Show that the various expressions for the entropy of mixing, derived in Section 1.5 , satisfy the following relations:

(a) For all \(N_{1}, V_{1}, N_{2}\), and \(V_{2}\),

\[
(\Delta S)_{1 \equiv 2}=\left\{(\Delta S)-(\Delta S)^{*}ight\} \geq 0
\]

the equality holding when and only when \(N_{1} / V_{1}=N_{2} / V_{2}\).

(b) For a given value of \(\left(N_{1}+N_{2}ight)\),

\[
(\Delta S)^{*} \leq\left(N_{1}+N_{2}ight) k \ln 2,
\]

the equality holding when and only when \(N_{1}=N_{2}\).