Redox equilibria, part IV titration curve equations for homogeneous and symmetrical redox reactions

Abstract

The earlier parts of this series have presented some results of investigations into the characteristic properties of potentiometric titration curves for symmetrical and homogeneous redox reactions. It was always assumed that only one redox form (e.g., the oxidized form, in the particular situation investigated) of the titrant couple was in the titrant and that the solution to be titrated contained initially only one redox form (e.g., the reduced form) of the other couple. For this idealized situation, a rigorously derived equation was obtained which readily permitted the calculation of the progress of the titration, f, as an explicit function of the potential, E. It has now been shown that for symmetrical redox reactions it is possible to solve this equation analytically so as to express E directly as a function of f. Furthermore, the theoretical treatment of redox titration curves has been extended beyond the idealized situation where initially only one form of each couple is present in each solution. The results for two cases of analytical interest have been presented. The solution to be titrated often contains some Ox2 as well as some Red2, which latter is to be oxidized by the Ox1 in the titrant. The effect of this initial amount of Ox2 on the nature of the titration curve is to cause the potential to be always more positive throughout the titration than it would be at corresponding locations, i.e., values of f, in the absence of some initially present Ox2. An immediate result of this is that now the theoretically derived equation indicates that at f=o, E is not negatively infinite but rather possesses a finite value. The value of f at which E=E*=(n1E10′+n2E20′)/(n1+n2) occurs at a value of f less than unity and depends upon ‡E0′ as well as upon the concentration of Ox2 initially present. Whereas in the idealized case the value of f at E=E20′ was always greater than 1/2 and the value of f at E=E10′ always less than 2, it is now possible at E=E20′ for f to be exactly equal to, or less than, 1/2 when α20(=COx20/CRed20) is less than k2, where k=exp[−(0.5 pF/RT)‡E0′]. The other case for which results are presented is that which occurs often in coulometric titrations, where the original solution to be titrated contains not only the reductant, Red2, but also a particular concentration of the reduced form of the titrant. The effect of this is to cause the potential to be always more negative throughout the titration than it would be at corresponding locations in the absence of some initially present Red1. At E=E* the value of f is greater than unity, and is dependent upon n1/n2 and k as well as upon the value of αR0(=CRed10/CRed20). For symmetrical reactions, the value of f at which E=E20′, is always greater than 1/2 whereas at E=E10′, the value of f may be equal to, or less than, 2 whenever αR0 is less than 2k2. In the course of the derivations, equations were obtained which explicitly express the variation in concentration of the various species during the titration. Each concentration may be written as the sum of two individual contributions: one arising from that which is produced during the titration, and the other arising from the initially present constituent of the appropriate redox couple.