Statistical theory of individual activity coefficients of electrolytes including multiple ionic charges

In previous work we developed a new statistical method for calculating the individual activities of ions including the association of ions. Here we study multi-particle electrostatic interactions connected within higher cluster integrals and identify the ionization constants of the mass action law of associating ion clusters. In contrast to Bjerrum and Fuoss, our concept of association is not based on spatial criteria, but instead on the strength of interaction measured in powers of the Bjerrum parameter ($e^2 / D_0 k_{\text{B}} T a$ ; $a$ is contact) and defined by asymptotic properties of the cluster integrals. For ion pair formation our mass action constant is the classical counterpart of Planck's famous hydrogenic partition function. As a rule, the new association constants are smaller than traditional expressions, e.g., by Fuoss and Kraus, in the interesting regions of interaction parameters about fifty percent. Several examples including CaCl$_2$, MgCl$_2$, Na$_2$SO$_4$, K$_2$SO$_4$, LaCl$_3$ and a model of seawater are studied. For several associating electrolytes and seawater, reasonable agreement with experiments and Monte Carlo results is achieved.