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Suppose that
and
are the are the two independent parameters that specify the system.
Because
![$\displaystyle T dS = d(T S)-S dT,$](img1028.png) |
(6.100) |
we can rewrite Equation (6.82) in the form
![$\displaystyle dF = -S dT - p dV,$](img1029.png) |
(6.101) |
where
![$\displaystyle F = E - T S$](img1030.png) |
(6.102) |
is termed the Helmholtz free energy.
Proceeding as before, we write
![$\displaystyle F= F(T,V),$](img1031.png) |
(6.103) |
which implies that
![$\displaystyle dF= \left(\frac{\partial F}{\partial T}\right)_V dT + \left(\frac{\partial F}{\partial V}\right)_T dV.$](img1032.png) |
(6.104) |
Comparison of Equations (6.101) and (6.104) yields
We also know that
![$\displaystyle \frac{\partial^{ 2} F}{\partial V \partial T} = \frac{\partial^{ 2}F}{\partial T \partial V},$](img1036.png) |
(6.107) |
or
![$\displaystyle \left(\frac{\partial}{\partial V}\right)_T\left(\frac{\partial F}...
...rac{\partial}{\partial T}\right)_V\left(\frac{\partial F}{\partial V}\right)_T.$](img1037.png) |
(6.108) |
Thus, it follows from Equations (6.105) and (6.106) that
![$\displaystyle \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V.$](img1038.png) |
(6.109) |
Next: Gibbs Free Energy
Up: Classical Thermodynamics
Previous: Enthalpy
Richard Fitzpatrick
2016-01-25