Deriving Helmholtz Thermodynamic Potential & Corresponding Maxwell Relation

1. Feb 22, 2010

Hart

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: $$F = U - TS$$

• Calculating the differential form:

For infinitesimal change: $$dF = dU - tdS - SdT$$

Then using: $$TdS = dU + PdV$$ ,

Therefore:

$$dF = -PdV - SdT$$

• Which then follows that can write: $$F = F(V,T)$$

Hence:

$$dF = \left(\frac{\partial F}{\partial V}\right)_{T}dV + \left(\frac{\partial F}{\partial T}\right)_{V}dT$$

• Comparing coefficients in the previous two stated expressions for $$dF$$:

$$P = -\left(\frac{\partial F}{\partial V}\right)_{T}$$

and

$$S = -\left(\frac{\partial F}{\partial T}\right)_{V}$$

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

$$\left(\frac{\partial P}{\partial T}\right)_{v} = \left(\frac{\partial S}{\partial V}\right)_{T}$$

Which is the Maxwell relation, derived as required.

2. Feb 22, 2010

Mapes

Looks good to me.

3. Feb 22, 2010

Hart

Brill.. .. just wanted to check it through with someone else!