total iron, and speciation of Fe2+ and Fe3+, in abandoned mine drainage (AMD) and other types of natural waters.

Table 3 gives results for determination of total iron in water from Pigeon Creek (Washington County, Pennsylvania, USA) during March and October 2007. The iron concentrations (± their standard deviations) given are averages of triplicate and quadruplicate determinations, and are reported as mol Fe L-1 and mg Fe L-1. Relative precision of replicates ranged from 2.2-10.2 percent, which is fairly good for these low part-per-million Fe levels. Calibration slopes given in Table 3 are in the range expected for Fe(Ferene S)32+, with R2 values approaching unity, indicating excellent linearity of the Fe calibration range (7.16 x 10-7 – 6.00 x 10-5 mol Fe L-1, or 0.04 – 3.35 mg Fe L-1) used in these determinations. The estimated limit of detection (LOD) was calculated to be 7.2 x 10-7 mol Fe L-1 (0.04 mg Fe L-1), using the standard deviation of the y intercept obtained from linear regression of the calibration data via the LINEST function in Microsoft® Excel™ (Billo, 2001) and Equation 19 (section 3.1).


Table 3. Results from determination of total iron in water from Pigeon Creek (Washington County, PA, USA) in 2007, using the colorimetric Ferene S method.

A precision study of the spectrophotometric Ferene S method for determination of iron in AMD was conducted by the authors on runoff water collected near an abandoned coal mine in the vicinity of Coal Bluff in Washington County, PA in October 2006. The mean total Fe concentration in this AMD sample was found to be 13.9 ± 0.09 mg Fe L-1 (n = 8 replicates), with a 95% confidence interval of ± 0.07 mg Fe L-1, suggesting good accuracy. A separate accuracy study for the Ferene S method was conducted in January 2007 in our laboratory on a series of water samples prepared in our laboratory and spiked with standard Fe solution to yield concentrations of 5.00 mg Fe L-1 in each sample. The samples were analyzed for total

Limiting Reactants in Chemical Analysis: Influences of Metals

Fe3+ to Fe2+.

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 329

hydroquinone. A second experiment, using 80.0 µL of 2% aqueous hydroquinone reducing agent and performed under the same conditions as before, required 3-5 minutes to achieve complete reduction of Fe3+ to Fe2+. All absorbances in both experiments were adjusted to the total solution volume in the cuvet after addition of hydroquinone. Sine wave-like fluctuations of the absorbance at the plateaus of both absorbance-time curves are attributed to insufficient mixing after addition of hydroquinone. Conversion of Fe3+ to Fe2+, expressed as the percent ratio of the concentration of Fe3+ initially present and the Fe2+ concentration determined at the end of the kinetic run (via the average absorbance at 590 nm at the plateau and Beer's Law), was found to be 103 and 107%, respectively, for the two experiments. Though the conversions are rather high, they do suggest essentially complete reduction of

Fig. 8. Determination of (a) the amount of hydroquinone reducing agent to add for conversion of 3.04 x 10-5 mol Fe3+ L-1 solution to Fe2+, and (b) the speciation of a synthetic

Further recovery experiments were conducted, involving speciation of iron in synthetic samples containing known concentrations of Fe(III) and Fe(II) and analyzed for Fe2+ by the procedures described in section 2.2.5. The purpose of these experiments was to check the iron recovery and efficiency of reduction of Fe3+ to Fe2+. Table 5 gives the added and found Fe2+, Fe3+, and total Fe concentrations (in mol Fe L-1) for three such recovery experiments, and Figure 8b illustrates the absorbance-time curve obtained for one of these recovery runs. The recoveries obtained for the forms of Fe in the samples are generally good, despite the high recoveries of 110.8 and 105.9%, and the rather low recovery of 84.3%. The recoveries suggest that complete Fe recoveries, and Fe(II)/Fe(III) speciation, are possible with this method. In Figure 8b, there is a very small leveling point prior to addition of hydroquinone, at which point apparent reduction of Fe3+ to Fe2+ was occurring, possibly promoted by

Average Fe(II) and Fe(III) results from the speciation of Fe2+ and Fe3+ in AMD samples collected from a mine runoff tributary of Gillespie Run, and analyzed for Fe by the procedure given in section 2.2.5, were found to be 33.7 ± 0.5 mg Fe2+ L-1 and 0.6 ± 0.2 mg Fe3+ L-1 for one sample, and 35.5 ± 1.0 mg Fe2+ L-1 and 0.7 ± 0.2 mg Fe3+ L-1 for a different sample. These results suggest that the form of iron in AMD is mainly Fe2+, with a very small Fe3+ concentration present. Calibration slopes associated with these determinations ranged from

sample containing 2.95 x 10-5 mol Fe2+ L-1 and 1.52 x 10-5 mol Fe3+ L-1.

Ferene S. Our group plans to further investigate this tendency.

iron (as Fe2+) by the common protocols outlined in section 2.2.1, with determination of the percent recovery of iron in each sample. Iron recoveries for the spiked water samples ranged from 100.6 – 101.4 percent, indicating essentially complete recovery of Fe and good accuracy. Additionally, our group has compared total Fe results for samples determined by the spectrophotometric Ferene S method to total Fe for the same samples determined by flame atomic absorption spectrophotometry (FAAS). Table 4 lists average Fe concentrations (in mg Fe L-1) found in water from clean and AMD-contaminated portions of Gillespie Run (Allegheny County, PA, USA), determined by the colorimetric Ferene S method and by FAAS. The results from Table 4 indicate good agreement between the spectrophotometric Ferene S and FAAS methods.


Table 4. Comparison of the colorimetric Ferene S method with flame atomic absorption spectrophotometry (FAAS) for determination of total iron in abandoned mine drainage (AMD).

Current work in our laboratory involves, among other things, the use of the spectrophotometric Ferene S method for speciation of iron (i.e., Fe2+, Fe3+, and total Fe) in runoff from abandoned mines and other natural waters. The experimental approach used in these studies is given in section 2.2.5. In the speciation experiments, a spectrophotometer or colorimeter with absorbance-versus-time measurement capability is used to monitor formation of Fe(Ferene S)32+ due to Fe2+ present in the sample. Upon chelation of Fe2+ initially present in the sample, the absorbance reaches a temporary plateau, at which time a reducing agent, e.g., hydroquinone, is added to reduce any Fe3+ present to Fe2+. Depending on the concentration of Fe3+, the absorbance at 590 nm (or 593 nm) increases accordingly until it reaches a maximum value and levels off, allowing for subsequent determination of total iron, and then Fe3+, in the sample. Figure 8a shows results from initial experiments that addressed the amount of hydroquinone reducing agent to add to achieve rapid, successful reduction of Fe(III) to Fe(II). For these investigations, 1120.0 µL of a 3.04 x 10-5 mol Fe3+ L-1 solution was pipetted into a 3.00-mL capacity, 1.00-cm path length plastic cuvet containing 800.0 µL of pH 5.00 KHP/NaOH buffer solution and 80.0 µL of 9.76 x 10-3 mol Ferene S L-1 chelating solution. A kinetic absorbance-time run at 590 nm, using a WPA Colourwave CO7500 B colorimeter, was initiated upon addition of the sample. At t = 240 s, 20.0 µL of 2% aqueous hydroquinone was pipetted into the cuvet (with brief mixing), followed by a significant increase and eventual leveling of the absorbance at 590 nm, indicating reduction of Fe3+ to Fe2+. This process required approximately 9 minutes, using 20.0 µL of 2%

iron (as Fe2+) by the common protocols outlined in section 2.2.1, with determination of the percent recovery of iron in each sample. Iron recoveries for the spiked water samples ranged from 100.6 – 101.4 percent, indicating essentially complete recovery of Fe and good accuracy. Additionally, our group has compared total Fe results for samples determined by the spectrophotometric Ferene S method to total Fe for the same samples determined by flame atomic absorption spectrophotometry (FAAS). Table 4 lists average Fe concentrations (in mg Fe L-1) found in water from clean and AMD-contaminated portions of Gillespie Run (Allegheny County, PA, USA), determined by the colorimetric Ferene S method and by FAAS. The results from Table 4 indicate good agreement between the spectrophotometric

Table 4. Comparison of the colorimetric Ferene S method with flame atomic absorption spectrophotometry (FAAS) for determination of total iron in abandoned mine drainage

Current work in our laboratory involves, among other things, the use of the spectrophotometric Ferene S method for speciation of iron (i.e., Fe2+, Fe3+, and total Fe) in runoff from abandoned mines and other natural waters. The experimental approach used in these studies is given in section 2.2.5. In the speciation experiments, a spectrophotometer or colorimeter with absorbance-versus-time measurement capability is used to monitor formation of Fe(Ferene S)32+ due to Fe2+ present in the sample. Upon chelation of Fe2+ initially present in the sample, the absorbance reaches a temporary plateau, at which time a reducing agent, e.g., hydroquinone, is added to reduce any Fe3+ present to Fe2+. Depending on the concentration of Fe3+, the absorbance at 590 nm (or 593 nm) increases accordingly until it reaches a maximum value and levels off, allowing for subsequent determination of total iron, and then Fe3+, in the sample. Figure 8a shows results from initial experiments that addressed the amount of hydroquinone reducing agent to add to achieve rapid, successful reduction of Fe(III) to Fe(II). For these investigations, 1120.0 µL of a 3.04 x 10-5 mol Fe3+ L-1 solution was pipetted into a 3.00-mL capacity, 1.00-cm path length plastic cuvet containing 800.0 µL of pH 5.00 KHP/NaOH buffer solution and 80.0 µL of 9.76 x 10-3 mol Ferene S L-1 chelating solution. A kinetic absorbance-time run at 590 nm, using a WPA Colourwave CO7500 B colorimeter, was initiated upon addition of the sample. At t = 240 s, 20.0 µL of 2% aqueous hydroquinone was pipetted into the cuvet (with brief mixing), followed by a significant increase and eventual leveling of the absorbance at 590 nm, indicating reduction of Fe3+ to Fe2+. This process required approximately 9 minutes, using 20.0 µL of 2%

Ferene S and FAAS methods.

(AMD).

hydroquinone. A second experiment, using 80.0 µL of 2% aqueous hydroquinone reducing agent and performed under the same conditions as before, required 3-5 minutes to achieve complete reduction of Fe3+ to Fe2+. All absorbances in both experiments were adjusted to the total solution volume in the cuvet after addition of hydroquinone. Sine wave-like fluctuations of the absorbance at the plateaus of both absorbance-time curves are attributed to insufficient mixing after addition of hydroquinone. Conversion of Fe3+ to Fe2+, expressed as the percent ratio of the concentration of Fe3+ initially present and the Fe2+ concentration determined at the end of the kinetic run (via the average absorbance at 590 nm at the plateau and Beer's Law), was found to be 103 and 107%, respectively, for the two experiments. Though the conversions are rather high, they do suggest essentially complete reduction of Fe3+ to Fe2+.

Fig. 8. Determination of (a) the amount of hydroquinone reducing agent to add for conversion of 3.04 x 10-5 mol Fe3+ L-1 solution to Fe2+, and (b) the speciation of a synthetic sample containing 2.95 x 10-5 mol Fe2+ L-1 and 1.52 x 10-5 mol Fe3+ L-1.

Further recovery experiments were conducted, involving speciation of iron in synthetic samples containing known concentrations of Fe(III) and Fe(II) and analyzed for Fe2+ by the procedures described in section 2.2.5. The purpose of these experiments was to check the iron recovery and efficiency of reduction of Fe3+ to Fe2+. Table 5 gives the added and found Fe2+, Fe3+, and total Fe concentrations (in mol Fe L-1) for three such recovery experiments, and Figure 8b illustrates the absorbance-time curve obtained for one of these recovery runs. The recoveries obtained for the forms of Fe in the samples are generally good, despite the high recoveries of 110.8 and 105.9%, and the rather low recovery of 84.3%. The recoveries suggest that complete Fe recoveries, and Fe(II)/Fe(III) speciation, are possible with this method. In Figure 8b, there is a very small leveling point prior to addition of hydroquinone, at which point apparent reduction of Fe3+ to Fe2+ was occurring, possibly promoted by Ferene S. Our group plans to further investigate this tendency.

Average Fe(II) and Fe(III) results from the speciation of Fe2+ and Fe3+ in AMD samples collected from a mine runoff tributary of Gillespie Run, and analyzed for Fe by the procedure given in section 2.2.5, were found to be 33.7 ± 0.5 mg Fe2+ L-1 and 0.6 ± 0.2 mg Fe3+ L-1 for one sample, and 35.5 ± 1.0 mg Fe2+ L-1 and 0.7 ± 0.2 mg Fe3+ L-1 for a different sample. These results suggest that the form of iron in AMD is mainly Fe2+, with a very small Fe3+ concentration present. Calibration slopes associated with these determinations ranged from

Limiting Reactants in Chemical Analysis: Influences of Metals

ratios and added acid.

transmittance measurements.

**4. Conclusions, future directions** 

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 331

Fe(Ferrozine)32+, the published pH range for optimum chelation is pH 3 – 9 (Gibbs, 1976; McBride, 1980). The tendency of both ligands to readily chelate Fe2+ even at low pH values was suggested by qualitative observations of formation of the indigo and magenta colors of Fe(Ferene S)32+ and Fe(Ferrozine)32+, respectively, further suggesting that the 1:3 chelates are formed preferentially, though incompletely, even at low pH. The pH titration curves for Fe(Ferene S)32+ and Fe(Ferrozine)32+ at varying Fe(II):ligand ratios (Figures 7a and 7b) provide some support for this hypothesis, as do the absorbance-pH formation curves (Figures 6a and 6b) from the pH-spectrophotometric formation constant experiments. The absorbance-pH formation curves show that for a typical Fe(II):ligand ratio, e.g., 1:4, that can be used in colorimetric determinations of iron using either of these ligands, that a significant absorbance (ca. 0.3 – 0.4) is obtained even at pH values as low as 1.20 (Figures 6a and 6b), which signifies extensive chelate formation at such low pH. The pH titration curves for Fe(Ferene S)32+ and Fe(Ferrozine)32+ at the 1:4 Fe2+:ligand ratio resemble strong acid-strong base titration curves, suggesting that competition between Fe2+ and H+ for the coordination site on the ligand occurs overwhelmingly in favor of Fe2+, leaving H+ to be titrated by standard base. The indigo and magenta colors of Fe(Ferene S)32+ and Fe(Ferrozine)32+, respectively, were already intense at the start of the titrations, even for the low Fe(II):ligand

The application our group selected was the spectrophotometric determination of iron in natural waters, using a ligand that would yield lower detection limits for iron, particularly in consideration of the US Environmental Protection Agency (USEPA) mandated limit of 0.3 mg Fe L-1 for water (Heakin, 2000). The idea of using Ferene S was appealing due to the high molar absorptivity (published value of 3.55 x 104 L mol-1 cm-1) (Higgins, 1981). The results given in Figures 8a and 8b, and Tables 3 – 5, suggest that the colorimetric Ferene S method is suitable for accurate and precise determinations of iron in natural waters, and that speciation of Fe2+ and Fe3+ in abandoned mine drainage is feasible with Ferene S. The portability of the Ferene S method has also made it useful in our attempts to determine iron in runoff from old coal mines on-site, in a "real-time" manner. The method has been microscaled so that solution mixing and absorbance measurements can be performed in the same 1-cm cuvet. The Ferene S reagent is not prohibitively expensive, is available from major chemicals suppliers, and is water-soluble. Portable, battery-operated clorimeters and spectrophotometers are inexpensive, low-maintenance, and yield reliable absorbance and

In this chapter, the concept of the limiting reactant in analytical chemistry was examined through studying the effects of the metal ion to ligand ratio on calibration curves, determinations of stoichiometries, and determination of formation constants for the iron(II) chelates of Ferene S and Ferrozine, two well known chelating agents for the spectrophotometric determination of iron in serum and other biomedical samples. The results of our studies for both chelates show that using insufficient ligand for spectrophotometric determination of iron produces premature deviations from Beer's Law that are not instrumental but due to the ligand becoming the limiting reactant. The results further indicate that the formation constant of each chelate is relatively unaffected by changes in ligand concentration, but that the extent of complexation is influenced by


3.50 x 104 to 3.60 x 104 L (mol Fe)-1, with excellent Beer's Law behavior (R2 ranged from 0.9993 to 1.000, and standard error of the estimate ranged from 0.0030 to 0.017) over the 1.00 x 10-6 – 3.94 x 10-5mol Fe L-1 (0.05 – 2.20 mg Fe L-1) concentration range.

Table 5. Recovery of Fe2+, Fe3+, and total Fe from synthetic water samples spiked with known concentrations of each form of Fe, using the colorimetric Ferene S method for speciation of iron.

### **3.5 Discussion**

The results obtained from our studies of the effect of metal ion to ligand on the Beer's Law behavior of calibration data (Figures 3a and 3b) support our initial hypothesis that insufficient ligand stunts the range of linearity of absorbance with analyte concentration due to the ligand becoming the limiting reactant rather than the Fe(II) analyte. There is also an instrumental limit on Beer's Law: the ability of the spectrophotometer to detect changes in transmittance (usually very small at high concentrations) with respect to large changes in analyte concentration (Skoog et al., 2007; Stauffer, 2007), usually associated with the highconcentration end of the calibration curve. Advances in spectrophotometer design over the past few decades have enabled more sophisticated instruments to extend Beer's Law linearity into the 2.0 – 3.0 absorbance unit region, which is unattainable with low-end photometers and simple colorimeters. Regardless, the limiting reactant concept must be kept in mind when performing spectrophotometric determinations of any analyte.

The results of our formation constant studies indicated that coordination of Fe2+ by Ferene S and Ferrozine is strong, based on the average log10β3 of 16.45 ± 0.25 and 17.46 ± 0.37 for Fe(Ferene S)32+ and Fe(Ferrozine)32+, respectively. As the formation constants for each chelate are means of quadruplicate determinations, their standard deviations suggest that LeChatelier's Principle is essentially followed. What does change, however, is the extent of chelation at low pH with changes in the ligand concentration used, which further emphasizes the importance of maintaining the analyte Fe2+ as the limiting reactant. Previous unpublished studies by our group regarding optimum formation of Fe(Ferene S)3 2+ as a function of pH indicated that the range for optimum chelation was pH 3 – 7. For Fe(Ferrozine)32+, the published pH range for optimum chelation is pH 3 – 9 (Gibbs, 1976; McBride, 1980). The tendency of both ligands to readily chelate Fe2+ even at low pH values was suggested by qualitative observations of formation of the indigo and magenta colors of Fe(Ferene S)32+ and Fe(Ferrozine)32+, respectively, further suggesting that the 1:3 chelates are formed preferentially, though incompletely, even at low pH. The pH titration curves for Fe(Ferene S)32+ and Fe(Ferrozine)32+ at varying Fe(II):ligand ratios (Figures 7a and 7b) provide some support for this hypothesis, as do the absorbance-pH formation curves (Figures 6a and 6b) from the pH-spectrophotometric formation constant experiments. The absorbance-pH formation curves show that for a typical Fe(II):ligand ratio, e.g., 1:4, that can be used in colorimetric determinations of iron using either of these ligands, that a significant absorbance (ca. 0.3 – 0.4) is obtained even at pH values as low as 1.20 (Figures 6a and 6b), which signifies extensive chelate formation at such low pH. The pH titration curves for Fe(Ferene S)32+ and Fe(Ferrozine)32+ at the 1:4 Fe2+:ligand ratio resemble strong acid-strong base titration curves, suggesting that competition between Fe2+ and H+ for the coordination site on the ligand occurs overwhelmingly in favor of Fe2+, leaving H+ to be titrated by standard base. The indigo and magenta colors of Fe(Ferene S)32+ and Fe(Ferrozine)32+, respectively, were already intense at the start of the titrations, even for the low Fe(II):ligand ratios and added acid.

The application our group selected was the spectrophotometric determination of iron in natural waters, using a ligand that would yield lower detection limits for iron, particularly in consideration of the US Environmental Protection Agency (USEPA) mandated limit of 0.3 mg Fe L-1 for water (Heakin, 2000). The idea of using Ferene S was appealing due to the high molar absorptivity (published value of 3.55 x 104 L mol-1 cm-1) (Higgins, 1981). The results given in Figures 8a and 8b, and Tables 3 – 5, suggest that the colorimetric Ferene S method is suitable for accurate and precise determinations of iron in natural waters, and that speciation of Fe2+ and Fe3+ in abandoned mine drainage is feasible with Ferene S. The portability of the Ferene S method has also made it useful in our attempts to determine iron in runoff from old coal mines on-site, in a "real-time" manner. The method has been microscaled so that solution mixing and absorbance measurements can be performed in the same 1-cm cuvet. The Ferene S reagent is not prohibitively expensive, is available from major chemicals suppliers, and is water-soluble. Portable, battery-operated clorimeters and spectrophotometers are inexpensive, low-maintenance, and yield reliable absorbance and transmittance measurements.

### **4. Conclusions, future directions**

330 Stoichiometry and Research – The Importance of Quantity in Biomedicine

3.50 x 104 to 3.60 x 104 L (mol Fe)-1, with excellent Beer's Law behavior (R2 ranged from 0.9993 to 1.000, and standard error of the estimate ranged from 0.0030 to 0.017) over the 1.00

Table 5. Recovery of Fe2+, Fe3+, and total Fe from synthetic water samples spiked with known concentrations of each form of Fe, using the colorimetric Ferene S method for

in mind when performing spectrophotometric determinations of any analyte.

The results obtained from our studies of the effect of metal ion to ligand on the Beer's Law behavior of calibration data (Figures 3a and 3b) support our initial hypothesis that insufficient ligand stunts the range of linearity of absorbance with analyte concentration due to the ligand becoming the limiting reactant rather than the Fe(II) analyte. There is also an instrumental limit on Beer's Law: the ability of the spectrophotometer to detect changes in transmittance (usually very small at high concentrations) with respect to large changes in analyte concentration (Skoog et al., 2007; Stauffer, 2007), usually associated with the highconcentration end of the calibration curve. Advances in spectrophotometer design over the past few decades have enabled more sophisticated instruments to extend Beer's Law linearity into the 2.0 – 3.0 absorbance unit region, which is unattainable with low-end photometers and simple colorimeters. Regardless, the limiting reactant concept must be kept

The results of our formation constant studies indicated that coordination of Fe2+ by Ferene S and Ferrozine is strong, based on the average log10β3 of 16.45 ± 0.25 and 17.46 ± 0.37 for Fe(Ferene S)32+ and Fe(Ferrozine)32+, respectively. As the formation constants for each chelate are means of quadruplicate determinations, their standard deviations suggest that LeChatelier's Principle is essentially followed. What does change, however, is the extent of chelation at low pH with changes in the ligand concentration used, which further emphasizes the importance of maintaining the analyte Fe2+ as the limiting reactant. Previous

unpublished studies by our group regarding optimum formation of Fe(Ferene S)3

function of pH indicated that the range for optimum chelation was pH 3 – 7. For

2+ as a

speciation of iron.

**3.5 Discussion** 

x 10-6 – 3.94 x 10-5mol Fe L-1 (0.05 – 2.20 mg Fe L-1) concentration range.

In this chapter, the concept of the limiting reactant in analytical chemistry was examined through studying the effects of the metal ion to ligand ratio on calibration curves, determinations of stoichiometries, and determination of formation constants for the iron(II) chelates of Ferene S and Ferrozine, two well known chelating agents for the spectrophotometric determination of iron in serum and other biomedical samples. The results of our studies for both chelates show that using insufficient ligand for spectrophotometric determination of iron produces premature deviations from Beer's Law that are not instrumental but due to the ligand becoming the limiting reactant. The results further indicate that the formation constant of each chelate is relatively unaffected by changes in ligand concentration, but that the extent of complexation is influenced by

Limiting Reactants in Chemical Analysis: Influences of Metals

471-39462-9, New York, New York, USA

*Scandinavica*, Vol. 2, pp. 297-318, ISSN 0001-5393

0-471-58526-2, New York, New York, USA

ISSN 0003-2697

0003-2700

15056032M, Copenhagen, Denmark

and Ligands on Calibration Curves and Formation Constants for Selected Iron-Ligand Chelates 333

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increasing the concentration of ligand used. The results of our studies for the two chelates also indicate good agreement of molar absorptivities and stoichiometries with published values, and some disagreement with the published formation constant for Fe(Ferrozine)32+ due most likely to experimental differences in the methods employed. Finally, an application of Ferene S to the colorimetric determination of total and speciated (Fe2+, Fe3+) iron in natural waters was presented and discussed.

Future plans for this research include continuation of efforts toward further development, implementation, and refinement of the field-portable colorimetric determination and speciation of iron in natural waters using Ferene S. Work on using Ferene S for spectrophotometric quantitation of iron in sample types other than natural waters continues in our laboratory. Additionally, other ferroin ligands besides the two examined in this chapter will be investigated for use in colorimetric quantitation of iron in various sample types. Finally, ferroin ligands and other types of iron(II) and even iron(III) complexing agents will be explored for pedagogical use in effective teaching of the limiting reactant concept in analytical chemistry courses.

### **5. Acknowledgements**

The authors gratefully acknowledge the moral and financial support of the University of Pittsburgh at Greensburg in this endeavor. The generous support of the McKenna Foundation, the Fisher Fund of the Pittsburgh Foundation, the Society for Analytical Chemists of Pittsburgh (SACP), and the Spectroscopy Society of Pittsburgh (SSP) is gratefully acknowledged for making this research possible. MTS thanks his wife, Resa, for her moral support and patience during this work, his colleagues in the Chemistry Department at Pitt-Greensburg for their support and input, his former and current research students as well as the many students who enrolled in his analytical chemistry and instrumental analysis courses over the past several years for their contributions to this research, and to Mrs. Cynthia Genard for helping with administrative functions during the last few weeks of manuscript preparation.

### **6. References**


increasing the concentration of ligand used. The results of our studies for the two chelates also indicate good agreement of molar absorptivities and stoichiometries with published values, and some disagreement with the published formation constant for Fe(Ferrozine)32+ due most likely to experimental differences in the methods employed. Finally, an application of Ferene S to the colorimetric determination of total and speciated (Fe2+, Fe3+)

Future plans for this research include continuation of efforts toward further development, implementation, and refinement of the field-portable colorimetric determination and speciation of iron in natural waters using Ferene S. Work on using Ferene S for spectrophotometric quantitation of iron in sample types other than natural waters continues in our laboratory. Additionally, other ferroin ligands besides the two examined in this chapter will be investigated for use in colorimetric quantitation of iron in various sample types. Finally, ferroin ligands and other types of iron(II) and even iron(III) complexing agents will be explored for pedagogical use in effective teaching of the limiting reactant

The authors gratefully acknowledge the moral and financial support of the University of Pittsburgh at Greensburg in this endeavor. The generous support of the McKenna Foundation, the Fisher Fund of the Pittsburgh Foundation, the Society for Analytical Chemists of Pittsburgh (SACP), and the Spectroscopy Society of Pittsburgh (SSP) is gratefully acknowledged for making this research possible. MTS thanks his wife, Resa, for her moral support and patience during this work, his colleagues in the Chemistry Department at Pitt-Greensburg for their support and input, his former and current research students as well as the many students who enrolled in his analytical chemistry and instrumental analysis courses over the past several years for their contributions to this research, and to Mrs. Cynthia Genard for helping with administrative functions during the

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**5. Acknowledgements** 

**6. References** 


**Part 6** 

**Biomedicine and Environment:** 

**The Future is Now?** 

