For an ideal gas, what is the relationship between Cp and Cv?

• A. Cp − Cv = R
• B. Cp = Cv + R
• C. Cp − Cv = nR
• D. Cp / Cv = γ

Answer: (A)

For an ideal gas, adding heat at constant pressure includes doing PV work as the gas expands, so the enthalpy H changes with temperature, while the internal energy U also changes with temperature. Enthalpy depends only on temperature for an ideal gas, and H = U + PV. Since PV = RT for one mole, differentiating with respect to temperature gives dH/dT = dU/dT + R, i.e., Cp = Cv + R. This means Cp − Cv = R per mole. If you have n moles, the relation becomes Cp − Cv = nR, which is the same idea in total terms. The ratio Cp/Cv = γ is related but is a separate descriptor, not the direct difference between Cp and Cv.