4.7 Exercise 36

(a) An infinitesimal amount of heat is added reversibly to a system. By combining the first and second laws, show that $d U = T d S - d W$. (b) When heat is added to an ideal gas, its temperature and volume change from $T_{1} \text{and} V_{1} \text{to} T_{2} \text{and} V_{2}$. Show that the entropy change of n moles of the gas is given by

$\Delta S = n C_{v} \text{ln} \frac{T_{2}}{T_{1}} + n R \text{ln} \frac{V_{2}}{V_{1}}$.