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In 1932 Onsager and Fouss found in a seminal work that the the conductivity of mixtures
of electrolytes depends on specific mixture effects which may be calculated by means
of a matrix theory. Here we formulate the matrix theory of conductivity for seawater. We solve that problem of seawater conductivity for a 6 - component model and show numerically, that the dominant effect is given by the mean mobility approximation. For practical calculations of the conductivity of seawater,
the mean mobility approximation, is as a rule, sufficiently accurate.

Based on statistical methods to calculate individual activity coefficients of ions in electrolyte solutions developed in parts I and II, we present here applications to two simplified seawater models (SWM 2+ and SWM 6+). As in the foregoing parts we start with exact results for the model of hard charged spheres (non-additive radii) obtained from the statistical cluster expansion theory. We develop analytical DHX-, and MSX-approximations beyond the traditional Debye-Hückel and Mean Spherical approximations for the calculation of individual activity coefficients which are compatible with the exact results but cover the concentration range of seawater systems with small or moderate salinities. Based on the tools of cluster theory tools we propose a set of formulae, which are generalizations of the known Debye-Hückel-and Mean Spherical Approximations which we name DHX and MSX-approximations. These nonlinear extensions include several higher order in e 2-contributions which for the model of hard-charged spheres with non-additive contact distances include the third order logarithmic contributions , relevant for unsymmetrical electrolytes and forth order contributions to the activities, responsible for association effects. The extensions are constructed in a way that the formulae are consistent with the exact statistical theory up to quadratic order in density and on the other hand so simple, to allow calculations on standard home computers. The main adaptable input parameters of our seawater model which provide in particular the individual ion activities are the smallest distances of ion pairs at contact, which are defined for each pair and must not satisfy rules of additvity. We show that this model is much more flexible as standard models of hard charged spheres with fixed additive radii and allows the modeling of individual properties. Beginning with the simplest seawater model we consider first seawater as a polluted NaCl solution (model 1 Preprint for RG and in part submitted to Trends in Phys. Chem., draft of Feb 20, 2020 1 SWM 2+). We study the activity of the remaining seawater ions in tracer concentrations, what means that their concentrations are so low that their interaction can be neglected. The activities of all other ions are studied only in this limit. In a second more realistic model (SWM 6+) we select the six most abundant components Na + , K + , Mg 2+ ,Ca 2+ ,Cl − , SO 2− 4 in concentration relations according to the Reference Composition of the IAFSO Standard and corresponding tables proposed first by Millero as the basic mother solution, the remaining ions H +-OH − , HCO − 3 , CO 2− 3 etc. are considered as tracers. We study the activities depending on milieu-concentrations changing from Baltic Sea situations to Atlantic milieus. We conclude with a discussion of the perspectives of applications to more realistic models of seawater.

Some new and exact results from cluster expansion incuding higher
virial coefficients, methods for estimation of potential parameters