What is free energy change? Show that the change in free energy is equal to useful work done.
or
prove that –∆G = w_{(useful work)}

(i) The overall criterion of a process to occur spontaneously i.e. driving force is expressed in terms of free energy change (∆G).
Free energy change (∆G) is related to enthalpy change (∆H) and entropy change (∆S) as ∆G = ∆H – T∆S [Gibbs-Helmoltz equation] For a spontaneous process, ∆G should have negative value.
(ii) According to first law of thermodynamics,

where q is heat absorbed by the system. ∆U is the change in internal energy and w is the work done on the system.
If we are to determine the work a system can do, then w should be taken as – w.

Now w includes expansion as well as non-expansion work. 

But work due to expansion at constant temperature is given by P∆V.

w_non-exp is also called useful work because this type of work can be used for useful effect e.g. electrical work.

Substituting this value in (2), we have,
                       
or     
       

This means that decrease in free energy of a system (–∆G) is a measure of the non-expansion or useful work done by the system in a process or –∆G = w_{useful work}