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!