Register to reply

Critical Pressure and Temperature of a van der Waals Gas

by e(ho0n3
Tags: critical, pressure, temperature, waals
Share this thread:
e(ho0n3
#1
Aug13-07, 11:36 AM
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

[tex]T_{cr} = \frac{8a}{27bR}[/tex]

[tex]P_{cr} = \frac{a}{27b^2}[/tex]

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

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

3. The attempt at a solution
The first and second derivative have powers of [itex]V[/itex] greater than 2. Unfortunately I don't have the skills to solve for [itex]dp/dt = 0[/itex] or [itex]d^2p/dt^2 = 0[/itex]. Perhaps there's a simpler way?
Phys.Org News Partner Science news on Phys.org
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice
e(ho0n3
#2
Aug20-07, 08:49 AM
P: 1,367
Quote Quote by e(ho0n3 View Post
The first and second derivative have powers of [itex]V[/itex] greater than 2. Unfortunately I don't have the skills to solve for [itex]dp/dt = 0[/itex] or [itex]d^2p/dt^2 = 0[/itex]. Perhaps there's a simpler way?
Err, that should be [itex]dP/dV = 0[/itex] and [itex]d^2P/dV^2 = 0[/itex].
e(ho0n3
#3
Aug20-07, 01:34 PM
P: 1,367
Just for reference,

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

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

cks
#4
Mar18-09, 02:51 PM
P: 167
Critical Pressure and Temperature of a van der Waals Gas

http://www.chem.arizona.edu/~salzman...t/vdwcrit.html

A good web site with complete calculation


Register to reply

Related Discussions
Maths problem: deriving expressions for the critical temperature/volume/pressure Introductory Physics Homework 4
Finding critical temperature from equation of state Biology, Chemistry & Other Homework 1
Air Pressure + Temperature Classical Physics 2
YBCO critical temperature not right Atomic, Solid State, Comp. Physics 3
Critical Temperature? Biology, Chemistry & Other Homework 2