The first step is to determine the equilibrium constant for the reaction.
CO+Cl2⇌COCl2
Keq=[COCl2][CO][Cl2]
Keq=0.4000.100×0.500=8.00
Now we can set up an ICE table to calculate the new concentrations.
1CO + 1Cl2 ⇌1COCl2
I/mol⋅L−1 0.400 0.500 0.400
C/mol⋅L−1 −x 1−x +x
E/mol⋅L−1 0.400−x 0.500−x 0.400+x
Keq=[COCl2][CO][Cl2]=0.400+x(0.400−x)(0.500−x)=8.00
We can't assume that x≪0.400, so we must solve a quadratic equation.
0.400+x=8.00(0.400+x)(0.500−x)=8.00(0.200−0.900x+x2)=1.60−7.20+8x2
8x2−8.20x+1.20=0
x2−1.025x+0.150=0
x=−b±√b2−4ac2a=1.025±√1.0252−4×1×0.1502×1=1.025−√1.051−0.6002=1.025±√0.4512=1.025±0.6712
x=0.177 or x=0.848
Since x cannot be greater than 0.400, x=0.177
The new concentrations are
[COCl2]=(0.400+x)mol/L=(0.400+0.177)mol/L=0.577 mol/L
[CO]=(0.400−x)mol/L=(0.400−0.177)mol/L=0.223 mol/L
[Cl2]=(0.500−x)mol/L=(0.500−0.177)mol/L=0.323 mol/L
Check:
0.5770.223×0.323=8.01.
Close enough!
