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We use free energy equations -- Helmholtz or Gibbs -- to predict whether or not a reaction is spontaneous. These equations depend on constant temperature.

This forum post describes a scenario where volumes of water at different temperatures are mixed. The $\Delta G$ appears to be positive. (The poster's textbook gives $\Delta G = 99.5 \, \mathrm{Cal}$, a result I've been able to duplicate.) This seems to show that $\Delta G$ doesn't predict spontaneity for non-isothermal reactions.

So what does?

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For an adiabatic spontaneous process in a closed system (like the one in your example), according to the Clausius Inequality, the change in entropy is greater than zero. If you have a closed system in which a chemical reaction can occur, if the system is adiabatic, the same criterion applies.

The criterion that, at constant temperature, if a reaction is spontaneous, $\Delta G^0$ in going from pure reactants to pure products at 1 bar will be negative is just a rule of thumb which, over the years, has confused students to no end. If you start out with pure reactants and mix them, spontaneous reaction will always occur to some extent. It's just that, if $\Delta G^0$ is positive, the reaction will tend to proceed to a lesser extent when equilibrating, and if $\Delta G^0$ is negative, it will tend to proceed to a greater extent when equilibrating. So, in this sense, every reaction is spontaneous.

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