All relationships between state functions can be obtained from six basic equations. \begin{eqnarray} dU & = & TdS - PdV\\ H & = & U +PV\\ A & = & U-TS\\ G & = & H-TS\\ C_V & = & \frac{dq_v} {dT}=\left(\frac{\partial U}{\partial T}\right)_v\\ C_P & = & \frac{dq_p}{dT}=\left(\frac{\partial H}{\partial T}\right)_P \end{eqnarray}

The equations of $C_v$ and $C_P$ can be expressed as derivatives of entropy by the equation $dq_{rev}=TdS$ \begin{eqnarray} C_V & = & T \left(\frac{\partial S}{\partial T}\right)_v\\ C_P & = & T\left(\frac{\partial S}{\partial T}\right)_p \end{eqnarray} Heat capacities allow us to calculate variations of U, H, and S with temperature. Note that the equation $dU=TdS-PdV$ is only valid for reversible processes