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Homework Help: Clairaut's theorem?

  1. Mar 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Given dU = TdS - PdV, prove that:
    1. (dT/dV)S = -(dP/dS)V
    2. (dS/dV)T = (dP/dT)V

    2. Relevant equations
    Clairaut's states that d^2U/dXdY = d^2U/dYdX is true, or in my case:
    [d/dX (dU/dY)X ]Y = [d/dY (dU/dX)Y ]X

    3. The attempt at a solution
    1. (dU/dS)V = T(S,V) and (dU/dV)S = -P(T,V)
    so [d/dS (dU/dV)S]V = [d/dV (dU/dS)V]S
    and that means [-P(S,V)/dS]V = [T(S,V)/dV]S
    and thus, if we differentiate again (-dP/dS)V = (dT/dV)S

    2. [d/dT (dU/dV)T]V = [d/dV (dU/dT)V]T
    (dU/dV)T = -P(S,V) and (dU/dT)V = dS(T) (unsure of this)
    so that means [-P(S,V)/dT]V = [dS(t)/dV)]T
    This is where I get stuck, because there's a dS on the right hand side, but it's only P on the left hand, not dP. Am I missing an important step here?

    Obviously we need to get the equation for dU into a form that depends not on S and V, but something that depends on T and V. I don't know how to do this, though..
    Last edited: Mar 15, 2009
  2. jcsd
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