I am trying to understand at the molecular level why a real gas with attractive forces between particles experiences an increase in internal energy when expanding isothermally. Here's my thinking:
For a gas, the path-independent differential form of change in internal energy is
\begin{equation} dU = \frac{\partial U}{\partial V}{ \delta V} + \frac{\partial U}{\partial T}{ \delta T} \end{equation}
For an isothermal expansion, \begin{equation} { \delta T} = 0 \end{equation}
so \begin{equation} dU = \frac{\partial U}{\partial V}{ \delta V} \end{equation}
(For an ideal gas, internal energy is proportional to temperature, so an isothermal expansion doesn't change the internal energy.)
The way I'm looking at the real gas situation is that:
When the real gas expands, say by free expansion into an evacuated chamber, the Δ U = Δ P.E. + Δ K.E.
The increase in volume causes the average distance between particles to increase and therefore increases the potential energy.
The increase in potential energy comes as the expense of a decrease in kinetic energy, because Δ U= 0 = Δ P.E. + Δ K.E., in the same way that throwing a ball upwards into the air causes it to lose K.E.
However, in the case of the gas, this is a very short-lived decrease in K.E., because the decrease in K.E. of the particles lasts no longer until they reach a wall.
Because the walls of the evacuated flask are maintaining an isothermal environment, they transfer heat energy to the particles inside, either by contact conduction, or radiation, thereby restoring K.E. to initial value.
U final = final P.E. (higher than initial conditions) + initial (restored) K.E., and is therefore increased.
(In the case of a volume expansion of a real gas by doing work against a piston, I believe that there would be an even larger transient drop in K.E., immediately counteracted by the introduction of correspondingly more heat because of the isothermal constraint. So the end-of-pathway conditions are still the same, increased P.E. and restored K.E.)
Is this reasoning correct? The version of Atkins I have asserts that for a real gas \begin{equation} \frac{\partial U}{\partial V} > 0 \end{equation}
but doesn't provide a molecular explanation.