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{{Short description|When the ratio of reactants to products of a chemical reaction is constant with time}}
{{TopicTOC-Chemistry}}

In a [[chemical reaction]], '''chemical equilibrium''' is the state in which both the [[Reagent|reactants]] and [[Product (chemistry)|products]] are present in [[concentration]]s which have no further tendency to change with time, so that there is no observable change in the properties of the [[Thermodynamic system|system]].<ref name=Atkins>{{cite book|first1=Peter |last1=Atkins |first2=Julio |last2=De Paula |title=Atkins' Physical Chemistry |url=https://archive.org/details/atkinsphysicalch00pwat |url-access=registration |edition=8th |publisher=W. H. Freeman |date=2006 |isbn=0-7167-8759-8 |pages=[https://archive.org/details/atkinsphysicalch00pwat/page/200 200–202]}}</ref> This state results when the forward reaction proceeds at the same rate as the [[Reversible reaction|reverse reaction]]. The [[reaction rate]]s of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known as [[dynamic equilibrium]].<ref name=aj/><ref>{{GoldBookRef|title=chemical equilibrium|file=C01023}}</ref>
It is the subject of study of ''[[equilibrium chemistry]]''.

== Historical introduction ==

The [[Concept learning|concept]] of chemical equilibrium was developed in 1803, after [[Claude Louis Berthollet|Berthollet]] found that some [[chemical reaction]]s are [[Reversible reaction|reversible]].<ref>{{cite book|last1=Berthollet|first1=C.L.|title=Essai de statique chimique|trans-title=Essay on chemical statics|date=1803|publisher=Firmin Didot|location=Paris, France|url=https://archive.org/details/essaidestatiquec01bert|language=fr}} On pp. 404–407, Berthellot mentions that when he accompanied Napoleon on his expedition to Egypt, he (Berthellot) visited Lake Natron and found sodium carbonate along its shores.  He realized that this was a product of the reverse of the usual reaction Na<sub>2</sub>CO<sub>3</sub> + CaCl<sub>2</sub> → 2NaCl + CaCO<sub>3</sub>↓ and therefore that the final state of a reaction was a state of equilibrium between two opposing processes.  From p. 405: ''" ... la décomposition du muriate de soude continue donc jusqu'à ce qu'il se soit formé assez de muriate de chaux, parce que l'acide muriatique devant se partager entre les deux bases en raison de leur action, il arrive un terme où leurs forces se balancent."'' ( ... the [[Chemical decomposition|decomposition]] of the sodium chloride thus continues until enough calcium chloride is formed, because the hydrochloric acid must be shared between the two bases in the ratio of their action [i.e., capacity to react]; it reaches an end [point] at which their forces are balanced.)</ref> For any reaction mixture to exist at equilibrium, the [[reaction rate|rates]] of the forward and backward (reverse) reactions must be equal. In the following [[chemical equation]], arrows point both ways to indicate equilibrium.<ref>The notation {{eqm}} was proposed in 1884 by the Dutch chemist [[Jacobus Henricus van 't Hoff]]. See: {{cite book|last1=van 't Hoff|first1=J.H.|title=Études de Dynamique Chemique|trans-title=Studies of chemical dynamics|date=1884|publisher=Frederik Muller & Co.|location=Amsterdam, Netherlands|pages=4–5|url=https://archive.org/stream/etudesdedynamiqu00hoff#page/4/mode/2up|language=fr}} Van 't Hoff called reactions that didn't proceed to completion "limited reactions".  From pp. 4–5:  ''"Or M. Pfaundler a relié ces deux phénomênes ... s'accomplit en même temps dans deux sens opposés."'' (Now Mr. Pfaundler has joined these two phenomena in a single concept by considering the observed limit as the result of two opposing reactions, driving the one in the example cited to the formation of sea salt [i.e., NaCl] and nitric acid, [and] the other to hydrochloric acid and sodium nitrate.  This consideration, which experiment validates, justifies the expression "chemical equilibrium", which is used to characterize the final state of limited reactions.  I would propose to translate this expression by the following symbol: 
:HCl + NO<sub>3</sub> Na {{eqm}} NO<sub>3</sub> H + Cl Na .
I thus replace, in this case, the = sign in the chemical equation by the sign {{eqm}}, which in reality doesn't express just equality but shows also the direction of the reaction.  This clearly expresses that a chemical action occurs simultaneously in two opposing directions.)</ref> A and B are [[reactant]] chemical species, S and T are product species, and [[Alpha (letter)|''α'']], [[Beta (letter)|''β'']], [[sigma|''σ'']], and [[Tau|''τ'']] are the [[stoichiometric coefficient]]s of the respective reactants and products:
:''α''&nbsp;A + ''β''&nbsp;B {{eqm}} ''σ''&nbsp;S + ''τ''&nbsp;T

The equilibrium concentration position of a reaction is said to lie "far to the right" if, at equilibrium, nearly all the reactants are consumed. Conversely the equilibrium position is said to be "far to the left" if hardly any product is formed from the reactants.

[[Cato Maximilian Guldberg|Guldberg]] and [[Peter Waage|Waage]] (1865), building on Berthollet's ideas, proposed the [[law of mass action]]:

:<math chem>\begin{align}
\text{forward reaction rate} &= k_{+} \ce{A}^\alpha\ce{B}^\beta \\
\text{backward reaction rate} &= k_{-} \ce{S}^\sigma\ce{T}^\tau
\end{align}</math>

where A, B, S and T are [[activity (chemistry)|active masses]] and ''k''<sub>+</sub> and ''k''<sub>−</sub> are [[rate constant]]s. Since at equilibrium forward and backward rates are equal:

:<math chem> k_+ \left\{ \ce A \right\}^\alpha \left\{\ce B \right\}^\beta = k_{-} \left\{\ce S \right\}^\sigma\left\{\ce T \right\}^\tau</math>

and the ratio of the rate constants is also a constant, now known as an [[equilibrium constant]].
:<math chem>K_c=\frac{k_+}{k_-}=\frac{\{\ce S\}^\sigma \{\ce T\}^\tau } {\{\ce A\}^\alpha \{\ce B\}^\beta}</math>

By convention, the products form the [[numerator]].
However, the [[law of mass action]] is valid only for concerted one-step reactions that proceed through a single [[transition state]] and is '''not valid in general''' because [[reaction rate#Rate equation|rate equations]] do not, in general, follow the [[stoichiometry]] of the reaction as Guldberg and Waage had proposed (see, for example, [[nucleophilic aliphatic substitution]] by S<sub>N</sub>1 or reaction of [[hydrogen]] and [[bromine]] to form [[hydrogen bromide]]). Equality of forward and backward reaction rates, however, is a [[Necessary and sufficient conditions|necessary condition]] for chemical equilibrium, though it is not [[Necessary and sufficient conditions|sufficient]] to explain why equilibrium occurs.

Despite the limitations of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the [[van 't Hoff equation]]. Adding a [[catalyst]] will affect both the forward reaction and the reverse reaction in the same way and will not have an effect on the equilibrium constant. The catalyst will speed up both reactions thereby increasing the speed at which equilibrium is reached.<ref name=aj>{{cite book|last1=Atkins |first1=Peter W. |last2=Jones |first2=Loretta  |title=Chemical Principles: The Quest for Insight |edition=2nd |isbn=978-0-7167-9903-0|year=2008 |publisher=W.H. Freeman }}</ref><ref>{{cite book|title=Chemistry: Matter and Its Changes |first=James E. |last=Brady |publisher=Fred Senese |edition=4th |isbn=0-471-21517-1|date=2004-02-04 }}</ref>

Although the [[macroscopic]] equilibrium concentrations are constant in time, reactions do occur at the molecular level. For example, in the case of [[acetic acid]] dissolved in water and forming [[acetate]] and [[hydronium]] ions,
:{{chem2|CH3CO2H + H2O <-> CH3CO2- + H3O+}}
a proton may hop from one molecule of acetic acid onto a water molecule and then onto an acetate anion to form another molecule of acetic acid and leaving the number of acetic acid molecules unchanged. This is an example of [[dynamic equilibrium]]. Equilibria, like the rest of thermodynamics, are statistical phenomena, averages of microscopic behavior.

'''[[Le Châtelier's principle]]''' (1884) predicts the behavior of an equilibrium system when changes to its reaction conditions occur. ''If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to partially reverse the change''. For example, adding more S (to the chemical reaction above) from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward (though the equilibrium constant will stay the same).

If [[mineral acid]] is added to the acetic acid mixture, increasing the concentration of hydronium ion, the amount of dissociation must decrease as the reaction is driven to the left in accordance with this principle. This can also be deduced from the equilibrium constant expression for the reaction:
:<math chem>K=\frac {\{\ce{CH3CO2-}\}\{\ce{H3O+}\}} \ce{\{CH3CO2H\}}</math>
If {H<sub>3</sub>O<sup>+</sup>} increases {CH<sub>3</sub>CO<sub>2</sub>H} must increase and {{chem2|CH3CO2-}} must decrease. The H<sub>2</sub>O is left out, as it is the solvent and its concentration remains high and nearly constant.

[[Josiah Willard Gibbs|J. W. Gibbs]] suggested in 1873 that equilibrium is attained when the "available energy" (now known as [[Gibbs free energy]] or Gibbs energy) of the system is at its minimum value, assuming the reaction is carried out at a constant temperature and pressure. What this means is that the derivative of the Gibbs energy with respect to [[reaction coordinate]] (a measure of the [[extent of reaction]] that has occurred, ranging from [[zero]] for all reactants to a maximum for all products) vanishes (because dG = 0), signaling a [[stationary point]]. This derivative is called the reaction Gibbs energy (or energy change) and corresponds to the difference between the [[chemical potential]]s of reactants and products at the composition of the reaction mixture.<ref name=Atkins/> This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (or [[Helmholtz energy]] at constant volume reactions) is the "driving force" for the composition of the mixture to change until equilibrium is reached. The equilibrium constant can be related to the standard [[Gibbs energy|Gibbs free energy]] change for the reaction by the equation

:<math>\Delta_rG^\ominus = -RT \ln K_\mathrm{eq}</math>

where ''R'' is the [[universal gas constant]] and ''T'' the [[temperature]].

When the reactants are [[Solution (chemistry)|dissolved]] in a medium of high [[ionic strength]] the quotient of [[activity coefficient]]s may be taken to be constant. In that case the '''concentration quotient''', ''K''<sub>c</sub>,
:<math chem>K_\ce{c}=\frac{[\ce S]^\sigma [\ce T]^\tau } {[\ce A]^\alpha [\ce B]^\beta}</math>
where [A] is the [[concentration]] of A, etc., is independent of the [[analytical concentration]] of the reactants. For this reason, equilibrium constants for [[Solution (chemistry)|solution]]s are usually [[Determination of equilibrium constants|determined]] in media of high ionic strength. ''K<sub>c</sub>'' varies with [[ionic strength]], temperature and pressure (or volume). Likewise ''K<sub>p</sub>'' for gases depends on [[partial pressure]]. These constants are easier to measure and encountered in high-school chemistry courses.

==Thermodynamics==

At constant temperature and pressure, one must consider the [[Gibbs free energy]], ''G'', while at constant temperature and volume, one must consider the [[Helmholtz free energy]], ''A'', for the reaction; and at constant internal energy and volume, one must consider the entropy, ''S'', for the reaction.

The constant volume case is important in [[geochemistry]] and [[atmospheric chemistry]] where pressure variations are significant. Note that, if reactants and products were in [[standard state]] (completely pure), then there would be no reversibility and no equilibrium. Indeed, they would necessarily occupy disjoint volumes of space. The mixing of the products and reactants contributes a large entropy increase (known as [[entropy of mixing]]) to states containing equal mixture of products and reactants and gives rise to a distinctive minimum in the Gibbs energy as a function of the extent of reaction.<ref name="AtkinsPChem">Atkins, P.; de Paula, J.; Friedman, R. (2014). ''Physical Chemistry – Quanta, Matter and Change'', 2nd ed., Fig. 73.2. Freeman.</ref> The standard Gibbs energy change, together with the Gibbs energy of mixing, determine the equilibrium state.<ref>{{cite journal|last1=Schultz|first1=Mary Jane|title=Why Equilibrium? Understanding Entropy of Mixing|journal=Journal of Chemical Education|volume=76|pages=1391|year=1999|doi=10.1021/ed076p1391|issue=10|bibcode = 1999JChEd..76.1391S }}</ref><ref>{{cite journal|last1=Clugston|first1=Michael J.|title=A mathematical verification of the second law of thermodynamics from the entropy of mixing|journal=Journal of Chemical Education|volume=67|pages=203|year=1990|doi=10.1021/ed067p203|issue=3|bibcode = 1990JChEd..67Q.203C }}</ref>

In this article only the '''constant pressure''' case is considered. The relation between the Gibbs free energy and the equilibrium constant can be found by considering [[chemical potential]]s.<ref name = Atkins/>

At constant temperature and pressure in the absence of an applied voltage, the [[Gibbs free energy]], ''G'', for the reaction depends only on the [[extent of reaction]]: ''ξ'' (Greek letter [[Xi (letter)|xi]]), and can only decrease according to the [[second law of thermodynamics]]. It means that the derivative of ''G'' with respect to ''ξ'' must be negative if the reaction happens; at the equilibrium this derivative is equal to zero.
:<math>\left(\frac {dG}{d\xi}\right)_{T,p} = 0~</math>:{{spaces|5}}equilibrium

In order to meet the thermodynamic condition for equilibrium, the Gibbs energy must be stationary, meaning that the derivative of ''G'' with respect to the extent of reaction, ''ξ'', must be zero. It can be shown that in this case, the sum of [[chemical potential]]s times the stoichiometric coefficients of the products is equal to the sum of those corresponding to the reactants.<ref name="MortimerBook"> Mortimer, R. G. ''Physical Chemistry'', 3rd ed., p. 305, Academic Press, 2008. </ref> Therefore, the sum of the Gibbs energies of the reactants must be the equal to the sum of the Gibbs energies of the products.
:<math> \alpha \mu_\mathrm{A} + \beta \mu_\mathrm{B} = \sigma \mu_\mathrm{S} + \tau \mu_\mathrm{T} \,</math>

where [[Mu (letter)|''μ'']] is in this case a partial molar Gibbs energy, a [[chemical potential]]. The chemical potential of a reagent A is a function of the [[Activity (chemistry)|activity]], {A} of that reagent.

:<math> \mu_\mathrm{A} = \mu_{A}^{\ominus} + RT \ln\{\mathrm{A}\} \,</math>
(where ''μ''{{su|b=A|p=<s>o</s>}} is the '''standard chemical potential''').

The definition of the [[Gibbs energy]] equation interacts with the [[fundamental thermodynamic relation]] to produce

:<math> dG = Vdp-SdT+\sum_{i=1}^k \mu_i dN_i </math>.

Inserting ''dN<sub>i</sub>''&nbsp;= ''ν<sub>i</sub>&nbsp;dξ'' into the above equation gives a [[stoichiometric coefficient]] (<math> \nu_i~</math>) and a differential that denotes the reaction occurring to an infinitesimal extent (''dξ''). At constant pressure and temperature the above equations can be written as

:<math>\left(\frac {dG}{d\xi}\right)_{T,p} = \sum_{i=1}^k \mu_i \nu_i = \Delta_\mathrm{r}G_{T,p}</math> 

which is the Gibbs free energy change for the reaction. This results in:

:<math> \Delta_\mathrm{r}G_{T,p} = \sigma \mu_\mathrm{S} + \tau \mu_\mathrm{T} - \alpha \mu_\mathrm{A} - \beta \mu_\mathrm{B} \,</math>.

By substituting the chemical potentials:

:<math> \Delta_\mathrm{r}G_{T,p} = ( \sigma \mu_\mathrm{S}^{\ominus} + \tau \mu_\mathrm{T}^{\ominus} ) - ( \alpha \mu_\mathrm{A}^{\ominus} + \beta \mu_\mathrm{B}^{\ominus} ) + ( \sigma RT \ln\{\mathrm{S}\} + \tau RT \ln\{\mathrm{T}\} ) - ( \alpha RT \ln\{\mathrm{A}\} + \beta RT \ln \{\mathrm{B}\} ) </math>,

the relationship becomes:

:<math> \Delta_\mathrm{r}G_{T,p}=\sum_{i=1}^k \mu_i^\ominus \nu_i + RT \ln \frac{\{\mathrm{S}\}^\sigma \{\mathrm{T}\}^\tau} {\{\mathrm{A}\}^\alpha \{\mathrm{B}\}^\beta} </math>
:<math>\sum_{i=1}^k \mu_i^\ominus \nu_i = \Delta_\mathrm{r}G^{\ominus}</math>:
which is the '''standard Gibbs energy change for the reaction''' that can be calculated using thermodynamical tables.
The [[reaction quotient]] is defined as:
:<math> Q_\mathrm{r} =  \frac{\{\mathrm{S}\}^\sigma \{\mathrm{T}\}^\tau} {\{\mathrm{A}\}^\alpha \{\mathrm{B}\}^\beta}  </math>

Therefore,
:<math>\left(\frac {dG}{d\xi}\right)_{T,p} = \Delta_\mathrm{r}G_{T,p}= \Delta_\mathrm{r}G^{\ominus} + RT \ln Q_\mathrm{r} </math>

At equilibrium:
:<math>\left(\frac {dG}{d\xi}\right)_{T,p} = \Delta_\mathrm{r}G_{T,p} = 0 </math>

leading to:

:<math> 0 = \Delta_\mathrm{r}G^{\ominus} + RT \ln K_\mathrm{eq} </math>
and
:<math> \Delta_\mathrm{r}G^{\ominus} = -RT \ln K_\mathrm{eq} </math>

Obtaining the value of the standard Gibbs energy change, allows the calculation of the equilibrium constant.

[[File:Diag eq.svg|thumb]]

===Addition of reactants or products===
For a reactional system at equilibrium: ''Q''<sub>r</sub>&nbsp;=&nbsp;''K''<sub>eq</sub>; ''ξ''&nbsp;=&nbsp;''ξ''<sub>eq</sub>.
* If the activities of constituents are modified, the value of the reaction quotient changes and becomes different from the equilibrium constant: ''Q''<sub>r</sub>&nbsp;≠&nbsp;''K''<sub>eq</sub> <math display="block">\left(\frac {dG}{d\xi}\right)_{T,p} = \Delta_\mathrm{r}G^{\ominus} + RT \ln Q_\mathrm{r}~</math> and <math display="block">\Delta_\mathrm{r}G^{\ominus} = - RT \ln K_{eq}~</math> then <math display="block">\left(\frac {dG}{d\xi}\right)_{T,p} = RT \ln \left(\frac {Q_\mathrm{r}}{K_\mathrm{eq}}\right)~</math>
In simplifications where the change in reaction quotient is solely due to the concentration changes, ''Q''<sub>r</sub> is referred to as the [[Mass–action ratio|mass-action ratio]], and the ratio ''Q''<sub>r</sub>/''K''<sub>eq</sub> is referred to as the disequilibrium ratio.{{cn|date=March 2025}}
* If activity of a reagent ''i'' increases <math display="block">Q_\mathrm{r} = \frac{\prod (a_j)^{\nu_j}}{\prod(a_i)^{\nu_i}}~,</math> the reaction quotient decreases. Then <math display="block">Q_\mathrm{r} < K_\mathrm{eq}~</math> and <math display="block">\left(\frac {dG}{d\xi}\right)_{T,p} < 0~</math> The reaction will shift to the right (i.e. in the forward direction, and thus more products will form).
* If activity of a product ''j'' increases, then <math display="block">Q_\mathrm{r} > K_\mathrm{eq}~</math> and <math display="block">\left(\frac {dG}{d\xi}\right)_{T,p} >0~</math> The reaction will shift to the left (i.e. in the reverse direction, and thus less products will form).

Note that activities and equilibrium constants are dimensionless numbers.

===Treatment of activity===

The expression for the equilibrium constant can be rewritten as the product of a concentration quotient, ''K''<sub>c</sub> and an [[activity coefficient]] quotient, ''Γ''.
:<math>K=\frac{[\mathrm{S}] ^\sigma [\mathrm{T}]^\tau ... } {[\mathrm{A}]^\alpha [\mathrm{B}]^\beta ...}
\times \frac{{\gamma_\mathrm{S}} ^\sigma {\gamma_\mathrm{T}}^\tau ... } {{\gamma_\mathrm{A}}^\alpha {\gamma_\mathrm{B}}^\beta ...} = K_\mathrm{c} \Gamma</math>

[A] is the concentration of reagent A, etc. It is possible in principle to obtain values of the activity coefficients, γ. For solutions, equations such as the [[Debye–Hückel equation]] or extensions such as [[Davies equation]]<ref>{{cite book|first=C. W. |last=Davies |title=Ion Association |publisher=Butterworths |date=1962}}</ref> [[Specific ion interaction theory]] or [[Pitzer equations]]<ref name="davies">{{cite web |first1=I. |last1=Grenthe |first2=H. |last2=Wanner |url=http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf |title=Guidelines for the extrapolation to zero ionic strength |access-date=2007-05-16 |archive-date=2008-12-17 |archive-url=https://web.archive.org/web/20081217001051/http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf |url-status=dead }}</ref> may be used. However this is not always possible. It is common practice to assume that ''Γ'' is a constant, and to use the concentration quotient in place of the thermodynamic equilibrium constant. It is also general practice to use the term ''equilibrium constant'' instead of the more accurate ''concentration quotient''. This practice will be followed here.

For reactions in the gas phase [[partial pressure]] is used in place of concentration and [[fugacity coefficient]] in place of activity coefficient. In the real world, for example, when making [[Haber process|ammonia]] in industry, fugacity coefficients must be taken into account. Fugacity, ''f'', is the product of partial pressure and fugacity coefficient. The chemical potential of a species in the [[real gas]] phase is given by
:<math>\mu = \mu^{\ominus} + RT \ln \left( \frac{f}{\mathrm{bar}} \right) = \mu^{\ominus} + RT \ln \left( \frac{p}{\mathrm{bar}} \right) + RT \ln \gamma </math>
so the general expression defining an equilibrium constant is valid for both solution and gas phases.{{Citation needed|date=September 2021}}

===Concentration quotients===

In aqueous solution, equilibrium constants are usually determined in the presence of an "inert" electrolyte such as [[sodium nitrate]], NaNO<sub>3</sub>, or [[potassium perchlorate]], KClO<sub>4</sub>. The [[ionic strength]] of a solution is given by
:<math> I = \frac12\sum_{i=1}^N c_i z_i^2 </math>
where ''c<sub>i</sub>'' and ''z<sub>i</sub>'' stand for the concentration and ionic charge of ion type ''i'', and the sum is taken over all the ''N'' types of charged species in solution. When the concentration of dissolved salt is much higher than the analytical concentrations of the reagents, the ions originating from the dissolved salt determine the ionic strength, and the ionic strength is effectively constant. Since activity coefficients depend on ionic strength, the activity coefficients of the species are effectively independent of concentration. Thus, the assumption that [[Gamma|''Γ'']] is constant is justified. The concentration quotient is a simple multiple of the equilibrium constant.<ref>{{cite book|first1=F. J. C. |last1=Rossotti |first2=H. |last2=Rossotti |title=The Determination of Stability Constants |publisher=McGraw-Hill |date=1961}}</ref>
:<math> K_\mathrm{c} = \frac{K}{\Gamma} </math>
However, ''K''<sub>c</sub> will vary with ionic strength. If it is measured at a series of different ionic strengths, the value can be extrapolated to zero ionic strength.<ref name="davies"/> The concentration quotient obtained in this manner is known, paradoxically, as a thermodynamic equilibrium constant.

Before using a published value of an equilibrium constant in conditions of ionic strength different from the conditions used in its determination, the value should be adjusted.

===Metastable mixtures===

A mixture may appear to have no tendency to change, though it is not at equilibrium. For example, a mixture of [[Sulfur dioxide|SO<sub>2</sub>]] and [[oxygen|O<sub>2</sub>]] is [[Metastability|metastable]] as there is a [[Activation energy|kinetic barrier]] to formation of the product, [[Sulfur trioxide|SO<sub>3</sub>]].
:2&nbsp;SO<sub>2</sub> + O<sub>2</sub> {{eqm}} 2&nbsp;SO<sub>3</sub>

The barrier can be overcome when a [[Catalysis|catalyst]] is also present in the mixture as in the [[contact process]], but the catalyst does not affect the equilibrium concentrations.

Likewise, the formation of [[bicarbonate]] from [[carbon dioxide]] and [[water]] is very slow under normal conditions
:{{chem2|CO2 + 2 H2O <-> HCO3- + H3O+}}
but almost instantaneous in the presence of the catalytic [[enzyme]] [[carbonic anhydrase]].

==Pure substances==

When pure substances (liquids or solids) are involved in equilibria their activities do not appear in the equilibrium constant<ref name=CEM>{{cite encyclopedia |encyclopedia=Concise Encyclopedia Chemistry |date=1994 |isbn=0-89925-457-8 |url-access=registration |url=https://archive.org/details/conciseencyclope00eagl|title=Biochemistry (2nd Ed.) |last1=Eagleson |first1=Mary }}</ref> because their numerical values are considered one.

Applying the general formula for an equilibrium constant to the specific case of a dilute solution of acetic acid in water one obtains
:CH<sub>3</sub>CO<sub>2</sub>H + H<sub>2</sub>O {{eqm}} CH<sub>3</sub>CO<sub>2</sub><sup>−</sup> + H<sub>3</sub>O<sup>+</sup>

:<math>K_\mathrm{c}=\frac \mathrm{[{CH_3CO_2}^-] [{H_3O}^+]} \mathrm{[{CH_3CO_2H}] [{H_2O}]}</math>

For all but very concentrated solutions, the water can be considered a "pure" liquid, and therefore it has an activity of one. The equilibrium constant expression is therefore usually written as
:<math>K=\frac \mathrm{[{CH_3CO_2}^-] [{H_3O}^+]} \mathrm{[{CH_3CO_2H}]} = K_\mathrm{c}</math>.

A particular case is the [[self-ionization of water]]

:2&nbsp;H<sub>2</sub>O {{eqm}} H<sub>3</sub>O<sup>+</sup> + OH<sup>−</sup>

Because water is the solvent, and has an activity of one, the self-ionization constant of water is defined as

:<math>K_\mathrm{w} = \mathrm{[H^+] [OH^-]}</math>

It is perfectly legitimate to write [H<sup>+</sup>] for the [[hydronium ion]] concentration, since the state of [[solvation]] of the proton is constant (in dilute solutions) and so does not affect the equilibrium concentrations. ''K''<sub>w</sub> varies with variation in ionic strength and/or temperature.

The concentrations of H<sup>+</sup> and OH<sup>−</sup> are not independent quantities. Most commonly [OH<sup>−</sup>] is replaced by ''K''<sub>w</sub>[H<sup>+</sup>]<sup>−1</sup> in equilibrium constant expressions which would otherwise include [[hydroxide ion]].

Solids also do not appear in the equilibrium constant expression, if they are considered to be pure and thus their activities taken to be one. An example is the [[Boudouard reaction]]:<ref name=CEM/>

:2&nbsp;CO {{eqm}} CO<sub>2</sub> + C

for which the equation (without solid carbon) is written as:

:<math>K_\mathrm{c}=\frac \mathrm{[CO_2]} \mathrm{[CO]^2}</math>

== Equilibria Among Multiple Reactions==

Consider the case of a dibasic acid H<sub>2</sub>A. When dissolved in water, the mixture will contain H<sub>2</sub>A, HA<sup>−</sup> and A<sup>2−</sup>. This equilibrium can be split into two steps in each of which one proton is liberated.
:<math chem>\begin{array}{rl}
\ce{H2A <=> HA^- + H+} : & K_1=\frac \ce{[HA-] [H+]} \ce{[H2A]} \\
\ce{HA- <=> A^2- + H+} : & K_2=\frac \ce{[A^{2-}] [H+]} \ce{[HA-]}
\end{array}</math>
''K''<sub>1</sub> and'' K''<sub>2</sub> are examples of ''stepwise'' equilibrium constants. The ''overall'' equilibrium constant, ''β''<sub>D</sub>, is product of the stepwise constants.
:{{spaces|5}}<math chem>\beta_\ce{D} = \frac \ce{[A^{2-}] [H^+]^2} \ce{[H_2A]}=K_1K_2</math>
Note that these constants are [[Acid dissociation constant|dissociation constants]] because the products on the right hand side of the equilibrium expression are dissociation products. In many systems, it is preferable to use association constants.
:<math chem>\begin{array}{ll}
\ce{A^2- + H+ <=> HA-}:&\beta_1=\frac \ce{[HA^-]} \ce{[A^{2-}] [H+]} \\
\ce{A^2- + 2H+ <=> H2A}:&\beta_2=\frac \ce{[H2A]} \ce{[A^{2-}] [H+]^2}
\end{array}</math>
''β''<sub>1</sub> and ''β''<sub>2</sub> are examples of association constants. Clearly {{math|1=''β''<sub>1</sub>&nbsp;=&nbsp;{{sfrac|''K''<sub>2</sub>}}}} and {{math|1=''β''<sub>2</sub>&nbsp;=&nbsp;{{sfrac|''β''<sub>D</sub>}}}}; {{math|1=log&nbsp;''β''<sub>1</sub>&nbsp;=&nbsp;p''K''<sub>2</sub>}} and {{math|1=log&nbsp;''β''<sub>2</sub>&nbsp;=&nbsp;p''K''<sub>2</sub>&nbsp;+&nbsp;p''K''<sub>1</sub>}}<ref>{{cite book|first1=M. T. |last1=Beck |title=Chemistry of Complex Equilibria|edition=2nd|first2=I. |last2=Nagypál |publisher=Akadémiai Kaidó |location=Budapest |date=1990}}</ref>
For multiple equilibrium systems, also see: theory of [[Response reactions]].

==Effect of temperature==

The effect of changing temperature on an equilibrium constant is given by the [[van 't Hoff equation]]
:<math>\frac {d\ln K} {dT} = \frac{\Delta H_\mathrm{m}^\ominus} {RT^2}</math>
Thus, for [[exothermic]] reactions (Δ''H'' is negative), ''K'' decreases with an increase in temperature, but, for [[endothermic]] reactions, (Δ''H'' is positive) ''K'' increases with an increase in temperature.<ref>{{cite book|first1=Peter |last1=Atkins |first2=Julio |last2=De Paula |title=Atkins' Physical Chemistry |url=https://archive.org/details/atkinsphysicalch00pwat |url-access=registration |edition=8th |publisher=W. H. Freeman |date=2006 |isbn=0-7167-8759-8 |page=212}}</ref> An alternative formulation is
:<math>\frac {d\ln K} {d(T^{-1})} = -\frac{\Delta H_\mathrm{m}^\ominus} {R}</math>
At first sight this appears to offer a means of obtaining the standard molar enthalpy of the reaction by studying the variation of ''K'' with temperature. In practice, however, the method is unreliable because error propagation almost always gives very large errors on the values calculated in this way.

==Effect of electric and magnetic fields==
The effect of electric field on equilibrium has been studied by [[Manfred Eigen]]<ref>{{Cite web|url=https://www.nobelprize.org/prizes/chemistry/1967/eigen/facts/|title=The Nobel Prize in Chemistry 1967|website=NobelPrize.org|language=en-US|access-date=2019-11-02}}</ref><ref>{{Cite web|url=https://www.nobelprize.org/uploads/2018/06/eigen-lecture.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.nobelprize.org/uploads/2018/06/eigen-lecture.pdf |archive-date=2022-10-09 |url-status=live|title=Immeasurably fast reactions|last=Eigen|first=Manfred|date=December 11, 1967|website=Nobel Prize|access-date=November 2, 2019}}</ref> among others.{{cln|reason=And what are the effects???|date=February 2025}}

==Types of equilibrium==
{{Quote box|title=[[Haber–Bosch process]]
|quote=<div>
# N<sub>2</sub> (g) {{eqm}} N<sub>2</sub> [[Adsorption|(adsorbed)]]
# N<sub>2</sub> (adsorbed) {{eqm}} 2&nbsp;N (adsorbed)
# H<sub>2</sub> (g) {{eqm}} H<sub>2</sub> (adsorbed)
# H<sub>2</sub> (adsorbed) {{eqm}} 2&nbsp;H (adsorbed)
# N (adsorbed) + 3&nbsp;H(adsorbed) {{eqm}} NH<sub>3</sub> (adsorbed)
# NH<sub>3</sub> (adsorbed) {{eqm}} NH<sub>3</sub> (g)
</div>
}}
Equilibrium can be broadly classified as heterogeneous and homogeneous equilibrium.<ref>{{Cite web|url=https://www.chemguide.co.uk/physical/equilibria/kc.html|title = Equilibrium constants – Kc}}</ref> Homogeneous equilibrium consists of reactants and products belonging in the same phase whereas heterogeneous equilibrium comes into play for reactants and products in different phases.  
* In the gas phase: [[rocket engine]]s<ref name="nasa">{{cite web|publisher=NASA |series=NASA Reference publication 1311|last1=Gordon|first1=Sanford|last2=McBride|first2=Bonnie J. |date=1994 |url=https://www.grc.nasa.gov/WWW/CEAWeb/RP-1311.pdf |archive-url=https://web.archive.org/web/20060421143635/http://www.grc.nasa.gov/WWW/CEAWeb/RP-1311.pdf |url-status=dead |archive-date=2006-04-21 |title=Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications}}</ref>
* The industrial synthesis such as [[ammonia]] in the [[Haber–Bosch process]] (depicted right) takes place through a succession of equilibrium steps including [[adsorption]] processes
* [[Atmospheric chemistry]]
* Seawater and other natural waters: [[chemical oceanography]]
* Distribution between two phases
** [[Partition coefficient|log ''D'' distribution coefficient]]: important for pharmaceuticals where lipophilicity is a significant property of a drug
** [[Liquid–liquid extraction]], [[Ion exchange]], [[Chromatography]]
** [[Solubility equilibrium|Solubility product]]
** Uptake and release of oxygen by [[hemoglobin]] in blood
* Acid–base equilibria: [[acid dissociation constant]], [[hydrolysis]], [[buffer solution]]s, [[pH indicator|indicators]], [[acid–base homeostasis]]
* Metal–ligand complexation: [[Chelation|sequestering agents]], [[chelation therapy]], [[Magnetic resonance imaging|MRI contrast reagents]], [[Schlenk equilibrium]]
* Adduct formation: [[host–guest chemistry]], [[supramolecular chemistry]], [[molecular recognition]], [[dinitrogen tetroxide]]
* In certain [[oscillating reaction]]s, the approach to equilibrium is not asymptotically but in the form of a damped oscillation .<ref name="CEM"/>
* The related [[Nernst equation]] in electrochemistry gives the difference in electrode potential as a function of redox concentrations.
* When molecules on each side of the equilibrium are able to further react irreversibly in secondary reactions, the final product ratio is determined according to the [[Curtin–Hammett principle]].

In these applications, terms such as stability constant, formation constant, binding constant, affinity constant, association constant and dissociation constant are used. In biochemistry, it is common to give units for binding constants, which serve to define the concentration units used when the constant's value was determined.

==Composition of a mixture==

When the only equilibrium is that of the formation of a 1:1 adduct as the composition of a mixture, there are many ways that the composition of a mixture can be calculated. For example, see [[ICE table]] for a traditional method of calculating the pH of a solution of a weak acid.

There are three approaches to the general calculation of the composition of a mixture at equilibrium.

# The most basic approach is to manipulate the various equilibrium constants until the desired concentrations are expressed in terms of measured equilibrium constants (equivalent to measuring chemical potentials) and initial conditions.
# Minimize the Gibbs energy of the system.<ref>{{cite book|first1=W. R. |last1=Smith |first2=R. W. |last2=Missen |title=Chemical Reaction Equilibrium Analysis: Theory and Algorithms|publisher= Krieger Publishing |location=Malabar, FL|date=1991 |edition=Reprinted}}</ref><ref>{{cite web|url=http://www.mathtrek.com|title=Mathtrek Systems}}</ref>
# Satisfy the equation of [[mass balance]]. The equations of mass balance are simply statements that demonstrate that the total concentration of each reactant must be constant by the law of [[conservation of mass]].

===Mass-balance equations===

In general, the calculations are rather complicated or complex. For instance, in the case of a dibasic acid, H<sub>2</sub>A dissolved in water the two reactants can be specified as the [[conjugate base]], A<sup>2−</sup>, and the [[hydronium|proton]], H<sup>+</sup>. The following equations of mass-balance could apply equally well to a base such as [[Ethylenediamine|1,2-diaminoethane]], in which case the base itself is designated as the reactant A:
:<math>T_\mathrm{A} = \mathrm{[A] + [HA] +[H_2A]} \,</math>
:<math>T_\mathrm{H} = \mathrm{[H] + [HA] + 2[H_2A] - [OH]} \,</math>

with T<sub>A</sub> the total concentration of species A. Note that it is customary to omit the ionic charges when writing and using these equations.

When the equilibrium constants are known and the total concentrations are specified there are two equations in two unknown "free concentrations" [A] and [H]. This follows from the fact that [HA]&nbsp;=&nbsp;''β''<sub>1</sub>[A] [H], [H<sub>2</sub>A]&nbsp;=&nbsp;''β''<sub>2</sub>[A] [H]<sup>2</sup> and [OH]&nbsp;=&nbsp;''K''<sub>w</sub>[H]<sup>−1</sup>

:<math> T_\mathrm{A} = \mathrm{[A]} + \beta_1\mathrm{[A] [H]} + \beta_2\mathrm{[A] [H]}^2 \,</math>
:<math> T_\mathrm{H} = \mathrm{[H]} + \beta_1\mathrm{[A] [H]} + 2\beta_2\mathrm{[A] [H]}^2 - K_w[\mathrm H]^{-1} \,</math>

so the concentrations of the "complexes" are calculated from the free concentrations and the equilibrium constants.
General expressions applicable to all systems with two reagents, A and B would be
:<math>T_\mathrm{A}=[\mathrm A]+\sum_i p_i \beta_i[\mathrm A]^{p_i}[\mathrm B]^{q_i}</math>
:<math>T_\mathrm{B}=[\mathrm B]+\sum_i q_i \beta_i[\mathrm A]^{p_i}[\mathrm B]^{q_i}</math>

It is easy to see how this can be extended to three or more reagents.

====Polybasic acids====
[[File:Al hydrolysis.svg|thumb|upright=1.5|Species concentrations during [[hydrolysis]] of the [[aluminium]].]]
The composition of solutions containing reactants A and H is easy to calculate as a function of [[pH|p[H]]]. When [H] is known, the free concentration [A] is calculated from the mass-balance equation in A.

The diagram alongside, shows an example of the hydrolysis of the [[aluminium]] [[Lewis acid]] Al<sup>3+</sup><sub>(aq)</sub><ref>The diagram was created with the program [http://www.hyperquad.co.uk/hyss.htm HySS]</ref> shows the species concentrations for a 5&nbsp;×&nbsp;10<sup>−6</sup>&nbsp;M solution of an [[aluminium salt]] as a function of pH. Each concentration is shown as a percentage of the total aluminium.

====Solution and precipitation====

The diagram above illustrates the point that a [[Precipitation (chemistry)|precipitate]] that is not one of the main species in the solution equilibrium may be formed. At pH just below 5.5 the main species present in a 5&nbsp;μM solution of Al<sup>3+</sup> are [[aluminium hydroxide]]s Al(OH)<sup>2+</sup>, {{chem2|AlOH2+}} and {{chem2|Al13(OH)32(7+)}}, but on raising the pH [[aluminium hydroxide|Al(OH)<sub>3</sub>]] precipitates from the solution. This occurs because Al(OH)<sub>3</sub> has a very large [[lattice energy]]. As the pH rises more and more Al(OH)<sub>3</sub> comes out of solution. This is an example of [[Le Châtelier's principle]] in action: Increasing the concentration of the hydroxide ion causes more aluminium hydroxide to precipitate, which removes hydroxide from the solution. When the hydroxide concentration becomes sufficiently high the soluble aluminate, {{chem2|Al(OH)4−}}, is formed.

Another common instance where precipitation occurs is when a metal cation interacts with an anionic ligand to form an electrically neutral complex. If the complex is [[hydrophobic]], it will precipitate out of water. This occurs with the [[nickel]] ion Ni<sup>2+</sup> and [[dimethylglyoxime]], (dmgH<sub>2</sub>): in this case the lattice energy of the solid is not particularly large, but it greatly exceeds the energy of [[solvation]] of the molecule Ni(dmgH)<sub>2</sub>.

===Minimization of Gibbs energy===

At equilibrium, at a specified temperature and pressure, and with no external forces, the [[Gibbs free energy]] ''G'' is at a minimum:

:<math>dG= \sum_{j=1}^m \mu_j\,dN_j = 0</math>

where μ<sub>j</sub> is the [[chemical potential]] of molecular species ''j'', and ''N<sub>j</sub>'' is the amount of molecular species ''j''. It may be expressed in terms of [[thermodynamic activity]] as:

:<math>\mu_j = \mu_j^{\ominus} + RT\ln{A_j}</math>

where <math>\mu_j^{\ominus}</math> is the chemical potential in the standard state, ''R'' is the [[gas constant]] ''T'' is the absolute temperature, and ''A<sub>j</sub>'' is the activity.

For a closed system, no particles may enter or leave, although they may combine in various ways. The total number of atoms of each element will remain constant. This means that the minimization above must be subjected to the constraints:

:<math>\sum_{j=1}^m a_{ij}N_j=b_i^0</math>

where ''a<sub>ij</sub>'' is the number of atoms of element ''i'' in molecule ''j'' and ''b''{{su|b=''i''|p=0}} is the total number of atoms of element ''i'', which is a constant, since the system is closed. If there are a total of ''k'' types of atoms in the system, then there will be ''k'' such equations. If ions are involved, an additional row is added to the a<sub>ij</sub> matrix specifying the respective charge on each molecule which will sum to zero.

This is a standard problem in [[Optimization (mathematics)|optimisation]], known as [[constrained minimisation]]. The most common method of solving it is using the method of [[Lagrange multipliers]]<ref name="nasa_cea">{{cite web |url=https://www.grc.nasa.gov/WWW/CEAWeb/ |archive-url=https://web.archive.org/web/20000901045039/http://www.grc.nasa.gov/WWW/CEAWeb/ |url-status=dead |archive-date=September 1, 2000 |title=Chemical Equilibrium with Applications |publisher=NASA |access-date=October 5, 2019}}</ref><ref name="nasa"/> (although other methods may be used).

Define:

:<math>\mathcal{G}= G + \sum_{i=1}^k\lambda_i\left(\sum_{j=1}^m a_{ij}N_j-b_i^0\right)=0</math>

where the ''λ<sub>i</sub>'' are the Lagrange multipliers, one for each element. This allows each of the ''N<sub>j</sub>'' and ''λ<sub>j</sub>'' to be treated independently, and it can be shown using the tools of [[multivariate calculus]] that the equilibrium condition is given by

:<math>0 = \frac{\partial \mathcal{G}}{\partial N_j} = \mu_j + \sum_{i=1}^k \lambda_i a_{ij} </math>
:<math>0 = \frac{\partial \mathcal{G}}{\partial \lambda_i} = \sum_{j=1}^m a_{ij}N_j-b_i^0</math>

(For proof see [[Lagrange multipliers]].) This is a set of (''m''&nbsp;+&nbsp;''k'') equations in (''m''&nbsp;+&nbsp;''k'') unknowns (the ''N<sub>j</sub>'' and the ''λ<sub>i</sub>'') and may, therefore, be solved for the equilibrium concentrations ''N<sub>j</sub>'' as long as the chemical activities are known as functions of the concentrations at the given temperature and pressure. (In the ideal case, [[thermodynamic activity|activities]] are proportional to concentrations.) (See [[Thermodynamic databases for pure substances]].) Note that the second equation is just the initial constraints for minimization.

This method of calculating equilibrium chemical concentrations is useful for systems with a large number of different molecules. The use of ''k'' atomic element conservation equations for the mass constraint is straightforward, and replaces the use of the stoichiometric coefficient equations.<ref name="nasa"/> The results are consistent with those specified by chemical equations. For example, if equilibrium is specified by a single chemical equation:,<ref name="K&K">{{cite book|author=C. Kittel, H. Kroemer|title=Thermal Physics|edition=2|publisher=W. H. Freeman Company|year=1980|isbn=0-7167-1088-9|chapter=9}}</ref>

:<math>\sum_{j=0}^m \nu_j R_j=0</math>

where ν<sub>j</sub> is the stoichiometric coefficient for the ''j'' th molecule (negative for reactants, positive for products) and ''R<sub>j</sub>'' is the symbol for the ''j'' th molecule, a properly balanced equation will obey:

:<math>\sum_{j=1}^m a_{ij} \nu_j =0</math>

Multiplying the first equilibrium condition by ν<sub>j</sub> and using the above equation yields:

:<math>0 =\sum_{j=1}^m \nu_j \mu_j +  \sum_{j=1}^m \sum_{i=1}^k \nu_j \lambda_i a_{ij} = \sum_{j=1}^m \nu_j \mu_j   </math>

As above, defining ΔG

:<math>\Delta G=\sum_{j=1}^m \nu_j \mu_j = \sum_{j=1}^m \nu_j (\mu_j^{\ominus} + RT \ln(\{R_j\})) = \Delta G^{\ominus} + RT \ln\left(\prod_{j=1}^m \{R_j\}^{\nu_j}\right) =  \Delta G^{\ominus} + RT \ln(K_c)</math>

where ''K<sub>c</sub>'' is the [[equilibrium constant]], and ΔG will be zero at equilibrium.

Analogous procedures exist for the minimization of other [[thermodynamic potentials]].<ref name="nasa"/>

==See also==
{{col div|colwidth=30em}}
* [[Acidosis]]
* [[Alkalosis]]
* [[Arterial blood gas]]
* [[Benesi–Hildebrand method]]
* [[Determination of equilibrium constants]]
* [[Equilibrium constant]]
* [[Henderson–Hasselbalch equation]]
* [[Mass-action ratio]]
* [[Michaelis–Menten kinetics]]
* [[pCO2|pCO<sub>2</sub>]]
* [[pH]]
* [[pKa|p''K''<sub>a</sub>]]
* [[Standard electrode potential|Redox equilibria]]
* [[Steady state (chemistry)]]
* [[Thermodynamic databases for pure substances]]
* [[Non-random two-liquid model]] (NRTL model) – Phase equilibrium calculations
* [[UNIQUAC]] model – Phase equilibrium calculations
{{colend}}

==References==
{{Reflist|30em}}

==Further reading==
{{Library resources box
|by=no nnkllkn
|onlinebooks=no
|others=no
|about=yes
|label=Chemical equilibrium}}
* {{cite book|first1=F. |last1=Van Zeggeren |first2=S. H. |last2=Storey |title=The Computation of Chemical Equilibria |publisher=Cambridge University Press |date=1970}} Mainly concerned with gas-phase equilibria.
* {{cite book|editor-first=D. J. |editor-last=Leggett |title=Computational Methods for the Determination of Formation Constants|publisher= Plenum Press|date= 1985}}
* {{cite book|first1=A. E. |last1=Martell |first2=R. J. |last2=Motekaitis |title=The Determination and Use of Stability Constants |publisher=Wiley-VCH |date=1992}}

==External links==
* {{Commons category-inline|Chemical equilibria}}

{{Chemical equilibria}}
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[[Category:Equilibrium chemistry|*]]
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