# Saturated vapor and liquid densities (VLC curve) from Helmholtz free energy equation of state

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 ($$F$$):

$$\min_{\varepsilon,\sigma} F=\sum_{i=1}^{N_{P_{sat}}} \Bigg[\frac{P_{sat,i}^{exp}(T_i)-P_{sat,i}^{calc}(T_i;\sigma,\varepsilon)}{P_{sat,i}^{exp}(T_i)} \Bigg]^2 + \sum_{j=1}^{N_{\rho_{satL}}} \Bigg[\frac{\rho_{satL,j}^{exp}(T_j)-\rho_{satL,j}^{calc}(T_j;\sigma,\varepsilon)}{\rho_{satL,j}^{exp}(T_j)} \Bigg]^2$$

$$P_{sat}$$: vapour pressure
$$\rho_{satL}$$: saturated liquid density
$$\varepsilon,\;\sigma$$: say, Lennard-Jones parameters

I know that the vapour pressure can be computed from the derivative w.r.t $$V$$, i.e. $$P_{sat}=-(\partial A/\partial V)_T$$ (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...)?

Without worrying too much about the details of your particular system, it might be helpful to consider the conditions under which a coexistence between a saturated liquid and a saturated vapour may exist.

In particular, we know that two coexisting phases in equilibrium should have

• Equal temperatures (otherwise the free energy could be lowered by heat exchange)
• Equal pressure (otherwise the free energy could be lowered by volume exchange)
• Equal chemical potential (otherwise the free energy could be lowered by particle exchange)

If you know the Helmholtz free energy of a system (at a given temperature) as a function of $$N$$ and $$V$$, then you can easily calculate the pressure and chemical potential at any state point. As you already wrote, $$P = -(\partial A/\partial V)_{N,T}$$. Similarly, the chemical potential $$\mu = (\partial F/\partial N)_{V,T}$$. In a single-component system in the thermodynamic limit, both these quantities only depend on the density $$\rho$$ and temperature $$T$$ of the system.

Finding the coexistence densities {$$\rho_{vapour}, \rho_{liquid}$$} then comes down to solving:

• $$P(\rho_{vapour}) = P_(\rho_{liquid})$$
• $$\mu(\rho_{vapour}) = \mu_(\rho_{liquid})$$

With two equations, and two unknowns, this is a solvable set of equations. Note that the pressure and chemical potential you find are then just $$P^{sat}$$ and $$\mu^{sat}$$.

This approach is equivalent to e.g. a Maxwell construction, or a common-tangent construction, which both cast the same problem into a slightly different form.

The Wikipedia page of the Van der Waals fluid has some discussion about methods for this as well.

• Thank you so much for this! Nov 15 '21 at 17:08