Equilibrium Concentrations

In Calculating Equilibrium Constant, we learned how you can use initial, equilibrium, and change in concentration to calculate the equilibrium constant for a particular reaction. But this also works the other way. If we know both initial concentrations and the equilibrium constant, we can calculate the equilibrium concentration of each species involved in the reaction.

• This article is about calculating equilibrium concentrations in chemistry.
• We'll start by introducing you to the formulae and units you'll use when calculating equilibrium concentrations.
• We'll then look at how you can use initial concentrations and the equilibrium constant to work out equilibrium concentrations of species in a reversible reaction.
• This will involve forming and solving a quadratic equation.

• You'll be able to practice your skills with the help of our worked examples.

Equilibrium concentrations formulae and units

To calculate equilibrium concentrations, we need to know two things:

• The value of the equilibrium constant K_c for this particular reaction.

• The formula for the equilibrium constant K_c.

• The initial concentration of each of the species involved in the expression for K_c. In a homogeneous reaction, this means all of the species in the reaction.

• It might be helpful to know the quadratic formula too.

The formula for the equilibrium constant K_c can be derived from the reaction equation. For example, take the reaction \(aA(g)+bB(g) \rightleftharpoons cC(g)+dD(g)\) . To find K_c, we'd use the following expression:

$$K_c=\frac{{[C]_{eqm}}^c\space {[D]_{eqm}}^d}{{[A]_{eqm}}^a\space {[B]_{eqm}}^b}\qquad K_p=\frac{{{(P_C)}_{eqm}}^c\space {{(P_D)}_{eqm}}^d}{{{(P_A)}_{eqm}}^a\space {{(P_B)}_{eqm}}^b}$$

Calculating equilibrium concentrations also involves solving a quadratic equation. Most scientific calculators can solve these for you, but you might instead want to use the quadratic formula. For the equation \(ax^2+bx+c=0\) , the quadratic formula looks a little something like this:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

It is typical for concentration to be expressed in M, which means moles per liter. This is equivalent to moles per decimeter cubed, or mol dm^-3. Don't be surprised if you see it written in this alternative form!

Calculating equilibrium concentrations from initial concentrations

Now that we've discussed the formulae and units used in calculating equilibrium concentrations, we're ready to move on to the method. But before we go any further, let's remind ourselves how you calculate the equilibrium constant from initial, equilibrium, and change in concentration. This will help us understand how we can adapt the process to work out equilibrium concentration from the initial concentration and the equilibrium constant:

1. Use the initial and equilibrium concentration of a species that you do know to calculate its change in concentration.
2. Use the balanced chemical equation to calculate the change in concentration and thus the equilibrium concentration of the remaining species involved in the reaction.
3. Substitute the equilibrium concentration values into the expression for the equilibrium constant to get your final answer.

So, how does this differ from calculating equilibrium concentrations? In the case of the latter, we don't know the equilibrium concentration or the change in concentration of any of the species- we only know their initial concentrations. However, we can represent the change in concentration of each of the species involved using multiples of an arbitrary letter, such as x. We then show the equilibrium concentration using the initial concentration and the change in x.

When we substitute the equilibrium concentrations into the formula for the equilibrium constant K_c, we end up with a quadratic equation. We can solve this to find x. Remember that x represents the change in concentration - and so we can therefore calculate the equilibrium concentration of each species using their initial concentrations and our value for x.

The process can seem a little tricky, but once you've got your head around it, it isn't too complicated. We'll tell you the steps and then walk through them with two worked examples:

1. Create a table with rows for the initial, change in, and equilibrium concentrations of the reactants and products. Fill in the initial concentrations of all of the species involved.
2. Use the balanced chemical equation to represent the change in concentration of each species in terms of x.
3. Write the equilibrium concentration of each species using its initial concentration and its change in x.
4. Substitute the equilibrium concentrations into the expression for K_c. You should end up with a fraction containing factors of x on one side, and a constant on the other.
5. Multiply the denominator of the fraction through both sides of the equation and rearrange to find a quadratic equation.
6. Solve the equation using your calculator or the quadratic formula to find two values for x. Discard the inappropriate value.
7. Go back to your table of initial, change in, and equilibrium concentrations of the reactants and products, and use your value of x to calculate the equilibrium concentration of each species.

Ready to give it a go? Here are two problems for you to try your hand at.

Calculating equilibrium concentrations examples

Hopefully, the process makes a little more sense now. Here's another example for you to try.

Calculating Equilibrium Partial Pressures

The same method can also be used to calculate equilibrium partial pressures. Instead of representing a change in concentration, the letter x simply represents a change in partial pressure and is measured in atm.

That's it for this article. By now, you should understand how to calculate equilibrium concentrations using initial concentrations and the equilibrium constant K_c. This involves representing the change in concentration in terms of x, and forming and solving a quadratic equation.