Rearranging variables Van Der Waals EoS into new variables

[KDS4]

The question I'm stuck on is:

P = NK_BT/(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 k_BT/(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 27bK_bT/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!

 

[DrClaude]

I managed to get P to appear as 27b²P/a

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

 

[KDS4]

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

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

http://imgur.com/a/Q29yi

 

[DrClaude]

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.

 

[Chestermiller]

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