Gibbs free energy, #G#, is just the enthalpy change of a reaction, #DeltaH#, minus the entropy change of the reaction system, #DeltaS_(sys)#, multiplied by the temperature of the reaction, #T#.

#DeltaG=DeltaH - TDeltaS_(sys)#

Helmholtz free energy, #DeltaA#, is the same, but uses the change in internal energy of the reaction system, #DeltaU#, instead of #DeltaH#.

#DeltaA=DeltaU-TDeltaS_(sys)#

For a reaction to occur, it needs to cause the total entropy of the reaction system, #S_(sys)#, and its surroundings, #S_(sur)#, to increase.

#DeltaS_(overall)=DeltaS_(sys)+DeltaS_(sur)>0#

But, because "the surroundings" is effectively the whole rest of the universe, it's quite difficult to accurately measure the surrounding entropy change.

Therefore, the free energy equations were invented, because if #DeltaG<0# or #DeltaA<0#, then that means that the total entropy change of a reaction is greater than zero, #DeltaS_(overall)>0#, and so the reaction will happen.

**Derivation of the Gibbs free energy equation:**

#DeltaS_(overall)=DeltaS_(sur)+DeltaS_(sys)>0#

#DeltaS_(sur)=-(DeltaH)/T#

#DeltaS_(overall)=(-DeltaH)/T+DeltaS_(sys)>0#

Multiplying through by #-T# gives

#DeltaG=-TDeltaS_(overall)=DeltaH-TDeltaS_(sys)<0#