What is Equilibrium - The Math

Our definition of equilibrium is simple: Equilibrium is when the rate of the forward reaction is equal to the rate of the backward reaction. Since rate has a mathematical statement (the rate law) then there must also be a mathematical way of describing equilibrium.

Let’s look at a sample, generic, reaction:

A + 2 B ⇄ C + 3 D

For the sake of this discussion, we’ll assume that both the forward and backward reactions are single step reactions and that the exponents in the rate law match the coefficients in the overall reaction. This is, of course, a horrible simplification, but it works mathematically in a way that will be shown later in the unit.

With that assumption, we can write the two rate laws like this:

where k_f is the forward rate constant and k_b is the backward rate constant. If the system is at equilibrium, we can say this:

It is important to remember that this math is ONLY true when the system is at equilibrium. That means that concentrations in the equation are NOT independent. They must be concentrations that give forward and backward rates that are the same.

Rearranging the equation above (putting the constants on the left and the concentrations on the right) give us this:

Since a constant divided by a constant is a constant, we can simplify further to this:

Where Keq is called the equilibrium constant and the fraction on the right is called either the "Equilibrium Expression" or the "Reaction Quotient (Q)".

So, our mathematical definition of equilibrium is simply:

A system is at equilibrium when the math above is true.

A few things to note:

1. Each reaction has its own equilibrium constant

2. Chemists pretend that equilibrium constants don't have units

3. Equilibrium constants change with temperature (but ONLY with temperature)

4. There are multiple combinations of concentrations that can be at equilibrium for the same reaction

That last one may need an example. Let’s say that (for the reaction above) the value of Keq is 100.

The system could be at equilibrium with [C] = 100. M, and [A]=[B]=[D] = 1.00 M. It could also be at equilibrium with [C]=1.00 M, [D] = 10.0 M, [A] = 10.0 M and [B] = 1.00 M. (Try the math if you don’t see it right away. 

The reason that this fact is so important is simple. Imagine that C is red, and that A is blue (and everything else is colorless). In the first example [C]=100M and [A]=1.00 M the solution will look red (the blue will be overwhelmed). In the second example, the solution will look purple (equal concentration of red C and blue A). The system will look VERY different, but both will be at equilibrium.