# Law of mass action

The law of mass action is a principle of chemistry stating that the velocity of a reaction depends on the active mass, that is, the concentrations of the reactants, and that when a reversible reaction reaches equilibrium at a given temperature the forward and backward rates are equal, so the concentrations of the reacting species bear a constant relation to each other described by the equilibrium constant.<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup> It was first recognized in 1864 as the kinetic law of mass action by [Cato Maximilian Guldberg](https://www.edgepedia.org/cato-maximilian-guldberg) and Peter Waage, two Norwegian chemists who also introduced the concept of dynamic equilibrium.<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup><sup> • </sup><sup>[2](https://iupac.org/wp-content/uploads/2025/02/Norway_NKS_2025.pdf)</sup>

| Key fact | Detail |
|---|---|
| Kinetic statement | For an elementary reaction \( \ce{A + B -> AB} \), the forward velocity is \( \nu_{1} = k_{1}[\ce{A}][\ce{B}] \); for the reverse dissociation, \( \nu_{2} = k_{2}[\ce{AB}] \). Valid only for elementary reactions.<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup> |
| Equilibrium statement | At equilibrium at a given temperature, \( \nu_{1} = \nu_{2} \), so \( K_{c} = \frac{[\ce{AB}]}{[\ce{A}][\ce{B}]} = \frac{k_{1}}{k_{2}} \).<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup> |
| General form | For \( a\ce{A} + b\ce{B} \rightleftharpoons c\ce{C} + d\ce{D} \), the mass action expression is \( K = \frac{a_{\ce{C}}^{c} \, a_{\ce{D}}^{d}}{a_{\ce{A}}^{a} \, a_{\ce{B}}^{b}} \), with each activity raised to its stoichiometric coefficient.<sup>[3](https://ch302.cm.utexas.edu/chemEQ/equilibrium/selector.php?name=equil-constant)</sup> |
| Thermodynamic form | The rigorous equilibrium constant is \( K = \prod_{i} a_{i}^{\nu_{i}} \) in terms of activities and fugacities, and is dimensionless.<sup>[4](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap10/Gibbs_and_K.htm)</sup> |
| Free energy link | \( \Delta G^{\circ} = -RT \ln K \) and \( \Delta G = \Delta G^{\circ} + RT \ln Q \).<sup>[5](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch21/gibbs.php)</sup> |
| History | Proposed in 1864 by Guldberg and Waage; a correct mathematical derivation accounting for reaction order came only after the work of Horstmann and van't Hoff.<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup> |
| Catalysts | A catalyst speeds both forward and reverse reactions equally and has no effect on the value of the equilibrium constant or the equilibrium concentrations.<sup>[6](https://openstax.org/books/chemistry-atoms-first-2e/pages/17-7-catalysis)</sup> |

## The kinetic statement

For an association reaction and its reverse, \( \ce{A + B -> AB} \) and \( \ce{AB -> A + B} \), the law gives the forward velocity as \( \nu_{1} = k_{1}[\ce{A}][\ce{B}] \), where \( k_{1} \) is the rate constant for association, and the dissociation velocity as \( \nu_{2} = k_{2}[\ce{AB}] \). IUPAC notes that this rate form is valid only for elementary reactions, those that occur in a single step.<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup> When the reversible reaction reaches equilibrium at a given temperature, the forward rate equals the backward rate, so \( k_{1}[\ce{A}][\ce{B}] = k_{2}[\ce{AB}] \), and rearranging gives one form of the equilibrium constant, \( K_{c} = \frac{[\ce{AB}]}{[\ce{A}][\ce{B}]} = \frac{k_{1}}{k_{2}} \).<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup> The same ratio of forward to reverse rate constants appears in general treatments of catalysis and equilibrium.<sup>[6](https://openstax.org/books/chemistry-atoms-first-2e/pages/17-7-catalysis)</sup>

## The equilibrium statement

In its modern general-chemistry formulation, the law says that when a reversible reaction has attained equilibrium at a given temperature, the reaction quotient remains constant.<sup>[7](https://openstax.org/books/chemistry/pages/13-key-terms)</sup> The reaction quotient \( Q \) is the ratio of the product of [molar concentrations](https://www.edgepedia.org/molar-concentration) (or pressures) of the products to that of the reactants, each raised to the power of its coefficient in the balanced equation.<sup>[7](https://openstax.org/books/chemistry/pages/13-key-terms)</sup> The constant value of \( Q \) at equilibrium is the equilibrium constant \( K \), written \( K_{c} \) when based on concentrations and \( K_{p} \) when based on partial pressures.<sup>[8](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-2-equilibrium-constants)</sup>

The constancy is independent of the starting composition. For the reaction \( \ce{2SO2 + O2 <=> 2SO3} \), a mixture begun with only reactants has \( Q_{c} = 0 \), which rises as the forward reaction proceeds, while a mixture begun with only product has an effectively infinite \( Q_{c} \), which falls as the reverse reaction proceeds; both scenarios reach the same value of \( K \).<sup>[8](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-2-equilibrium-constants)</sup> Comparing \( Q \) with \( K \) predicts the direction of net reaction: if \( Q < K \) the reaction proceeds forward, and if \( Q > K \) it proceeds in reverse.<sup>[8](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-2-equilibrium-constants)</sup>

As a worked example, for \( \ce{2NO2 <=> N2O4} \) at 25 °C with equilibrium concentrations \( [\ce{NO2}] = 0.016 \ \mathrm{M} \) and \( [\ce{N2O4}] = 0.042 \ \mathrm{M} \), the law gives \( K_{c} = \frac{0.042}{(0.016)^{2}} = 1.6 \times 10^{2} \).<sup>[9](https://openstax.org/books/chemistry-atoms-first/pages/13-2-equilibrium-constants)</sup>

## Activities and the rigorous form

Using raw concentrations is a convenient simplification that can fail. Equilibria involving aqueous ions often show equilibrium constants that vary significantly at high solution concentrations, and this is avoided by computing \( K \) from activities, which are dimensionless measures of effective concentration.<sup>[9](https://openstax.org/books/chemistry-atoms-first/pages/13-2-equilibrium-constants)</sup> For relatively dilute solutions, activity and molar concentration are roughly equal; for pure solids and liquids the activity is 1, so those species do not appear in the equilibrium expression.<sup>[9](https://openstax.org/books/chemistry-atoms-first/pages/13-2-equilibrium-constants)</sup>

The thermodynamically rigorous equilibrium constant is written as \( K = \prod_{i} a_{i}^{\nu_{i}} \), a product over the activities of all participants, each raised to its stoichiometric number, positive for products and negative for reactants.<sup>[4](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap10/Gibbs_and_K.htm)</sup><sup> • </sup><sup>[10](https://goldbook.iupac.org/terms/view/E02177)</sup> Because activities and fugacities are dimensionless, this thermodynamic \( K \) is dimensionless.<sup>[4](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap10/Gibbs_and_K.htm)</sup> In dilute solutions and low-pressure gases, concentrations and partial pressures give excellent approximations to it.<sup>[4](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap10/Gibbs_and_K.htm)</sup> The activity of a species is defined through its chemical potential, \( \mu_{i} = \mu_{i}^{\circ} + RT \ln a_{i} \), and takes phase-dependent forms: \( a_{i} = p_{i}/p^{\circ} \) for an ideal gas, \( a_{i} = f_{i}/p^{\circ} \) with the fugacity \( f_{i} = \phi_{i} p_{i} \) for a real gas, and \( a_{i} = \gamma_{i} b_{i}/b^{\circ} \) for an aqueous solute.<sup>[11](https://chem.academy/notes/standard-states.html)</sup> An equilibrium constant must always be accompanied by its corresponding balanced equation to be interpreted.<sup>[3](https://ch302.cm.utexas.edu/chemEQ/equilibrium/selector.php?name=equil-constant)</sup>

## Connection with thermodynamics

The law of mass action follows from the minimization of the Gibbs energy at equilibrium. From \( \mu = \mu^{\circ} + RT \ln a \), the condition that the stoichiometric sum of chemical potentials vanishes at equilibrium yields \( \Delta G^{\circ} = -RT \ln K \), with \( K = \exp(-\Delta G^{\circ}/RT) \).<sup>[12](https://publications.iupac.org/pac/pdf/2011/pdf/8305x1031.pdf)</sup><sup> • </sup><sup>[5](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch21/gibbs.php)</sup> More generally, \( \Delta G = \Delta G^{\circ} + RT \ln Q \); as a reaction proceeds and \( Q \) approaches \( K \), \( \Delta G \) falls to zero.<sup>[5](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch21/gibbs.php)</sup><sup> • </sup><sup>[13](https://openstax.org/books/chemistry-atoms-first/pages/13-introduction)</sup> The sign of \( \Delta G^{\circ} \) fixes the position of equilibrium: a negative value gives \( K > 1 \) and products favored, a positive value gives \( K < 1 \) and reactants favored.<sup>[4](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap10/Gibbs_and_K.htm)</sup>

Because \( K = k_{1}/k_{2} \) and the rate constants vary with temperature according to the [Arrhenius equation](https://www.edgepedia.org/arrhenius-equation), the equilibrium constant likewise varies with temperature.<sup>[14](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-3-shifting-equilibria-le-chateliers-principle)</sup> A stress on concentration shifts the composition without changing \( K \), while a temperature shift re-establishes equilibrium with a different value of \( K \).<sup>[14](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-3-shifting-equilibria-le-chateliers-principle)</sup>

## Working with coupled equilibria

Because the law allows equilibrium expressions to be derived directly from balanced equations, the \( K \) of a system of coupled equilibria can be related to the \( K \) values of the individual reactions through three manipulations: reversing an equation inverts its constant, multiplying the coefficients by a factor \( x \) raises \( K \) to the power \( x \), and adding equations multiplies their constants.<sup>[15](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-2-equilibrium-constants?modal=MH)</sup> These rules underpin the standard equilibrium calculations, in which the law is used to find an equilibrium constant from known equilibrium concentrations, a missing concentration from \( K \), or equilibrium concentrations from initial ones.<sup>[16](https://openstax.org/books/chemistry-atoms-first/pages/13-4-equilibrium-calculations)</sup>

## History

Guldberg and Waage discovered the relationship between what they called the "active masses" experimentally and reported it in 1864, just before the theoretical work of [Josiah Willard Gibbs](https://www.edgepedia.org/josiah-willard-gibbs).<sup>[12](https://publications.iupac.org/pac/pdf/2011/pdf/8305x1031.pdf)</sup> IUPAC credits them with pioneering the idea of dynamic equilibrium, though they erred in supposing that the rates followed directly from the stoichiometric equation; a sound mathematical derivation of reaction rates that took the reaction order into account had to await the contributions of August Georg Horstmann and Jacobus Henricus van't Hoff.<sup>[1](https://goldbook.iupac.org/terms/view/08184)</sup> The Norwegian Chemical Society, the national society of the law's originators, has awarded a Guldberg-Waage Medal since its founding in 1964.<sup>[2](https://iupac.org/wp-content/uploads/2025/02/Norway_NKS_2025.pdf)</sup>

## References

1. [IUPAC Gold Book: mass action, law of (08184)](https://goldbook.iupac.org/terms/view/08184)
2. [The Norwegian Chemical Society, IUPAC](https://iupac.org/wp-content/uploads/2025/02/Norway_NKS_2025.pdf)
3. [Equilibrium Constant, CH302, University of Texas at Austin](https://ch302.cm.utexas.edu/chemEQ/equilibrium/selector.php?name=equil-constant)
4. [Chem 351: ΔG° and Kp, California State University East Bay](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap10/Gibbs_and_K.htm)
5. [Gibbs Free Energy, Purdue University Chemistry](https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch21/gibbs.php)
6. [17.7 Catalysis, Chemistry: Atoms First 2e, OpenStax](https://openstax.org/books/chemistry-atoms-first-2e/pages/17-7-catalysis)
7. [Ch. 13 Key Terms, Chemistry, OpenStax](https://openstax.org/books/chemistry/pages/13-key-terms)
8. [13.2 Equilibrium Constants, Chemistry: Atoms First 2e, OpenStax](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-2-equilibrium-constants)
9. [13.2 Equilibrium Constants, Chemistry: Atoms First, OpenStax](https://openstax.org/books/chemistry-atoms-first/pages/13-2-equilibrium-constants)
10. [IUPAC Gold Book: equilibrium constant (E02177)](https://goldbook.iupac.org/terms/view/E02177)
11. [Standard States: Conventions and Corrections, Chem.Academy](https://chem.academy/notes/standard-states.html)
12. [Computational thermodynamics, Pure and Applied Chemistry 83(5), 1031](https://publications.iupac.org/pac/pdf/2011/pdf/8305x1031.pdf)
13. [Ch. 13 Introduction, Chemistry: Atoms First, OpenStax](https://openstax.org/books/chemistry-atoms-first/pages/13-introduction)
14. [13.3 Shifting Equilibria: Le Châtelier's Principle, Chemistry: Atoms First 2e, OpenStax](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-3-shifting-equilibria-le-chateliers-principle)
15. [13.2 Equilibrium Constants (coupled equilibria), Chemistry: Atoms First 2e, OpenStax](https://openstax.org/books/chemistry-atoms-first-2e/pages/13-2-equilibrium-constants?modal=MH)
16. [13.4 Equilibrium Calculations, Chemistry: Atoms First, OpenStax](https://openstax.org/books/chemistry-atoms-first/pages/13-4-equilibrium-calculations)

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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Thermodynamics and equilibrium › Chemical equilibrium › Equilibrium constant*

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