Ideal-gas identity: For an ideal gas, the difference between heat capacities at constant pressure and constant volume (Cp − Cv) equals what?

Introduction / Context:The Mayer relation links the constant-pressure and constant-volume heat capacities of an ideal gas: Cp − Cv = R. This identity follows from the ideal-gas equation of state combined with definitions of enthalpy and internal energy.

Given Data / Assumptions:

• Ideal-gas behavior.
• Standard thermodynamic definitions: H = U + pV; for ideal gas, U = U(T) and H = H(T).

Concept / Approach:Starting with dH = Cp dT and dU = Cv dT, and using H = U + pV and pV = RT per mole, differentiation yields Cp − Cv = R. This holds independent of the number of degrees of freedom for an ideal gas (which instead affects numerical values of Cp and Cv individually and thus gamma = Cp/Cv).

Step-by-Step Solution:Per mole, pV = RT (ideal gas).H = U + pV → dH = dU + d(pV) = Cv dT + R dT.But dH = Cp dT, so Cp dT = Cv dT + R dT.Therefore, Cp − Cv = R.

Verification / Alternative check:For a monatomic ideal gas: Cv = 3R/2, Cp = 5R/2; difference equals R, confirming the relation.

Why Other Options Are Wrong:Fractions like R/2, multiples like 2R or 3R, and gamma*R are not general identities.

Common Pitfalls:Mixing up Cp/Cv (gamma) with Cp − Cv. Gamma varies with molecular structure; Cp − Cv stays equal to R for any ideal gas.

Final Answer:R