# Rearranging variables Van Der Waals EoS into new variables

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1. Sep 27, 2016

### KDS4

• Moved from a technical forum, so homework template missing
The question I'm stuck on is:

P = NKBT/(V-Nb) - aN2/(V2) -----> (1)

Re-arrange variables in the Van Der Waals equation of state, Eq. (1), so that V always appears in the equation as V/(3Nb) and P appears as 27b2P/a. Then T should appear in the combination 27b kBT/(8a). Call these dimensionless combinations v, p, and t (also called reduced variables), and express the van der Waals equation in terms of v, p, and t.

I managed to get P to appear as 27b2P/a and T to appear as 27bKbT/8a using algebra and getting some pretty ugly terms along the way that are floating around but with V appearing multiple times in the initial equation I can't for the life of me get it to appear the way the question wants me to.

Any insight would be greatly appreciated!

2. Sep 28, 2016

### Staff: Mentor

That should be $27 b^2 P / (8a)$ as the critical pressure is $p_c = 8a/(27 b^2)$.

3. Sep 28, 2016

### KDS4

My assignment says otherwise but it also doesn't reference critical pressure.

http://imgur.com/a/Q29yi

4. Sep 28, 2016

### Staff: Mentor

That'll teach me to look up things on the internet

Indeed, the source I used was wrong, and there is no 8 there. You should look up "critical constants" (maybe here http://cbc.arizona.edu/~salzmanr/480a/480ants/vdwcrit/vdwcrit.html) to get a deeper understanding of that rescaling of the VdW equation.

As for you problem, you'll have to post some details of the derivation if you want more help.

5. Sep 30, 2016

### Staff: Mentor

The easiest way to work this problem is to substitute:$$V=3Nbv$$
$$P=\frac{a}{27b^2}p$$
$$T=\frac{8a}{27bK_B}t$$