## Theory: thermodynamic properties – isothermal compressibility

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

Statistical mechanical expression for the excess isothermal compressibility is (*Phys Rev A* **1977**, *16*, 2153-2168)

with being the total particle density, sums going over all components of the system, being the compressibility of the ideal gas, and being short-ranged direct correlation function.

Note that this short-ranged is different from the function used for the Ng-procedure. Long range part of is given by . 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 :

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

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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