What Direction Will a System Shift in Order to Reach Equilibrium

If a system is not at equilibrium, then it must either be running forward faster than backwards or backwards faster than forward. Determining which of these is a simple matter. 

We’ll use the following problem to explore this: 

The system A + 2 B ⇄ C + D has a K = 4.12. If [A] = 1.00M and all of the other concentrations are 4.12 M, the system is not at equilibrium. What direction must the system shift in order to reach equilibrium?

We know that at equilibrium, the following math must be true:

If we plug in the current values for the expression Q, we get the following:

Which is definitely NOT 4.12. 

So, which direction will it go? There are two different ways to find the answer. The first requires some mathematical logic, the other requires only careful set-up.

The Logical Way

Right now, we know that we are not at equilibrium:

But we can work out what needs to happen, again using this equation:

we know that the right side of this equation is currently too small (it's less than the value of K). In order for the fraction to be larger, the numerator of the fraction must get larger (or the denominator must get smaller...the logic works either way). The top of the fraction is comprised of the products of the reaction. So, to reach equilibrium we need more products, which means that the reaction must run to the right.

The Simple Way

We know that since K = 4.12 and Q = 1.00, that K > Q. Now all we have to do is turn the ">" sign into an arrow →. That arrow will always point in the direction that the reaction is shifting. So, in those situations where K>Q, the reaction will shift to the right (→). When K<Q, the reaction will shift to the left (←).

A word of warning

The simple way ONLY works if you put K on the left and Q on the right. If you reverse their order, you will always be exactly wrong.

A Note About Zeros

Lots of equilibrium problems start with one or more compounds missing - that is with a concentration of 0 M. That zero can mess with your math.  For instance, if you reached a point where the math looked like this:

Your math instructor would tell you that the problem is unsolvable, since dividing by zero is undefined. 

However, this is not a math class. We are not trying to solve the equation, merely to determine the direction that the system will shift. That means we can logic our way out of the mathematical trouble. 

If a system is missing one or more products, then the reaction CANNOT go backwards. (If the products don't collide, then the backward reaction can't occur.) In the same way, if the system is missing one or more reactants, it CANNOT run forward. 

That leaves us with a simple rule:

A system will ALWAYS shift toward the side that has a 0 concentration.