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Question

Derive the following general equation,which is similar to the Gibbs-Helmholtz equation:(∂/∂T)(F/T)=−U/T^2

Expert Solution

Step 1

The Helmholtz free energy is a thermodynamic potential which is a measure of the useful work done by a system at a constant temperature. Mathematically, Helmholtz free energy is given by the relation

$F = U - T S$

where $U, T$ and $S$ are the internal energy, temperature, and entropy of a system respectively.

Step 2

The Helmholtz free energy is given by the relation

$F = U - T S$

Taking differential of the above expression we get

$d F = d U - T d S - S d T$

By the combined first and second law of thermodynamics

$T d S = d U + P d V$

Substituting this in the above equation we get

$d F = d U - d U - P d V - S d T \Rightarrow d F = - P d V - S d T$

Now when we consider Helmholtz free energy as a function of temperature and volume which are considered as independent variables, we have

$F = F (T, V)$

Taking differential of the above equation we get

$d F = {(\frac{\partial F}{\partial T})}_{V} d T + {(\frac{\partial F}{\partial V})}_{T} d V$

Comparing the above expression with $d F = - P d V - S d T$ we get

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

Now from the definition of Helmholtz free energy

$F = U - T S$

Substituting the expression for entropy $S$ , we have

$F = U + T {(\frac{\partial F}{\partial T})}_{V} \Rightarrow U = F - T (\frac{\partial F}{\partial T}) \Rightarrow U = - T^{2} {\{\frac{\partial}{\partial T} (\frac{F}{T})\}}_{V} \Rightarrow {\{\frac{\partial}{\partial T} (\frac{F}{T})\}}_{V} = - \frac{U}{T^{2}}$

This is the required result

${\{\frac{\partial}{\partial T} (\frac{F}{T})\}}_{V} = - \frac{U}{T^{2}}$

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