Why does Gibbs free energy have to be negative?

1 Answer
Mike G.
Sep 26, 2014

For a reaction to happen spontaneously, the total entropy of the system and surroundings must increase:

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

The entropy of the system changes by #(DeltaH_(sys))/T#, and because #DeltaH_(sys)=-DeltaH_(sur)#, entropy change of the surroundings can be calculated from the equation

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

Substituting this for #DeltaS_(sur)# gives

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

Multiplying through by #-T# gives

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

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