Equilibrium and the Reaction Gibbs Energy

It is the tendency of chemical reactions to reach a condition of dynamic equilibrium, where the rate of the forward reaction (conversion of reactants into products) is equal to the rate of the reverse reaction (conversion of products back into reactants). The composition of a reaction mixture that has reached equilibrium thus does not change over time.

Often, the concentration of products at equilibrium is overwhelmingly greater than the concentration of reactants. In such a case the reaction is said to have gone to completion – for all practical purposes it can be assumed that complete conversion of reactants to products has occurred. (If the equilibrium composition is composed overwhelmingly of reactants, with a negligible amount of product, then the reaction is said not to go, and it may be considered that no conversion of  reactants to products has taken place.)

However, in many reactions, there is a significant concentration of both reactants and products at equilibrium, and in such a situation knowledge of the equilibrium composition is needed to allow accurate prediction of the properties of the mixture. As will be shown, the equilibrium composition of  any reaction may be predicted successfully under any conditions, from thermodynamic data about the reaction.

The basic idea that at constant temperature and pressure change occurs in the direction of decreasing Gibbs energy should be familiar. From this, it logically follows that if one plots out the Gibbs energy of a reaction mixture at different compositions (i.e. different stages of reaction from no reaction through to complete reaction), then the minimum in the Gibbs energy corresponds to the equilibrium composition of the mixture. (At the Gibbs energy minimum, movement in either direction results in a Gibbs energy increase, which is not a permitted process. Thus at this position there is no driving force for the reaction to proceed in either direction, which is an acceptable definition of equilibrium.):

The quantity “stage of reaction” indicates the proportion of reactants that have been converted to products.

For the diagram above, therefore, the equilibrium mixture contains significant concentrations of both reactants and products (as the stage of reaction is approximately 0.5 . A stage of reaction of 0.5 would indicate that half of the reactant has been converted to product. Thus the further to the right the Gibbs energy minimum is on such a diagram, the closer to completion the reaction gets, and the less reactant remains at equilibrium). The precise stage of reaction at equilibrium is given by the x coordinate of the point Φ, at the minimum point of the curve.

Further discussion of this idea is simplified by the introduction of a thermodynamic quantity known as the reaction Gibbs energy, ΔGr. We start by defining a quantity called the extent of reaction, xIt has units of moles (amount of substance). Consider a reaction A ↔B , then if an infinitesimal amount dx of A turns into B, then the change in the amount of A is given by dnA = -dx and the change in the amount of B is dnB = +dx. Finite changes in the extent of reaction are represented by Δx. Thus for example, if in the conversion of A to B we start with 5 mol of pure A, then when Δx = 3 mol, 3 moles of A have been converted to B, so there are 2 mole of A and 3 moles of B present.

The reaction Gibbs energy is defined as the slope of a plot of the Gibbs energy against the extent of reaction:
We normally encounter the symbol Δ indicating a difference between two values, but in this case it signifies a derivative. However, a link to the more common usage does exist, as it may be shown that the reaction Gibbs energy is equal to the difference between the chemical potential of the products and the reactants at the specified composition.This graph is essentially the same as the one above – motion from left to right along the horizontal axis still represents progress of the reaction – but the labeling of the horizontal axis has been altered to use the quantity extent of reaction, x. Unlike our previous measure, the stage of reaction, this quantity is not restricted to values between zero and one, but may take any value from zero to the amount of reactant initially present (which corresponds to complete reaction).

Consider the simple reaction outlined above, the interconversion of A and B. The change in Gibbs energy when the reaction advances by dx at constant temperature and pressure is as follows:

from which it follows that:

This simple proof may be extended to more complex reactions by using the chemical potentials of all the species present, weighted by the appropriate stoichiometric coefficient.

If ΔGr < 0 for a reaction, then the forward reaction as written is spontaneous.

Such a reaction is described as being exergonic, as it may be used to do non-expansion work. (eg the reactions in electrochemical cells). From the above equation, we can see that this is will be true when the chemical potential of the reactants is higher than that of the products.

The tendency is thus from high to low potential.

If ΔGr > 0 for a reaction, then the reverse reaction will be spontaneous (this will be when the chemical potential of the products of the reaction as written is higher than that of the reactants).

Such a reaction is called endergonic, and can be made to go only by doing work upon it. For example, electrolysing water will cause it to decompose, the reverse of its spontaneous formation reaction.

If ΔGr = 0 for a reaction, then the reaction is at equilibrium.

This is when the chemical potentials of reactants and products are precisely equal (which is another good definition of equilibrium). Reactions at equilibrium are neither endergonic nor exergonic; they are not spontaneous in either direction.