How does Gibbs free energy change relate to work?

The #DeltaG# for a reversible process is equal to the maximum non-PV work that can be performed at constant temperature and pressure on a conservative system.

Consider the differential relationship between the Gibbs' free energy, enthalpy, and entropy:

#dG = dH - d(TS)#

From the definition of enthalpy, #H = U + PV#, where #U# is the internal energy. As a result,

#dG = dU + d(PV) - d(TS)#

From the first law of thermodynamics, #dU = deltaq + deltaw#, where #delta# indicates a path function.

#dG = deltaq + deltaw + PdV + VdP - TdS - SdT#

Work can be defined as

#deltaw = deltaw_"PV" + deltaw_"non-PV"#,

where #"PV"# work defined from the perspective of the system is #deltaw_"PV" = -PdV#. Non-PV work can be, e.g. electrical work (think electrochemistry).

From this, assuming that the process performed is reversible (in thermal equilibrium the whole way through), #q_(rev) = TdS#, so:

#color(green)(dG) = overbrace(cancel(TdS))^(q_(rev)) + deltaw_"non-PV" overbrace(- cancel(PdV))^(w_(rev,"PV")) + cancel(PdV) + VdP - cancel(TdS) - SdT#

#= color(green)(-SdT + VdP + deltaw_"non-PV")#

In the end, we find that at constant temperature and pressure, the Gibbs' free energy corresponds to the maximum non-compression and non-expansion work that can be performed:

#color(blue)(dG = deltaw_"non-PV", " const T & P")#