Page 209 - Physical Chemistry

lev38627_ch06.qxd 3/3/08 10:07 AM Page 190

190
Chapter 6 EXAMPLE 6.6 Equilibrium composition in isomerization
Reaction Equilibrium in Ideal Gas
Mixtures
Suppose the gas-phase isomerization reactions A ∆ B, A ∆ C, and B ∆ C
reach equilibrium at a fixed T. Express the equilibrium mole fractions of A, B,
and C in terms of equilibrium constants.
Let K denote K° for A ∆ B, and let K denote K° for A ∆ C. We have
B/A P C/A P
P >P° x P>P° x B x C
B
B
K B>A and K C>A (6.44)
P >P° x P>P° x A x A
A
A
The sum of the mole fractions is 1, and use of (6.44) gives
x x x 1
C
B
A
x x K B>A x K C>A 1
A
A
A
1
x (6.45)
A
1 K B>A K C>A
From x K B/A A C C/A A
x , we get
x and x K
B
K B>A K C>A
x and x (6.46)
B
C
1 K B>A K C>A 1 K B>A K C>A
Using these equations, one finds (Prob. 6.35) the equilibrium mole fractions
in a gas-phase mixture (assumed ideal) of pentane, isopentane, and neopentane
Figure 6.7 at various temperatures to be as shown in Fig. 6.7.
Mole fractions versus T in a Exercise
gas-phase equilibrium mixture A 300-K gas-phase equilibrium mixture of the isomers A, B, and C contains 0.16
of the three isomers of pentane
(n-pentane, isopentane, and mol of A, 0.24 mol of B, and 0.72 mol of C. Find K B/A and K C/A at 300 K.
neopentane). (Answers: 1.5, 4.5.)
Since the standard pressure P° appears in the definition of K°, the 1982 change of
P
P° from 1 atm to 1 bar affects K° values slightly. See Prob. 6.39.
P
When G° is large, K° is very large or very small. For example, if G° 137
P 298
kJ/mol, then K° 10 24 . From this value of K°, we might well calculate that at
P,298 P
equilibrium only a few molecules or even only a fraction of one molecule of a prod-
uct is present. When the number of molecules of a species is small, thermodynamics
is not rigorously applicable and the system shows continual fluctuations about the
thermodynamically predicted number of molecules (Sec. 3.7).
Tables of data often list H°and G°values to 0.01 kJ/mol. However, experi-
f f 1
mental errors in measured H°values typically run to 2 kJ/mol, although they may
f 2
be substantially smaller or larger. An error in G° of 2 kJ/mol corresponds to a fac-
298
tor of 2 in K°. Thus, the reader should take equilibrium constants calculated from
P
thermodynamic data with a grain of NaCl(s).
In the preceding examples, the equilibrium composition for a given set of condi-
tions (constant T and V or constant T and P) was calculated from K° and the initial
P
composition. For a system reaching equilibrium while T and P are constant, the
equilibrium position corresponds to the minimum in the system’s Gibbs energy G.
Figure 6.8 plots the conventional values (Chapter 5) of G, H, and TS (where G
H TS) versus extent of reaction for the ideal-gas reaction N 3H ∆ 2NH run at
2 2 3
the fixed T and P of 500 K and 4 bar with initial composition 1 mol N and 3 mol H .
2 2
At equilibrium, j 0.38. The plot is made using the fact that each of G, H, and S of
eq
the ideal gas mixture is the sum of contributions from each pure gas (Sec. 6.1). See
Prob. 6.61 for details.