# Finding a Helmholtz free energy from the equation of state

When we try to find the heat capacity which will be used to calculate the inner energy, it would not be very hard when neglecting the force interactions between atoms, such as ideal gas. But if we consider the interactions, is it possible to find out the expression of Helmholtz free energy from the EOS?

As an example, I found the equation of state (EOS) of gaseous and solid Helium is

$$P/\rho kT= A+B\rho+C\rho^2+D\rho^3,$$

where $$A, B, C$$ and $$D$$ are constants, $$P, T$$ and $$\rho$$ represent the pressure, temperature and helium atom density respectively, and $$k$$ is the Boltzmann constant.

Is there any way to calculate the Helmholtz free energy from the above EOS?

• Are you trying to find the heat capacity of a real gas using the equation of state, or are you trying to find the Helmholtz free energy? Nov 16, 2019 at 12:22
• In fact I'm trying to find the Helmholtz free energy, but I thought it is necessary to find out the capacity before get the Helmholtz free energy. Nov 16, 2019 at 12:26
• Are you looking for the Helmholtz free energy relative to a specified reference state, or the absolute Helmholtz free energy? What are the two end states (in terms of the temperature, pressure, and volume)? Nov 16, 2019 at 12:30
• Do you at least know the heat capacity of the gas in the ideal gas limit? If so, then, contrary to what @GiorgioP says, this can be done. Nov 16, 2019 at 23:05

No way, without additional information. The Helmholtz free energy per particle $$f=F/N$$, is related to the pressure P by the relation:
$$P = \rho^2 \left.\frac{\partial{f}}{\partial{\rho}}\right|_T$$
Since the partial derivative is taken at constant $$T$$, it could be integrated only within an unknown function of $$T$$.
Additional information, like the specific heat, has to be used in order to find the full $$\rho$$ and $$T$$ dependence.