Proof that Heat Capacity is independent of Pressure for a new Equation of State

[Thyferra2680]

For a gas that obeys the equation of State P(v-b) = RT, where b is a constant, show that Cp is independent of Pressure, i.e., ($$\delta$$Cp/$$\delta$$p) at constant T is equal to zero

Homework Equations

Maxwell Relations
H = U+PV
dh = TdS + PdV
dh/dT at constant P is defined as Cp

The Attempt at a Solution

Unfortunately I can't simply say that since Cp is defined as existing at a constant pressure state, that it's independent of pressure; would have made the problem much simpler.

I figure that I'm supposed to prove that ($$\delta$$(dH/dT)/$$\delta$$p) at constant T is equal to zero, but I'm having trouble figuring out which maxwell relation is the best fit.

d/dP (dH/dT) = d[((TdS)/dT + (VdP)/dT) at constant P]/dP at constant T

My problem is the dS portion of the equation. It's defined in terms of Cp, among other things, and that doesn't really help me in any way... I think.

I have a similar problem when I attempt it with dH = d(U+PV). dU is defined in terms of Cv, or if I break it apart with U = Q+W, I get U = TdS- PdV. Again, not too helpful.

Is it an issue of which relation I'm using to start? Or can the Cv and Cp actually help me?

Thanks for the help

 

[Mapes]

This is a good start. Now how about writing $\partial/\partial P[(\partial S/\partial T)_P]_T$ in a way that let's you use a Maxwell relation?

 

[Thyferra2680]

Oh... Is it possible to switch the order of differentiation here? I suppose this is derived from an exact differential? If that's so, then (dS/dP) at constant T is equal to (dv/dT) at constant P...

 

[Mapes]

Yep! Should be no problem now. (Don't forget any sign changes!)

 

[Thyferra2680]

Right! thanks a bunch