Specific Heats And Mayer's Relation

Connecting the two principal molar specific heats of an ideal gas, Mayer's relation states $C_p-C_v=R$, where $R$ is the universal gas constant. The physical reason is that heating a gas at constant volume raises only its internal energy, while heating at constant pressure also requires the gas to do work $P\Delta V$ on its surroundings as it expands. The extra heat needed per mole per kelvin for this expansion work is exactly $R$. Therefore the difference equals $R$. The option $R/2$ wrongly associates the difference with a single degree of freedom. The option $2R$ doubles the expansion work, which has no physical basis. The option $3R/2$ confuses the difference with the constant-volume specific heat of a monatomic gas. This is the standard NCERT result derived from the first law of thermodynamics applied to ideal gases. As a check, both $C_p$ and $C_v$ depend on degrees of freedom, yet their difference is universally $R$ for every ideal gas, regardless of molecular structure.