Page 43 - Thermodynamics of Biochemical Reactions

3.1 Derivation of the Expression for the Equilibrium Constant 37

To discuss equilibrium in a chemical reaction system, it is convenient to
introduce the activity a, of a species to replace the chemical potential of a species
because a, is more closely related to partial pressures and concentrations of
species. The activity of a species is defined by

pi = py + RTlna, (3.1-7)

where $, which is referred to as the standard chemical potential, is the chemical
potential when ai = 1. A superscript zero is used to designate a standard property,
and so py is the standard chemical potential of species i. For ideal mixtures of ideal
gases, ui can be replaced by P,/PO, where Po is the standard state pressure, and
for ideal solutions a, can be replaced by ci/cO, where co is the standard
concentration. Note that the activity a, of species i is dimensionless. We will use
molar concentrations, but measurements in physical chemistry are frequently
based on molal concentrations (mol kg- I). Molal concentrations mi have the
advantage that they do not change with temperature.
From one point of view nothing is gained by using equation 3.1-7 to define
the activity ai of a species and using it to replace the chemical potential pi of the
species. The difference between pi and ai is that pi of an ideal gas goes from - 00
to m, whereas a, goes from 0 to co. Thus the activity of a species in solution is
more closely related to its concentration than pi is. However, the activity of a
species in solution is directly proportional to its concentration only for ideal
solutions. In general, the activity of a species in solution is given by a, = yici,
where yi is the activity coefficient of species i. The activity coefficient of a solute is
a function of the concentration, especially for ions. Strictly speaking (Mills et a].,
1993), this relation should be written a, = yici/cO, where c0 is the standard
concentration (1 M). However, c0 is omitted in this book to simplify the
equations. Thus equation 3.1-7 is written

0
pi = pi + RTlnyici (3.1-8)
When using the molar concentration scale, the convention is that the activity
coefficient of a species approaches unity as the concentration of the species
approaches zero. In discussing biochemical reactions in dilute aqueous solutions,
effects on activity coefficients arise primarily because of electrostatic interactions
between charged species and depend on the ionic strength (see Section 1.2 and
Section 3.6). Since the ionic strength is under the control of the investigator and
is nearly constant during the approach to equilibrium when a biochemical
reaction is carried out in dilute aqueous solution with a buffer, we can postpone
discussing the effects of ionic strength to Section 3.6 by making the following
observation: Equation 3.1-8 can be written

pi = ,UP + RTIny, + RTlnci (3.1-9)

In discussing biochemical thermodynamics, however, it is convenient to write this
equation as

p, = + RTlnc, (3.1 - 10)

where pj and are functions of the ionic strength. In equations 3.1-9 and 3.1-10,
p, and p: have been used in two different ways, but in the rest of the book
equation 3.1-10 will always be used. In other words, chemical potentials and other
thermodynamic properties of species in dilute aqueous solutions will be taken to
be functions of the ionic strength. This will allow us to avoid including y, in many
places (even though the effect of ionic strength is taken into account) and to treat
solutions at a specified ionic strength as “ideal solutions,” that is as solutions
following equation 3.1-10. We have already seen an example of this in the
treatment of pH in Section 1.2.