Material equilibrium

Let us consider a closed system of constant composition that goes from a state $(P_1, T_1)$ to $(P_2, T_2)$, regardless of the trajectory followed or irreversibility. Consider the entropy of this system as a function of T,P, $S=S(T,P)$. Differentiating: \begin{equation} dS=\left(\frac{\partial S}{\partial T}\right)_PdT+\left(\frac{\partial S}{\partial P}\right)_TdP=\frac {C_P}{T}dT-\alpha VdP \end{equation} Integrating from an initial state (1) to the final state (2) we obtain the entropy change. \begin{equation} \Delta S=\int_{T_1}^{T_2}\frac{C_p}{T}dT-\int_{P_1}^{P_2}\alpha VdP \end{equation} Since entropy is a As a state function, the entropy change between two states does not depend on the path followed. The first integral is calculated at the pressure $P_1$ while the second is evaluated at the temperature $T_2$