Reaction Equilibrium Constraint

Several sub-steps are involved in the procedure outlined in Section 17.1 steps (1) and (2) to find the equilibrium constant. In this section, we derive the equilibrium constraint, and then show how the thermodynamic properties are used to simplify to Eqn. 17.1. At reaction equilibria, the total Gibbs energy is minimized. If the composition of a system is changing, the change in the Gibbs energy is given by:

The fact that species are being created or consumed by a reaction does not alter this equation. At constant temperature and pressure, the first two terms on the right-hand side drop out:¹

Substituting the definition of reaction coordinate from Eqn. 3.39,

Because G is minimized at equilibrium at fixed T and P, the derivative with respect to reaction coordinate is zero:

Now there is one unknown, ζ, in terms of which we can determine the changes in moles for all of the components. We make a further manipulation before we apply the equilibrium constraint. In phase equilibria, we found fugacity to be a convenient property to use because it simplified to the partial pressure for a component in an ideal gas mixture. We can rewrite Eqn. 17.9 in terms of fugacities. We recall our definition of fugacity dG = RT dln f. Integrating from the standard state to the mixture state of interest (cf. generalizing Eqn. 10.48),

where G_i^o is the standard state Gibbs energy of species i and f_i^o is the standard state fugacity. A standard state is introduced for liquids in Section 11.3, and now we generalize the approach. The standard state is at the reaction temperature, but a specified composition (often pure) and pressure P°. Substitution of Eqn. 17.10 into Eqn. 17.9,

or

We will need to calculate both summations appearing in Eqn. 17.12, and then combine the results. Qualitatively, this equation indicates how atoms should be arranged within molecules to minimize Gibbs energy. The connection between Eqn. 17.12 and Eqn. 17.1 will become obvious as we move through the next sections. Let us work on the two summations separately. We will show that the second summation is related to the product of partial pressures for gas phase species which we define as the equilibrium constant. The first summation will relate to the negative numerical value of the equilibrium constant because the two terms of Eqn. 17.12 are equal and opposite.