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Inversion Curve for a gas obeying Dieterici's equation of state

  1. Nov 23, 2012 #1
    1. The problem statement, all variables and given/known data

    For a gas obeying Dieterici's equation of state:

    P(V-b) = RTexp(-a/RTV)

    for one mole, prove that the equation of the inversion curve is

    P = ((2a/b^2) - (RT/b)) * exp((1/2) - (a/(RTb)))

    and hence find the maximum inversion temperature.


    2. Relevant equations

    N/A

    3. The attempt at a solution

    So, I know that for the inversion curve, the condition is (dV/dT) = V/T (where the derivative is evaluated at constant pressure). But this would need implicit differentiation to find dV/dt ... and it seems completely intractable - is there something I'm missing?
     
    Last edited: Nov 23, 2012
  2. jcsd
  3. Nov 23, 2012 #2

    haruspex

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    Do the partial differentiation.
    To arrive at the target equation, you need to eliminate ∂V/∂T (for which you have an equation) and V (for which you have the original equation).
    If you're still stuck, please post your working.
     
  4. Nov 24, 2012 #3
    Hi, thanks for the help!

    I've tried the partial differentiation, but when I try and eliminate V and ∂V/∂T between the equation I obtain and the equation of state I just get a horrible mess - I'm not sure if I'm doing the differentiation right...

    I rearranged the equation to get:

    [itex]\frac{-a}{RTV}[/itex] = ln(p(V-b)) - ln(RT)

    So, differentiating wrt T:

    [itex]\frac{a}{RT^{2}V}[/itex] + [itex]\frac{a}{RTV^{2}}[/itex]*[itex]\frac{∂V}{∂T}[/itex] = [itex]\frac{1}{V-b}[/itex]*[itex]\frac{∂V}{∂T}[/itex] - [itex]\frac{1}{T}[/itex]

    There's not really any point me posting any of the further work / manipulation I've done - I've tried a load of different things and nothing gets anywhere...

    Is the initial differentiation correct?

    Cheers!
     
  5. Nov 24, 2012 #4

    haruspex

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    Yes, that looks good. Substitute for ∂V/∂T and get it into the form V = ....
     
  6. Nov 24, 2012 #5
    Awesome, thanks! Got it now, much less nasty than I'd thought :-) (I was just messing up the cancellation of terms - which made me doubt I'd got the differentiation right in the first place because the whole thing looked such a mess!)
     
  7. Nov 24, 2012 #6
    And the maximum inversion temperature is just found by setting P = 0, right? So that gives T = 2a/bR ?
     
  8. Nov 24, 2012 #7

    haruspex

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    If you say so. I know nothing about this subject matter.
     
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