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Chapter 11 11.5 REACTION EQUILIBRIUM IN NONIDEAL GAS MIXTURES
Reaction Equilibrium
in Nonideal Systems
The activity a of component i of a nonideal gas mixture is [Eqs. (10.96) and (10.99)]
i
a f >P° f P>P° f x P>P° where P° 1 bar (11.28)
i
i
i i
i i
where f , f , P , and x are the fugacity, fugacity coefficient, partial pressure, and mole
i
i
i
i
n
fraction of gas i, and P is the pressure of the mixture. Substitution into K° ß (a ) i
i
i
[Eq. (11.6)] gives at equilibrium in a gas-phase reaction with stoichiometric coeffi-
cients n i
f i n i f x P n i
i i
K° q a b q a b (11.29)
i P° i P°
The standard state for each gas has the pressure fixed at 1 bar, so G° depends only
on T. Hence the equilibrium constant K°, which equals exp( G°/RT) [Eq. (11.4)],
depends only on T. Using the identity ß (a b ) ß a ß b , we rewrite (11.29) as
i i
i
i
i
i
i
K° x P n i
i
q a b (11.30)
q 1f 2 n i i P°
i
i
To calculate the equilibrium composition at a given T and P of a reacting nonideal
gas mixture, the following approximate procedure is often used. Tables of G° for the
T
f
reacting gases are used to calculate G° for the reaction. The equilibrium constant K°
T
is then calculated from G°. The fugacity coefficients f*(T, P) of the pure gases are
T
i
found using either law-of-corresponding-states charts of f* as a function of reduced
i
temperature and pressure (Sec. 10.10) or tabulations of f*(T, P) for the individual
i
gases. The Lewis–Randall rule f f*(T, P) (Sec. 10.10) is then used to estimate f i
i
i
for each gas in the mixture. The quantity on the left side of (11.30) is calculated, and
(11.30) is then used to find the equilibrium composition by the procedures of Sec. 6.4.
A better, but more complicated, procedure is to use an equation of state for the
mixture. One initially sets all the f ’s equal to 1 and solves (11.30) for the initial
i
estimate of the equilibrium composition. One uses the mixture’s equation of state to
calculate each f from Eq. (10.101) at this composition. These f ’s are used in (11.30)
i
i
to solve for an improved estimate of the equilibrium composition, which is then used
with the equation of state to find improved f ’s; and so on. One continues until no
i
further change in composition is found. For an example, see H. F. Gibbard and M. R.
Emptage, J. Chem. Educ., 53, 218 (1976).
11.6 COMPUTER PROGRAMS FOR
EQUILIBRIUM CALCULATIONS
A natural-water system such as a lake or stream might contain one or two dozen
dissolved chemicals, which can react with one another to form hundreds of possible dis-
solved species or solid precipitates. To deal with such a complex system, a computer
program is essential. There are two common methods of computer solution of multiple-
equilibria problems at constant T and P. One approach uses equilibrium constants and
finds the species amounts that satisfy the equilibrium-constant expressions and the stoi-
chiometry (conservation of matter) requirements. An alternative approach writes G of
each phase as G n m , where each m is expressed as a function of composition. One
i i i i
then minimizes G of the system by varying the composition subject to the stoichiome-
try requirements (see W. R. Smith and R. W. Missen, Chemical Reaction Equilibrium
Analysis, Krieger, 1991, for details). Most of the following programs have a built-in
database of free-energy data and parameters for estimating ionic activity coefficients.