 #1
baseballfan_ny
 91
 23
 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?
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?