Which expression correctly represents the gas constant R in relation to Cp and Cv for ideal gases?

Which expression correctly represents the gas constant R in relation to Cp and Cv for ideal gases?

• A. Cp - Cv
• B. Cv
• C. Cp
• D. Cp + Cv

Answer: (A)

For ideal gases, the difference between heat capacities at constant pressure and constant volume equals the gas constant. Enthalpy H equals U + PV, and for an ideal gas PV = RT per mole, so H = U + RT. Since internal energy U depends only on temperature for an ideal gas, (dH/dT)_P = (dU/dT)_P + R = Cv + R. Therefore Cp = Cv + R, which rearranges to Cp − Cv = R. This reflects that heating at constant pressure includes not only raising the temperature (the Cv part) but also doing PV work to expand, contributing the R term. So the correct expression is Cp minus Cv.

For ideal gases, the difference between heat capacities at constant pressure and constant volume equals the gas constant. Enthalpy H equals U + PV, and for an ideal gas PV = RT per mole, so H = U + RT. Since internal energy U depends only on temperature for an ideal gas, (dH/dT)_P = (dU/dT)_P + R = Cv + R. Therefore Cp = Cv + R, which rearranges to Cp − Cv = R. This reflects that heating at constant pressure includes not only raising the temperature (the Cv part) but also doing PV work to expand, contributing the R term. So the correct expression is Cp minus Cv.