Equilibrium Constant Kc

Kc is the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of its stoichiometric coefficient.

K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}

Kc = [C]^c [D]^d / [A]^a [B]^b

What it means

For the general reaction aA + bB ⇌ cC + dD, the equilibrium constant Kc is the product concentrations multiplied together (each raised to its coefficient) divided by the reactant concentrations multiplied together (each raised to its coefficient). A large Kc means the equilibrium mixture is mostly products; a small Kc means it is mostly reactants; Kc near 1 means sizable amounts of both. Kc depends only on temperature: concentrating, diluting, or adding one species shifts the mixture, but the ratio always returns to the same Kc.

Variables

Symbol	Name	Unit	Meaning
$[C], [D]$	Equilibrium concentrations of the products	mol/L (M)	The molar concentrations of each product after the reaction has settled into equilibrium.
$[A], [B]$	Equilibrium concentrations of the reactants	mol/L (M)	The molar concentrations of each reactant after the reaction has settled into equilibrium.
$a, b, c, d$	Stoichiometric coefficients	dimensionless	The numbers that balance the chemical equation; each concentration is raised to the power of its own coefficient.
$Kc$	Equilibrium constant (concentration basis)	dimensionless	A single number summarizing the equilibrium position: Kc much larger than 1 favors products, much smaller than 1 favors reactants.

Assumptions

The concentrations in the expression are the equilibrium values, not values measured earlier in the reaction.
Temperature is held constant, because Kc changes whenever the temperature changes.
Concentrations stand in for activities, which is a good approximation for dilute solutions and ideal gases.
Pure solids and pure liquids are omitted from the expression because their concentrations do not change.

Derivation

At equilibrium the forward and reverse rates are equal: kf[A]^a[B]^b = kr[C]^c[D]^d. Rearranging gives [C]^c[D]^d/[A]^a[B]^b = kf/kr, and the ratio of the two rate constants is itself a constant, Kc. More fundamentally, thermodynamics gives ΔG° = −RT ln K, so Kc = e^(−ΔG°/RT): a negative standard free-energy change produces Kc greater than 1.

Worked example

Problem

For the gas-phase reaction N2O4(g) ⇌ 2 NO2(g), the equilibrium concentrations are [N2O4] = 0.0200 M and [NO2] = 0.0300 M. Calculate Kc.

Given:

[N2O4] = 0.0200 M at equilibrium
[NO2] = 0.0300 M at equilibrium
Balanced reaction: N2O4(g) ⇌ 2 NO2(g), so Kc = [NO2]^2 / [N2O4]

Write the expression: Kc = [NO2]^2 / [N2O4].
Substitute the equilibrium values: Kc = (0.0300)^2 / 0.0200.
Square the numerator: (0.0300)^2 = 0.000900.
Divide: 0.000900 / 0.0200 = 0.0450.

Kc = 0.0450 (dimensionless)

Because Kc is much smaller than 1, the equilibrium mixture favors the reactant N2O4 over the product NO2 at this temperature.