A mixture of 0.5540 M CO and 0.3810 M Cl, is enclosed in a vessel and heated to 1000 K.

Question

A mixture of 0.5540 M CO and 0.3810 M Cl, is enclosed in a vessel and heated to 1000 K.

CO(g) + Cl2 (g) - COCI2 (g)

K. = 255.0 at 1000 K

Calculate the equilibrium concentration of each gas at 1000 K.

CO= __M

Cl2 = __ M

COCl2 = __M

Answer 1

It may be easiest to set up an ICE table.

CO(g) + Cl₂(g) ==> COCl₂(g)

0.5540...0.3810..........0................Initial

-x............-x................+x...............Change

0.5540-x...0.3810-x....x................Equilibrium

So, @ equilibrium, we have 0.5540-x M of CO, 0.3810-x M of Cl₂ and x M of COCl₂. Now, we use the K that is provided and solve for x:

K = [COCl₂] / [CO][Cl₂]

255 = x / (0.5540-x)(0.3810-x)

255 = x / x² - 0.9350x + 0.2111

x = 255x² - 238.4x + 53.83

255x² - 239.4x + 53.83 = 0

x = 0.5660 M and 0.3730 M

Eliminate 0.566 as it exceeds the initial concentrations. Thus, we take x = 0.3730 M

Go back to the ICE table and plug in this value of x to solve for equilibrium concentrations:

[CO] = 0.5540 - 0.3730 = 0.1810 M

[Cl₂] = 0.3810 - 0.3730 = 0.0080 M

[COCl₂] = x = 0.3730 M

Answer 2

We are given that the starting concentrations of CO and Cl₂ are

[CO]₀ = 0.5540 M

[Cl₂]₀ = 0.3810 M

and that the equilibrium constant for the reaction at 1000 K is K_eq = 255.0

So we start with

K_eq = [COCl₂]/([CO][Cl₂]) = 255.0

and our goal is to find the equilibrium concentrations of [CO], [Cl₂], and [COCl₂]

If we let x = [COCl₂] then we know that the concentrations of [CO] and [Cl₂] can be expressed as:

[CO] = 0.5540 – x and [Cl₂] = 0.3810 – x

In both cases the concentration is the initial concentration minus the amount that reacted to form the product.

So K_eq = x/[(0.5540 – x)(0.3810 – x)] = 255

Multiply both sides by (0.5540 – x)(0.3810 – x) to get

x = 255(0.5540 – x)(0.3810 – x) = 255(0.211074 – 0.935x + x²) which leads to

x = 53.82387 – 238.425x + 255x² or 255x² – 238.425x + 53.82387 = x

Subtract x from both sides gives the quadratic equation we need to solve for x:

255x² – 239.425x + 53.82387 = 0

The two solutions to the quadratic equation are

x = 0.5660 and x = 0.3729236

We throw out the solution x = 0.5660 because x = [COCl₂] can't be greater than the initial concentration of either of the reactants that it is formed from.

So our concentrations are

[COCl₂] = 0.3729 M

[CO] = 0.5540 – 0.3729 = 0.1811 M

[Cl₂] = 0.3810 – 0.3729236 = 0.008076 M

It's a good idea to check whether these concentrations

are consistent with the equilibrium constant K_eq = 255.0

K_eq = [COCl₂]/([CO][Cl₂]) = (0.3729)/[(0.1811)(0.008076)] = 254.96 which rounds off to 255.0

And we're grateful (and relieved!) to see that they are.

Answers

[CO] = 0.1811 M

[Cl₂] = 0.0081 M

[COCl₂] = 0.3729 M

A note about significant figures:

Since each number of our data is given in four significant figures, I initially gave all final concentrations in four significant figures. This is correct for the [CO] and [COCl₂] final concentrations, but the [Cl₂] should be rounded to [Cl₂] = 0.0081 M since it is derived by subtracting from a number that has 4 digits after the decimal point.]