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Recall that in physical chemistry, the spontaneity of a reaction at constant pressure and a given temperature $T$ can be quantified by the Gibbs free energy

$$\Delta G = \Delta H - T \Delta S$$

A reaction is spontaneous if $\Delta G < 0$. For an endothermic reaction to be spontaneous, $\Delta H < 0$, $\Delta S > 0$ and $|\Delta H|< |T\Delta S|$

Now suppose we monitor the rate of change of $\Delta G$ as the reaction progressed. This means

\begin{align} \frac{\mathrm d\Delta G}{\mathrm dt} & = \frac{\mathrm d\Delta H}{\mathrm dt} - \frac{\mathrm d(T \Delta S)}{\mathrm dt}\\ \frac{\mathrm d\Delta G}{\mathrm dt} & = \frac{\mathrm dT}{\mathrm dt}\left(\frac{\partial \Delta H}{\partial T} - T\frac{\partial \Delta S}{\partial T}- \Delta S\right) \end{align}

Now suppose for simplicity $\Delta H$ is roughly constant with temperature. Then:

$$\frac{\mathrm d\Delta G}{\mathrm dt} = \frac{\mathrm dT}{\mathrm dt}\left( - T\frac{\partial \Delta S}{\partial T}- \Delta S\right)$$

Thus for $\frac{\mathrm dT}{\mathrm dt} < 0$, a reaction to be self sustaining indefinitely if the following holds for all $t > 0$

\begin{align} - T\frac{\partial \Delta S}{\partial T}- \Delta S & > 0\\ - T\frac{\partial \Delta S}{\partial T} & > \Delta S \end{align}

In all closed systems, entropy will eventually maximises according to the second law, thus there will be a point where $\frac{\partial\Delta S}{\partial T} < 0$ (i.e. the increase in entropy as the reaction progressed will slow down over time) until the reaction stops (thus explaining why both exothermic and endothermic reactions will eventually go to completion)

Therefore, my question on self sustaining endothermic reaction that last for hours is boiled down to finding a time $t>0$ such that the above inequality holds in the order of hours, given a $\Delta H$ as positive as physically realisable

Any example of an endothermic reactions that has $\Delta H > 50{-}100\ \mathrm{kJ\ mol^{-1}}$ and can sustain itself in the order of hours?

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    $\begingroup$ Thermodynamics does not inform us about the time-scales of processes but only about the starting and ending conditions. Reaction rates are controlled by the activation energies between reactants and products for both endothermic and exothermic reactions. see answer at chemistry.stackexchange.com/questions/10115/… $\endgroup$ – porphyrin Nov 20 '16 at 10:54
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    $\begingroup$ How about a large block of ice melting? Latent heat is $80\,\mathrm{kcal\, mol^{-1}}$ if memory serves. $\endgroup$ – Zhe Nov 20 '16 at 15:41
  • $\begingroup$ @Zhe A large block of ice melting is not a chemical reaction. It is a phase change that is controlled time-wise by the heat transfer rate (not reaction kinetics). $\endgroup$ – Chet Miller Nov 20 '16 at 16:59
  • $\begingroup$ @ChesterMiller True, but thermodynamics clearly governs this process in the same way as a chemical reaction. I agree though that I didn't quite read the question right. This isn't quite "self-sustaining" as the OP wanted... $\endgroup$ – Zhe Nov 20 '16 at 17:07
  • $\begingroup$ It depends on what you mean by "in the same way." Conductive heat transfer is certainly fundamentally different from molecules interacting to form different molecules. $\endgroup$ – Chet Miller Nov 20 '16 at 22:01

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