# Why is there no volume dependence on internal energy for kinetic gas?

The textbook I am currently reading (An Introduction to Thermal Physics by Schroeder) states that the total thermal energy of a system is given by:

$U_{thermal}=nf\cdot&space;\frac{1}{2}kT$

where n is the total number of molecules and f is the degrees of freedom of the molecules.

Correct me if I'm wrong, but the thermal energy of a system consists of both the kinetic energies of molecules as well as their potential energies due to the intermolecular forces associated with phase of the molecules. And as far as I know, the phase of the molecules (and hence the potential energies of the molecules) depends on both temperature AND the pressure of the system. This leads me to question: why there isn't a term in this equation that contains the pressure of the system? Does this equation assume that the system is an ideal gas which then there would be no potential energies involved?

• simple answer: we say in kinetic gas theory that molecules don't know each other Jun 5, 2021 at 23:56
• So the model that this formula is based on assumes the molecules don't interact with one another (and hence potential energy doesn't need to be considered)? Jun 6, 2021 at 0:03
• Yes. An ideal gas is noninteracting, except for instantaneous collisions, so there is no intermolecular potential.
– Buzz
Jun 6, 2021 at 0:06
• That makes sense, thank you! Jun 6, 2021 at 0:19

The formula $$U_{thermal}=nf\cdot\frac{1}{2}kT$$ can be derived from the equipartition theorem and holds for systems of molecules such that their Hamiltonian is only made by $$f$$ quadratic terms. In practice, it is valid for a very dilute gas (in the limit of the perfect gas, i.e. negligible inter-molecular interaction) of rigid molecules or non-rigid molecules, provided the intra-molecular interactions could be treated within harmonic approximation. As noticed in the question, in the case of real systems the internal energy should also depend on the density (or pressure) as a consequence of the inter-molecular interactions.