Thermodynamics

DG AND THE EQUILIBRIUM CONSTANT

The preceding sections have covered how the free energy change of a reaction, DG, indicates reaction spontaneity. In other words, knowing the value of DG, and whether it is positive or negative, tells us in which direction a chemical reaction will spontaneously proceed.

It is important to remember that the DG value for any reaction depends on the actual conditions under which that reaction takes place, including the temperature and the concentrations of the chemicals. Therefore, to make comparisons between reactions easier, chemists decided on standard conditions under which DG values for various reactions would be calculated and reported in tables. These standard reaction conditions are when all the substrates and products of the reaction are present at a concentration of 1M. The standard state free energy change is designated as DG°.

This convention, however, creates a problem for biochemists, since almost all biochemical reactions take place near pH 7, where the [H⁺] is about 10^–7 M. (Remember that a [H⁺] of 1 M would mean a pH of 0; almost no biochemical enzymes can function at such a low pH!) Therefore, biochemists came up with slightly different reaction conditions, where all the reactant and products except H⁺ are present at 1 M, and H⁺ is present at a physiologically relevant concentration (usually near pH 7). This biochemical standard free energy change is indicated as DG°´.

It seems logical that a reaction’s free energy change (DG) under nonstandard conditions (whatever the pressure, temperature and reactant concentrations happen to be when the DG is measured) is related to its standard free energy change, DG°´. After all, the only difference between the two values is that for DG°´, the atmospheric pressure is set at 1 atm, the reactant concentrations (except [H⁺]) are all at 1 M, and the temperature of the reaction is 25°C (298 K). Indeed, the two values are related by the differences in pressure, temperature, and reactant concentrations. This relationship can be described with the following equation:

DG = DG°´ + RT ln [C][D] [A][B] Where: R is the gas constant (8.3145 J mol–1K–1) T is the temperature (in Kelvin)

Remember that the DG value tells us about physical processes, including chemical reactions.

In other words, when the DG is zero, the reaction does not spontaneously proceed in any direction–the concentrations of reactants and products don’t change at all, and the reaction is said to be at equilibrium.

A + B C + D

The equilibrium constant is defined as (remember that the brackets indicate the molar concentration of each substance):

Keq = [C]eq[D]eq [A]eq[B]eq

If we place the value of DG = 0 into the above equation, we obtain the following:

DG = DG°´ + RT ln [C][D] [A][B]   0 = DG°´ + RT ln [C]eq[D]eq [A]eq[B]eq   DG°´ = – RT ln [C]eq[D]eq [A]eq[B]eq Substituting the Keq expression: DG°´= –RT ln Keq

Here, DG°´ is defined as the force that drives a reaction toward equilibrium. In other words, when the components of a chemical reaction are not in equilibrium, they experience a force, DG°´ that drives them to reach their equilibrium values.

Example 6: Equilibrium

How would you describe equilibrium?

Answer

When imagining how a chemical reaction proceeds, it is natural to think that the reaction just keeps on going until the reactants have all been converted to products. It is actually rare, however, for a reaction to continue to such an extreme end. Most chemical reactions reach a state where no more product is formed, despite the fact that more reactants are still available. This state is known as equilibrium. Keep in mind that the reaction does not actually stop. Rather, the equilibrium point is where the rate of the forward reaction is exactly the same as the rate of the reverse reaction. Thus, for a chemical reaction at equilibrium:

A + B C + D

The reaction A + B C + D is occurring at the same rate as the reaction C + D A + B.

The concentrations of the reactants and products at equilibrium are different for each chemical reaction, since it depends on the chemical nature of the products and reactants (as well as the temperature at which the reaction proceeds). The equilibrium constant K_eq is defined by the concentrations of each of the reaction components when the chemical reaction is at equilibrium. Thus, for the above reaction the equilibrium constant is defined as

Keq = [C]eq[D]eq [A]eq[B]eq

(Remember that the brackets indicate the molar concentration of each substance.)