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Theory: thermodynamic properties-excess chemical potential

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Evaluation of the excess chemical potential \mu_i^\mathrm{ex} of a component i in the solution is generally a complex task, but in the HNC approximation, it actually can be done really easily (Phys Rev A 1977, 16, 2153-2168):

\displaystyle \beta \mu_i^\mathrm{ex} = \sum_j \rho_j \int_0^\infty \left( \frac{h_{ij}\gamma_{ij}}{2}-c_{ij}^s\right) 4 \pi r^2 dr

with the sum going over all components of the mixture. The short-ranged direct correlation function c_{ij}^s is the same as used for the evaluation of the isothermal compressibilities. Exponentiation of \beta\mu^\mathrm{ex} yields directly the activity coefficient \gamma.

Mean activity coefficients

Remeber that the chemical potential can be written as a sum of ideal part and excess part (involving activity coefficient \gamma.

\displaystyle \mu_i = \mu_i^\circ + kT \ln{a_i} = \mu_i^\circ + kT \ln{x_i \gamma_i} = \mu_i^{\mathrm{id}} + \mu_i^\mathrm{ex}

The activity coefficient of ion i is then

\displaystyle \gamma_i = \mathrm{e}^{\beta \mu_i^\mathrm{ex}}

It is not experimentally possible to measure the activity coefficient of an individual ion (the whole system has to be electrically neutral). As a solution to this problem, the so called mean activity coefficients were introduced, combining the contribution of an anion and a cation at the same time.

For a binary electrolyte, we can write

\displaystyle \gamma_\pm = (\gamma_i^{\nu_i} \gamma_j^{\nu_j})^{1/(\nu_i + \nu_j)}

with \nu_x being the respective stoichiometric coefficient describing the dissociation of the salt. This leads to the well know expression for the mean activity coefficient of a symmetric binary electrolyte.

\displaystyle \gamma_\pm = \sqrt{\gamma_i \gamma_j}

pyOZ is currently capable of evaluating excess chemical potentials/activity coefficient only when using the HNC closure relation.

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