(adsbygoogle = window.adsbygoogle || []).push({}); 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] ,

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.

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# Homework Help: Deriving Helmholtz Thermodynamic Potential & Corresponding Maxwell Relation

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