Blundell Thermal Physics · Problem 11.2

The key is recognizing that $\gamma = C_p / C_V$ is defined by the ratio of heat capacities, and that the first law of thermodynamics gives you a direct relationship between them. For an ideal gas, $C_p - C_V = R$ always holds—this comes from the work done during constant-pressure heating. Once you use this constraint along with the definition of $\gamma$, you can algebraically manipulate these two equations to isolate $R$ in terms of $C_V$ and $C_p$, yielding both identities.

The key is recognizing that $\gamma = C_p / C_V$ is defined by the ratio of heat capacities, and that the first law of thermodynamics gives you a direct relationship between them. For an ideal gas, $C_p - C_V = R$ always holds—this comes from the work done during constant-pressure heating. Once you use this constraint along with the definition of $\gamma$, you can algebraically manipulate these two equations to isolate $R$ in terms of $C_V$ and $C_p$, yielding both identities.