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Homework Help: Proof the internal pressure is 0 for an ideal gas and ((n^2)a)/(v^2) for a Van der Wa

  1. Feb 28, 2008 #1
    Like in the other problem I posted- This is the other question that I missed and just can't find a solution for.

    1. The problem statement, all variables and given/known data
    Prove the internal pressure is 0 for an ideal gas and ((n^2)a)/(v^2) for a Van der Waals gas.

    2. Relevant equations
    1. VdQ Eqn: p= (nRT)/(v-b) - ((n^2)a)/(v^2)
    2. (partial S/partial V) for constant T = (partial p/partial T) for contant V.
    3. dU = TdS - pdV
    4. pi sub t (internal pressure) = (partial U/partial V) for constant T


    3. The attempt at a solution

    a) Ideal Gas
    0 = (partial U/partial V) for const T
    int 0 dv = int du
    0 = int (TdS - pdV)
    int p dv = int T ds
    int (nRT/v) dv = int (Pv/nR) dS
    nRT x int(1/V) dv = pv/nR x int 1 dS
    ... and I get kind of lost here, though I know that what I've already done is wrong.. :(

    b) VdW gas
    I actual have to get going to school, but I'll come back and type up what i've done (incorrectly :( for this part afterwards).
     
  2. jcsd
  3. Feb 28, 2008 #2
    Hello Jennifer,

    I suppose you are given the so called thermal equation of state [itex]p=p(T,V,n)[/itex] for both

    the ideal gas

    [tex]p=\frac{nRT}{V}[/tex]

    and the Van der Waals gas

    [tex]p=\frac{nRT}{V-nb}-\frac{n^2a}{V^2}[/tex]

    The inner pressure [itex]\left(\frac{\partial U}{\partial V}\right)_T[/itex] can be calculated after finding the so called caloric equation of state [itex]U=U(T,V,n)[/itex] for both cases.

    Another straightforward method would be to use the following identity which shows that the caloric and thermal equations of state are not independent of each other:

    [tex]\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial p}{\partial T}\right)_V-p[/tex]

    Do you know how to derive this identity?
     
  4. Feb 28, 2008 #3
    I think so..
    [tex]\Pi[/tex]t = [tex]\partial[/tex]u/[tex]\partial[/tex]v for constant t
    = ( 1/[tex]\partial[/tex]v[tex]\times[/tex](Tds - pdv) )
    = T [tex]\times[/tex] ([tex]\partial[/tex]p/[tex]\partial[/tex]t) - p

    I think that's right. I still don't know how to get from that Maxwell relation to the ideal gas and Van der Waals eqn, though.
     
    Last edited: Feb 28, 2008
  5. Feb 28, 2008 #4
    Ok, I think I figured it out, from my previous post (sorry- im still getting used to using the tools for math on this board) I replace the vanderwaals eqn into P in my partial p and then just solve from there.
     
  6. Feb 28, 2008 #5
    Thanks everyone!
     
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