# Verifying properties of Van der Waals Gas

baseballfan_ny
Homework Statement:
(a) Show that the entropy of the VDW gas is ##\sigma = N \{ \ln \left [ \frac { n_Q(V - Nb) } {N} \right] + \frac 5 2 \}##

(b) Show that the energy of the VDW gas is ##U = \frac 3 2 N \tau - \frac {N^2 a} {V} ##

(c) Show that ## H(\tau, V) = \frac 5 2 N \tau + \frac {N^2 b \tau} {V} - \frac {2N^2 a} {V} ##
and ## H(\tau, P) = \frac 5 2 N \tau + Nbp - \frac {2Nap} {\tau} ## where ## H = U + pV##

All results arc given to first order in the van der Waals correction terms a, b.
Relevant Equations:
Van der waals equation: ## \left( p + \frac {N^2 a} {V^2} \right) \left( V - Nb \right) = N \tau ##
So a and b were pretty straightforward. Got stuck on part c.

The question says they approximated Van der Waals in first order in a and b. So I started with that by rewriting Van der Waals eqn as ## p = \frac { N \tau } { V - Nb } - \frac {N^2a} {V^2} ## and I then Taylor approximated ## \frac {1} {V - Nb} \approx \frac 1 V + \frac {N} {V^2} b ##.

Then p becomes
$$p = N\tau \left( \frac 1 V + \frac {N} {V^2} b \right) - \frac {N^2a} {V^2}$$

and subbing this into ## H = U + pV## gave me ## H(\tau, V) = \frac 5 2 N \tau + \frac {N^2 b \tau} {V} - \frac {2N^2 a} {V} ##.

Now I'm stuck on getting ## H(\tau, P) ##. I'm pretty sure all I need to do is rewrite V in terms of P, but I'm not able to do that. Just trying to write V from the 1st order approximation I got above of Van der Waals equation gives:
$$p = N\tau \left( \frac 1 V + \frac {N} {V^2} b \right) - \frac {N^2a} {V^2}$$
$$\frac {p} {N \tau} = \frac 1 V + \frac {Nb} {V^2} - \frac {Na} {V^2 \tau}$$

And that's still first order in a and b but I'm not sure how to solve it. Is there some approximation I'm missing?

Homework Helper
Gold Member
Suggestion: ## PV=N \tau ## to zeroth order. Substitute ## V=N \tau/P ## in the denominator of the ## a N^2/V^2 ## term, (Edit: and/or the denominator of the ## H(\tau, V ) ## terms=it's very simple), and you then can get ## V ## in terms of ## P ## to first order in ## a ## and ## b ##.

Last edited:
BvU and baseballfan_ny
baseballfan_ny
Suggestion: ## PV=N \tau ## to zeroth order. Substitute ## V=N \tau/P ## in the denominator of the ## a N^2/V^2 ## term, (Edit: and/or the denominator of the ## H(\tau, V ) ## terms=it's very simple), and you then can get ## V ## in terms of ## P ## to first order in ## a ## and ## b ##.
Works! Thank you!