Chapter 20, Problem 51P

(III) Consider an ideal gas of n moles with molar specific heats C_V and C_P. (a) Starting with the first law, show that when the temperature and volume of this gas are changed by a reversible process, its change in entropy is given by

$dS=n{C}_V\frac{dT}{T}+nR\frac{dV}{V}.$

(b) Show that the expression in part (a) can be written as

$dS=n{C}_V\frac{dP}{P}+n{C}_P\frac{dV}{V}.$

(c) Using the expression from part (b), show that if dS = 0 for the reversible process (that is, the process is adiabatic), then PV^γ = constant, where γ = C_P/C_V.