7.1. Titration in K2Cr2O7 (C, V) ) KI (C0)+H2SO4 (C01) V0 system

Note that V<sup>0</sup> mL of titrand D composed of KI (N03)+H2SO4 (N04)+H2O (N05) is titrated with V mL of K2Cr2O7 (N01)+H2O (N02) as titrant T added up to a defined point of the titration. In V<sup>0</sup> + V mL of D + T mixture, we have the following species:

H2OðN1Þ, <sup>H</sup>þ<sup>1</sup> <sup>ð</sup>N2, n2<sup>Þ</sup> OH�<sup>1</sup>ðN3, n3Þ, <sup>K</sup>þ<sup>1</sup> ðN4, n4Þ, HSO4 �<sup>1</sup>ðN5, n5Þ, SO4 �<sup>2</sup>ðN6, n6Þ, I �<sup>1</sup>ðN7, n7Þ, I<sup>3</sup> �<sup>1</sup>ðN8, n8Þ, I2ðN9, n9Þ,I<sup>2</sup>ð<sup>s</sup>ÞðN10, n10Þ, HIO <sup>ð</sup>N11, n11Þ, IO�<sup>1</sup>ðN12, n12Þ, HIO3ðN13, n13Þ, IO3 �<sup>1</sup>ðN14, n14Þ, H5IO6ðN15, n15Þ, H4IO6 �<sup>1</sup>ðN16, n16Þ, H3IO6 �<sup>2</sup>ðN17, n17Þ, HCr2O7 �<sup>1</sup>ðN18, n18Þ, Cr2O7 �<sup>2</sup>ðN19, n19Þ, H2CrO4ðN20, n20Þ, HCrO4 �<sup>1</sup>ðN21, n21Þ, CrO4 �<sup>2</sup>ðN22, n22Þ, Crþ<sup>3</sup> <sup>ð</sup>N23, n23Þ, CrOH<sup>þ</sup><sup>2</sup> ðN24, n24Þ, CrðOHÞ<sup>2</sup> <sup>þ</sup><sup>1</sup>ðN25, n25Þ, CrðOHÞ<sup>4</sup> �<sup>1</sup>ðN26, n26Þ, CrSO4 <sup>þ</sup><sup>1</sup>ðN27, n27Þ:

On this basis, we formulate f<sup>1</sup> = f(H), f<sup>2</sup> = f(O) and then:

BrO3

narrowed (Figure 9, column c).

36 Redox - Principles and Advanced Applications

For Φ > 0.2, an increase of efficiency of the competitive reaction

7. Redox systems with two electron-active elements

2Br�<sup>1</sup> <sup>þ</sup> BrO3

is noted. A growth of C<sup>03</sup> value causes a small extension of the potential range in the jump region, on the side of higher E values (Figure 9, column b). With an increase in the C<sup>03</sup> value, the graphs of pH vs. Φ resemble two almost straight-line segments intersecting at Φeq = 0.2 (Figure 9, column c). However, the pH ranges covered by the titration curves are gradually

Figure 9. Plots for the KBrO3 (C, V) ) NaBr (C0)+H2SO4 (C03) V0 system: speciation diagrams (column a); E = E(Φ) (column b) and pH = pH(Φ) (column c), at V<sup>0</sup> = 100, C<sup>0</sup> = 0.01, C = 0.1 and indicated C<sup>03</sup> [mol/L] values for H2SO4.

In the redox systems considered above, one electron-active element was involved in disproportionation or symproportionation reactions affected by NaOH (in dynamic systems) or water (in static systems). In the HCl (C, V) ) NaIO (C0, V0) system, where HCl was used as disproportionating reagent, the possibility of oxidation of Cl�<sup>1</sup> ions was allowed a priori, in which HCl was used as a disproportionating reagent, i.e. chlorine in HCl was treated as an

�<sup>1</sup> <sup>þ</sup> 5Br�<sup>1</sup> <sup>þ</sup> 6Hþ<sup>1</sup> <sup>¼</sup> 3Br2 <sup>þ</sup> 3H2O (62)

�<sup>1</sup> <sup>þ</sup> 3Hþ<sup>1</sup> <sup>¼</sup> 3HBrO (63)

$$\begin{aligned} f\_{12} &= 2 \cdot f(\mathbf{O}) \cdot f(\mathbf{H}) \cdot \mathbf{N}\_2 + \mathbf{N}\_3 + 7\mathbf{N}\_5 + 8\mathbf{N}\_6 + \mathbf{N}\_{11} + 2\mathbf{N}\_{12} + 5\mathbf{N}\_{13} + 6\mathbf{N}\_{14} \\ &+ 7\mathbf{N}\_{15} + 8\mathbf{N}\_{16} + 9\mathbf{N}\_{17} + 13\mathbf{N}\_{18} + 14\mathbf{N}\_{19} + 6\mathbf{N}\_{20} + 7\mathbf{N}\_{21} + 8\mathbf{N}\_{22} + \mathbf{N}\_{24} \\ &+ 2\mathbf{N}\_{25} + 4\mathbf{N}\_{26} + 8\mathbf{N}\_{27} = 14\mathbf{N}\_{01} + 6\mathbf{N}\_{04} \end{aligned} \tag{64}$$

$$\begin{aligned} f\_0 &= \text{ChB} : \text{N}\_2 - \text{N}\_3 + \text{N}\_4 - \text{N}\_5 - 2\text{N}\_6 - \text{N}\_7 - \text{N}\_8 - \text{N}\_{12} - \text{N}\_{14} - \text{N}\_{16} - 2\text{N}\_{17} - \text{N}\_{18} \\ &- 2\text{N}\_{19} - \text{N}\_{21} - 2\text{N}\_{22} + 3\text{N}\_{23} + 2\text{N}\_{24} + \text{N}\_{25} - \text{N}\_{26} + \text{N}\_{27} = 0 \end{aligned} \tag{65}$$

$$f\_{\mathfrak{J}} = f(\mathbf{K}):\quad \mathcal{N}\_{\mathfrak{k}} = 2\mathcal{N}\_{01} + \mathcal{N}\_{03})\tag{66}$$

$$f\_4 = f(\mathbf{S}) = f(\mathbf{SO\_4}):\ \ N\_5 + N\_6 + N\_{27} = N\_{04} \tag{67}$$

$$2\cdot f(\mathcal{O}) - f(H) + \text{ChB} - f(\mathcal{K}) - 6\cdot f(\mathcal{S}) = 0 \Rightarrow\tag{68}$$

$$\begin{aligned} &-\text{N}\_7 - \text{N}\_8 + \text{N}\_{11} + \text{N}\_{12} + 5\text{N}\_{13} + 5\text{N}\_{14} + 7\text{N}\_{15} + 7\text{N}\_{16} + 7\text{N}\_{17} + 12\text{N}\_{18} + 12\text{N}\_{19} + 6\text{N}\_{20} \\ &+ 6\text{N}\_{21} + 6\text{N}\_{22} + 3\text{N}\_{23} + 3\text{N}\_{24} + 3\text{N}\_{25} + 3\text{N}\_{26} + 3\text{N}\_{27} = 12\text{N}\_{01} - \text{N}\_{03} \Rightarrow \end{aligned} \tag{68a}$$

$$\begin{aligned} (-1) \cdot N\_7 + (-1/3) \cdot 3N\_8 + 0 \cdot (N\_9 + N\_{10}) + (+1) \cdot (N\_{11} + N\_{12}) + (+5) \cdot (N\_{13} + N\_{14}) \\ + (+7) \cdot (N\_{15} + N\_{16} + N\_{17}) + (+6) \cdot (2N\_{18} + 2N\_{19} + N\_{20} + N\_{21} + N\_{22}) + (+3) \cdot \\ (N\_{23} + N\_{24} + N\_{25} + N\_{26} + N\_{27}) = (+6) \cdot 2N\_{01} + (-1) \cdot N\_{03} \end{aligned} \tag{68b}$$

$$\begin{aligned} &(-1)\cdot[\Gamma^{-1}]+(-1/3)\cdot3[I\_{3}^{-1}]+0\Big(2[I\_{2}]+2[I\_{2(s)}]\Big)+(+1)([\text{HIO}][\text{IO}^{-1}])+(+5)\cdot([\text{HIO}\_{3}]+[\text{IO}\_{3}^{-1}]) \\ &+(+7)\cdot([\text{H}\_{5}\text{IO}\_{6}]+[\text{H}\_{4}\text{IO}\_{6}^{-1}]+[\text{H}\_{3}\text{IO}\_{6}^{-2}])+(+6)\cdot(2[\text{HCr}\_{2}\text{O}\_{7}^{-1}])+2[\text{Cr}\_{2}\text{O}\_{7}^{-2}]+[\text{H}\_{2}\text{CrO}\_{4}] \\ &+[\text{HCrO}\_{4}^{-1}]+[\text{CrO}\_{4}^{-2}])+(+3)\cdot([\text{Cr}^{3}]+[\text{CrOH}^{+2}]+[\text{Cr(OH)}\_{2}]^{+1})+[\text{Cr(OH)}\_{4}^{-1}] \\ &+[\text{CrSO}\_{4}^{+1}]) = ((+6)\cdot2\text{CV}+(-1)\cdot\text{C}\_{0}V\_{0})/(V\_{0}+V) \end{aligned} \tag{69}$$

where C0V<sup>0</sup> = 103 ∙N03/NA, CV = 103 ∙N01/NA. Note that Eq. (69) was obtained only from linear combination of electron-non-active elements (fans) in this system. The balances for electronactive elements are as follows:

$$f\_{5} = f(\mathbf{I}):\ N\_{7} + 3\mathbf{N}\_{8} + 2\mathbf{N}\_{9} + 2\mathbf{N}\_{10} + \mathbf{N}\_{11} + \mathbf{N}\_{12} + \mathbf{N}\_{13} + \mathbf{N}\_{14} + \mathbf{N}\_{15} + \mathbf{N}\_{16} + \mathbf{N}\_{17} = \mathbf{N}\_{03} \tag{70}$$

$$\begin{aligned} \left( \left[ I^{-1} \right] + 3 \left[ I\_3^{-1} \right] + 2 \cdot \left( \left[ I\_2 \right] + \left[ I\_{2(s)} \right] \right) + \left( \left[ \text{HIO} \right] + \left[ \text{IO}^{-1} \right] \right) + 5 \cdot \left( \left[ \text{HIO}\_3 \right] + \left[ \text{IO}\_3^{-1} \right] \right) + 7 \cdot \left( \left[ \text{H}\_5 \text{IO}\_6 \right] \right) \\ + \left[ \left[ \text{H}\_4 \text{IO}\_6^{-1} \right] + \left[ \text{H}\_5 \text{IO}\_6^{-2} \right] \right) = \text{Co} \, V\_0 / (V\_0 + V) \end{aligned} \tag{71}$$

$$f\_6 = f(\mathbf{Cr}): \ 2\mathbf{N}\_{18} + 2\mathbf{N}\_{19} + \mathbf{N}\_{20} + \mathbf{N}\_{21} + \mathbf{N}\_{22} + \mathbf{N}\_{23} + \mathbf{N}\_{24} + \mathbf{N}\_{25} + \mathbf{N}\_{26} + \mathbf{N}\_{27} = 2\mathbf{N}\_{01} \Rightarrow \tag{72}$$

$$\begin{aligned} &2[\text{Cr}\_2\text{O}\_7^{-2}] + [\text{H}\_2\text{CrO}\_4] + [\text{HCrO}\_4^{-1}] + [\text{CrO}\_4^{-2}]) + [\text{Cr}^{+3}] + [\text{CrOH}^{+2}] \\ &+ [\text{Cr(OH)}\_2^{+1}] + [\text{Cr(OH)}\_4^{-1}] + [\text{CrSO}\_4^{+1}] = \text{CV/(V}\_0 + \text{V}) \end{aligned} \tag{73}$$

Subtraction of 3∙f(Cr) (Eq. (72)) from Eq. (68a) and further operations give

$$\begin{aligned} \left[\text{I}^{-1}\right] + \left[\text{I}\_{3}^{-1}\right] - \left[\text{HIO}\right] + \left[\text{IO}^{-1}\right] \left|-\text{5}\right\rangle \left(\left[\text{HIO}\_{3}\right] + \left[\text{IO}\_{3}^{-1}\right]\right) - \mathcal{T} \left(\left[\text{H}\_{5}\text{IO}\_{6}\right] + \left[\text{H}\_{4}\text{IO}\_{6}^{-1}\right] + \left[\text{H}\_{3}\text{IO}\_{6}^{-2}\right]\right) \\ - 3 \cdot \left(2\left[\text{HCr}\_{2}\text{O}\_{7}^{-1}\right] + 2\left[\text{Cr}\_{2}\text{O}\_{7}^{-2}\right] + \left[\text{H}\_{2}\text{CrO}\_{4}\right] + \left[\text{HCrO}\_{4}^{-1}\right] + \left[\text{CrO}\_{4}^{-2}\right]\right) = \left(\text{C}\_{0}V\_{0} - 6\text{CV}\right)/\left(V\_{0} + V\right) \end{aligned} \tag{74}$$

The simplest/shortest form of GEB, obtained from Eqs. (68), (70) and (72) is the relation

$$2\cdot f(\mathbf{O}) - f(\mathbf{H}) + \,\,\mathbf{ChB} - f(\mathbf{K}) - 6\cdot f(\mathbf{S}) + f(\mathbf{I}) - 6\cdot f(\mathbf{C}\mathbf{r}) = 0 \,\Rightarrow\tag{75}$$

$$\begin{aligned} \mathbf{N\_8} + \mathbf{N\_9} + \mathbf{N\_{10}} + \mathbf{N\_{11}} + \mathbf{N\_{12}} + \mathbf{3} \cdot (\mathbf{N\_{13}} + \mathbf{N\_{14}}) + \mathbf{4} \cdot (\mathbf{N\_{15}} + \mathbf{N\_{16}} + \mathbf{N\_{17}}) \\ \mathbf{r} = \mathbf{1.5} \cdot (\mathbf{N\_{23}} + \mathbf{N\_{24}} + \mathbf{N\_{25}} + \mathbf{N\_{26}}) \Rightarrow \end{aligned} \tag{75a}$$

$$\begin{aligned} \left( \left[ I\_3^{-1} \right] + \left[ I\_2 \right] + \left[ \mathbf{I}\_{2(s)} \right] + \left[ \mathbf{H} \mathbf{O} \right] + \left[ \mathbf{I} \mathbf{O}^{-1} \right] + 3 \cdot \left( \left[ \mathbf{H} \mathbf{O}\_3 \right] + \left[ \mathbf{I} \mathbf{O}\_3^{-1} \right] \right) + 4 \cdot \left( \left[ \mathbf{H}\_5 \mathbf{O}\_6 \right] + \left[ \mathbf{H}\_4 \mathbf{O}\_6^{-1} \right] \right) \\ + \left[ \mathbf{H}\_3 \mathbf{O} \mathbf{O}\_6^{-2} \right] \left[ \right] &= 1.5 \cdot \left( \left[ \mathbf{C} \mathbf{r}^{+3} \right] + \left[ \mathbf{C} \mathbf{r} \mathbf{O} \mathbf{H}^{+2} \right] + \left[ \mathbf{C} \mathbf{r} \left( \mathbf{O} \mathbf{H} \right)\_2^{+1} \right] \left[ \mathbf{C} \mathbf{r} \left( \mathbf{O} \mathbf{H} \right)\_4^{-1} \right] + \left[ \mathbf{C} \mathbf{r} \mathbf{O}\_4^{+1} \right] \end{aligned} \tag{75b}$$

Note that the numbers (N01, N03) of components forming the system are not involved in Eqs. (75a) and (75b).

Applying the atomic numbers: ZI = 53 for I and ZCr = 24 for Cr, we obtain the linear combination

$$\begin{split} &\mathbf{Z}\_{\mathrm{I}} \cdot f(\mathbf{I}) + \mathbf{Z}\_{\mathrm{Cr}} \cdot f(\mathbf{C}\mathbf{r}) - \Big( 2 \cdot f(\mathbf{O}) \cdot f(\mathbf{H}) + \mathbf{C}\mathbf{h} \mathbf{B} - f(\mathbf{K}) - \mathbf{6} \cdot f(\mathbf{S}) \Big) = \mathbf{0} \Rightarrow & (\mathbf{Z}\_{\mathrm{I}} + 1)\mathbf{N}\_{\mathrm{I}} + (3\mathbf{Z}\_{\mathrm{I}} + 1)\mathbf{N}\_{\mathrm{R}} \\ &+ 2\mathbf{Z}\_{\mathrm{I}}(\mathbf{N}\_{\mathrm{R}} + \mathbf{N}\_{\mathrm{I}0}) + (\mathbf{Z}\_{\mathrm{I}} - \mathbf{1})(\mathbf{N}\_{\mathrm{I}1} + \mathbf{N}\_{\mathrm{I}2}) + (\mathbf{Z}\_{\mathrm{I}} - \mathbf{5})(\mathbf{N}\_{\mathrm{I}3} + \mathbf{N}\_{\mathrm{I}4}) + (\mathbf{Z}\_{\mathrm{I}} - \mathbf{7})(\mathbf{N}\_{\mathrm{I}5} + \mathbf{N}\_{\mathrm{I}6} + \mathbf{N}\_{\mathrm{I}7}) \\ &+ (\mathbf{Z}\_{\mathrm{Cr}} - \mathbf{6})(2\mathbf{N}\_{\mathrm{R}} + 2\mathbf{N}\_{\mathrm{I}9} + \mathbf{N}\_{\mathrm{20}} + \mathbf{N}\_{\mathrm{21}} + \mathbf{N}\_{\mathrm{22}}) + (\mathbf{Z}\_{\mathrm{Cr}} \cdot \mathbf{3})(\mathbf{N}\_{\mathrm{23}} + \mathbf{N}\_{\mathrm{24}} + \mathbf{N}\_{\mathrm{25}} + \mathbf{N}\_{\mathrm{26}}) \\ &= (\mathbf{Z}\_{\mathrm{I}} + 1)\mathbf{N}\_{\mathrm{03}} + 2(\mathbf{Z}\_{\mathrm{Cr}} \cdot \mathbf{6})\mathbf{N}\_{\mathrm{01}$$

ðZ<sup>I</sup> þ 1Þ½I �1 �þð3ZI þ 1Þ½I �1 <sup>3</sup> � þ 2ZIð½I2�þ½I2�Þ þ ðZI–1Þð½HIO�þ½IO�<sup>1</sup> �Þ þ ðZI–5Þð½HIO3� þ½IO�<sup>1</sup> <sup>3</sup> �Þ þ ðZI–7Þð½H5IO6�þ ½H4IO�<sup>1</sup> <sup>6</sup> �þ½H3IO�<sup>2</sup> <sup>6</sup> �Þ þ ðZCr–6Þð2½HCr2O�<sup>1</sup> <sup>7</sup> � þ <sup>2</sup>½Cr2O�<sup>2</sup> <sup>7</sup> � þ½H2CrO4�þ½HCrO�<sup>1</sup> <sup>4</sup> �þ½CrO�<sup>2</sup> <sup>4</sup> �Þ þ ðZCr � <sup>3</sup>Þð½Crþ<sup>3</sup> �þ½CrOH<sup>þ</sup><sup>2</sup> �þ½CrðOHÞ þ1 <sup>2</sup> � þ½CrðOH�<sup>1</sup> <sup>4</sup> Þ� þ ½CrSOþ<sup>1</sup> <sup>4</sup> �Þ ¼ ððZI þ 1Þ � C0V<sup>0</sup> þ 2ðZCr–6Þ � CVÞ=ðV<sup>0</sup> þ VÞ (77)

(76)

Eqs. (69), (74), (75b) and (77) (and other linear combinations, as well) are equivalent forms of GEB for this system. Note that Eq. (77) is identical with the one obtained immediately on the basis of the approach I to GEB [4]. The E = E(Φ) and pH = pH(Φ) and some speciation curves for iodine and chromium species are plotted in Figures 10 and 11, where Φ is the fraction titrated.

For C<sup>01</sup> = 0.01, I2, I2(s) and I<sup>3</sup> �<sup>1</sup> are formed in reactions:

where C0V<sup>0</sup> = 103

þ ½H4IO�<sup>1</sup>

�<sup>3</sup> � ð2½HCr2O�<sup>1</sup>

½I �1 � þ 3½I �1

½I �1 �þ½I �1

> ½I �1

and (75b).

ðZ<sup>I</sup> þ 1Þ½I

þ½CrðOH�<sup>1</sup>

þ½IO�<sup>1</sup>

þ½H3IO�<sup>2</sup>

Z<sup>I</sup> � fðIÞ þ ZCr � fðCrÞ–

�1

þ½H2CrO4�þ½HCrO�<sup>1</sup>

¼ ðZ<sup>I</sup> þ 1ÞN<sup>03</sup> þ 2ðZCr–6ÞN<sup>01</sup> )

�þð3ZI þ 1Þ½I

<sup>4</sup> Þ� þ ½CrSOþ<sup>1</sup>

<sup>3</sup> �Þ þ ðZI–7Þð½H5IO6�þ ½H4IO�<sup>1</sup>

�1

<sup>4</sup> �þ½CrO�<sup>2</sup>

active elements are as follows:

38 Redox - Principles and Advanced Applications

<sup>6</sup> �þ½H3IO�<sup>2</sup>

<sup>2</sup>½Cr2O�<sup>2</sup>

þ ½CrðOHÞ

<sup>3</sup> ��½HIO�þ½IO�<sup>1</sup>

<sup>3</sup> �þ½I2�þ½I<sup>2</sup>ð<sup>s</sup>Þ�þ½HIO�þ½IO�<sup>1</sup>

<sup>6</sup> �Þ ¼ <sup>1</sup>:<sup>5</sup> � ð½Crþ<sup>3</sup>

<sup>7</sup> � þ <sup>2</sup>½Cr2O�<sup>2</sup>

þ1

∙N03/NA, CV = 103

<sup>3</sup> � þ <sup>2</sup> � ð½I2�þ½I<sup>2</sup>ð<sup>s</sup>Þ�Þ þ ð½HIO�þ½IO�<sup>1</sup>

<sup>7</sup> �þ½H2CrO4�þ½HCrO�<sup>1</sup>

Subtraction of 3∙f(Cr) (Eq. (72)) from Eq. (68a) and further operations give

�þ½CrOHþ<sup>2</sup>

�1

�Þ � <sup>5</sup> � ð½HIO3�þ½IO�<sup>1</sup>

<sup>7</sup> �þ½H2CrO4�þ½HCrO�<sup>1</sup>

The simplest/shortest form of GEB, obtained from Eqs. (68), (70) and (72) is the relation

N<sup>8</sup> þ N<sup>9</sup> þ N<sup>10</sup> þ N<sup>11</sup> þ N<sup>12</sup> þ 3 � ðN<sup>13</sup> þ N14Þ þ 4 � ðN<sup>15</sup> þ N<sup>16</sup> þ N17Þ

� þ <sup>3</sup> � ð½HIO3�þ½IO�<sup>1</sup>

�þ½CrðOHÞ

Note that the numbers (N01, N03) of components forming the system are not involved in Eqs. (75a)

Applying the atomic numbers: ZI = 53 for I and ZCr = 24 for Cr, we obtain the linear combination

<sup>3</sup> � þ 2ZIð½I2�þ½I2�Þ þ ðZI–1Þð½HIO�þ½IO�<sup>1</sup>

<sup>4</sup> �Þ ¼ ððZI þ 1Þ � C0V<sup>0</sup> þ 2ðZCr–6Þ � CVÞ=ðV<sup>0</sup> þ VÞ

<sup>6</sup> �þ½H3IO�<sup>2</sup>

<sup>4</sup> �Þ þ ðZCr � <sup>3</sup>Þð½Crþ<sup>3</sup>

2 � fðOÞ–fðHÞ þ ChB–fðKÞ– 6 � fðSÞ

þðZCr–6Þð2N<sup>18</sup> þ 2N<sup>19</sup> þ N<sup>20</sup> þ N<sup>21</sup> þ N22ÞþðZCr–3ÞðN<sup>23</sup> þ N<sup>24</sup> þ N<sup>25</sup> þ N26Þ

þ2ZIðN<sup>9</sup> þ N10ÞþðZI–1ÞðN<sup>11</sup> þ N12ÞþðZI–5ÞðN<sup>13</sup> þ N14ÞþðZI–7ÞðN<sup>15</sup> þ N<sup>16</sup> þ N17Þ

<sup>2</sup> �þ½CrðOHÞ

∙N01/NA. Note that Eq. (69) was obtained only from linear

<sup>3</sup> �Þ þ 7 � ð½H5IO6�

�

<sup>6</sup> �þ½H3IO�<sup>2</sup>

<sup>4</sup> �Þ ¼ ðC0V0–6CVÞ=ðV<sup>0</sup> þ VÞ

<sup>4</sup> �Þ

¼ 0 ) ðZ<sup>I</sup> þ 1ÞN7 þ ð3Z<sup>I</sup> þ 1ÞN<sup>8</sup>

�Þ þ ðZI–5Þð½HIO3�

�þ½CrðOHÞ

<sup>7</sup> � þ <sup>2</sup>½Cr2O�<sup>2</sup>

þ1 <sup>2</sup> � <sup>7</sup> �

<sup>6</sup> �Þ

<sup>6</sup> �

(74)

(75b)

(76)

(77)

�þ½CrOH<sup>þ</sup><sup>2</sup>

<sup>4</sup> � ¼ CV=ðV0 <sup>þ</sup> <sup>V</sup><sup>Þ</sup> (73)

<sup>3</sup> �Þ þ <sup>4</sup> � ð½H5IO6�þ½H4IO�<sup>1</sup>

<sup>4</sup> �þ½CrSO<sup>þ</sup><sup>1</sup>

�1

�Þ þ <sup>5</sup> � ð½HIO3�þ½IO�<sup>1</sup>

<sup>4</sup> �Þ þ ½Crþ<sup>3</sup>

<sup>3</sup> �Þ � <sup>7</sup>ð½H5IO6�þ½H4IO�<sup>1</sup>

<sup>4</sup> �þ½CrO�<sup>2</sup>

2 � fðOÞ � fðHÞ þ ChB � fðKÞ � 6 � fðSÞ þ fðIÞ � 6 � fðCrÞ ¼ 0 ) (75)

<sup>¼</sup> <sup>1</sup>:<sup>5</sup> � ðN<sup>23</sup> <sup>þ</sup> <sup>N</sup><sup>24</sup> <sup>þ</sup> <sup>N</sup><sup>25</sup> <sup>þ</sup> <sup>N</sup>26Þ ) (75a)

þ1 <sup>2</sup> �½CrðOHÞ

<sup>6</sup> �Þ þ ðZCr–6Þð2½HCr2O�<sup>1</sup>

�þ½CrOH<sup>þ</sup><sup>2</sup>

<sup>6</sup> �Þ ¼ <sup>C</sup>0V0=ðV<sup>0</sup> <sup>þ</sup> <sup>V</sup><sup>Þ</sup> (71)

combination of electron-non-active elements (fans) in this system. The balances for electron-

f <sup>5</sup> ¼ fðIÞ : N<sup>7</sup> þ 3N<sup>8</sup> þ 2N<sup>9</sup> þ 2N<sup>10</sup> þ N<sup>11</sup> þ N<sup>12</sup> þ N<sup>13</sup> þ N<sup>14</sup> þ N<sup>15</sup> þ N<sup>16</sup> þ N<sup>17</sup> ¼ N<sup>03</sup> (70)

f <sup>6</sup> ¼ fðCrÞ : 2N<sup>18</sup> þ 2N<sup>19</sup> þ N<sup>20</sup> þ N<sup>21</sup> þ N<sup>22</sup> þ N<sup>23</sup> þ N<sup>24</sup> þ N<sup>25</sup> þ N<sup>26</sup> þ N<sup>27</sup> ¼ 2N<sup>01</sup> ) (72)

<sup>4</sup> �þ½CrSO<sup>þ</sup><sup>1</sup>

<sup>4</sup> �þ½CrO�<sup>2</sup>

$$\text{Cr}\_2\text{O}\_7^{-2} + (6,6,9)\text{I}^{-1} + 14\text{H}^{+1} = 2\text{Cr}^{+3} + 3(I\_{2(s)}, I\_2, I\_3^{-1}) + 2\text{Cr}^{+3} + 7\text{H}\_2\text{O} \tag{78}$$

$$\text{Cr}\_2\text{O}\_7^{-2} + (6,6,9)\text{I}^{-1} + 12\text{H}^{+1} + 2\text{HSO}\_4^{-1} = 2\text{Cr}^{+3} + 3(I\_{2(s)}I\_{2}I\_3^{-1}) \ + 2\text{CrSO}\_4^{+1} + 7\text{H}\_2\text{O} \tag{79}$$

$$\text{Cr}\_2\text{O}\_7^{-2} + (6,6,9)\text{I}^{-1} + 14\text{H}^{+1} + 2\text{SO}\_4^{-2} = 2\text{Cr}^{+3} + 3(\text{I}\_{2(s)}, \text{I}\_2, \text{I}\_3^{-1}) + 2\text{CrSO}\_4^{+1} + 7\text{H}\_2\text{O} \tag{80}$$

where predominating products are involved. Binding the H+1 ions corresponds to the pH increase, which is largest for low C<sup>01</sup> value (relatively low buffer capacity of the solution, compare with [53, 54, 58–60]). In reactions (78)–(80), protons are consumed and then dpH/dΦ > 0; moreover, dE/dΦ > 0 for Φ > 0.

At C<sup>01</sup> = 0.02, E = E(Φ) and [I2(s)] pass through maximum at Φ ca. 0.2. The plot of pH = pH(Φ) shows a slight distortion of the course at Φ = 1/6, and DpH/DΦ > 0 for Φ > 0. The [IO3 �1 ] is comparable with [I2(s)] and [I2]; [I<sup>3</sup> �1 ] is small here because [I �] < 10�<sup>6</sup> . The reactions

$$\text{G(I\_2, I\_{2(s)}) + 5Cr\_2O\_7^{-2} + 34H^+ = 6IO\_3^{-1} + 10Cr^{+3} + 17H\_2O \tag{81}$$

$$\text{S(I\_2, I\_{2(s)}) } + \text{ 5Cr\_2O}\_7^{-2} + 24\text{H}^{+1} + 10\text{HSO}\_4^{-1} = 6\text{IO}\_3^{-1} + 10\text{CrSO}\_4^{+1} + 17\text{H}\_2\text{O} \tag{82}$$

Figure 10. The plots K2Cr2O7 (C, V) ) KI (C0)+H2SO4 (C01) V<sup>0</sup> system for (a) E = E(Φ) and (b) pH = pH(Φ) functions, at V<sup>0</sup> = 100, C<sup>0</sup> = 0.01 and C<sup>01</sup> values indicated at the corresponding curves.

Figure 11. Speciation diagrams plotted for iodine (a1, a2, a3) and chromium (b1, b2, b3) species, at indicated C<sup>01</sup> values.

$$\text{-S(I\_2, I\_{2(s)}) + 5Cr\_2O\_7^{-2} + 34H^+ + 10SO\_4^{-2} = 6IO\_3^{-1} + 10CrSO\_4^{+1} + 17H\_2O \tag{83}$$

are clearly indicated.

At C<sup>01</sup> = 0.05, the function E = E(Φ) has a complex course: it first increases, reaches a maximum at Φ = 0.295, then decreases, reaches a minimum at Φ = 1, increases again, passes through a flat maximum at Φ ca. 1.6 and decreases. The curve pH = pH(Φ) breaks at Φ = 1/6 and Φ = 1. The I2(s) exists as the equilibrium solid phase at 0.0934 < Φ < 0.762. A decrease in [I2] value is more expressed at Φ > 1. The ratio [HIO3]/[IO3 �1 ] grows with growth of C<sup>01</sup> value. The stoichiometry at Φ = Φeq2 = 1 is described by the reaction

$$\text{Cr}\_2\text{O}\_7^{-2} + \text{I}^{-1} + 8\text{H}^{+1} = 2\text{Cr}^{+3} + \text{IO}\_3^{-1} + 4\text{H}\_2\text{O} \tag{84}$$

and by reactions where HSO4 �1 , SO4 �2 , CrSO4 +1 and HIO3 are involved.

At C<sup>01</sup> ≥ 0.1, a jump on the E = E(Φ) curve at Φ = 1/6 is clearly marked. The growth of jump at Φ = 1 results from a more significant decrease in the [I2] value at Φ > 1. The E-range covered by the jump at Φ = 1 extends with an increase in the C<sup>01</sup> value (Figure 10a). For more details, see Ref. [4].

#### 7.2. A comment

<sup>3</sup>ðI2,I<sup>2</sup>ð<sup>s</sup>ÞÞ þ 5Cr2O�<sup>2</sup>

40 Redox - Principles and Advanced Applications

expressed at Φ > 1. The ratio [HIO3]/[IO3

at Φ = Φeq2 = 1 is described by the reaction

and by reactions where HSO4

Ref. [4].

Cr2O�<sup>2</sup> <sup>7</sup> þ I

> �1 , SO4 �2

are clearly indicated.

<sup>7</sup> <sup>þ</sup> 34Hþ<sup>1</sup> <sup>þ</sup> 10SO�<sup>2</sup>

�1

, CrSO4

At C<sup>01</sup> = 0.05, the function E = E(Φ) has a complex course: it first increases, reaches a maximum at Φ = 0.295, then decreases, reaches a minimum at Φ = 1, increases again, passes through a flat maximum at Φ ca. 1.6 and decreases. The curve pH = pH(Φ) breaks at Φ = 1/6 and Φ = 1. The I2(s) exists as the equilibrium solid phase at 0.0934 < Φ < 0.762. A decrease in [I2] value is more

Figure 11. Speciation diagrams plotted for iodine (a1, a2, a3) and chromium (b1, b2, b3) species, at indicated C<sup>01</sup> values.

�<sup>1</sup> <sup>þ</sup> 8Hþ<sup>1</sup> <sup>¼</sup> 2Crþ<sup>3</sup> <sup>þ</sup> IO�<sup>1</sup>

At C<sup>01</sup> ≥ 0.1, a jump on the E = E(Φ) curve at Φ = 1/6 is clearly marked. The growth of jump at Φ = 1 results from a more significant decrease in the [I2] value at Φ > 1. The E-range covered by the jump at Φ = 1 extends with an increase in the C<sup>01</sup> value (Figure 10a). For more details, see

<sup>4</sup> <sup>¼</sup> 6IO�<sup>1</sup>

<sup>3</sup> <sup>þ</sup> 10CrSO<sup>þ</sup><sup>1</sup>

] grows with growth of C<sup>01</sup> value. The stoichiometry

+1 and HIO3 are involved.

<sup>3</sup> þ 4H2O (84)

<sup>4</sup> þ 17H2O (83)

Redox systems with two electron-active elements were widely discussed from the GATES/GEB viewpoint. From the earlier literature, one can recall the simulated systems: KMnO4 (C, V) ) VSO4 (C0)+H2SO4 (C01) V<sup>0</sup> [18]; K2Cr2O7 (C, V) ) VSO4 (C0)+H2SO4 (C01) V<sup>0</sup> [18]; KMnO4 (C, V) ) FeSO4 (C0)+H2SO4 (C01) [18]; Ce(SO4)2 (C)+H2SO4 (C1) V ) FeSO4 (C0)+H2SO4 (C01) V<sup>0</sup> [18]; Cl2 (C, V) ) KI (C0, V0) [19]; KMnO4 (C, V) ) KBr (C0), H2SO4 (C01) V<sup>0</sup> [19]. Except the NaOH ) Br2 and NaOH ) HBrO systems, in [16] were simulated also the systems: NaOH (C) + CO2 (C1) V ) I2 (C0) + KI (C01) + CO2 (C02); NaOH (C) + CO2 (C1) V ) I2 (C0) + KI (C01) + CO2 (C02) V<sup>0</sup> + CCl4 (V\*) (dynamic liquid-liquid extraction system). In [16], an interesting qualitative reaction where MgSO4 solution was added into I<sup>2</sup> + KI solution previously alkalinized with an excess of NaOH. This addition causes precipitation of Mg(OH)2 and a decrease of pH value, a growth of solubility of this precipitate and a shift of equilibrium (symproportionation) involved with formation of iodide (I2) that adsorbes on the Mg(OH)2 precipitate turning it red-brown (similarity with starch action in visual iodometric titrations). The dynamic solubility curve was plotted, i.e. the MgSO4 addition is formally treated as titration. The speciation diagrams in all the systems considered were plotted. Among others, it is stated that the disproportionation of iodine (I2(s), I2) occurs according to the scheme (A) 3(I2(s), I2) + 6OH�<sup>1</sup> = IO3 �<sup>1</sup> + 5I�<sup>1</sup> + 3H2O, not the scheme (B) (I2(s), I2) + 2OH�<sup>1</sup> = IO�<sup>1</sup> + I�<sup>1</sup> + H2O, as stated in almost all contemporary textbooks. One can calculate [16] that the yield of the reaction (A) is 2.5∙109 times higher than the yield of reaction (B) of the same stoichiometry, 3:6 = 1:2. It is interesting to add that 80 years ≈ 2.5∙109 s. Thus, the yield of the reaction B relative to A is in such a ratio as 1 s in relation to the average length of human's life.

It should also be added that GATES/GEB enables to verify experimental results. Such a case was mentioned in Ref. [18] to the system, where KIO3 (C0) + HCl (C01)+H2SeO3 (C02) + HgCl2 (C03) was titrated with ascorbic acid (C6H8O6). Authors of the work [61] cited in there committed, among other things, a simple mistake resulting from improper recalculation of numerical values of potentials; for more details, see Refs. [1, 17, 18].

In all instances, full attainable (quantitative+qualitative) physicochemical knowledge was involved in the related algorithms. The quantitative knowledge was related to the complete set of equilibrium constants values, whereas qualitative knowledge was helpful in aspect of metastable and kinetic systems.

#### 7.3. An illustrative presentation of the approach I to GEB

A redox reaction can also participate two or more electron-active elements. In this convention, the approach I to GEB can be perceived as the card game, with electron-active elements as 'players', electron-non-active elements—as 'fans', and electrons—as 'money' transferred between 'players', see the picture from Ref. [62]. The 'players' provide a common pool of their own electrons into the system in question. The 'money' is transferred between 'players'. As a result of the game, the players' accounts are changed, while the fans' accounts remain intact/ unchanged, in this convention.

In C<sup>0</sup> mol/L aqueous solution of Br2, bromine as one electron-acitve element in Br2 can be perceived as a 'distributor' of its own electrons, and H, O as 'fans'. For example, Eq. (42) involves H and O as 'fans', Eq. (43) involves Na as 'fan', whereas Eq. (44) does not involve 'fans'. Generally, 'fans' are eliminated after the proper (i.e. indicated above) combination of 2∙f(O) – f(H) with the balances for elements considered as 'fans'. Further simplification can also result from further combination with the balance for 'player(s)'. But, in any case, a linear combination of balances related to a redox system does not lead to the identity 0 = 0.

As usually happens in the 'card game' practice, the players devote to the game only a part of their cash resources. Similarly, in redox reactions, electrons may participate from the valence shells of atoms of electron-active elements; the electrons from the valence shell of the reductant atoms are transferred onto the valence shell of the oxidant atoms. However, this restriction to the valence electrons is not required here. For example, replacing ZBr in Eq. (46a) by ζBr (ζBr < ZBr), we have

$$\begin{aligned} & (\zeta\_{\text{Br}} - \mathfrak{S})([\text{HBrO}\_3] + [\text{BrO}\_3^{-1}]) + (\zeta\_{\text{Br}} - 1)([\text{HBrO}] + [\text{BrO}^{-1}]) + 2\zeta\_{\text{Br}}[\text{Br}\_2] + (3\zeta\_{\text{Br}} + 1)[\text{Br}\_3^{-1}] \\ & + (\zeta\_{\text{Br}} + 1)[\text{Br}^{-1}] = 2\zeta\_{\text{Br}}\mathbb{C}\_0 V\_0 / (V\_0 + V) \end{aligned}$$

(85)

Replacing Z<sup>I</sup> by ζ<sup>I</sup> and ZCr by ζCr (ζ<sup>I</sup> < ZI, ζCr < ZCr) in Eq. (77), we have

$$\begin{aligned} & (\zeta\_{\text{I}} + 1)[I^{-}] + (3\zeta\_{\text{I}} + 1)[I\_{3}^{-1}] + 2\zeta\_{\text{I}}([I\_{2}] + [I\_{2}]) + (\zeta\_{\text{I}} - 1)([\text{H}\text{O}] + [\text{IO}^{-1}]) + (\zeta\_{\text{I}} - 5)([\text{H}\text{O}\_{3}] \\ & + [\text{IO}\_{3}^{-1}]) + (\zeta\_{\text{I}} - 7)([\text{H}\_{5}\text{O}\_{6}] + [\text{H}\_{4}\text{O}\_{6}^{-1}] + [\text{H}\_{3}\text{IO}\_{6}^{-2}]) + (\zeta\_{\text{Cr}} - 6)(2[\text{HCr}\_{2}\text{O}\_{7}^{-1}] + 2[\text{Cr}\_{2}\text{O}\_{7}^{-2}] \\ & + [\text{H}\_{2}\text{CrO}\_{4}] + [\text{HCrO}\_{4}^{-1}] + [\text{CrO}\_{4}^{-2}]) + (\zeta\_{\text{Cr}} - 3)([\text{Cr}^{+3}] + [\text{CrOH}^{+2}] + [\text{Cr(OH)}\_{2}^{+1}] \\ & + [\text{Cr(OH)}\_{4}^{-1}] + [\text{CrSO}\_{4}^{+1}]) = ((\zeta\_{\text{I}} + 1)\zeta\_{\text{I}}V\_{0} + 2(\zeta\_{\text{Cr}} - 6)\text{CV})/(V\_{0} + \text{V}) \end{aligned} \tag{86}$$

In particular, we can put ζCr = ζCr = 0 in Eq. (86). Obviously, we get the relation

½I �1 �þ½I �1 <sup>3</sup> ��ðHIOÞþ½IO�<sup>1</sup> � � <sup>5</sup>ð½HIO3�Þ þ ½IO�<sup>1</sup> <sup>3</sup> � � <sup>7</sup>ð½H5IO6�þ½H4IO�<sup>1</sup> <sup>6</sup> �þ½H3IO�<sup>2</sup> <sup>6</sup> �Þ �6ð2½HCr2O�<sup>1</sup> <sup>7</sup> �Þ þ <sup>2</sup>½Cr2O�<sup>2</sup> <sup>7</sup> �þ½H2CrO4�þ½HCrO�<sup>1</sup> <sup>4</sup> �þ½CrO�<sup>2</sup> <sup>4</sup> � � <sup>3</sup>ð½Crþ<sup>3</sup> �Þ þ ½CrOH<sup>þ</sup><sup>2</sup> � þ½CrðOHÞ þ1 <sup>2</sup> �þ½CrðOHÞ �1 <sup>4</sup> �þ½CrSO<sup>þ</sup><sup>1</sup> <sup>4</sup> � ¼ C0V<sup>0</sup> � 12CVÞ=ðV<sup>0</sup> þ VÞ�½I �1 ��½I �1 <sup>3</sup> � þ ð½HIO� þ½IO�<sup>1</sup> �Þ þ <sup>5</sup>ð½HIO3�þ½IO�<sup>1</sup> <sup>3</sup> �Þ þ <sup>7</sup>ð½H5IO6�þ½H4IO�<sup>1</sup> <sup>6</sup> �þ½H3IO�<sup>2</sup> <sup>6</sup> �Þ þ <sup>12</sup>ð½HCr2O�<sup>1</sup> <sup>7</sup> � þ½Cr2O�<sup>2</sup> <sup>7</sup> �Þ þ <sup>6</sup>ð½H2CrO4�þ½HCrO�<sup>1</sup> <sup>4</sup> �þ½CrO�<sup>2</sup> <sup>4</sup> �Þ � <sup>3</sup>ð½Crþ<sup>3</sup> �þ½CrOH<sup>þ</sup><sup>2</sup> �þ½CrðOHÞ þ1 <sup>2</sup> �Þ þ½CrðOHÞ �1 <sup>4</sup> �þ½CrSO<sup>þ</sup><sup>1</sup> <sup>4</sup> �Þ ¼ ð12CV � C0V0Þ=ðV<sup>0</sup> þ VÞ (87)

identical to the one obtained from Eq. (68a). If we put ζBr = 0 in Eq. (85), we get the relation

$$\begin{aligned} &-5([\text{HBrO}\_3] + [\text{BrO}\_3^{-1}]) - ([\text{HBrO}] + [\text{BrO}^{-1}]) + [\text{Br}\_3^{-1}] + [\text{Br}^{-1}] = 0\\ &5([\text{HBrO}\_3] + [\text{BrO}\_3^{-1}]) + ([\text{HBrO}] + [\text{BrO}^{-1}]) - [\text{Br}\_3^{-1}] - [\text{Br}^{-1}] = 0 \end{aligned} \tag{88}$$

identical to the ones obtained from Eq. (44). This way, we recall the card game without 'live cash' but with 'debt of honor'—in not accidental reference to the title of the thriller novel by T. Clancy; btw, nota bene, the "Debt of honor" was published in 1994, like the papers [15, 16, 18].

#### 8. Conclusions

In C<sup>0</sup> mol/L aqueous solution of Br2, bromine as one electron-acitve element in Br2 can be perceived as a 'distributor' of its own electrons, and H, O as 'fans'. For example, Eq. (42) involves H and O as 'fans', Eq. (43) involves Na as 'fan', whereas Eq. (44) does not involve 'fans'. Generally, 'fans' are eliminated after the proper (i.e. indicated above) combination of 2∙f(O) – f(H) with the balances for elements considered as 'fans'. Further simplification can also result from further combination with the balance for 'player(s)'. But, in any case, a linear

As usually happens in the 'card game' practice, the players devote to the game only a part of their cash resources. Similarly, in redox reactions, electrons may participate from the valence shells of atoms of electron-active elements; the electrons from the valence shell of the reductant atoms are transferred onto the valence shell of the oxidant atoms. However, this restriction to the valence electrons is not required here. For example, replacing ZBr in Eq. (46a) by ζBr (ζBr <

<sup>3</sup> �Þ þ ðζBr � <sup>1</sup>Þð½HBrO�þ½BrO�1�

� ¼ 2ζBrC0V0=ðV<sup>0</sup> þ VÞ

�Þ þ <sup>2</sup>ζBr½Br2�þð3ζBr <sup>þ</sup> <sup>1</sup>Þ½Br�<sup>1</sup>

<sup>3</sup> �

(85)

ZBr), we have

<sup>ð</sup>ζBr � <sup>5</sup>Þð½HBrO3�þ½BrO�<sup>1</sup>

42 Redox - Principles and Advanced Applications

þ ðζBr <sup>þ</sup> <sup>1</sup>Þ½Br�<sup>1</sup>

combination of balances related to a redox system does not lead to the identity 0 = 0.

The GEB formulated according to approach I was named first as electron prebalance and presented, in totally mature form, in three papers issued in 1994 [15, 16, 18] and then followed by further articles and other communications. Currently, in context with the approach II to GEB, it is named as the approach I to GEB, fully equivalent to the approach II to GEB, explicitly related to the law of conservation of H and O.

The linear combination 2�f(O) – f(H) is a keystone for the overall thermodynamic knowledge on electrolytic systems. It can be formulated both for non-redox and redox systems, in aqueous, non-aqueous and mixed-solvent systems, with amphiprotic (co)solvent(s) involved. The 2�f(O) – f(H) is linearly independent on ChB and other balances, for elements/cores f(Ym) 6¼ H, O, in any redox system. For any non-redox system, 2�f(O) – f(H) is linearly dependent on those balances. Then, the linear independency/dependency of 2�f(O) – f(H) on the other balances is the general criterion distinguishing between redox and non-redox systems. The equation for 2�f (O) – f(H), considered as the primary form of GEB, pr � GEB ¼ 2 � fðOÞ � fðHÞ, is the basis of GEB formulation for redox systems according to approach II to GEB.

Ultimately, GEB, ChB and elemental/core balances are expressed in terms of molar concentrations—to be fully compatible/congruent with expressions for equilibrium constants, interrelating molar concentrations of defined species, on the basis of the mass action law applied to correctly written reaction equation. The mass action law is the one and only chemical law applied in GATES. A complete set of independent (non-contradictory [22]) relations for the equilibrium constants is needed for this purpose.

When compared with the approach I, the approach II to GEB offers several advantages. Although derivation of GEB according to the approach II is more laborious, it enables to formulate this balance without prior knowledge of oxidation numbers for the elements involved frequently in complex components and species of the system. Only the composition (expressed by chemical formula) of components forming the system and composition of the species formed in the system, together with their external charges, are required, i.e. it provides an information sufficient to formulate the GEB; it is the paramount advantage of the approach II to GEB. Anyway, the oxidation number, representing the degree of oxidation of an element in a compound or a species, is commonly perceived as a contractual concept. In this regard, formulation of GEB according to approach II is more useful than the approach I when applied to complex organic species in redox systems of biological origin [63–67]. The approach II to GEB is advantageous/desired, inter alia, for redox systems where radical and ion-radical species are formed. What is more, the 'players' and 'fans', as ones perceived from the approach I to GEB viewpoint, are not indicated a priori within the approach II to GEB. The approach I, considered as a 'short' version of GEB, is more convenient when oxidation numbers for all elements of the system are known beforehand. Within the approaches I and II to GEB, the roles of oxidants and reductants are not ascribed a priori to particular components forming the redox system and to the species formed in this system. In other words, full 'democracy' is established a priori within GATES/GEB, where the oxidation number, oxidant, reductant, equivalent mass and stoichiometric reaction notation are the redundant concepts only.

The 1 + K balances composed of ChB and K concentration balances related to equations (not equalities, see e.g. Eq. (40)) for elemental balances form a complete set of equations related to a non-redox system. The set of 1 + K variables in x = (pH, pX1,…, pXK) <sup>T</sup> is involved in the algorithm applied for calculation purposes. For example, 1 + K = 1 + 2 = 3 balances are related to non-redox systems presented in Examples 1 and 2, with independent variables represented by components of the vector x = (pH, pCu, pSO4) T . In this context, Eq. (35) is considered as equality, not equation.

The 2 + K balances composed of GEB, ChB and K concentration balances related to equations (not equalities) for elemental balances form a complete set of equations related to the redox system. The set of 2 + K variables in x = (E, pH, pX1,…, pXK) <sup>T</sup> is involved in the algorithm applied for calculation purposes. In this chapter, some results of the simulations of electrolytic redox systems, made according to the GATES/ GEB principles, with use of iterative computer programs, are graphically presented and discussed. The computer simulation realized within GATES with use of iterative computer programs, e.g. MATLAB, provides quite a new quality in knowledge gaining. It enables to follow the details of the process, registered with use of measurable quantities, such as pH and/or potential E. All these calculations are made under assumption that the relevant reactions take place in quasi-static manner under isothermal conditions. The reactions proceeding in the respective systems were formulated under assumption that all equilibrium constants found in the relevant tables and then used in the calculations are correct.

balances. Then, the linear independency/dependency of 2�f(O) – f(H) on the other balances is the general criterion distinguishing between redox and non-redox systems. The equation for 2�f (O) – f(H), considered as the primary form of GEB, pr � GEB ¼ 2 � fðOÞ � fðHÞ, is the basis of

Ultimately, GEB, ChB and elemental/core balances are expressed in terms of molar concentrations—to be fully compatible/congruent with expressions for equilibrium constants, interrelating molar concentrations of defined species, on the basis of the mass action law applied to correctly written reaction equation. The mass action law is the one and only chemical law applied in GATES. A complete set of independent (non-contradictory [22]) relations for the equilibrium

When compared with the approach I, the approach II to GEB offers several advantages. Although derivation of GEB according to the approach II is more laborious, it enables to formulate this balance without prior knowledge of oxidation numbers for the elements involved frequently in complex components and species of the system. Only the composition (expressed by chemical formula) of components forming the system and composition of the species formed in the system, together with their external charges, are required, i.e. it provides an information sufficient to formulate the GEB; it is the paramount advantage of the approach II to GEB. Anyway, the oxidation number, representing the degree of oxidation of an element in a compound or a species, is commonly perceived as a contractual concept. In this regard, formulation of GEB according to approach II is more useful than the approach I when applied to complex organic species in redox systems of biological origin [63–67]. The approach II to GEB is advantageous/desired, inter alia, for redox systems where radical and ion-radical species are formed. What is more, the 'players' and 'fans', as ones perceived from the approach I to GEB viewpoint, are not indicated a priori within the approach II to GEB. The approach I, considered as a 'short' version of GEB, is more convenient when oxidation numbers for all elements of the system are known beforehand. Within the approaches I and II to GEB, the roles of oxidants and reductants are not ascribed a priori to particular components forming the redox system and to the species formed in this system. In other words, full 'democracy' is established a priori within GATES/GEB, where the oxidation number, oxidant, reductant, equivalent mass and stoichiometric reaction notation are the redundant

The 1 + K balances composed of ChB and K concentration balances related to equations (not equalities, see e.g. Eq. (40)) for elemental balances form a complete set of equations related to a

algorithm applied for calculation purposes. For example, 1 + K = 1 + 2 = 3 balances are related to non-redox systems presented in Examples 1 and 2, with independent variables represented

The 2 + K balances composed of GEB, ChB and K concentration balances related to equations (not equalities) for elemental balances form a complete set of equations related to the redox

applied for calculation purposes. In this chapter, some results of the simulations of electrolytic redox systems, made according to the GATES/ GEB principles, with use of iterative computer

T

<sup>T</sup> is involved in the

. In this context, Eq. (35) is considered as

<sup>T</sup> is involved in the algorithm

non-redox system. The set of 1 + K variables in x = (pH, pX1,…, pXK)

by components of the vector x = (pH, pCu, pSO4)

system. The set of 2 + K variables in x = (E, pH, pX1,…, pXK)

GEB formulation for redox systems according to approach II to GEB.

constants is needed for this purpose.

44 Redox - Principles and Advanced Applications

concepts only.

equality, not equation.

The number of electron-active elements (considered as 'players', in terms of the approach I to GEB) in a redox system, is practically unlimited and adapted according to current needs; among others, the systems with one, two, three [13] or four [1, 18] 'players' were considered.

The GATES and GATES/GEB (in particular) can be applied for thermodynamic resolution of systems of any degree of complexity. An example is the four-step process involved with iodometric method of copper(+2) determination, considered in detail in Refs. [14, 68], where 47 species are involved in the system with three electron-active elements, eight equations and two equalities, interrelated in 35 independent expressions for equilibrium constants. The systems with similar complexity were resolved. The complexity is limited, however, by factors of the qualitative and quantitative nature, that is, the knowledge of species the knowledge of equilibrium constants interrelating concentration of complex species with concentrations of their constituting parts. The physicochemical data are incomplete or not reliable, in many cases. This problem was raised lately in Ref. [12].

Each of the components of x = (pH, pX1,…, pXK) <sup>T</sup> is informally ascribed to the corresponding balance: ChB, and the corresponding concentration balances for elements/cores 6¼ H, O. Each of the components of x = (E, pH, pX1,…,pXk) <sup>T</sup> is informally ascribed to the corresponding balance: GEB, ChB, and the corresponding concentration balances for elements/cores 6¼ H, O. A unequivocal solution of equations is obtained when the number of independent equations equals to the number of independent variables; it is the 'iron rule' obligatory in mathematics; pe = – log[e �1 ] = <sup>A</sup>∙E, where 1/<sup>A</sup> <sup>=</sup> RT/F∙ln10, pH = –log[H+1], pXi <sup>=</sup> –log½Xzi i �.

GATES avoids the necessity of quantitative inferences based on fragile/rachitic chemical reaction notation, involving only some of the species existing in the system; it is only a faint imitation of a true, algebraic notation. From the GATES viewpoint, the 'stoichiometry' can be perceived as a mnemonic term only. In calculations, the metastable state is realized by omission of potential products in the related balances, whereas 'opening' a reaction pathway in metastable state is based on insertion of possible (from equilibrium viewpoint) products in the related balances [17, 18]. One can also test the interfering effects of different kinds.

All the inferences made within GATES/GEB are based on firm, mathematical (algebraic) foundations. The proposed approach allows us to understand far better all physicochemical phenomena occurring in the system in question and improve some methods of analysis. All the facts testify very well about the potency of simulated calculations made, according to GATES, on the basis of all attainable physicochemical knowledge. Testing the complex redox and non-redox systems with use of iterative computer programs deserves wider popularization among physicochemists and chemists-analysts.

The GATES/GEB is put in context with constructivistic and deterministic principles, and GEB is perceived as the general law of nature, referred to as electrolytic (aqueous media) redox systems. It is proved that stoichiometry of reactions is not a primary concept in chemistry, and its application provides false results, for obvious reasons. From the GATES viewpoint, the stoichiometric reactions are only the basis to formulate the related equilibrium constants. GATES/GEB referred to modeling of redox titration curves in context with earlier approaches to this problem. The GATES/GEB is also presented in three other chapters issued in 2017 within InTech [68–70].

The dependency/independency criteria ascribed to 2∙f(O) – f(H) distinguishing between the relevant (non-redox and redox) systems are the properties of the equations obtained from the linear combination of the balances for H and O. Namely, the resulting equation is not independent of non-redox systems, since it is a linear combination of the remaining (charge and concentration) balances, whereas in the case of redox systems, this equation is linearly independent of those balances. This is a general property of nature, independent of the complexity of the system under consideration, which is the electrolytic system. GATES and GATES/GEB, in particular, are clear confirmation of the fact that the nature is mathematically designed and the true laws of nature are mathematical. In other words, the quantitative, mathematical method became the essence of science. To paraphrase a Chinese proverb, one can state that 'the lotus flower, lotus leaf and lotus seed come from the same root' [2]. Similarly, the three kinds of balances: GEB, charge and concentration balances come from the same family of fundamental laws of preservation. This compatibility is directly visible from the viewpoint of the approach II to GEB. The equivalent equations for GEB, based on a reliable law of the matter conservation, are equally robust as equations for charge and concentration balances. The complementarity of the GEB (approaches I and II) to other balances is regarded as the expression of harmony of nature and GATES/GEB as an example of excellent epistemological paradigm.

All earlier (dated from the 1960s) efforts made towards formulation of electrolytic redox systems were only clumsy attempts of resolution of the problem in question, as stated in review papers [2, 10–14]. These approaches were slavishly related to the stoichiometric reaction notations, involving only two pairs of indicated species participating in redox reaction; there were usually minor species of the system considered. The species different from those involved in the reaction notation were thus omitted in considerations. Moreover, the charge balance and concentration balances for accompanying substances were also omitted. Theoretical considerations were related to virtual cases, not to real, electrolytic redox systems.
