5. Criterion for the quantitative titration and the influence of side-reactions

The principle of redox titrations is that the solution of a reducing agent is titrated with a solution of an oxidizing agent (or vice versa)


On the Titration Curves and Titration Errors in Donor Acceptor Titrations of Displacement and Electronic… http://dx.doi.org/10.5772/intechopen.68750 137


Table 4. Titration curves of V(II) with MnO4 �.

$$y\text{Oxr} + z\text{Reds} \leftrightarrow y\text{Redr} + z\text{Oxs} \tag{59}$$

At the equivalence point

$$p e\_{\rm eq} = \frac{1}{z+y} (\log K\_{\rm S} + \log K\_{\rm T}) \tag{60}$$

The criterion of the quantitative titration can be deduced if we consider that the substance to be determined must be oxidized (or reduced) during the titration to an extent of 99.9% [23]. This means that the amount of determinant remaining untitrated at the equivalence point should not exceed 0.1% of that originally present, i.e.,

$$\log \frac{[O\chi\_S]}{[\text{Red}\_S]} \gtrsim 3\tag{61}$$

and

T1 T2 T3 T SUM E pX 0.0001 0 0 0.0001 0.5 8.333 0.0006 0 0 0.0006 0.45 7.5 0.0038 0 0 0.0038 0.4 6.667 0.0254 0 0 0.0254 0.35 5.833 0.151 0 0 0.151 0.3 5 0.5478 0 0 0.5478 0.25 4.167 0.8919 0 0 0.8919 0.2 3.333 0.9825 0 0 0.9825 0.15 2.5 0.9974 0 0 0.9974 0.1 1.667 0.9996 0 0 0.9996 0.05 0.833 0.9999 0 0 0.9999 0 0 0 0 1 0.05 0.833 0.0001 0 1.0001 0.1 1.667 0.0008 0 1.0008 0.15 2.5 0.0052 0 1.0052 0.2 3.333 0.0343 0 1.0343 0.25 4.167 0.1947 0 1.1947 0.3 5 0.6222 0 1.6222 0.35 5.833 0.9182 0 1.9182 0.4 6.667 0.9871 0 1.9871 0.45 7.5 0.9981 0 1.9981 0.5 8.333 0.9997 0 1.9997 0.55 9.167 1 0 2 0.6 10 1 0 2 0.65 10.833 1 0 2 0.7 11.667 1 0.0001 2.0001 0.75 12.5 1 0.0004 2.0004 0.8 13.333 1 0.003 2.003 0.85 14.167 1 0.0203 2.0203 0.9 15 1 0.1238 2.1238 0.95 15.833 1 0.4904 2.4904 1 16.667 1 0.8677 2.8677 1.05 17.5 1 0.9781 2.9781 1.1 18.333 1 0.9967 2.9967 1.15 19.167 1 0.9995 2.9995 1.2 20 1 0.9999 2.9999 1.25 20.833

136 Redox - Principles and Advanced Applications

$$p e\_{\rm eq} > (p e\_{\rm eq}{}^0)\_S + 3 \frac{1}{y} \tag{62}$$

$$(\left(\mathrm{pe}\_{\mathrm{eq}}\right)\_{\mathcal{S}} = \frac{1}{y} \log K\_{\mathcal{S}} \tag{63}$$

From Eqs. (60) and (63) also we obtain

$$y \log K\_T - z \log K\_S > \Im(z+y) \tag{64}$$

If both oxidation-reduction systems in the titration involve two electrons, the difference between the log K values must be greater than 6.

In addition, if the oxidized or reduced product present in the solution containing the redox system takes part in a side-reaction, and the equilibrium position of this reaction can be kept constant, by maintaining suitable experimental conditions, the conditional oxidation-reduction constant, K', can be deduced and used similarly to those used in complex chemistry

$$K' = \frac{[\text{Red}']}{[\text{Ox}'][e^-]^\sharp} \tag{65}$$

[Red'] and [Ox'] are analytical concentrations without any respect to side-reactions. The connection between the conditional and real constants is the following one

$$K' = K \frac{\alpha\_{\text{Red}}(\text{A})}{\alpha\_{\text{Ox}}(\text{B})} \tag{66}$$

where αRed(A) and αOx(B) are the side-reaction functions, and A and B denote the substances reacting with the reduced and oxidized substance, respectively.

$$\alpha\_{\mathbb{R}\text{ ed}}(A) = 1 + [L]\beta\_1 + [L]^2 \beta\_2 + \dots \tag{67}$$

$$\alpha\_{\odot \star}(B) = 1 + [L]\beta\_1^\* + [L]\beta\_2^\* + \dots \tag{68}$$

[A] and [B] are concentrations of the species reacting with the reduced and oxidized form, respectively, β's and β\*'s are complex products or protonization constants products.

In all calculations concentration constants can be used if they are corrected to the corresponding ionic strength.

In practice, the most important side-reactions are complex formation and protonation. The oxidized and reduced form of a metal ion may form complexes of different stabilities with the complexing ligand L.

So, if the criterion of the quantitative determination is not fulfilled by a suitable pH change or by the use of a complexing agent that shifts the values of the conditional constants, the titration may be realized. For example, according to Vydra and Pribil [101], cobalt (II) can be titrated with iron (III) ions if 1,10-phenanthroline is added to the solution, and the pH is adjusted to 3 even though KCo >>> KFe.

On the other hand, if another component present in the solution has similar oxidizing or reducing properties, then interfering species can be masked, so that the conditional redox constant of the interfering system is changed to such an extent that it no longer interferes with the main reaction.

#### 5.1. Practical examples

(1) Calculate the pH necessary for the accurate direct titration of potassium hexacyanoferrate (III) with ascorbic acid, given that log KFe(CN)6 ¼ 6.1; the protonation constants of hexacyanoferrate(II) are log K<sup>1</sup> ¼ 4.17, log K<sup>2</sup> ¼ 2.22, log K<sup>3</sup> < 1, log K<sup>4</sup> < 1; the logarithms of all the protonation constants of hexacyanoferrate(III) are >1. The equilibrium constant of the dehydroascorbic acid-ascorbinate redox system is log K<sup>A</sup> ¼ �2.5; the protonation constants of the ascorbinate ion are log K<sup>1</sup> ¼ 11.56 and 4.17.

The criterion for the feasibility of the titration, according to Eq. (64) is:

$$2\log K'\_{\text{Fe(CN)}\_6} - \log K'\_A > \Im(1+2) = 9\tag{69}$$

From the protonation constants of hexacyanoferrate(II), if the pH > 5.5, then αFe(CN)6(H) ¼ 1 and

$$\log K'\_{\text{Fe(CN)}\_6} = \log K\_{\text{Fe(CN)}\_6} = 6.1. \tag{70}$$

Therefore 2 x 6.1 � log K<sup>0</sup> <sup>A</sup> > 9; or log K<sup>0</sup> <sup>A</sup> < 3.2. If the pH <sup>¼</sup> 6, then <sup>α</sup>A(H) <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>10</sup>�<sup>6</sup> x 1011.57 <sup>þ</sup> <sup>10</sup>�<sup>12</sup> x 1015.73 <sup>¼</sup> <sup>10</sup>5.58

$$
\log K\_A' = \text{3.1} \tag{71}
$$

Thus, if the pH > 6, the titration can be performed with adequate accuracy.

(2) Calculate the [L] maximum for the accurate direct titration of Ceþ<sup>4</sup> with Fe2þ, being L the organic complexant (acetylacetone) present in the solution and given that log KT ¼ 24 and log KS ¼ 11.3 (see Eqs. (28)–(30)); the global constants of Fe-(L)(II) are log β<sup>1</sup> ¼ 5.07; log β<sup>2</sup> ¼ 8.67.

The reactions involved would be

<sup>K</sup><sup>0</sup> <sup>¼</sup> Red<sup>0</sup> ½ � Ox<sup>0</sup> ½ � e

[Red'] and [Ox'] are analytical concentrations without any respect to side-reactions. The con-

where αRed(A) and αOx(B) are the side-reaction functions, and A and B denote the substances

[A] and [B] are concentrations of the species reacting with the reduced and oxidized form,

In all calculations concentration constants can be used if they are corrected to the

In practice, the most important side-reactions are complex formation and protonation. The oxidized and reduced form of a metal ion may form complexes of different stabilities with the

So, if the criterion of the quantitative determination is not fulfilled by a suitable pH change or by the use of a complexing agent that shifts the values of the conditional constants, the titration may be realized. For example, according to Vydra and Pribil [101], cobalt (II) can be titrated with iron (III) ions if 1,10-phenanthroline is added to the solution, and the pH is adjusted to 3

On the other hand, if another component present in the solution has similar oxidizing or reducing properties, then interfering species can be masked, so that the conditional redox constant of the interfering system is changed to such an extent that it no longer interferes with

(1) Calculate the pH necessary for the accurate direct titration of potassium hexacyanoferrate (III) with ascorbic acid, given that log KFe(CN)6 ¼ 6.1; the protonation constants of hexacyanoferrate(II) are log K<sup>1</sup> ¼ 4.17, log K<sup>2</sup> ¼ 2.22, log K<sup>3</sup> < 1, log K<sup>4</sup> < 1; the logarithms of all the protonation constants of hexacyanoferrate(III) are >1. The equilibrium constant of the dehydroascorbic acid-ascorbinate redox system is log K<sup>A</sup> ¼ �2.5; the protonation constants of

<sup>α</sup>R edð Þ <sup>A</sup> <sup>α</sup>Oxð Þ <sup>B</sup>

<sup>1</sup> þ ½ � L β�

K<sup>0</sup> ¼ K

<sup>α</sup>R edð Þ <sup>A</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ½ � <sup>L</sup> <sup>β</sup><sup>1</sup> <sup>þ</sup> ½ � <sup>L</sup> <sup>2</sup>

<sup>α</sup>Oxð Þ <sup>B</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ½ � <sup>L</sup> <sup>β</sup>�

respectively, β's and β\*'s are complex products or protonization constants products.

nection between the conditional and real constants is the following one

reacting with the reduced and oxidized substance, respectively.

corresponding ionic strength.

138 Redox - Principles and Advanced Applications

complexing ligand L.

even though KCo >>> KFe.

the main reaction.

5.1. Practical examples

the ascorbinate ion are log K<sup>1</sup> ¼ 11.56 and 4.17.

� ½ �<sup>z</sup> <sup>ð</sup>65<sup>Þ</sup>

β<sup>2</sup> þ … ð67Þ

<sup>2</sup> þ … ð68Þ

ð66Þ

$$\begin{aligned} \text{Ce}^{4+} + \text{Fe}^{2+} &\leftrightarrow \text{Ce}^{3+} + \text{Fe}^{3+} \\ \text{Fe}^{2+} + L &\leftrightarrow \text{Fe}L^{2+} \\ \text{Fe}L^{2+} + L &\leftrightarrow \text{Fe}L\_2^{2+} \end{aligned} \tag{72}$$

The criterion for the feasibility of the titration, according to Eq. (64) is

$$
\log K\_T - \log K\_S' > 3(1+1) = 6 \tag{73}
$$

$$
\log K\_T - 6 > \log K'\_{S'} \quad \log K'\_S < 18 \tag{74}
$$

$$K\_S' = K\_S a\_{\left[\text{FeL}\_2\right]^{+2}} \quad a\_{\left[\text{FeL}\_2\right]^{+2}} < 10^{6.7} \tag{74}$$

$$
\alpha\_{\left[\text{FeL}\_2\right]^{+2}} = 1 + \left[L\right]\beta\_1 + \left[L\right]^2 \beta\_2 \tag{75}
$$

Thus, if the [L] < 0.1 M, the titration can be performed with adequate accuracy.

(3) Calculate the [SO4 �2 ] maximum present in the solution for the accurate direct titration of Ce4<sup>þ</sup> with Tlþ, given that log KT <sup>¼</sup> 24 and log KS <sup>¼</sup> 41.6 (see Eqs. (28), (42), (43)); the global constants of Ce-SO4(IV) are log β<sup>1</sup> ¼ 3.5; log β<sup>2</sup> ¼ 8.0; log β<sup>3</sup> ¼ 10.4.

The criterion for the feasibility of the titration, according to (64) is:

$$2\log K\_T' - \log K\_S > 3(1+2) = 9\tag{76}$$

$$\log K\_T' > 25.3$$

$$K\_T ' = K\_T \frac{1}{a\_{\text{Ce}-\text{SO}\_4}}, \quad a\_{\text{Ce}-\text{SO}\_4} < 10^{7.2} \tag{77}$$

$$\alpha\_{\left[\text{FeL}\_{2}\right]^{\ast 2}} = 1 + \left[\text{SO}\_{4}{}^{-2}\right] \beta\_{1} + \left[\text{SO}\_{4}{}^{-2}\right]^{2} \beta\_{2} + \left[\text{SO}\_{4}{}^{-2}\right]^{3} \beta\_{3} \tag{78}$$

Thus, [SO4 �2 ] should be < 0.39 M, to perform the titrimetry with accuracy.

#### 6. Final comments

As a matter of fact redox titrations play a prominent role in volumetric analysis of redox actives species. A systematic study of the bibliography is undertaken in order to ascertain the state of the art concerning to redox titration curves. A method for the determination of titration error in donor/acceptor titrations of displacement and electronic transference reactions has been devised; a hyperbolic sine expression being derived for the titration error, applicable to symmetrical reactions (no polynuclear species being involved in one side of a half-reaction). The hyperbolic sine expression developed is compact and allows calculating the entire titration curve without piecemeal approximations, as usually occurs by dividing the titration curve in three parts: before, in, and beyond the equivalence point regions.

The method has been applied to some experimental systems characteristics of redox titration reactions. The method proposed is also applicable to mixtures of analytes, e.g., Fe(II) þ Tl(I), as well as to multistep redox titrations, e.g., V(II)/V(III)/V(IV)/V(V) system. The forms of the redox titration curves are independent of the concentrations. However, when the concentrations involved are very low the responses of the electrodes are not appropriate. All calculations involved have been checked with the method proposed by "de Levie" [62] for the sake of comparison, and no differences were found in the numerical values obtained by both methods. A diagram for the titration error in function of the difference between the end and equivalence point (pX) is drawn in order to facilitate the graphical calculation of titration error.

Automatic titrators enable recording automatically the change with potential (E) or pH in titre during a given titration. The accuracy of the measurements can increase with the help of online microcomputer for the control and data acquisition, allowing among the possibility for curve-smoothing and differentiation.

The extension of the method to nonhomogeneous systems of the type Cr2O7 <sup>2</sup>�/Cr3<sup>þ</sup>, I2/I�, or S4O6 <sup>2</sup>�/S2O3 <sup>2</sup>� remains a challenge, this being a complex problem involving a complete reformulation of the presented equations, which implies a higher level of difficulty.

At the end of the chapter an appendix including a detailed study of the propagation of systematic and random errors on redox titration error has been carried out and spite of the complex expression obtained first on differentiation, the final expressions formulated were very compact. This topic is still under study and it will be dealt in further calculus.
