According to law of equipartition of energy, the molar specific heat of a diatomic gas at constant volume where the molecule has one additional vibrational mode is 

Diatomic gas molecules will have total seven degrees of freedom in since they have one vibrational mode (two extra degrees of freedom) and three translational degrees of freedom, and two rotational degrees of freedom.

$\mathrm{f}=3+2+2=7$

${\mathrm{C}}_{\mathrm{v}}=\frac{\mathrm{fR}}{2}=\frac{7\mathrm{R}}{2}$

or,

${\mathrm{C}}_{\mathrm{V}}=\frac{1}{\mathrm{n}}\frac{\mathrm{dU}}{\mathrm{dT}}\Rightarrow \frac{1}{\mathrm{n}}\frac{\mathrm{d}}{\mathrm{dT}}\left[\frac{5}{2}\mathrm{nRT}+\mathrm{nRT}\right]=\frac{7\mathrm{R}}{2}$