Irodov Problem 2.51

This problem explores a polytropic process where internal energy follows a specific volume dependence—the key is recognizing that this constraint links pressure, volume, and temperature in a particular way. Since $U = aV^{\alpha}$ and internal energy depends only on temperature for an ideal gas, you can connect $T$ to $V$, then use the first law ($\Delta U = Q - W$) and the ideal gas law to find work and heat. The molar heat capacity emerges by treating this as a custom process with its own effective "polytropic index," distinct from standard isobaric or isochoric processes.

This problem explores a polytropic process where internal energy follows a specific volume dependence—the key is recognizing that this constraint links pressure, volume, and temperature in a particular way. Since $U = aV^{\alpha}$ and internal energy depends only on temperature for an ideal gas, you can connect $T$ to $V$, then use the first law ($\Delta U = Q - W$) and the ideal gas law to find work and heat. The molar heat capacity emerges by treating this as a custom process with its own effective "polytropic index," distinct from standard isobaric or isochoric processes.