Q.Derive Meyer's relation for an ideal gas.

Step 1. Consider $\mu$ moles of an ideal gas. Heated at CONSTANT VOLUME so its temperature rises by $dT$, no work is done ($dV=0$), so all the supplied heat raises internal energy: $dU=\mu C_v\,dT$, where $C_v$ is the molar specific heat capacity at constant volume.

Step 2. Now heated at CONSTANT PRESSURE by the same $dT$: if $Q$ is the heat supplied and $C_p$ the molar specific heat capacity at constant pressure, $Q=\mu C_p\,dT$. If $W=P\,dV$ is the work done in this expansion, the first law gives $Q=dU+W$, so $\mu C_p\,dT=\mu C_v\,dT+P\,dV.$

Step 3. Since internal energy is a state variable, $dU=\mu C_v\,dT$ holds for BOTH cases (only its origin differs). Differentiating the ideal gas equation of state $PV=\mu RT$ at constant pressure gives $P\,dV=\mu R\,dT$ (since $P$ is fixed, $dP=0$, and $PdV+VdP=\mu R\,dT$ reduces to $P\,dV=\mu R\,dT$). …

Compare $Q=dU$ (constant V) against $Q=dU+PdV$ (constant P), using $PdV=\mu R …

• Forgetting that $dU=\mu C_v\,dT$ applies in BOTH the constant-V and constant-P cases (since $U$ is a state variable) -- this …