C_v is given by

Definition of $${C_V}$$ : when we add the heat to a system which is undergoing only expansion work or $$PdV$$ work at constant volume than the amount of heat required to raise the temperature by an infinitesimal amount is called as $${C_V}$$.

$${C_V} = \mathop {\lim }\limits_{\delta T \to \,0} \frac{{\partial Q}}{{\partial T}}$$

By first law of thermodynamics: $$\delta Q = dU + \delta W\left( {PdV} \right)$$     Since here $$dV=0$$
$$\delta Q = dU$$
So, $${C_V} = {\left( {\frac{{dU}}{{dT}}} \right)_V}.$$   Where $$U=$$  internal energy.