Equilibrium constant (derivation)

The equilibrium constant of a chemical reaction is the reaction quotient of the reaction at dynamic equilibrium.

The derivation of the equilibrium constant involves the following steps:

1. 1. Derive the expression for the Gibbs energy of a multi-component reaction.
   2. Derive the chemical potential of a multi-component system.
   3. Combine the two expressions.

Step 1

From eq14 of the article on the Nernst equation, the reaction Gibbs energy for a reaction is

and at standard state

Step 2

From eq24 of the article on the Nernst equation, the chemical potential of a multi-component system is

Step 3

Combining eq1 and eq3

Substitute eq2 in eq4

Since lnx^a +lnx^b +… = ln(x^ax^b …), eq5 becomes

Let , where Q is the reaction quotient.

At equilibrium, a reversible reaction is spontaneous in neither direction. Hence the change in Gibbs energy with respect to the change in the extent of the reaction, i.e. the reaction Gibbs energy , is zero. Eq7 becomes

where K is the reaction quotient at equilibrium, i.e.

Since the value K of does not change for a particular reaction at constant temperature, we call it the equilibrium constant.

 

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