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Ideal gas partial differential calculus

  1. Oct 19, 2011 #1
    1. The problem statement, all variables and given/known data
    Use partial differential calculus to show that if 3 quantities p, V, T are related to each other by some
    unknown but smooth (which means all derivatives are well defined) equation of state f (P, V, T ) = 0. Then the
    partial derivatives must satisfy the relation


    ∂p/∂T = - (∂V /∂T ) / ( (∂V /∂p) )



    2. Relevant equations

    Not sure any would help in this case.


    3. The attempt at a solution

    I'm not even sure where to start since I haven't a proper understanding of the problem.

    If p, V, and T are related then

    how does f (P, V, T ) = 0 help?
     
  2. jcsd
  3. Oct 19, 2011 #2

    George Jones

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    This requires some playing around with the multivariable chain rule and the implicit function theorem.
     
  4. Oct 20, 2011 #3

    George Jones

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    I'll try and provide more of a start.

    The equation of state [itex]f \left( P , V , T \right) = 0[/itex] picks out a surface in [itex]\left( P , V , T \right)[/itex] space, and the implicit function theorem says that (locally) on this surface, any of the three variables can be written as a function of the other two, e.g., [itex]P = P \left( V , T \right)[/itex]. Define a new (related) function
    [tex]\tilde{f} \left( V , T \right) = f \left( P \left( V , T \right) , V , T \right) [/tex]
    Use this and the chain rule to find [itex]\partial \tilde{f} / \partial T[/itex].

    This is just a start.
     
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