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. 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

\Delta_rG=\sum_j\mu_jv_j\; \; \; \; \; \; \; \; 1

and at standard state

\Delta_rG^{\: o}=\sum_j\mu_j^{\; \; o}v_j\; \; \; \; \; \; \; \; 2

Step 2

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

\mu_j=\mu_j^{\; \, o}+RTlna_j\; \; \; \; \; \; \; \; 3

Step 3

Combining eq1 and eq3

\Delta_rG=\sum_j\left ( \mu_j^{\; o}+RTlna_j \right )v_j=\sum_j\mu_j^{\; o}v_j+RT\sum_jv_jlna_j\; \; \; \; \; \; \; \; 4

Substitute eq2 in eq4

\Delta_rG=\Delta_rG^{\; o}+RT\sum_jlna_j^{\: v_j}\; \; \; \; \; \; \; \; 5

Since lnxa +lnxb +… = ln(xaxb …), eq5 becomes

\Delta_rG=\Delta_rG^{\; o}+RTln\prod _ja_j^{\: v_j}\; \; \; \; \; \; \; \; 6

Let Q=\prod_ja_j^{\; v_j}, where Q is the reaction quotient.

\Delta_rG=\Delta_rG^{\; o}+RTlnQ\; \; \; \; \; \; \; \; 7

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 \Delta_rG=\left ( \frac{\partial G}{\partial\xi} \right )_{T,p}, is zero. Eq7 becomes

\Delta_rG^{\circ}=-RTlnK\; \; \; \; \; \; \; \; 8

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

K=\left ( \prod _ja_j^{\; v_j} \right )_{eqm}\; \; \; \; \; \; \; \; 9

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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