# Equation of state (EoS) for liquid density, in terms of pressure, temperature and concentration

I'm currently in need of an equation to calculate density of aqueous solutions such as acids, bases, and salts. The equations to calculate liquid density that I have found so far do not account for all three parameters that I'm interested in: pressure, temperature, and concentration in the same equation.

I considered the modified Rackett equation and another equation described in the following papers, respectively:

1. Spencer CF, Danner RP. Improved Equation for Prediction of Saturated Liquid Density. J Chem Eng Data. 1972;17(2):236-241. doi:10.1021/je60053a012

$$1/\rho_s = \frac{RT_c}{P_c}\times Z_c^{[1+(1-T_r)^{2/7}]},$$

1. Laliberté M, Cooper WE. Model for Calculating the Density of Aqueous Electrolyte Solutions. J Chem Eng Data. 2004;49(5):1141-1151. doi:10.1021/je0498659

$$\bar v_{app,i}=\frac{w_i+c_2+c_3t}{(c_0w_i+c_1)e^{0.000001(t+c_4)^2}}.$$

Here, $\bar v_{app,i}$ is specific volume, $w_i$ is mass fraction, $t$ is temperature, and $c_0,...,c_4$ are empirical constants.

However, I'm not sure about the accuracy of, and how to include other variables in these equations to have density as a function of pressure, temperature, and concentration.

Do I need an EoS for each solution that I'm using, i.e. one for (aqueous) acids, one for (aqueous) bases, and one for the salt? Or can I find a more general equation, perhaps 'equation of state for electrolytes (or polar)'?

• I imagine you'd end up with a generic equation of state for aqueous electrolytes with species-dependent parameters---something like this? – a-cyclohexane-molecule Jul 9 '18 at 20:45
• Thank you for the suggestion. I see that the equation of state is of $A(T,V,\bar n)$, or Helmholtz free energy. The author did mention how it can be used to calculate density, but it's not explicitly described in the paper. Any ideas? – sam o Jul 9 '18 at 22:54
• Sorry, I'm not sure and don't have the time to find out (else I'd have posted it as an answer!)---I just saw that they did calculate it in one of the figures. It may be possible to rescale the Helmholtz free energy by the volume of solution, hence obtaining the Helmholtz free energy density $A/V = a(T, \rho)$, and then numerically invert to extrct the density. – a-cyclohexane-molecule Jul 10 '18 at 3:11
• No worries. I couldn't figure it out the first time I looked it up, but I'll find out more. Thanks for all your help. – sam o Jul 10 '18 at 14:14