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Critical Pressure and Temperature of a van der Waals Gas 
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#1
Aug1307, 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? 


#2
Aug2007, 08:49 AM

P: 1,367




#3
Aug2007, 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] 


#4
Mar1809, 02:51 PM

P: 167

Critical Pressure and Temperature of a van der Waals Gas



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