Critical Pressure and Temperature of a van der Waals Gas

[e(ho0n3]

Homework Statement

From the van der Waals equation of state, show that the critical temperature and pressure are given by

T_{cr} = \frac{8a}{27bR}

P_{cr} = \frac{a}{27b^2}

Hint: Use the fact that the P versus V curve has an inflection point at the critical point so that the first and second derivatives are zero.

Homework Equations

P = \frac{RT}{V/n - b} - \frac{a}{(V/n)^2}

The Attempt at a Solution

The first and second derivative have powers of V greater than 2. Unfortunately I don't have the skills to solve for dp/dt = 0 or d^2p/dt^2 = 0. Perhaps there's a simpler way?

 

[e(ho0n3]

The first and second derivative have powers of V greater than 2. Unfortunately I don't have the skills to solve for dp/dt = 0 or d^2p/dt^2 = 0. Perhaps there's a simpler way?

Err, that should be dP/dV = 0 and d^2P/dV^2 = 0.

 

[e(ho0n3]

Just for reference,

\frac{dP}{dV} = \frac{-RT}{n(V/n - b)^2}

\frac{d^2P}{dV^2} = \frac{2RT}{n^2(V/n - b)^3}

 

[cks]

http://www.chem.arizona.edu/~salzmanr/480a/480ants/vdwcrit/vdwcrit.html

A good website with complete calculation