Squares and Cubes in Equilibrium Problems

Let’s look at an equilibrium problem:

At a certain temperature, the reaction H2 (g) + Cl2 (g) → 2 HCl(g) has a K=426. If a 4.00 L container holds 2.00 moles of hydrogen, 2.00 moles of chlorine and 1.00 mole of hydrogen chloride, what will the concentrations be when equilibrium is reached.

The equilibrium expression for this system is:

You will need to first determine what direction this reaction is shifting and then to create an ICE table. Here is the finished ICE table, if you don’t understand where it came from, go back and review those sections of the text.

We can then substitute into the equilibrium expression, which will give us this:

This equation is a quadratic (x2 is the highest power of x in the equation). You could, therefore, solve this with the quadratic equation. That process will absolutely work, but it involves a fair amount of algebra to get the equilibrium expression into a form that will allow you to use the quadratic equation.

There is, for us, an easier way. If we look carefully at the bottom of the equation, we can see a way to simplify the formula like this:

Now we can see that the right side is a perfect square. We can take the square root of both sides, leaving us this formula:

Now we can easily solve the algebra equation

Of course we aren't QUITE done yet. The question doesn't ask for x, it asks for the final concentrations of all species in the system. So, now we need to take our value of x (0.445) and plug it back into the "E" row of the ice table, like this:

This process (taking the root of both sides) works for any reaction where the equilibrium expression (the fraction) is a perfect square or a perfect cube. These problems aren't terribly common in the real world, but they represent a notable percentage of the problems you might find on a chem test.

A Warning

Be aware that this method will NOT work in math class. That’s because we are losing a root (we got one answer rather than the two answers that the quadratics would give us). But in chemistry, only one root is ever valid (in this case the negative root would be nonsensical) and our simplification loses the meaningless root.