In chemistry, the **law of mass action** states that the rate of a chemical reaction is directly proportional to the product of the concentrations of the reactants. The law gives an equation for calculating the **equilibrium constant**. The law of mass action is also known as the equilibrium law or law of chemical equilibrium.

### Law of Mass Action Equation

At equilibrium, the rates of the forward and reverse chemical reactions are equal:

aA + bB ⇌ cC + dD

The ratio between the concentrations of the products and reactants is a constant, known as the equilibrium constant, K_{c}:

**K _{c} = [C]^{c}[D]^{d}/[A]^{a}[B]^{b}**

In this equation, the square brackets indication concentration of the chemical species. The exponents are the coefficients from the chemical equation.

The equilibrium constant for the reverse reaction, K’_{c}, is given by the following:

K’_{c} = 1/K_{c} = [A]^{a}[B]^{b}/[C]^{c}[D]^{d}

### When to Use the Law of Mass Action

Remember, the law of mass action only applies in cases of dynamic equilibrium. Regardless of the arrows in a chemical equation, make sure the following statements are true:

- The chemical equation represents the reaction of a closed system. That is, there is no heat or mass entering or leaving the system.
- Temperature remains a constant. At equilibrium, temperature does not change. Similarly, the equilibrium constant for a reaction depends on temperature. It’s value at one temperature may differ from K
_{c}at another temperature.

### Equation Using Mole Fractions

When expressing concentration using mole fraction, the law of mass action gives the following expression for the equilibrium constant K_{x}:

K_{x} = [X_{C}]^{c}[K_{D}]^{d}/[X_{A}]^{a}[X_{B}]^{b}

### Law of Mass Action for Gases

For gases, use partial pressures instead of concentration values. The equilibrium constant using partial pressures is Kp:

K_{p} = p^{c}_{C}p^{d}_{D}/P^{a}_{A}p^{b}_{B}

### Law of Mass Action Examples

For example, write the equilibrium constant expression for the dissociation of sulfuric acid into hydrogen and sulfate ions:

H_{2}SO_{4} ⇌ 2H^{+} + SO_{4}^{2-}

Answer: Kc = [H^{+}]^{2}[SO_{4}^{2-}]/[H_{2}SO_{4}]

For example, if you know K_{c} is 5×10^{5} for the reaction:

HCOOH + CN^{−} ⇌ HCN + HCOO^{−}

Calculate the equilibrium constant for the reaction:

HCN + HCOO^{−} ⇌ HCOOH + CN^{−}

Answer: The second equation is the reverse of the first equation.

K’_{c} = 1/K_{c} = 1/(5 x 10^{5}) = 2 x 10^{-6}

### History

Cato Gulberg and Peter Waage proposed the law of mass action in 1864 based on “chemical activity” or “reaction force” rather than reactant mass or concentration. They realized that, at equilibrium, the reaction force for the forward reaction equaled the reaction force of the reverse reaction. Setting the reaction rates of the forward and reverse reactions equal, Guldberg and Waage found the equilibrium constant formula. The big difference between their original equation and the one in use today is that they used “chemical activity” in place of concentration.

### Law of Mass Action in Other Disciplines

The law of mass action applies to other disciplines besides chemistry. For example:

- In semiconductor physics, the product of electron and hole densities is a constant at equilibrium. The constant depends on the Boltzmann constant, temperature, band gap, and effective density of the valence and conduction band states.
- In condensed matter physics, the diffusion process relates to absolute reaction rates.
- The Lotka-Volterra equations in mathematical ecology apply the law of mass action to predator-prey dynamics. The rate of predation is proportional to the rate of predator-prey interactions. The concentration of prey and predators works in place of reactant concentration.
- Sociophysics applies the law of mass action in describing social and political behavior of people.
- In mathematical epidemiology, the law of mass action acts as a model for disease spread.

### References

- Érdi, Péter; Tóth, János (1989).
*Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models*. Manchester University Press. ISBN 978-0-7190-2208-1. - Guggenheim, E.A. (1956). “Textbook errors IX: More About the Laws of Reaction Rates and of Equilibrium”.
*J. Chem. Educ*. 33 (11): 544–545. doi:10.1021/ed033p544 - Guldberg, C.M.; Waage, P. (1879). “Ueber die chemische Affinität” [On chemical affinity].
*Journal für praktische Chemie*. 2nd series (in German). 19: 69–114. doi:10.1002/prac.18790190111 - Lund, E.W. (1965). “Guldberg and Waage and the law of mass action.”
*J. Chem. Educ*. 42(10): 548. doi:10.1021/ed042p548 - Waage, P.; Guldberg, C.M. (1864). “Studier over Affiniteten” [Studies of affinities].
*Forhandlinger I Videnskabs-selskabet I Christiania*(Transactions of the Scientific Society in Christiania) (in Danish): 35–45.