2. For a pure simple substance, entropy S is a function of three independent variables. In the present problem independent variables will be taken as temperature T, volume V and the number of mols N. Partial derivatives of S(T, V, N) satisfy the relations (as/at)v,n = Cv/T and (as/av)t.n = (@P/BT)v.n. Here Cy is the heat capacity at constant volume. a) Show that, for fixed N, ds dT + d . (2) T ar = - +(?) V.N b) For an ideal gas, the following equations of state hold: E = CNRT and P = NRT/V, where R is the universal gas constant and c a constant (c = 3/2 for monatomic gases, while c=5/2 for diatomic gases at room like temperatures). Find Cy and (8P/8T)v,n for the ideal gas and show that when N mols of the gas goes from the initial state at T;,V; to final state at Tf, Vf, the change AS in entropy is given by AS = NcR In(T/T;)], + NR In(V/V)