Law of Mass Action

The Law of Mass Action, formulated by Norwegian chemists Cato Guldberg and Peter Waage in 1864, is one of the foundational principles of chemical equilibrium. It states that the rate of a chemical reaction is directly proportional to the product of the 'active masses' (molar concentrations) of the reacting substances, each raised to a power equal to its stoichiometric coefficient in the balanced equation. This law elegantly connects kinetics and thermodynamics, providing a quantitative description of how concentrations drive reactions forward and in reverse. When a reversible reaction reaches dynamic equilibrium, the rate of the forward reaction equals the rate of the reverse reaction. At this point, the concentrations of reactants and products no longer change with time — though molecules continue to interconvert at equal rates. The Law of Mass Action allows us to define the equilibrium constant expression, K_c, a dimensionless ratio of product concentrations to reactant concentrations at equilibrium. This constant depends only on temperature, not on the initial concentrations or the path taken to reach equilibrium. Understanding K_c is essential for predicting the direction of a reaction, calculating equilibrium concentrations, and designing industrial chemical processes. A large K_c (much greater than 1) indicates that products are favoured at equilibrium, while a small K_c (much less than 1) indicates that reactants are favoured. The Law of Mass Action thus gives chemists a precise mathematical tool to describe, predict, and control the outcome of reversible reactions under varying conditions.

Key takeaways

Writing the Balanced Equation
Applying the Law of Mass Action to Rate Expressions
Setting Forward and Reverse Rates Equal
Writing the K_c Expression
Interpreting the Value of K_c

Writing the Balanced Equation

Begin with a general reversible reaction: $a\ce{A} + b\ce{B} \rightleftharpoons c\ce{C} + d\ce{D}$ , where A and B are reactants, C and D are products, and a, b, c, d are stoichiometric coefficients. The double arrow (⇌) denotes that the reaction proceeds in both the forward and reverse directions simultaneously.

Applying the Law of Mass Action to Rate Expressions

The Law of Mass Action states that the rate of the forward reaction is: $r_f = k_f[\text{A}]^a[\text{B}]^b$ , and the rate of the reverse reaction is: $r_r = k_r[\text{C}]^c[\text{D}]^d$ . Here, $k_f$ and $k_r$ are the rate constants for the forward and reverse reactions, and square brackets denote molar concentration in mol/L.

Setting Forward and Reverse Rates Equal

At equilibrium, $r_f = r_r$ , so: $k_f[\text{A}]^a[\text{B}]^b = k_r[\text{C}]^c[\text{D}]^d$ . Rearranging gives: $\dfrac{k_f}{k_r} = \dfrac{[\text{C}]^c[\text{D}]^d}{[\text{A}]^a[\text{B}]^b}$ . This ratio of rate constants is itself a constant at a given temperature — this is the equilibrium constant $K_c$ .

Writing the K_c Expression

The equilibrium constant expression is: $K_c = \dfrac{[\text{C}]^c[\text{D}]^d}{[\text{A}]^a[\text{B}]^b}$ . Important rules: (1) Pure solids and pure liquids are omitted because their concentrations are constant and incorporated into K_c. (2) Only aqueous (aq) and gaseous (g) species appear in the expression. (3) The concentrations used are equilibrium values, not initial values.

Interpreting the Value of K_c

If $K_c \gg 1$ (e.g., $10^{10}$ ): equilibrium lies far to the right; products strongly favoured. If $K_c \approx 1$ : significant amounts of both reactants and products present at equilibrium. If $K_c \ll 1$ (e.g., $10^{-10}$ ): equilibrium lies far to the left; reactants strongly favoured. For a reaction run in reverse, the new equilibrium constant is $K_c' = 1/K_c$ .

The Key Equation

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

K_c — Equilibrium constant (concentration) (dimensionless): Ratio of product to reactant concentrations at equilibrium; depends only on temperature
[\text{C}], [\text{D}] — Equilibrium concentrations of products (mol/L (mol dm⁻³)): Molar concentrations of products C and D measured at equilibrium
[\text{A}], [\text{B}] — Equilibrium concentrations of reactants (mol/L (mol dm⁻³)): Molar concentrations of reactants A and B measured at equilibrium
a, b, c, d — Stoichiometric coefficients (unitless): Powers to which each concentration is raised, taken directly from the balanced chemical equation

Worked Example

Problem. At 500°C, the following equilibrium is established: $\ce{N2(g) + 3H2(g) <=> 2NH3(g)}$ . At equilibrium, the concentrations are: [N₂] = 0.500 mol/L, [H₂] = 0.800 mol/L, [NH₃] = 0.150 mol/L. (a) Write the K_c expression. (b) Calculate K_c at this temperature.

K_c = \frac{[\ce{NH3}]^2}{[\ce{N2}][\ce{H2}]^3} = \frac{(0.150)^2}{(0.500)(0.800)^3} = 0.0879

Answer. K_c = 0.0879 at 500°C; equilibrium favours reactants.

Figure

Diagram

Origins · 1 minCato Maximilian Guldberg and how this was discovered

Cato Maximilian Guldberg

1836–1902 · Norwegian

Together with Peter Waage, Guldberg co-developed the Law of Mass Action in 1864, providing the first quantitative mathematical description of chemical equilibrium. Their work introduced the concept that reaction rates depend on the concentrations (active masses) of reacting substances, and they derived the equilibrium constant expression — a cornerstone of physical chemistry. Guldberg was also a gifted mathematician who made contributions to the theory of analytic functions. Interestingly, his collaboration with Waage began partly because Waage married Guldberg's sister, making them brothers-in-law as well as scientific partners!

Le Chatelier's Principle — Le Chatelier's Principle describes how a system at equilibrium responds to disturbances in concentration, temperature, or pressure — directly building on the equilibrium position defined by K_c.
Reaction Quotient (Q) — The reaction quotient Q uses the same expression as K_c but with non-equilibrium concentrations, allowing prediction of which direction a reaction will proceed to reach equilibrium.
Rate Law and Rate Constant — The Law of Mass Action originated from reaction kinetics; understanding rate laws and rate constants k_f and k_r is essential for appreciating how K_c = k_f/k_r is derived.
Relationship Between K_p and K_c — For gas-phase equilibria, K_p (expressed in terms of partial pressures) is related to K_c by the expression K_p = K_c(RT)^Δn, extending the Law of Mass Action to pressure-based systems.