(a) An amount of substance equal to 2 kmols of an ideal gas at temperature \(T\) and pressure \(p\) is contained in a compartment. In an adjacent compartment is an amount of substance equal to \(1 \mathrm{kmol}\) of an ideal gas at temperature \(2 T\) and pressure \(p\). The gases mix adiabatically but do not react chemically when a partition separating the compartments is withdrawn. Show that, as a result of the mixing process, the entropy increases by

\[\Re\left(\ln \frac{27}{4}+\frac{\kappa}{\kappa-1} \ln \frac{32}{27}\right)\]

provided that the gases are different and that \(\kappa\), the ratio of specific heats, is the same for both gases and remains a constant in the temperature range \(T\) to \(2 T\).

(b) What would be the entropy change if the gases being mixed were of the same species?

\(\left[\frac{\kappa \Re}{\kappa-1} \ln \frac{32}{27}\right]\)