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Deriving Helmholtz Thermodynamic Potential & Corresponding Maxwell Relation

  1. Feb 22, 2010 #1
    1. The problem statement, all variables and given/known data

    To state the differential form of the Helmholtz thermodynamic potential and
    derive the corresponding Maxwell's relation.

    2. Relevant equations

    Stated within the solution attempt.

    3. The attempt at a solution

    • Helmholtz function: [tex]F = U - TS[/tex]

    • Calculating the differential form:

      For infinitesimal change: [tex]dF = dU - tdS - SdT[/tex]

      Then using: [tex]TdS = dU + PdV[/tex] ,


      [tex]dF = -PdV - SdT[/tex]

    • Which then follows that can write: [tex]F = F(V,T)[/tex]


      [tex]dF = \left(\frac{\partial F}{\partial V}\right)_{T}dV + \left(\frac{\partial F}{\partial T}\right)_{V}dT[/tex]

    • Comparing coefficients in the previous two stated expressions for [tex]dF[/tex]:

      [tex]P = -\left(\frac{\partial F}{\partial V}\right)_{T}[/tex]


      [tex]S = -\left(\frac{\partial F}{\partial T}\right)_{V}[/tex]

    • Then as F is a function of state, then dF is an exact differential and
      the condition for an exact differential gives:

      [tex]\left(\frac{\partial P}{\partial T}\right)_{v} = \left(\frac{\partial S}{\partial V}\right)_{T}[/tex]

      Which is the Maxwell relation, derived as required.
  2. jcsd
  3. Feb 22, 2010 #2


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    Looks good to me.
  4. Feb 22, 2010 #3
    Brill.. :biggrin: .. just wanted to check it through with someone else!
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