If you have an endothermic reaction with a negative entropy change, is it still possible to induce the reaction in some way despite the fact that the Gibbs free energy change is positive or all temperatures?
Answer
Yes, $\Delta G = -RT\ln K$.
If a reaction is endothermic with a negative entropy change, $\Delta G$ is positive.
That $\Delta G$ is positive only means that $K<1$.
$\Delta G$ would need to approach infinity for $K$ to approach zero.
Therefore, at equilibrium, there will always be some products in principle no matter how unfavorable the reaction.
As a practical matter, the reaction would need to be only slightly endothermic and having only a slight negative entropy change to get a significant amount of product.
However, if the small amount of product is continuously removed from the system, as for example by precipitation from a liquid phase, or gas bubbling out of a liquid phase, or by liquid-liquid extraction, one can keep the reaction going forward.
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