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I'm not really sure what you mean by minimum temperature (minimum temperature of what?), but delta G has to be zero at phase changes (ex. at boiling point and melting point) because this is the point at which the process goes from being spontaneous (negative delta G) to not spontaneous (positive).
We set deltaG equal to 0 when we want to find the temperatures at which a reaction is spontaneous. This is because at equilibrium deltaG is 0; neither process, forward or reverse, products or reactants is favored. If we have T for which deltaG is 0, then we know that any slight change in T from that value leads to a favoring of either the forward or reverse reaction (T for which deltaG is 0 is the minimum T for spontaneity for processes where deltaS is positive and deltaH is positive and the maximum T for spontaneity for processes where deltaS is negative and deltaH is negative).
I think what you are asking what is the minimum temperature when the reaction is spontaneous? The reason that ∆G is 0 because when ∆G is negative, the reaction is spontaneous, therefore, to figure out the minimum temperature, we set the limit:∆G=0, so any temperature above the limit will have a ∆G above 0 and the reaction will be spontaneous.
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