Q.Derive the relation $C_p - C_v = R$ (Mayer's relation) for an ideal gas, starting from the first law of thermodynamics.

By the first law of thermodynamics, $dQ = dU + dW$, where $dW = P\,dV$ is the work done by the gas.

At constant volume, $dW=0$, so all heat goes into internal energy: $dQ = dU = nC_v\,dT$ for $n$ moles, giving the definition of $C_v$. Since internal energy $U$ of an ideal gas depends only on temperature (not on volume or pressure separately), $dU=nC_v\,dT$ holds generally, whatever process is used to change $T$.

At constant PRESSURE, $n$ moles absorb heat $dQ = nC_p\,dT$ (defining $C_p$), and by the first law,

From the ideal gas equation $PV=nRT$, at constant pressure, $P\,dV = nR\,dT$ (differentiating with $P$ held fixed). Substituting,

Dividing through by $n\,dT$,

$\boxed{C_p - C_v = R}$ …

Apply the first law separately at constant volume (defines $C_v$) and constant pressure (defines $C_p$), then use …

• Forgetting that $dU=nC_v\,dT$ holds for ANY process for an ideal gas (since $U$ depends only on $T$), not only for a constant-volume process -- this is what lets the same $C_v$ term be reused in the constant-pressure equation. …