# Using Helmholtz Free Energy to Calculate Liquid Density

My objective is to find an equation of state (EoS) for density, i.e. density as a function of pressure, temperature and concentration, for aqueous acids, bases and salts.

A StackExchange user suggested that I look into the following paper:

1. Myers JA, Sandler SI, Wood RH. An equation of state for electrolyte solutions covering wide ranges of temperature, pressure, and composition. Ind Eng Chem Res. 2002;41(13):3282–97.

The paper describes the EoS as "the total change in the Helmholtz free energy for forming the electrolyte solution on this path" in which they combine contributions from Peng-Robinson model, Born model, and MSA (Mean Spherical Approximation) model:

$$A(T,V,\bar n)-A^{IGM}(T,V,\bar n)=\Delta A^{PR}+\Delta A^{Born}+\Delta A^{MSA},$$

where $$T$$ is the temperature of the system, $$V$$ is the system volume, $$\bar n$$ is the vector of the number of moles of each component of the mixture, and $$A^{IGM}$$ is the Helmholtz free energy an ideal gas mixture.

In the paper, they included graphs of density vs. molality of aqueous $$NaCl, NaBr, CaCl_2$$. So it seems like this EoS is what I need, but the paper doesn't explicitly describe how to calculate density from the EoS.

I've seen how taking partial derivatives of Helmholtz free energy results in equations for pressure, entropy, and $$\mu$$, but not for density. I'm wondering if someone has already figured out how exactly one can calculate density from Helmholtz free energy.

UPDATE: Also, I don't think the Helmholtz free energy I described above (from the paper) is the same as this.

• Can you give a few more details of the setup you are considering? Are you working at constant pressure or constant volume? Is the total amount of material fixed? Can the various components of your solution react with each other? Depending on the answers to these questions it may well be that the Helmholtz free energy is not the relevant potential. For example for systems at a fixed pressure it is the Gibbs free energy that is important. Jul 10, 2018 at 16:26
• I'm considering a set up where certain amount of acid and base are introduced to a porous material in water. The amount of acid and base will be predetermined, and the acid and base will get mixed either in, or prior to, the region in the material where they're injected. Jul 12, 2018 at 13:42
• @BySymmetry thanks for the reply. The volume of each component will change during reaction (acid + base -> salt + H2O + heat). If the heat is high enough, the water may vaporize as well. If the set up is in an open environment, pressure would be ~1 atm regardless of these changes, wouldn't it? Jul 12, 2018 at 13:51