For an ideal gas, Cp minus Cv equals which constant?

For an ideal gas, Cp minus Cv equals which constant?

• A. R
• B. Cv
• C. Cp
• D. 0

Answer: (A)

The difference between the heat capacities at constant pressure and constant volume for an ideal gas is a fixed amount, equal to the gas constant R. This comes from how enthalpy behaves for an ideal gas. Enthalpy H is U + PV, and for an ideal gas PV = nRT, so H = U(T) + nRT. If you differentiate H with respect to temperature at constant pressure, you get Cp = dH/dT at constant P = dU/dT + nR. Since dU/dT is the molar heat capacity at constant volume (Cv), Cp = Cv + nR. For one mole (n = 1), Cp − Cv = R. This Mayer’s relation explains why Cp is always larger than Cv by exactly R for ideal gases. In real gases, deviations can occur due to interactions, but the ideal-gas result is Cp − Cv = R.

The difference between the heat capacities at constant pressure and constant volume for an ideal gas is a fixed amount, equal to the gas constant R. This comes from how enthalpy behaves for an ideal gas. Enthalpy H is U + PV, and for an ideal gas PV = nRT, so H = U(T) + nRT. If you differentiate H with respect to temperature at constant pressure, you get Cp = dH/dT at constant P = dU/dT + nR. Since dU/dT is the molar heat capacity at constant volume (Cv), Cp = Cv + nR. For one mole (n = 1), Cp − Cv = R. This Mayer’s relation explains why Cp is always larger than Cv by exactly R for ideal gases. In real gases, deviations can occur due to interactions, but the ideal-gas result is Cp − Cv = R.