Activity coefficients on different concentration scales and reference states

Abstract

The activity coefficient measures the degree of departure of a substance's behavior from ideal or ideally dilute behavior and is a very important thermodynamic parameter. Its numerical values depend on concentration scales and reference states at fixed temperature and pressure. However, its dependency on reference states chosen have not been considered fully. Some activity coefficients using different definitions have been obtained by various experimental and/or theoretical methods. Therefore, the conversions of activity coefficients from one concentration scale and/or from one reference state to another are very significant, not only in theoretical researches but also in applications. However, these conversions are tricky and confusing. Here we report the relationship between activity coefficients and some methods for their conversions.Consider a ternary system, as an example: electrolyte (E)-nonelectrolyte (N)-water (W). Generally, there are two molality scales: one in which the electrolyte is regarded as the solute, whereas the N-W mixture as the mixed solvent, and thus the molality of E (m E M ), is defined as the number of moles of E per kilogram (kg) of mixed solvent; the other in which the electrolyte and nonelectrolyte are regarded as two solutes and pure water as the solvent, molalities of E (m E W ) and N (m N W ) are defined as the number of moles of E and N per kg of pure water, respectively. They are called "the water-molality" and are denoted usually by m E and m N in the following. When m N ϭ 0, that is, for the binary E-W system, a superscript "0 " will be used (m E 0 ). It is easy to deduce 1where M N and w N denote the molar mass of N and the mass fraction of N in the mixed solvent (N-W), respectively. With respect to m E M , the molality-scale mean activity coefficient of E is denoted by ␥ Ϯ m(M) , which is referred to unity at infinite dilution in a given mixed solvent (the subscript " Ϯ" denotes the mean ionic quantity and will be omitted in the following). Similarly, ␥ m(W) denotes the water-molality-scale mean activity coefficient of E with respect to m E W , which is referred to unity at infinite dilution in pure water. Based on the thermodynamic theory, the following relationship was derived:where ␥ ϱ m(W) is the limiting mean activity coefficient of E at a given m N at infinite dilution of E. Experimentally, its value can be obtained from measured values offor the sucrose-NaCl-water system are presented in Figure 1, where A and B are the Debye-Hückel constants, a°is the ion size parameter, C is the ion-interaction parameter, and M NW is the mean molar mass of the mixed solvent (N-W). Method 2. Based on the Pitzer equation 3 and Eq. 2, we have where A i represents the empirical parameters. Method 4. From the values of standard Gibbs free energies of transfer, ␥ ϱ m(W) can been calculated bywhere ⌬ t G°m (M) is the standard Gibbs energy of transfer of E from water to the W-N mixture and the infinite dilution of E in the given N-W mixed solvent is taken as the reference state on the m E M ...