aBetterChemText: Solving Little x Problems

Let’s look at a specific example:

The system 2 R + D ⇄ 2 T has a Keq = 5.61x10-17. If the [R] = 2.00 M, [D] = 1.50 M and [T] = 0, what will the concentrations be when the system reaches equilibrium?

The equilibrium expression for this system will be:

and the ICE table will look like this:

Substituting in will give us:

If you are feeling the temptation to solve this algebraically, I strongly urge you to rethink. This is a cubic equation (highest exponent is x3). There is a cubic equation (like the quadratic equation) that could give you the three roots of the equation, but to use the cubic equation, you’d have to do a LOT of algebra just to get the equation ready. Then, of course, the cubic equation involves imaginary numbers.

There is a much better way, although it might make your math teacher pull their hair out. We’re going to do significant figures BEFORE we solve the equation.

Looking at this equation, we can determine that x must be small. If you need to review the reasoning, look here. If x is small, then it is insignificant when added to, or subtracted from a given concentration.

In this case those situations occur in the denominator. What this means for us is this:

2.00-2x ≈ 2.00 and 1.50 - x ≈ 1.50

That makes our equilibrium expression just:

Multiplying both sides by 22 and 1.5 give us:

Finally, solving the equation gives x = 2.90x10-9.

Substituting that back into the ICE table gives us the final concentrations:

Let's take a quick look at one more example:

The system L + T ⇄ 3 A has a Keq = 1.88x1026. If the [A] = 2.15 M, the [L] = 0.55 M and the [T] = 0, what will the concentration be when the system reaches equilibrium.

The equilibrium expression will be

and the ICE table will look like this:

Our equilibrium expression now looks like this:

We know that x will be small (check the reasoning here if you aren’t sure why). So, since x is insignificant when added to or subtracted from a given concentration, that means that

0.55 + x ≈ 0.55 and 2.15 - 3x ≈ 2.15

so the equilibrium expression simplifies to: