Deriving Helmholtz Thermodynamic Potential & Corresponding Maxwell Relation

[Hart]

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: $$F = U - TS$$
  
  
  
• 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.
  

 

[Mapes]

Looks good to me.

 

[Hart]

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