Page 42 - Thermodynamics of Biochemical Reactions

36 Chapter 3 Chemical Equilibrium in Aqueous Solutions

dG < 0 at constant 7; P, and {ncl), where there are C components with amounts
{ncl). Components are discussed in Section 3.3, and the various choices of
components that can be used will become clearer in Chapter 5 on matrices.
In this chapter we will find that when isomers are in chemical equilibrium, it
is convenient to treat isomer groups like species in order to reduce the number of
terms in the fundamental equation. We will also discuss the effect of ionic strength
and temperature on equilibrium constants and thermodynamic properties of
species. More introductory material on the thermodynamics of chemical reactions
is provided in Silbey and Alberty (2001).

3.1 DERIVATION OF THE EXPRESSION FOR THE
EQUILIBRIUM CONSTANT

When a chemical reaction occurs in a system, the changes in the amounts n, of
species are not independent because of the stoichiometry of the reaction that
occurs. A single chemical reaction can be represented by the reaction equation
N,
1 viBi = 0 (3.1-1)
i= 1
where Bi represents species i and N, is the number of different species. Chemical
reactions balance the atoms of all elements and electric charge. The stoichiometric
numbers vi are positive for products and negative for reactants. The amount ni of
species i at any stage in a reaction is given by

ni = nio + vij' (3.1 -2)
where nio is the initial amount of species i and j' is the extent of reaction. It is
evident from this definition of 4 that it is an extensive property. Stoichiometric
numbers are dimensionless, and so the extent of reaction is expressed in moles.
The differential of the amount of species i is given by

dni = vid( (3.1-3)

When a single chemical reaction occurs in a closed system, the differential of
the Gibbs energy (see equation 2.5-5) is given by

dG = -SdT+ VdP + C pivi d( (3.1-4)
ti=, ,)
This form of the fundamental equation applies at each stage of the reaction. The
rate of change of G with extent of reaction for a closed system with a single
reaction at constant T and P is given by

(3.1-5)

where A,G is referred to as the reaction Gibbs energy. The Gibbs energy of the
system is at a minimum at equilibrium, where (i?G/d()T,p = 0. At the minimum
Gibbs energy, the equilibrium condition is

(3.1-6)
i= 1
Notice that this relation has the same form as the chemical equation (equation
3.1- 1).