D21.1 Concentration Equilibrium Constants

A concentration equilibrium constant (Kc ) is a ratio of equilibrium concentrations of products and reactants that is constant for a given reaction at a given temperature. For example, consider this generic reaction:

m A + n B ⇌ x C + y D

For this reaction, the concentration equilibrium constant, Kc, is:

 K_c = \dfrac{[\text{C}]^{x}[\text{D}]^{y}}{[\text{A}]^{m}[\text{B}]^{n}}

This mathematical expression is called the equilibrium constant expression. Square brackets represent concentration in M (mol/L), so “[A]” represents the equilibrium concentration of reactant A. Each concentration is raised to the power of its coefficient. Sometimes, a subscript “e” (as in [A]e) can be used to explicitly indicate that the concentration correspond to concentration at equilibrium.

In general, Kc expressions include only substances whose concentrations change as a reaction proceeds. In other words, Kc expressions do not contain terms for pure solids or pure liquids. For example, consider the equilibrium constant expression for the following reaction:

2 HgO(s) ⇌ 2 Hg(ℓ) + O2(g)          Kc = [O2]

Because HgO is a solid, the number of HgO per unit volume depends only on the density of HgO at the temperature of the reaction. In other words, HgO(s) concentration is the same throughout the reaction, and does not change when HgO are consumed or produced. Similarly, the number of Hg atoms per unit volume of Hg(ℓ) is constant throughout the reaction. Thus, these concentrations are not included in the Kc expression. It is necessary for some HgO(s) and some Hg(ℓ) to be present for the equilibrium to be maintained, but the total quantity of each does not affect the equilibrium.

In addition, for dilute solutions, the concentration of the solvent remains constant throughout the reaction and is also not included in the Kc expression, even though the solvent sometimes appear in the reaction equation.

A homogeneous equilibrium is one in which all of the reactants and products are present in a single phase. Examples of homogeneous equilibria are reactions in the gas phase and reactions in liquid solutions. For example:

C2H2(g) + 2 Br2(g) ⇌ C2H2Br4(g)           K_c = \dfrac{[\text{C}_2\text{H}_2\text{Br}_4]}{[\text{C}_2\text{H}_2] [\text{Br}_2]^{2}}
I2(aq) + I⁻(aq) ⇌ I3⁻(aq)          K_c = \dfrac{[\text{I}_3^{-}]}{[\text{I}_2] [\text{I}^{-}]}
Hg22+(aq) + NO3⁻(aq) + 3 H3O+(aq) ⇌ 2 Hg2+(aq) + HNO2(aq) + 4H2O(ℓ)       K_c = \dfrac{[\text{Hg}^{2+}]^{2}[\text{HNO}_2]}{[\text{Hg}_2^{2+}] [\text{NO}_3^{-}] [\text{H}_3\text{O}^{+}]^{3}}
HF(aq) + H2O(ℓ) ⇌ H3O+(aq) + F⁻(aq)           K_c = \dfrac{[\text{H}_3\text{O}^{+}] [\text{F}^{-}]}{[\text{HF}]}
NH3(aq) + H2O(ℓ) ⇌ NH4+(aq) + OH⁻(aq)           K_c = \dfrac{[\text{NH}_4^{+}] [\text{OH}^{-}]}{[\text{NH}_3]}

In the aqueous equilibrium systems, H2O(l) is the solvent. Its concentration does not appear in the Kc expression.

A heterogeneous equilibrium is a system in which reactants and products are found in two or more phases. Some heterogeneous equilibria involve chemical changes, for example:

PbCl2(s) ⇌ Pb2+(aq) + 2Cl⁻(aq)          Kc = [Pb2+][Cl⁻]2
CaO(s) + CO2(g) ⇌ CaCO3(s)           K_c = \dfrac{1}{[\text{CO}_2]}
C(s) + 2S(g) ⇌ CS2(g)           K_c = \dfrac{[\text{CS}_2]}{[\text{S}]^{2}}

Other heterogeneous equilibria involve phase changes, for example:

Br2(ℓ) ⇌ Br2(g)          Kc = [Br2(g)]

Working with Kc

There are a few things to keep in mind when working with equilibrium constant expressions.

When all the coefficients in a balanced chemical equation are multiplied by some factor n, then the new Kc is the original Kc raised to the nth power. For example:

2 NO2(g) ⇌ N2O4(g)           K_{c,1} = \dfrac{[\text{N}_2\text{O}_4]}{[\text{NO}_2]^{2}}
NO2(g) ⇌ ½ N2O4(g)           K_{c,2} = \dfrac{[\text{N}_2\text{O}_4]^{1/2}}{[\text{NO}_2]} = (K_{c,1})^{1/2}

When a reaction’s direction is reversed, the Kc for the new reaction is the reciprocal (inverse) of the original reaction Kc. For example:

A + 2B ⇌ AB2           K_{c,1} = \dfrac{[\text{AB}_2]}{[\text{A}] [\text{B}]^{2}}
AB2 ⇌ A + 2B           K_{c,2} = \dfrac{[\text{A}] [\text{B}]^{2}}{[\text{AB}_2]} = \dfrac{1}{K_{c,1}}

When two reactions occur sequentially to yield a new overall reaction, the Kc for the overall reaction is the product of the Kc values for the individual steps. For example:

A + C ⇌ AC           K_{c,1} = \dfrac{[\text{AC}]}{[\text{A}] [\text{C}]}
AC + C ⇌ AC2            K_{c,2} = \dfrac{[\text{AC}_2]}{[\text{AC}] [\text{C}]}
A + 2 C ⇌ AC2           K_{c,3} = \dfrac{[\text{AC}_2]}{[\text{A}] [\text{C}]^{2}} = K_{c,1}\times K_{c,2}

Exercise: Properties of Chemical Equilibrium

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