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Looking at a decomposition reaction, let's say A --> B + C, if the reaction has a negative delta G, then the decomposition reaction is spontaneous, or favorable. This means that A has a tendency to decompose into B and C; in other words, we can say that A is unstable.
I believe because a negative delta g indicates that a reaction is spontaneous (more likely to occur or favorable), and if that compound is unstable it means that its more likely to revert back to its pure elements (hence, decomposition).
If delta G is negative, then the forward reaction is favored, which means that the reactants are unstable and will decompose into the products spontaneously.
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