Theoretical Analysis of System’s Composition Changes in the Course of Electrolysis of Acidic Chloride Aqueous Solution

Abstract

Variations of the indicator electrode potential under zero current condition, E, and of the (quasi)equilibrium composition of the aqueous solution (which initially contained 0.5 M chloride anion) in the anode chamber of a model electrolytic cell were calculated, provided that the pH of the solution is kept constant (pH 0) and that the same total number of Cl atoms in all chlorine compounds (including the gas phase above it) is maintained. Theoretical analysis is carried out for five different hypotheses regarding the possible depth of electrolysis and the nature of the occurring processes. The hypotheses are: (a) no formation of chlorine compounds with oxidation states above +1, i.e., the electrolysis produced molecular chlorine Cl₂ as a solute and in the gas space above the solution ${\text{Cl}}_{{\text{2}}}^{{{\text{gas}}}}{\text{,}}$ trichloride anion ${\text{Cl}}_{3}^{ - },$ as well as dissolved HClO and ClO^– (in addition to the retaining amount of Cl^–); (b) in addition to the compounds indicated for case (a), chlorine compounds are formed in solution in the +3 oxidation state, i.e. HClO₂ and ${\text{ClO}}_{2}^{ - }{\text{;}}$ (c) in addition to the compounds listed for case (b), chlorine compounds are formed in the +4 oxidation state, i.e. dissolved ClO₂ and ${\text{ClO}}_{{\text{2}}}^{{{\text{gas}}}}$ in the gas phase; (d) the process proceeds with formation of both the chlorate ion $\left( {{\text{ClO}}_{3}^{ - }} \right)$ and the chlorine compounds of lower oxidation states in solution and in gas phase indicated above for case (c) (Cl^–, ${\text{Cl}}_{3}^{ - },$ Cl₂, ${\text{Cl}}_{{\text{2}}}^{{{\text{gas}}}}{\text{,}}$ HClO, ClO^–, HClO₂, ${\text{ClO}}_{2}^{ - }{\text{,}}$ ClO₂, and ${\text{ClO}}_{{\text{2}}}^{{{\text{gas}}}}$); (e) in addition to the components indicated for variant (d), perchlorate anion $\left( {{\text{ClO}}_{4}^{ - }} \right)$ can also be formed. All electrochemical and chemical reactions involving the chlorine-containing species which are taken into account within the framework of each of the options (a), (b), (c), (d) or (e) are assumed to be in (quasi)equilibrium state. Predictions for all these five hypotheses on the relationship between the redox-charge of the system, Q (or the average oxidation state of chlorine atoms, x), and the indicator electrode potential, E, as well as on the dependence of the system’s composition (concentrations of all compounds) on either x or E are derived. Approaches for the experimental determining of the variant for the evolution of the Cl-containing anolyte composition in the course of electrolysis are proposed.

APPENDIX

The system compositions are calculated basing on thermodynamical relations (А1)–(А11) for equilibria in electrochemical, chemical, and physical transformations of Table 2.

In these relations, we used the following notation: Х for Cl-containing component in aqueous solution (Cl^–, Cl₂, ${\text{Cl}}_{3}^{ - },$ HClO, ClO^–, HClO₂, ${\text{ClO}}_{2}^{ - },$ ${\text{ClO}}_{3}^{ - },$ ${\text{ClO}}_{4}^{ - }$), {X} for its activity, ${\text{Cl}}_{{\text{2}}}^{{{\text{gas}}}}$ and ${\text{ClO}}_{{\text{2}}}^{{{\text{gas}}}}$ for molecular chlorine and its dioxide in gas phase, $\left\{ {{\text{Cl}}_{{\text{2}}}^{{{\text{gas}}}}} \right\}$ and $\left\{ {{\text{ClO}}_{{\text{2}}}^{{{\text{gas}}}}} \right\}$ for their activities (in the molecular concentration unities: mol/dm³), V^sol and V^gas for the volumes of solution and the gas phase above it, $E_{i}^{^\circ }$ for the standard potential of the corresponding electrochemical reaction, K_i for the equilibrium constant of the corresponding chemical transformation or for the transfer to the gas phase (K^{gas, с}). The dimensionless constants $K_{3}^{{{\text{gas,c}}}}$ and $K_{9}^{{{\text{gas,c}}}}$ for the equilibria (3) and (9) between the gas phase and the Cl₂ and ClO₂ solutes are calculated by using the values of the corresponding Henry constants K_H3 and K_H9 [M/atm] [46]. We denote the common logarithm as “log”. The parameter A = F/(RT ln 10) = 16.92 V^–1 (at a temperature of 298 К).

For the components’ concentrations [X_i] all expressions (А1)–(А11) are valid provided the activity of each component {X_i} has been replaced by [X_i]; the standard potential of each reaction$E_{i}^{^\circ }$, by the formal potential ${E}_{i}^{{^\circ }'},$; the equilibrium constant for any transformation K_i, by the apparent equilibrium constant $K_{i}^{'}$ [36]. Further calculations were carried out under the assumption that the components’ activities in all relations in Table 2 have been replaced by the concentrations, while it is thermodynamical quantities of the parameters that were used in these relations.

The Method of Calculation of the System’s Component Concentrations

The calculations were carried out for the known values of the solution рН (рН 0), the chlorine atom summary concentration in the system c_tot = 0.5 M, and the volume ratio for the solution and gas phase (in the calculations, we assumed the value V^gas/V^sol = 0.8) for the set of values of the potential Е. This set of the potential values is chosen in such a way that the system’s mean oxidation value therein х varied from –1 to the maximal possible value: up to +7 or up to +5, or up to +4, or up to +3, or up to +1, respectively, in the variants (e), (d), (c), (b), or (а).

For the variant (e), a nonlinear set of 12 algebraic equations (14), (А1)–(А11) is solved for 12 unknown concentrations of the chlorine compounds in Table 1. The calculations for the variants (а), (b), (c), and (d) are carried out by the same scheme, however, with a decreased number of chlorine-containing compounds and upon removing of the same number of their determining conditions of thermodynamical equilibrium. In particular, for the variant (d), in which no perchlorate-anion is formed, its concentration is zeroed, and equation (А11) is excluded from the analysis. For the variant (c), the chlorate-anion ${\text{ClO}}_{3}^{ - }$ and equation (А10) are also excluded; for the variant (b), also ClO₂ and ${\text{ClO}}_{{\text{2}}}^{{{\text{gas}}}}$ and equations (А9) and (А8); for the variant (а), also HClO₂ and ${\text{ClO}}_{2}^{ - }$, as well as equations (А7) and (А6) for their concentrations.

In all cases, the equations for the concentrations (А1)–(А11) we used in which the рН potential values are assumed to be known, allow expressing all concentrations с_i in terms of one of the concentrations, for example, [Cl^–]. The substituting of these expressions into equation (14) for the summary number of chlorine atoms brings about to nonlinear algebraic cubic equation for this concentration. The equation was solved numerically for each value of the potential E. Then, using formulas (А1)–(А11) we calculated the concentrations of the rest of components, the system’s redox-charge Q, and the chlorine-atom mean oxidation state х by formulas (15) for the chosen value of the potential Е.

The numerical solution of the nonlinear equation for the chloride-anion concentration [Cl^–] (see the preceding paragraph) was carried out as follows. The right-hand part of equation (14) (∑n_ic_i) is a sum of concentrations of all system’s components c_i (depending on the variant of the calculations, part of concentrations was zeroed, that is, excludes from the summation over i) with positive coefficients n_i, where each quantity c_i is connected with [Cl^–] by a corresponding equation (А1)–(А11) from Table 2. At that, all concentrations c_i increased monotonously with the increasing of the concentration [Cl^–]. This means that (at any fixed values of рН and Е) the sum of concentrations ∑n_ic_i also increased monotonously with the increasing of the concentration [Cl^–]. Moreover, the sun is 0 at zero concentration [Cl^–] for any values of рН and Е. Wherefrom we derived a conclusion: at any value of the system’s global parameters (рН, Е, and с_tot) there exists—moreover, the only possible—solution of equation (14)for [Cl^–], that is, the sum of concentrations ∑n_ic_i equals с_tot. As at any nonzero (positive) value of [Cl^–] the rest of the concentrations also is positive, the above-mentioned only possible solution falls within the interval: 0 < [Cl^–] < с_tot, because at its left-hand boundary the sum ∑n_ic_i is 0; at its right-hand boundary, a fortiori it exceeds с_tot.

To find the solution of equation (14) in this interval, we used the bisection method, that is, the halving of the interval and the calculating of the sum ∑n_ic_i for this mid-value of [Cl^–], for example, at [Cl^–] = 1/2с_tot in the first step. When the obtained value of the sum is larger than с_tot, the solution of the equation should be looked for at the left-hand half of the interval; when the sum is less than с_tot, the root is located at the right-hand half of the interval. This procedure is further repeated for the required half of the interval, and so on, until the preset accuracy of determination of [Cl^–] or deviation of the summary concentration, that is, the difference ∑n_ic_i – с_tot, has been reached.

For the other set of global parameters (рН, Е, and с_tot), for instance, for the other value of the potential Е, the same procedure is to be repeated. Eventually, the system’s composition has been found, that is, all considered concentrations c_i, as well as the system’s redox-charge Q and the chlorine-atom mean oxidation state х is found using equations (15) as a function of the potential Е at given values of the rest of parameters (рН and с_tot).

At sufficiently high positive potentials, the concentration [Cl^–] becomes so small that the above-described calculations became impossible because of limited accuracy of the numerical operations. In this case, we used in the calculation the same approach, yet, we chose as a basic unknown the concentration of the compound with positive oxidation degree, e.g., $\left[ {{\text{ClO}}_{3}^{ - }} \right]$ for the variants (e) and (d) or [ClО₂] for the variant (c). There exists a broad interval of intermediate potentials in which we succeeded in the calculating of both unknown quantities whose comparison allowed verifying the identity of the obtained results.

The correctness of the calculations was controlled by the calculating of the summary concentration of chlorine atoms, as well as by verifying the fulfilment of equations (А1)–(А11).

When performing the calculations (at fixed values of рН and c_tot) for a set of various potential values, we succeeded in the predicting of the dependence of х on Е (or, which is the same thing, Е on х, see Fig. 1), as well as the system composition on the electrode potential Е (Fig. 3) or on the mean oxidation degree х (Fig. 2) for each evolution variant (а)–(e).