Derive heat capacity at constant pressure

[tjlaxs]

Homework Statement

For temperatures $$T >> T_C$$ (critical temperature) derive the heat capacity at constant pressure $$C_P$$ from van der Waals equation.

Homework Equations

Critical temperature:
$$T_C = \frac{2N(V - Nb)^2}{kV^2}$$

$$T_C$$ is derived from the fact that it exist at the point in which
$$(\frac{\mathrm{d}P}{\mathrm{d}V})_T = 0$$ but I'm pretty certain that this is not needed in this derivation.

Van der Waals equation:
$$(P + aN^2/V^2)(V - Nb) = NkT$$

Heat capacity:
$$C_P = (\frac{\mathrm{d}U}{\mathrm{d}T})_P + P(\frac{\mathrm{d}V}{\mathrm{d}T})_P$$

The Attempt at a Solution

I've tried to get the point in this. The first term in the equation of $$C_P$$ is easy, but the problem is the second term.

If I try to solve for the $$V$$ in the van der Waals equation I get a long equation set to derive. And I don't think this is what is the point of the exercise.

Is there another approach or something to simplify the van der Waals equation before the derivation?

 

[Redbelly98]

It looks like you get a cubic equation in V to solve, which I agree is probably not what you're expected to do.

Just taking an educated guess here, but it's probably the case that

V >> Nb​

and

P >> aN²/V²​

.
It may be a reasonable approximation to replace the "V²" term with whatever the ideal gas equation gives for V, since it appears as part of a term that is small to begin with.

 

[Nembrook]

You're using the wrong equation for hear capacity, you should use Cp = dh/dt

where h is enthaply