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 Sci Advisor HW Helper PF Gold P: 2,532 I don't know if this will get you anywhere, but the change in $C_P$ with pressure is a classic problem in thermo (I remember getting it on an exam before I was familiar with it): $$\left(\frac{\partial C_P}{\partial P}\right)_T=\frac{\partial}{\partial P}\left[\left(T\frac{\partial S}{\partial T}\right)_P\right]_T=T\frac{\partial}{\partial P}\left[\left(\frac{\partial S}{\partial T}\right)_P\right]_T=T\frac{\partial}{\partial T}\left[\left(\frac{\partial S}{\partial P}\right)_T\right]_P$$ Then we use a Maxwell relation to get $$T\frac{\partial}{\partial T}\left[\left(-\frac{\partial V}{\partial T}\right)_P\right]_P=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P$$ Thus, the change you're looking for is related to the second derivative of volume with temperature. This is zero for an ideal gas, but it may get you somewhere if you have a constitutive equation for a real gas.