next up previous
Next: General Relation Between Specific Up: Classical Thermodynamics Previous: Helmholtz Free Energy

Gibbs Free Energy

Suppose, finally, that $ T$ and $ p$ are the are the two independent parameters that specify the system. Because

$\displaystyle T dS = d(T S)-S dT,$ (6.110)

we can rewrite Equation (6.91) in the form

$\displaystyle dG = -S dT + V dp,$ (6.111)

where

$\displaystyle G = H - T S = E-T S+p V$ (6.112)

is termed the Gibbs free energy.

Proceeding as before, we write

$\displaystyle G= G(T,p),$ (6.113)

which implies that

$\displaystyle dG= \left(\frac{\partial G}{\partial T}\right)_p dT + \left(\frac{\partial G}{\partial p}\right)_T dp.$ (6.114)

Comparison of Equations (6.111) and (6.114) yields

$\displaystyle \left(\frac{\partial G}{\partial T}\right)_p$ $\displaystyle = -S,$ (6.115)
$\displaystyle \left(\frac{\partial G}{\partial p}\right)_T$ $\displaystyle = V.$ (6.116)

We also know that

$\displaystyle \frac{\partial^{ 2} G}{\partial p \partial T} = \frac{\partial^{ 2}G}{\partial T \partial p},$ (6.117)

or

$\displaystyle \left(\frac{\partial}{\partial p}\right)_T\left(\frac{\partial G}...
...rac{\partial}{\partial T}\right)_p\left(\frac{\partial G}{\partial p}\right)_T.$ (6.118)

Thus, it follows from Equations (6.115) and (6.116) that

$\displaystyle -\left(\frac{\partial S}{\partial p}\right)_T = \left(\frac{\partial V}{\partial T}\right)_p.$ (6.119)

Equations (6.89), (6.99), (6.109), and (6.119) are known collectively as Maxwell relations.


next up previous
Next: General Relation Between Specific Up: Classical Thermodynamics Previous: Helmholtz Free Energy
Richard Fitzpatrick 2016-01-25