Calculating the Equilibrium Constant

The equilibrium constant for a reaction is calculated from the equilibrium concentrations (or pressures) of its reactants and products. If these concentrations are known, the calculation simply involves their substitution into the K_c expression.

For example, gaseous nitrogen dioxide forms dinitrogen tetroxide according to this equation:

When 0.10 mol NO₂ is added to a 1.0-L flask at 25 °C, the concentration changes so that at equilibrium, [NO₂] = 0.016 M and [N₂O₄] = 0.042 M. The value of the equilibrium constant for the reaction can be calculated as follows:

A slightly more challenging example is provided next, in which the reaction stoichiometry is used to derive equilibrium concentrations from the information provided. The basic strategy of this computation is helpful for many types of equilibrium computations and relies on the use of terms for the reactant and product concentrations initially present, for how they change as the reaction proceeds, and for what they are when the system reaches equilibrium. The acronym ICE is commonly used to refer to this mathematical approach, and the concentration terms are usually gathered in a tabular format called an ICE table.

Calculation of an Equilibrium Constant

Iodine molecules react reversibly with iodide ions to produce triiodide ions.

If a solution with the concentrations of I₂ and I^? both equal to 1.000 × 10^?3 M before reaction gives an equilibrium concentration of I₂ of 6.61 × 10^?4 M, what is the equilibrium constant for the reaction?

To calculate the equilibrium constants, equilibrium concentrations are needed for all the reactants and products:

The initial concentrations of the reactants and the equilibrium concentration of the product are provided. This information can be used to derive terms for the equilibrium concentrations of the reactants, presenting all the information in an ICE table.

At equilibrium the concentration of I₂ is 6.61 × 10^?4 M so that

The ICE table may now be updated with numerical values for all its concentrations:

Finally, the equilibrium concentrations can be substituted into the K_c expression and solved:

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

The equilibrium constant, K_c, can be determined by substituting the corresponding values into the equilibrium constant expression if the concentrations of all the reactants and products at equilibrium are known.

A gaseous mixture of sulfur dioxide and oxygen at 530 °C is allowed to react according to the reaction shown. At equilibrium, the mixture contains 0.10 molar of sulfur dioxide, 0.15 molar of oxygen, and 10.88 molar of sulfur trioxide. 

By substituting the values into the equilibrium expression, the K_c equals 7.9 × 10⁴. 

The K_c can also be calculated as long as the initial concentration of all of the components and the equilibrium concentration of at least one compound is known.

The unknown equilibrium concentrations can then be calculated using the reaction stoichiometry. An ICE table is used to organize the information for the initial, change, and equilibrium concentrations of the reaction.

When a reaction mixture containing 0.11 molar nitrogen and 0.36 molar hydrogen is allowed to reach equilibrium at 500 °C, it produces 0.020 molar ammonia at equilibrium. To calculate the K_c, the equilibrium concentrations of the nitrogen and hydrogen need to be determined.

The stoichiometry of the reaction shows that 1 mole of nitrogen gas and 3 moles of hydrogen gas are required to produce 2 moles of ammonia gas.

The change, x, when multiplied by the coefficients of the reactants and products, denotes the concentration of the reactants consumed and the concentration of the product produced to reach equilibrium. 

Since 2x equals 0.020, x equals 0.010. The equilibrium concentration of nitrogen and hydrogen can then be determined by subtracting the respective concentration change from their initial concentration which equals 0.10 and 0.33 molar, respectively.

Substituting equilibrium concentrations in the K_c expression, the K_c equals 0.11.

The K_p for reactions involving gases can be calculated using an ICE table and the equilibrium expression written with partial pressures.