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Theory: thermodynamic properties – isothermal compressibility

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Isothermal compressibility \chi_T gives the response of the system volume V to the applied pressure p

\displaystyle\chi_T = -\frac{1}{V}\left(\ \frac{\partial V}{\partial p} \right)_T

Statistical mechanical expression for the excess isothermal compressibility \chi_T^{\mathrm{ex}} is (Phys Rev A 1977, 16, 2153-2168)

\displaystyle (\chi_T^{\mathrm{ex}})^{-1} = \frac{\chi_T^\circ}{\chi_T} = 1-\sum_i \sum_j \frac{\rho_i \rho_j}{\rho} \int_0^\infty c_{ij}^s 4\pi r^2 dr

with \rho = \sum_n \rho_n being the total particle density, sums going over all components of the system, \chi_T^{\circ} = \beta/\rho being the compressibility of the ideal gas, and c_{ij}^s being short-ranged direct correlation function.

c_{ij}^s = c_{ij} + \beta U_{ij}^{\mathrm{coul}}

Note that this short-ranged c_{ij}^s is different from the function used for the Ng-procedure. Long range part of c_{ij} is given by -\beta U_{ij}^\mathrm{coul}. It is sufficient to take only the short-ranged part of the direct correlation function into account in the integration, since the long-ranged contribution vanishes due to electroneutrality.

Compressibility route to the osmotic coefficient

An alternative approach to the osmotic coefficients utilizes the so called compressibility route. The osmotic coefficient is also given by the osmotic pressure \Pi:

\displaystyle\Phi = \frac{\beta\Pi}{\rho_S}

with \rho_S being the particle density (undissociated salt). \Pi can be calculated from the isothermal compressibility using the following expression:

\displaystyle \chi_T = \left( \rho_S \frac{\partial \Pi}{\partial \rho_S} \right)^{-1}

The evaluation of the osmotic coefficient then essentially involves numerical integration of the combined expression from zero up to the target density.

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