The internal energy of a real gas

Question

I apologise in advance, but I am asking for clarification.
I learnt from a first year undergraduate physics book that in the case of ideal monatomic gases, the internal energy is given by $E_{int} = n times C_V times T$. But this represents the rms kinetic energy of a molecule.
Q1. In the real gases I think the potential energy should contribute. I read a book called Physical Chemistry by Atkins, and he says

In thermodynamics, the total energy of a system is called its internal
energy, U. The internal energy is the total kinetic and potential
energy of the molecules in the system.

Is another term representing electrical (and perhaps gravitational?) potential energy there?
Q2. What if the gas is diatomic? Halliday, in his book Fundamentals of Physics, says

Let us also assume that the internal energy Eint is the sum of the
translational kinetic energies of the atoms. (Quantum theory disallows
rotational kinetic energy for individual atoms.)

What is this actually about?
Thank you.

Chet Miller · Accepted Answer

The potential energy they are talking about here is the energy of interaction between the molecules which is usually approximated by the so-called Leonard-Jones 6-12 potential. Because of this, for a real gas, the internal energy is a function of both temperature and specific volume, given by $$dU=nC_vdT-left[P-Tleft(frac{partial P}{partial T}right)_Vright]dV$$Note that, for an ideal gas, whose equation of state is such that pressure is directly proportional to temperature at constant volume, the 2nd term vanishes, and U is a function only of T. And, of course, at low pressures, a real gas approaches ideal gas behavior.

Correct answer by Chet Miller on June 29, 2021

The internal energy of a real gas

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