4.3. Fe2<sup>þ</sup> <sup>þ</sup> Tl<sup>þ</sup> titration curve with Ce4<sup>þ</sup>

In this particular case, the total ΔT function (or total T) is additive, i.e., ΔT is the sum of the values of ΔT (T) corresponding to the individual titration of Feþ<sup>2</sup> with Ceþ<sup>4</sup> and Tl<sup>þ</sup> with Ceþ<sup>4</sup> . The reactions and equations involved have been previously described in Sections 4.1. and 4.2.. Thus, in this case we should only sum the values given by Eqs. 31 (Feþ<sup>2</sup> with Ceþ<sup>4</sup> ) and 48 (Tl<sup>þ</sup> with Ceþ<sup>4</sup> ). The corresponding titration curve of a mixture of Fe2<sup>þ</sup> and Tl<sup>þ</sup> with Ce4<sup>þ</sup> calculated in this way is shown in Figure 4.

#### 4.4. Fe2<sup>þ</sup> titration curve with MnO4 � as titrant

For the half-reaction MnO4� <sup>þ</sup> 5 e <sup>þ</sup> 8 H<sup>þ</sup> <sup>¼</sup> Mn2<sup>þ</sup> <sup>þ</sup> 4 H2O (E0<sup>T</sup> <sup>¼</sup> 1.51 v)

$$K\_T = \frac{\left[\text{Mn}^{2+}\right]}{\left[\text{MnO}\_4^{-}\right]\left[\text{e}\right]^5\left[\text{H}^+\right]^8} \qquad pe = \frac{\log K\_T}{5} - \frac{8}{5}pH + \frac{1}{5}\log\frac{\left[\text{MnO}\_4^{-}\right]}{\left[\text{Mn}^{2+}\right]} \tag{49}$$

$$E = E\_0^T + \frac{0.06}{5} \log \frac{\left[\text{MnO}\_4^-\right] \left[\text{H}^+\right]^8}{\left[\text{Mn}^{2+}\right]} = E\_0^T - \frac{0.06 \cdot 8}{5} pH + \frac{0.06}{5} \log \frac{\left[\text{MnO}\_4^-\right]}{\left[\text{Mn}^{2+}\right]} \tag{50}$$

and thus following the previous procedure

<sup>Δ</sup><sup>T</sup> <sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi KS=KT p 1 þ ½ � e 2 KS ½ � <sup>e</sup> <sup>2</sup>�<sup>1</sup>

Table 1. Titration curve of Fe(II) with Ce(IV): hyperbolic sine method.

half-reactions exchanging a single electron, VO2�/VO2

v), and Fe <sup>3</sup>þ/Fe2<sup>þ</sup> (E<sup>0</sup> <sup>¼</sup> 0.68 v).

Ce(IV) þ Fe(II) ¼ Ce(III) þ Fe(III)

130 Redox - Principles and Advanced Applications

¼1.000Eþ24 KS<sup>0</sup>

KT<sup>0</sup>

<sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi KS=KT p

<sup>1</sup> <sup>þ</sup> <sup>10</sup>�<sup>2</sup>peKS

ffiffi <sup>e</sup> <sup>p</sup> sinh <sup>3</sup>

<sup>2</sup> sinh ln 10 2ð Þ <sup>þ</sup> <sup>1</sup> pe � peeq

2

The titration curve E ¼ f (T) is shown in Figure 3, together with those corresponding to other

E0T¼1.44 neT¼1 nHT¼0 f(pHT)¼ 0 pH¼�0.3 1M H2SO4

Ws [X]^(a�b)/2 W sine h ΔT T E X pX ΔpX LOG

0.091 1 8.439E�08 �1.077Eþ07 �0.9091 0.0909 0.62 4.642E�11 10.333 �7.333 �0.041 0.128 1 1.188E�07 �7.339Eþ06 �0.8720 0.1280 0.63 3.162E�11 10.500 �7.167 �0.059 0.177 1 1.645E�07 �5.000Eþ06 �0.8227 0.1773 0.64 2.154E�11 10.667 �7.000 �0.085 0.240 1 2.230E�07 �3.406Eþ06 �0.7597 0.2403 0.65 1.468E�11 10.833 �6.833 �0.119 0.317 1 2.943E�07 �2.321Eþ06 �0.6830 0.3170 0.66 1.000E�11 11.000 �6.667 �0.166 0.405 1 3.762E�07 �1.581Eþ06 �0.5948 0.4052 0.67 6.813E�12 11.167 �6.500 �0.226 0.500 1 4.642E�07 �1.077Eþ06 �0.5000 0.5000 0.68 4.642E�12 11.333 �6.333 �0.301 0.595 1 5.521E�07 �7.339Eþ05 �0.4052 0.5948 0.69 3.162E�12 11.500 �6.167 �0.392 0.683 1 6.340E�07 �5.000Eþ05 �0.3170 0.6830 0.70 2.154E�12 11.667 �6.000 �0.499 0.760 1 7.053E�07 �3.406Eþ05 �0.2403 0.7597 0.71 1.468E�12 11.833 �5.833 �0.619 0.823 1 7.638E�07 �2.321Eþ05 �0.1773 0.8227 0.72 1.000E�12 12.000 �5.667 �0.751 0.872 1 8.095E�07 �1.581Eþ05 �0.1280 0.8720 0.73 6.813E�13 12.167 �5.500 �0.893 0.909 1 8.439E�07 �1.077Eþ05 �0.0909 0.9091 0.74 4.642E�13 12.333 �5.333 �1.041 0.936 1 8.691E�07 �7.339Eþ04 �6.378E�02 0.9362 0.75 3.162E�13 12.500 �5.167 �1.195 0.956 1 8.871E�07 �5.000Eþ04 �4.436E�02 0.9556 0.76 2.154E�13 12.667 �5.000 �1.353 0.969 1 8.999E�07 �3.406Eþ04 �3.065E�02 0.9693 0.77 1.468E�13 12.833 �4.833 �1.514 0.979 1 9.087E�07 �2.321Eþ04 �2.109E�02 0.9789 0.78 1.000E�13 13.000 �4.667 �1.676 0.986 1 9.149E�07 �1.581Eþ04 �1.447E�02 0.9855 0.79 6.813E�14 13.167 �4.500 �1.840 0.990 1 9.191E�07 �1.077Eþ04 �9.901E�03 0.9901 0.80 4.642E�14 13.333 �4.333 �2.004 0.993 1 9.220E�07 �7.339Eþ03 �6.767E�03 0.9932 0.81 3.162E�14 13.500 �4.167 �2.170 0.995 1 9.240E�07 �5.000Eþ03 �4.620E�03 0.9954 0.82 2.154E�14 13.667 �4.000 �2.335

¼2.154Eþ11 Wc¼9.283E�07 pXeq¼17.667 E0S'¼0.68

¼1.44

(ABS(ΔT))

E0S¼0.68 neS¼1 nHS¼0 f(pHS)¼ 0 E0T<sup>0</sup>

� � � �

ln 10ð Þ pe � 21:889 � �

2

<sup>þ</sup> (E<sup>0</sup> ¼ 1.001 v), NO3

ð48Þ

�/NO2 (E<sup>0</sup> ¼ 0.80

$$\frac{E}{0.06} = \frac{E\_0^T}{0.06} - \frac{8}{5}pH + \frac{1}{5}\log\frac{[\text{MnO}\_4^-]}{[\text{Mn}^{2+}]} \quad pe = pe\_0^T - \frac{8}{5}pH + \frac{1}{5}\log\frac{[\text{MnO}\_4^-]}{[\text{Mn}^{2+}]} \tag{51}$$

$$pe = \frac{E}{0.06} \tag{52}$$

$$p e\_0^T = \frac{E\_0^T}{0.06} = \frac{\log K\_T}{5} \qquad \log K\_T = 5 \frac{E\_0^T}{0.06} \quad K\_T = 10^{\frac{E\_0^T}{0.06}} \tag{53}$$

Figure 4. Titration curve of a mixture of Fe2<sup>þ</sup> and Tl<sup>þ</sup> with Ce4<sup>þ</sup>.

$$p e\_0^{T'} = p e\_0^T - \frac{8}{5} p H = \frac{\log K\_T}{5} - \frac{8}{5} p H = \frac{\log K\_T'}{5} \qquad \log K' r = \log K r - 8p H \tag{54}$$

The terms of Eq. (54) are conditional (pH-dependent). In the equivalence point

$$p e'\_{\text{eq}} = \frac{\log K'\_T + \log K'\_S}{a + b} \tag{55}$$

The expression for the valuation error will be given by (a ¼ 1, b ¼ 5)

$$\Delta T = \frac{2\sqrt{K\_S'/K\_T'}}{1 + [\mathbf{e}]K\_S'} [\mathbf{e}]^{\frac{1-5}{2}} \sinh\left(\ln 10(5+1)\left(\frac{\Delta p e}{2}\right)\right) = \frac{2\sqrt{K\_S'/K\_T'}}{1 + [\mathbf{e}]K\_S'} [\mathbf{e}]^{-2} \sinh(3\ln 10(pe - pe'\_{\text{eq}})) \tag{56}$$

The curve and the titration error are shown in Figures 5 and 6, respectively, at different pH values. Part of the necessary calculations, at pH ¼ 0, is shown in Table 2. When T ¼ 2 and pH ¼ 0, E ¼ 1.51 v, normal MnO4 �/Mn2<sup>þ</sup> system potential.

#### 4.5. Titration curve of V2<sup>þ</sup> with MnO4 �

Oxido-reductor systems involving the vanadium oxidation states 2þ, 3þ, 4þ, and 5þ are shown in Table 3. The V2<sup>þ</sup> ion undergoes various successive ionizations at 3þ, 4þ, and 5<sup>þ</sup> when MnO4 � is added, being appreciated in the curve (Figure 7), the three corresponding jumps. The total ΔT function (or total T) is additive. The titration curve is the sum of the contributions of each individual reaction

$$
\Delta T\_{\text{SUM}} = \Delta T\_{32} + \Delta T\_{43} + \Delta T\_{54} \tag{57}
$$

$$T\_{\rmSUM} = \Delta T\_{\rmSUM} - \mathfrak{J} = T\_{32} + T\_{43} + T\_{54} \tag{58}$$

On the Titration Curves and Titration Errors in Donor Acceptor Titrations of Displacement and Electronic… http://dx.doi.org/10.5772/intechopen.68750 133

Figure 5. Titration curve of Fe2<sup>þ</sup> with MnO4 �, at different pH values.

peT´ <sup>0</sup> <sup>¼</sup> peT

132 Redox - Principles and Advanced Applications

<sup>Δ</sup><sup>T</sup> <sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K0 S=K<sup>0</sup> T

when MnO4

1 þ ½ � e K<sup>0</sup> S ½ �e 1�5

pH ¼ 0, E ¼ 1.51 v, normal MnO4

4.5. Titration curve of V2<sup>þ</sup> with MnO4

contributions of each individual reaction

p

Figure 4. Titration curve of a mixture of Fe2<sup>þ</sup> and Tl<sup>þ</sup> with Ce4<sup>þ</sup>.

pH <sup>¼</sup> log KT

The expression for the valuation error will be given by (a ¼ 1, b ¼ 5)

<sup>2</sup> sinh ln 10 5ð Þ <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>pe´

<sup>5</sup> � <sup>8</sup> 5

The terms of Eq. (54) are conditional (pH-dependent). In the equivalence point

� � � �

�

pe´eq <sup>¼</sup> log <sup>K</sup><sup>0</sup>

2

The curve and the titration error are shown in Figures 5 and 6, respectively, at different pH values. Part of the necessary calculations, at pH ¼ 0, is shown in Table 2. When T ¼ 2 and

�/Mn2<sup>þ</sup> system potential.

Oxido-reductor systems involving the vanadium oxidation states 2þ, 3þ, 4þ, and 5þ are shown in Table 3. The V2<sup>þ</sup> ion undergoes various successive ionizations at 3þ, 4þ, and 5<sup>þ</sup>

jumps. The total ΔT function (or total T) is additive. The titration curve is the sum of the

� is added, being appreciated in the curve (Figure 7), the three corresponding

ΔTSUM ¼ ΔT<sup>32</sup> þ ΔT<sup>43</sup> þ ΔT<sup>54</sup> ð57Þ

TSUM ¼ ΔTSUM � 3 ¼ T<sup>32</sup> þ T<sup>43</sup> þ T<sup>54</sup> ð58Þ

pH <sup>¼</sup> log <sup>K</sup><sup>0</sup>

T <sup>5</sup> log <sup>K</sup><sup>0</sup>

<sup>T</sup> þ log K<sup>0</sup>

S

<sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K0 S=K<sup>0</sup> T

1 þ ½ � e K<sup>0</sup> S ½ �<sup>e</sup> �<sup>2</sup>

p

<sup>T</sup> ¼ log KT � 8pH ð54Þ

sinhð3 ln 10ðpe � pe´eqÞÞ

ð56Þ

<sup>a</sup> <sup>þ</sup> <sup>b</sup> <sup>ð</sup>55<sup>Þ</sup>

Figure 6. Titration error diagram ΔT ¼ f(E) at different pH values.

The head of the spreadsheet in EXCEL with all the necessary elements to carry out the numerical calculations applying the equations of the type of Eq. (22) is shown in Table 3. From the corresponding EXCEL sheet we have extracted a few columns, those corresponding to the



Table 3. Head of the spreadsheet in EXCEL of the titration curve of V(II) with MnO4.

Figure 7. Titration curve of V2<sup>þ</sup> with MnO4 �, at pH 0, 1 and 2.

values of E, pX, T1, T2, T3 and TSUM (1, 2, 3 refer to 32, 43, and 54, respectively, in Eqs. 57 and

pH¼0 MnO4

E0T¼1.51 neT¼5 nHT¼8 f(pHT)¼ 0 E0T<sup>0</sup>

E0S¼0.77 neS¼1 nHS¼0 f(pHS)¼ 0 E0S<sup>0</sup>

¼6.81292Eþ12 Wc¼6.32456E�57 pXeq¼23.111 Eeq¼1.387

Ws [X]^(a�b)/2 W sine h ΔT T E X pX ΔpX LOG(ABS

0.0909 4.642Eþ23 2.669E�34 �3.406Eþ33 �0.9091 0.0909 0.71 1.468E�12 11.833 �11.278 �4.139E�02 0.1280 1.000Eþ24 8.095E�34 �1.077Eþ33 �0.8720 0.1280 0.72 1.000E�12 12.000 �11.111 �5.948E�02 0.1773 2.154Eþ24 2.415E�33 �3.406Eþ32 �0.8227 0.1773 0.73 6.813E�13 12.167 �10.944 �8.473E�02 0.2403 4.642Eþ24 7.053E�33 �1.077Eþ32 �0.7597 0.2403 0.74 4.642E�13 12.333 �10.778 �1.193E�01 0.3170 1.000Eþ25 2.005E�32 �3.406Eþ31 �0.6830 0.3170 0.75 3.162E�13 12.500 �10.611 �1.656E�01 0.4052 2.154Eþ25 5.521E�32 �1.077Eþ31 �0.5948 0.4052 0.76 2.154E�13 12.667 �10.444 �2.256E�01 0.5000 4.642Eþ25 1.468E�31 �3.406Eþ30 �0.5000 0.5000 0.77 1.468E�13 12.833 �10.278 �3.010E�01 0.5948 1.000Eþ26 3.762E�31 �1.077Eþ30 �0.4052 0.5948 0.78 1.000E�13 13.000 �10.111 �3.923E�01 0.6830 2.154Eþ26 9.306E�31 �3.406Eþ29 �0.3170 0.6830 0.79 6.813E�14 13.167 �9.944 �4.989E�01 0.7597 4.642Eþ26 2.230E�30 �1.077Eþ29 �0.2403 0.7597 0.80 4.642E�14 13.333 �9.778 �6.193E�01 0.8227 1.000Eþ27 5.203E�30 �3.406Eþ28 �0.1773 0.8227 0.81 3.162E�14 13.500 �9.611 �7.514E�01 0.8720 2.154Eþ27 1.188E�29 �1.077Eþ28 �0.1280 0.8720 0.82 2.154E�14 13.667 �9.444 �8.928E�01 0.9091 4.642Eþ27 2.669E�29 �3.406Eþ27 �0.0909 0.9091 0.83 1.468E�14 13.833 �9.278 �1.041Eþ00 0.9362 1.000Eþ28 5.921E�29 �1.077Eþ27 �0.0638 0.9362 0.84 1.000E�14 14.000 �9.111 �1.195Eþ00 0.9556 2.154Eþ28 1.302E�28 �3.406Eþ26 �0.0444 0.9556 0.85 6.813E�15 14.167 �8.944 �1.353Eþ00 0.9693 4.642Eþ28 2.846E�28 �1.077Eþ26 �0.0307 0.9693 0.86 4.642E�15 14.333 �8.778 �1.514Eþ00 0.9789 1.000Eþ29 6.191E�28 �3.406Eþ25 �0.0211 0.9789 0.87 3.162E�15 14.500 �8.611 �1.676Eþ00 0.9855 2.154Eþ29 1.343E�27 �1.077Eþ25 �0.0145 0.9855 0.88 2.154E�15 14.667 �8.444 �1.840Eþ00 0.9901 4.642Eþ29 2.907E�27 �3.406Eþ24 �0.0099 0.9901 0.89 1.468E�15 14.833 �8.278 �2.004Eþ00 0.9932 1.000Eþ30 6.282E�27 �1.077Eþ24 �0.0068 0.9932 0.90 1.000E�15 15.000 �8.111 �2.170Eþ00 0.9954 2.154Eþ30 1.356E�26 �3.406Eþ23 �0.0046 0.9954 0.91 6.813E�16 15.167 �7.944 �2.335Eþ00

(III) þ 4 H2O

¼1.51

¼0.77

� þ 5 Fe(II) þ 8 H<sup>þ</sup> ¼ Mn(II) þ 5 Fe

(ΔpX))

Potential values varying from 0.05 in 0.05 units have been varied in this table in order to cover the entire valuation curve. Figure 7 has been drawn instead with potential variations of only

þ), which are

58, that is, the oxidations of V2<sup>þ</sup> to V3þ, from this to VO2<sup>þ</sup> and from this to VO2

shown in Table 4.

Table 2. Titration curve of Fe(II) with MnO4(�).

0.01 v.

KT<sup>0</sup>

¼6.8129Eþ125 KS<sup>0</sup>

134 Redox - Principles and Advanced Applications
