14.6: Calculating Equilibrium Concentrations

Being able to calculate equilibrium concentrations is essential to many areas of science and technology—for example, in the formulation and dosing of pharmaceutical products. After a drug is ingested or injected, it is typically involved in several chemical equilibria that affect its ultimate concentration in the body system of interest. Knowledge of the quantitative aspects of these equilibria is required to compute a dosage amount that will solicit the desired therapeutic effect.

A more challenging type of equilibrium calculation can be one in which equilibrium concentrations are derived from initial concentrations and an equilibrium constant. For these calculations, a four-step approach is typically useful:

1. Identify the direction in which the reaction will proceed to reach equilibrium.
2. Develop an ICE table.
3. Calculate the concentration changes and, subsequently, the equilibrium concentrations.
4. Confirm the calculated equilibrium concentrations.

Calculation of Equilibrium Concentrations

Under certain conditions, the equilibrium constant K_c for the decomposition of PCl₅(g) into PCl₃(g) and Cl₂(g) is 0.0211. The above procedure can be used to determine the equilibrium concentrations of PCl₅, PCl₃, and Cl₂ in a mixture that initially contained only PCl₅ at a concentration of 1.00 M.

Step 1. Determine the direction the reaction proceeds.

The balanced equation for the decomposition of PCl₅ is

Because only the reactant is present initially, Q_c = 0, and the reaction will proceed to the right.

Step 2. Develop an ICE table.

Step 3. Solve for the change and the equilibrium concentrations.

Substituting the equilibrium concentrations into the equilibrium constant equation gives

An equation of the form ax² + bx + c = 0 can be rearranged to solve for x:

In this case, a = 1, b = 0.0211, and c = −0.0211. Substituting the appropriate values for a, b, and c yields:

The two roots of the quadratic are, therefore,

For this scenario, only the positive root is physically meaningful (concentrations are either zero or positive), and so x = 0.135 M. The equilibrium concentrations are

Step 4. Confirm the calculated equilibrium concentrations.

Substitution into the expression for K_c (to check the calculation) gives

The equilibrium constant calculated from the equilibrium concentrations is equal to the value of K_c given in the problem (when rounded to the proper number of significant figures).

This text has been adapted from Openstax, Chemistry 2e, Section 13.4 Equilibrium Calculations.

Unknown equilibrium concentrations can be determined from the initial concentration of the reactants and the equilibrium constant, K, for the reaction.

Consider the reaction where 0.30 molar nitrogen gas and 0.40 molar oxygen gas react to produce nitric oxide gas and where K is 0.10. To calculate the equilibrium concentrations, the known values are tabulated in an ICE table.

For the change in concentration, the increase or decrease for each product or reactant, respectively, is denoted by x times its stoichiometric coefficient. The sum or difference is used to find the equilibrium concentrations, which are then substituted into the equilibrium expression.

To solve for x, the expression is expanded and all the terms are put on one side to convert it into the form: ax² + bx + c. This equation can be solved using the quadratic formula.

Solving results in two values for x: 0.047 and −0.065. As the negative concentration of a substance is impossible, the value can be rejected.

Using 0.047 for x, the equilibrium concentrations of nitrogen, oxygen, and nitric oxide equal 0.25, 0.35, and 0.094 molar, respectively.

The perfect square condition is one situation where a shortcut can be used to avoid the quadratic formula.

For example, if the initial concentrations of nitrogen and oxygen in the above reaction were 0.30 molar each, the equation becomes a perfect square. In such cases, the equation can be simplified by taking the square root of both sides to solve for x.