4. Suppose that n moles of an ideal gas initially is confined within a volume V and held at temperature T. The gas is then expanded to a total volume of aV, where a is a constant by (a) a reversible isothermal expansion and (b) removing a partition and allowing a free expansion into the vacuum, as shown in below. (I) Derive an expression for the change of entropy of the gas in each case. (II) Repeat the above calculation for case (a), assuming that the gas obeys the van na. der Waals equation of state (P+ -)(v-nb) = nRT for which C, is a constant. 2 (III) Show that the temperature of the van der Waals gas falls by an amount proportional to (a- 1)/a for case (b). (a) gas (a-1)V (b) gas partition (a-1)V vacuum