# Homework Help: Critical Pressure and Temperature of a van der Waals Gas

1. Aug 13, 2007

### e(ho0n3

1. The problem statement, all variables and given/known data
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.

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

3. 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?

2. Aug 20, 2007

### e(ho0n3

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

3. Aug 20, 2007

### e(ho0n3

Just for reference,

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

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

4. Mar 18, 2009