# 12.4: Gibbs Free Energy

- Page ID
- 7288

The Gibbs free energy G is defined as

\[G=H-T S\]

or, what amounts to the same thing,

\[G=A+P V.\]

As when we first defined enthalpy, this doesn't seem to mean much until we write it in differential form:

\[d G=d H-T d S-S d T\]

or

\[d G=d A+P d V+V d P.\]

Then, either from equations 12.1.5 (*dH* = *TdS* + *VdP* + ∑*XdY*) and 12.5.3 or from equation 12.4.3 (*dA* = −*SdT* − *PdV* + ∑*XdY*) and 12.5.4, we obtain

\[d G=-S d T+V d P+\sum X d Y\]

That is to say that, if the temperature and pressure are constant, the increase in the Gibbs function of a system is equal to the reversible work (other than *PdV* work of compression) done on it. Conversely, if the temperature and pressure are held constant, and a machine is used to do external work (which may include but is not limited to *PdV* work of expansion), the Gibbs function decreases by the amount of reversible (i.e.useful) work done by the machine other than the *PdV* work of expansion.