The Equilibrium Constant Expression: Kc
The equilibrium constant Kc is the ratio of the product concentrations to the reactant concentrations, each raised to the power of its coefficient: for aA + bB ⇌ cC + dD, Kc = [C]c[D]d / ([A]a[B]b). In brief: for N2 + 3H2 ⇌ 2NH3, Kc = [NH3]2/([N2][H2]3), and equilibrium concentrations of 0.20, 0.50 and 0.30 M give Kc = 3.0.
The equilibrium constant is a single number that sums up where an equilibrium lies at a given temperature. It lets chemists predict the composition of an equilibrium mixture and decide whether a reaction is worth carrying out. This article explains how to write the expression, what the size of K means, how K changes when the equation is modified and how to calculate K from data.
How do you write the Kc expression?
For the general reaction aA + bB ⇌ cC + dD at equilibrium:
Kc = [C]c [D]d / ([A]a [B]b)
- Write the balanced equation.
- Put the products in the numerator and the reactants in the denominator.
- Raise each concentration (in mol/L) to the power of its coefficient.
- Leave out pure solids and pure liquids, and the solvent, since their concentrations are constant (see the article on heterogeneous equilibria).
The subscript c shows that concentrations are used. The value of Kc is the same for any set of equilibrium concentrations at a fixed temperature, whatever the starting amounts. Formally, K is dimensionless, since concentrations are divided by the standard concentration of 1 mol/L, and units are usually not written.
What does the size of K tell you?
| Value of K | Equilibrium mixture | Description |
|---|---|---|
| K > 103 | Mostly products | Reaction goes (nearly) to completion |
| 10−3 < K < 103 | Significant amounts of both | Neither side is strongly favored |
| K < 10−3 | Mostly reactants | Hardly any reaction |
K says nothing about the speed. A reaction with a large K can be extremely slow, as in the reaction of diamond with oxygen at room temperature.
How does K change when the equation changes?
| Change to the equation | New equilibrium constant | Example (K = 54.3 for H2 + I2 ⇌ 2HI) |
|---|---|---|
| Reverse the reaction | 1/K | 2HI ⇌ H2 + I2: K = 1/54.3 = 0.0184 |
| Multiply coefficients by n | Kn | ½H2 + ½I2 ⇌ HI: K = (54.3)1/2 = 7.37 |
| Add two reactions | K1 × K2 | Overall K is the product |
Worked examples
Example 1: writing the expression
- 2SO2(g) + O2(g) ⇌ 2SO3(g): Kc = [SO3]2 / ([SO2]2[O2]).
- CH4(g) + 2O2(g) ⇌ CO2(g) + 2H2O(g): Kc = [CO2][H2O]2 / ([CH4][O2]2).
- PCl5(g) ⇌ PCl3(g) + Cl2(g): Kc = [PCl3][Cl2] / [PCl5].
Example 2: calculating Kc from concentrations
At equilibrium for N2 + 3H2 ⇌ 2NH3, [N2] = 0.50 M, [H2] = 0.30 M and [NH3] = 0.20 M. Then Kc = (0.20)2 / (0.50 × 0.303) = 0.040 / (0.50 × 0.027) = 0.040 / 0.0135 = 3.0.
Example 3: hydrogen iodide
At 430 °C the equilibrium concentrations for H2 + I2 ⇌ 2HI are [H2] = [I2] = 0.0222 M and [HI] = 0.163 M. Kc = (0.163)2 / (0.0222 × 0.0222) = 0.02657 / 4.928 × 10−4 = 53.9, which is close to the accepted value of 54.3.
Example 4: using K to find an unknown concentration
For the equilibrium PCl5 ⇌ PCl3 + Cl2, Kc = 0.040 at a certain temperature. At equilibrium, [PCl3] = [Cl2] = 0.10 M. Then [PCl5] = [PCl3][Cl2] / Kc = 0.010 / 0.040 = 0.25 M.
Example 5: combining reactions
Given N2 + O2 ⇌ 2NO (K1) and 2NO + O2 ⇌ 2NO2 (K2), the overall reaction N2 + 2O2 ⇌ 2NO2 has K = K1 × K2, since the NO terms cancel.
What does K depend on?
K depends only on the temperature (and on the form of the balanced equation). It does not depend on the initial concentrations, the pressure (for ideal systems), the volume or the presence of a catalyst. When the temperature rises, K increases for an endothermic reaction and decreases for an exothermic one. Any statement about K therefore has to specify the temperature.
Where does the expression come from?
Cato Guldberg and Peter Waage proposed the law of mass action in 1864, by considering the equilibrium as a balance between the rates of the forward and reverse reactions. They defined the “active mass” of a substance as its concentration. Jacobus van ‘t Hoff derived the law from thermodynamics in 1884. The modern definition uses activities instead of concentrations, so that K is dimensionless and is related directly to the standard Gibbs energy change.
Which mistakes are common?
- Writing reactants over products: products always go in the numerator.
- Forgetting the exponents: the coefficients become powers.
- Including pure solids or liquids: they are left out.
- Using initial concentrations: K needs the equilibrium values.
- Reusing K for another temperature or for a rewritten equation: K changes in both cases.