I have been reading about the statistical associating fluid theory (SAFT) which computes the Helmholtz free energy (A) from the SAFT EoS (*a hell lot of equations*) for a molecule of interest, and is then used in the estimation of intermolecular potential parameters by minimising a relative least-squares objective function.
I know that the vapour pressure can be computed from the derivative w.r.t volume, i.e. P(sat)= - (dA/dV) (although I have not tried to actually do it yet, by hand or code).
But I am struggling to understand how is the saturated liquid (or vapour) density computed from a Helmholtz free energy derivative (similar to that of vapour pressure). I read that ancillary equations have been used to compute saturated properties (https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=918040). However, SAFT EoS have been developed to predict properties below the critical temperature.
Can anyone please help me to understand how this works (theory, calculation, equation, code, etc.), or point me to any reference paper that explains about this (because I have not been able to find any...)?
You need the residual Helmholtz free energy and the fugacities of the liquid and the vapor. See for example: Article An EOS for the Lennard-Jones fluid: A virial expansion approach
(eqs. 25-27).
At the end you have 2 equations, which must be simultanious true.
The solution gives you the vapor and liquid densities.
You can read: An Equation of State for the Thermodynamic Properties of Cyclohexane
Article An Equation of State for the Thermodynamic Properties of Cyclohexane
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