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
