Gibbs Free Energy change under non-constant temperature

Question

The Gibbs Free Energy is defined as $G=H-TS$
Therefore, $\Delta G=\Delta H - T\Delta S - S\Delta T$
When temperature is assumed to be constant, $\Delta G = \Delta H - T\Delta S$, which yields the Gibbs free energy change corresponding to the maximum non-expansion work that can be done by a system and indicating the spontaneity of a process.

However, how can the Gibbs free energy change be interpreted for conditions in which the temperature is not constant? In exothermic reactions, for example, heat is evolved from the system. Doesn't this then disrupt the thermodynamic equilibrium between the system and the surroundings, meaning that the above definition of $\Delta G$ can no longer be applied? If so, how can $\Delta G$ under non-constant temperatures be interpreted?

Answer 1

The Gibbs energy is useful in analyzing processes under the condition that $T$, $P$ and $N_i$ is constant (in the case of reactions $N_i$ refers to atoms, not molecules). Reactions, exothermic or endothermic, are analyzed on the condition that $T$ is constant.

This is not a contradiction: Conduct the reaction starting at $T$, $P$ and record the heat. If at the end of the reaction the temperature is $T'$ and the pressure is $P'$, bring both back to their initial conditions noting that the composition of the products and reactants will be changing as you do this. Once your products are at the same temperature and pressure as your reactants, so you can go ahead and use the Gibbs energy.

If you do not bring the products to the same $T$ and $P$ of the reactants you can still calculate a $\Delta G$ between the initial and final states but this does not say anything useful about the process.