$$pV\,=\,A(T)\,+\,B(T)p\,+\,C(T)p^2 $$
Find $C_p(T,p)$ in terms of $C_p(T,p_0)$ and $p\,,\,p_0$ ( initial and the final pressures) and $A(T)\,,\,B(T)\,,\,C(T)$ and their derivatives with respect to temperature $T$. [ Write an appropriate expression for
$$\frac{\partial C_p}{\partial p} $$ and use it to obtain $C_p$]
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