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Hart
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Homework Statement
To state the differential form of the Helmholtz thermodynamic potential and
derive the corresponding Maxwell's relation.
Homework Equations
Stated within the solution attempt.
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] ,
Therefore:
[tex]dF = -PdV - SdT[/tex]
- Which then follows that can write: [tex]F = F(V,T)[/tex]
Hence:
[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]
and
[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.