Equilibrium and using ICE tables

Question

For the reaction below, Kc = 4.60 × 10⁻⁶. Note Kc is sometimes called K. What is the equilibrium concentration of B if the reaction begins with 0.700 M A? A (g) ⇌ 3 B (g)

Answer 1

When finding equilibrium concentrations using an equilibrium constant and initial concentrations, it's often helpful to write out the reaction and construct an ICE chart, in which we write out the initial concentrations, changes in concentrations, and the concentrations at equilibrium. This is useful because our equilibrium constant K involves the last part - concentrations at equilibrium.

A <-> 3B

Initial 0.7 M 0

Change - x +3x

Equilibrium 0.7-x 3x

This was constructed using the fact that we start with 0.7 M of only A (so zero of B). If we lose -x moles of A, then using the mole ratios from the coefficients we must then gain 3 times as much B, so +3x. The equilibrium concentrations are then the initial amounts plus the change.

We are given that our equilibrium constant K_c=4.60*10^-6. K_c is also calculated using the concentrations of our reactants and products at equilibrium, where the coefficients in the reaction show up as exponents:

K_c = [B]^3/[A] We plug in the equilibrium concentrations from our ICE chart above to get:

K_c = (3x)^3 / (0.7-x) = 4.60*10^-6

27x^3 / (0.7-x) = 4.60*10^-6

Multiplying the 0.7-x over to the right side:

27x^3 = 4.60*10^-6 * (0.7-x)

We need to note something here. This equation involves a cubic, which makes it more difficult to solve for x. We can't use the quadratic equation as we would do in the case of x^2.

Some calculators will let you write out both sides of the equation as separate functions and solve for where they overlap. You can also graph each side as separate functions and see where they overlap.

If those aren't an option, often the best approach is to take a shortcut and assume that our change in concentration x is very small. This is reasonable if our equilibrium constant K is very small (in this case 10^-6) because small equilibrium constants mean that very little product will actually be formed and most of the solution will remain as reactant.

If x is very small, then instead of writing 0.7 - x in our equation we can simply write 0.7 (because if x is small, subtracting it from 0.7 will still give a result VERY close to 0.7)

Now our equation looks like:

27x^3 = 4.60*10^-6 * (0.7)

27x^3 = 3.22*10^-6

x^3 = 1.19*10^-7

x = 0.0049 M

Note that x is indeed very small compared to 0.7. If we divide x by 0.7 we get:

0.0049 M / 0.7 M = 0.007

which corresponds to 0.7%. The "5% Rule" says that the shortcut works fairly well so long as x is no more than 5% of the starting concentration.

If x = 0.0049 M, then going back to our ICE chart we find that the equilibrium concentration of B will be:

3x = 3*0.0049 M = 0.0147 M

And if we needed to find A, we would do 0.7-x.