(Cp − Cv) for a non-ideal gas differs from (Cp − Cv) for a perfect gas by the expression

Concept:-

For non-ideal gas,

\((C_P - C_V)_{non-ideal}=T(\frac{\partial P}{\partial T})_{V}\:(\frac{\partial V}{\partial T})_{P}\)

Whereas, for a perfect gas,

\((C_P-C_V)_{perfect}=nR\)

We know that,

\((\frac{\partial U}{\partial V})_{T}=T.(\frac{\partial P}{\partial T})_{V}-P\)-----------(1)

Explanation:-

Putting equation (1) in (C_p-C_v)_non-ideal we get,

 (Cp-Cv)non-ideal    =\([(\frac{\partial U}{\partial V})_{T} +P] (\frac{\partial V}{\partial T})_{P}\)

                          =\((\frac{\partial U}{\partial V})_{T}. (\frac{\partial V}{\partial T})_{P}+ P (\frac{\partial V}{\partial T})_{P}\)

                         =\((\frac{\partial U}{\partial V})_{T}. (\frac{\partial V}{\partial T})_{P}+ P \times \frac{nR}{P}\) (as 

                         =\((\frac{\partial U}{\partial V})_{T}. (\frac{\partial V}{\partial T})_{P}+ nR\)

\((C_P-C_V)_{non-ideal}-(C_P-C_V)_{perfect}=(\frac{\partial U}{\partial V})_{T}. (\frac{\partial V}{\partial T})_{P}+ nR -nR\)

\((C_P-C_V)_{non-ideal}-(C_P-C_V)_{perfect}=(\frac{\partial U}{\partial V})_{T}. (\frac{\partial V}{\partial T})_{P}\)             

Conclusion:-

 Hence  (Cp − Cv) for a non-ideal gas differs from (Cp − Cv) for a perfect gas by the expression \(\rm\left(\frac{\partial V}{\partial T}\right)_P\left(\frac{\partial U}{\partial V}\right)_T\). So the option 2 is correct.