Entropy of mixing

Consider two gasses separated by a barrier, then allowed to mix. Since entropy is expansive, $S_i = S_A + S_B$ . First consider the simpler example where the two gasses are distinguishable but are at equilibrium.

We determine the entropy before and after mixing from the Sackur-Tetrode equation:

\begin{align*} S_i &= S_A + S_B = k N_A \left[ \ln \frac{V_A}{N_A} + \frac 32 \ln \frac{U_A}{N_A} + C_A \right] \\ & \phantom{= S_A + S_B} + k N_B \left[ \ln \frac{V_B}{N_B} + \frac 32 \ln \frac{U_B}{N_B} + C_B \right]. \end{align*}

to do.