# Critical Pressure and Temperature of a van der Waals Gas

by e(ho0n3
Tags: critical, pressure, temperature, waals
 P: 1,367 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?
 Quote by 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 $d^2P/dV^2 = 0$.
 P: 1,367 Just for reference, $$\frac{dP}{dV} = \frac{-RT}{n(V/n - b)^2}$$ $$\frac{d^2P}{dV^2} = \frac{2RT}{n^2(V/n - b)^3}$$