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?

 

[Charles Link]

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 $b$.

 

[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!