A vessel of volume $V$ contains $N$ molecules of an ideal gas held at temperature $T$ and pressure $P_1$. The energy of a molecule may be written in the form
$$
E_k\left(p_x, p_y, p_z\right)=\frac{1}{2 m}\left(p_x^2+p_y^2+p_z^2\right)+\varepsilon_k,
$$
where $\varepsilon_k$ denotes the energy levels corresponding to the internal states of the molecules of the gas.
(a) Evaluate the free energy $F=-k T \ln Z$, where $Z$ is the partition function and $k$ is Boltzmann's constant. Explicitly display the dependence on the volume $V_1$.
Now consider another vessel, also at temperature $T$, containing the same number of molecules of an identical gas held at pressure $P_2$.
(b) Give an expression for the total entropy of the two gases in terms of $P_1, P_2, T, N$.
(c) The vessels are then connected to permit the gases to mix without doing work. Evaluate explicitly the change in entropy of the system. Check whether your answer makes sense by considering the special case $V_1=$ $V_2\left(\right.$ i.e., $\left.P_1=P_2\right)$.