Verifying properties of Van der Waals Gas

• baseballfan_ny
In summary, the question is asking for an equation for Van der Waals in terms of 1st order in a and b, and they get: p = N\tau \left( \frac 1 V + \frac {N} {V^2} b \right) - \frac {N^2a} {V^2} where PV=N \tau. They are then stuck on getting H(\tau, V) which is just pV=N\tau. To solve it, they zeroth order the equation and get V=N \tau/P.
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?

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
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!

1. What is Van der Waals gas?

Van der Waals gas is a theoretical model of a gas that takes into account the intermolecular forces between gas molecules. It is named after Dutch scientist Johannes Diderik van der Waals.

2. What properties are typically verified in Van der Waals gas?

The properties that are typically verified in Van der Waals gas include the compressibility factor, the equation of state, and the critical point.

3. How is the compressibility factor measured in Van der Waals gas?

The compressibility factor is measured by comparing the actual volume of the gas to the ideal gas volume at the same temperature and pressure. The difference between the two volumes is then divided by the ideal gas volume to calculate the compressibility factor.

4. What is the equation of state for Van der Waals gas?

The equation of state for Van der Waals gas is given by: (P + a/V^2)(V - b) = RT, where P is the pressure, V is the volume, T is the temperature, and a and b are constants that take into account the intermolecular forces between gas molecules.

5. How is the critical point of Van der Waals gas determined?

The critical point of Van der Waals gas can be determined by plotting the isotherms (lines of constant temperature) on a P-V diagram and finding the point where the isotherms intersect at a maximum. At this point, the gas is in a state of critical temperature and pressure, and the volume is known as the critical volume.

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