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A negative Gibbs Free Energy means that the energy of the products is less than the energy of the reactants. If it were the reverse, an input of energy would be needed, which is not favorable. Therefore, a negative Gibbs free energy is favored because it drives the reaction forward spontaneously.
For Michael Cheng, when Gibbs Free Energy is negative it is associated with a spontaneous reaction. When gibbs is negative, the system is releasing energy (exergonic), and the definition of spontaneous is when a reaction proceeds without any outside force.
A negative deltaG is favored because then the reaction will proceed spontaneously and exothermically, releasing heat and increasing the entropy of its surroundings. Reactions that increase entropy are usually more favorable than reactions that require energy and do the opposite.
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