Chapter 4, Problem 83CP

(a) infinitesimal amount of heat is added reversibly to a system. By combining the first and second laws, show that $dU=TdS-dW.$ (b) When heat is added to an ideal gas, its temperature and volume change from $T_1$ and $V_1$ to $T_2$ and $V_2$ . Show that the entropy change of n moles of the gas is given by $\Delta S=C{}_v\ln \frac{T_2}{T_1}nR\ln \frac{V_2}{V_1}$ .