Q.Derive the relationship between Cp and Cv for an ideal gas.

Heat capacity $c=q/(m\Delta T)$ measures how much heat is needed to raise a system's temperature; per mole it becomes molar heat capacity, measured either at constant volume ($C_V$) or constant pressure ($C_P$). From the first law, $C_V=(\partial U/\partial T)_V$ -- the rate of change of internal energy with temperature at fixed volume; similarly $C_P=(\partial H/\partial T)_P$. For an ideal gas, starting from $H=U+nRT$ and differentiating gives the key relation $C_P-C_V=nR$ (per mole, $C_P-C_V=R$): $C_P$ is always LARGER than $C_V$, because at constant pressure the system must additionally do expansion work against its surroundings as it warms, needing extra heat for the same temperature rise that a constant-volume p …