Two systems \(A\) and \(B\), of identical composition, are brought together and allowed to exchange both energy and particles, keeping volumes \(V_{A}\) and \(V_{B}\) constant. Show that the minimum value of the quantity \(\left(d E_{A} / d N_{A}ight)\) is given by

\[
\frac{\mu_{A} T_{B}-\mu_{B} T_{A}}{T_{B}-T_{A}}
\]

where the \(\mu\) 's and the \(T\) 's are the respective chemical potentials and temperatures.