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EXAMPLE 6.5 Equilibrium composition at fixed T and V Section 6.4
Ideal-Gas Equilibrium Calculations
K° 6.51 at 800 K for the ideal-gas reaction 2A B ∆ C D. If 3.000 mol
P
3
of A, 1.000 mol of B, and 4.000 mol of C are placed in an 8000-cm vessel at
800 K, find the equilibrium amounts of all species.
Proceeding to step 3 of the preceding scheme, we suppose that x moles of B
react to reach equilibrium. Then at equilibrium,

n 11 x2 mol, n 13 2x2 mol, n 14 x2 mol, n x mol
D
A
C
B
The reaction is run at constant T and V. Using P n RT/V according to step 4(b)
i i
and substituting into K°, we get
P
1P >P°21P >P°2 1n RT>V21n RT>V2P° n n VP°
C
C
D
D
C D
K°
P
2
2
2
1P >P°2 1P >P°2 1n RT>V2 1n RT>V2 n n RT
A
A B
B
B
A
where P° 1 bar. Use of 1 atm 760 torr, 1 bar 750.06 torr, and R
3
1
3
82.06 cm atm mol 1 K 1 gives R 83.14 cm bar mol 1 K . Substitution
for the n ’s gives
i
3
14 x2x mol 2 8000 cm bar
6.51
1
3
3
2
1
13 2x2 11 x2 mol 183.14 cm bar mol K 21800 K2
2
3
x 3.995x 5.269x 2.250 0 (6.43)
3
where we divided by the coefficient of x . We have a cubic equation to solve. The
formula for the roots of a cubic equation is quite complicated. Moreover, equa-
tions of degree higher than quartic often arise in equilibrium calculations, and
there is no formula for the roots of such equations. Hence we shall solve (6.43)
by trial and error. The requirements n 0 and n 0 show that 0 x 1. For
B D
x 0, the left side of (6.43) equals 2.250; for x 1, the left side equals 0.024.
Hence x is much closer to 1 than to 0. Guessing x 0.9, we get 0.015 for the
left side. Hence the root is between 0.9 and 1.0. Interpolation gives an estimate
of x 0.94. For x 0.94, the left side equals 0.003, so we are still a bit high.
Trying x 0.93, we get 0.001 for the left side. Hence the root is 0.93 (to two
places). The equilibrium amounts are then n 1.14 mol, n 0.07 mol, n
A B C
4.93 mol, and n 0.93 mol.
D
Exercise
K° 3.33 at 400 K for the ideal-gas reaction 2R 2S ∆ V W. If 0.400 mol
P
of R and 0.400 mol of S are placed in an empty 5.000-L vessel at 400 K, find the
equilibrium amounts of all species. (Hint: To avoid solving a quartic equation,
take the square root of both sides of the equation.) (Answer: 0.109 mol of R,
0.109 mol of S, 0.145 mol of V, 0.145 mol of W.)
Some electronic calculators can automatically find the roots of an equation. Use
of such a calculator gives the roots of Eq. (6.43) as x 0.9317. . . and two imaginary
numbers.
Examples 6.4 and 6.5 used general procedures applicable to all ideal-gas equi-
librium calculations. Example 6.6 considers a special kind of ideal-gas reaction:
isomerization.