Critical Pressure and Temperature of a van der Waals Gas

  1. 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. jcsd
  3. Err, that should be [itex]dP/dV = 0[/itex] and [itex]d^2P/dV^2 = 0[/itex].
  4. 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]
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