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I've read an article that equilibrium exists when $G$ (Gibbs free energy) becomes minimum. If formation of products decreases Gibbs free energy, why won't the reaction proceed to completion so that Gibbs energy becomes minimum.

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Just because the Gibbs energy change of reaction ($\Delta _r G$) is negative, doesn't mean that it reaches a minimum when all reactants have been converted to products.

Consider the reaction: $$\ce{A <=> B}$$

We can plot the total Gibbs energy of the system against the percentage conversion from $\ce{A}$ to $\ce{B}$ and we get a graph like this:

enter image description here

Technically the curve should be a parabola but it was much easier to draw an arc in the program I used. As you can see, the minimum Gibbs energy is reached when there is a mixture of products and reactants. This is in fact true for all chemical reactions: the minimum on the Gibbs energy graph cannot be at the point where only reactants or products exist. It can be extremely close to that point, in which case we tend to call the reaction irreversible (if the minimum is close to 100% products) or we say that the reaction doesn't happen (if the minimum is close to 100% reactants).

It turns out that there is a relationship between $\Delta _r G$ and the equilibrium constant which is expressed as: $$\Delta _r G = -RT\ln K$$

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This is because the Gibbs free energy also depends on the concentration of the reactants and products. Hence, for a reaction, if the energy change is negative it moves forwards, forming more product, which also changes the concentrations of the species involved. The Gibbs energy change at this new concentration will be less negative than that in the initial case, but the reaction still proceeds since the reaction is accompanied by a negative change of free energy.

A point comes, where the concentrations of the reactions and the products is such, that the Gibbs energy change now becomes zero for the particular reaction. This is the equilibrium condition (minimum free energy, maximum entropy). If the energy change is positive, the reverse reaction becomes favorable, again inching towards the zero energy change condition, i.e. equilibrium. So, the reaction is already at the minimum Gibbs energy when it is at equilibrium. Further proceeding of the reaction will be accompanied by a positive change, and hence a rise in energy.

For how it depends on the concentrations, see these questions

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