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DeltaG is 0 at equilibrium because neither the forward nor the reverse reactions are favored. They occur at equal rates. If deltaG were to be negative and spontaneous for one reaction (either forward or reverse), that would mean the opposite reaction (either reverse or forward, respectively), would have to be positive and nonspontaneous.
We can also say that an endergonic process has a positive delta g and is non spontaneous, an exergonic process has a negative delta g and is spontaneous so it makes sense that a process at equilibrium has a delta g of zero.
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