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Equilibrium constant

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</style><div role="note" class="hatnote navigation-not-searchable">For experimental methods and computational details, see <a href="Determination_of_equilibrium_constants" title="Determination of equilibrium constants">Determination of equilibrium constants</a>.</div>

The equilibrium constant of a chemical reaction is the value of its <a href="Reaction_quotient" title="Reaction quotient">reaction quotient</a> at <a href="Chemical_equilibrium" title="Chemical equilibrium">chemical equilibrium</a>, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the <a href="Chemical_equilibrium#Composition_of_a_mixture" title="Chemical equilibrium">composition of the system at equilibrium</a>. However, reaction parameters like temperature, solvent, and <a href="Ionic_strength" title="Ionic strength">ionic strength</a> may all influence the value of the equilibrium constant.

A knowledge of equilibrium constants is essential for the understanding of many chemical systems, as well as the biochemical processes such as oxygen transport by <a href="Hemoglobin" title="Hemoglobin">hemoglobin</a> in blood and <a href="Acid%E2%80%93base_homeostasis" title="Acid–base homeostasis">acid–base homeostasis</a> in the human body.

<a href="Stability_constants_of_complexes" title="Stability constants of complexes">Stability constants</a>, formation constants, <a href="Binding_constant" title="Binding constant">binding constants</a>, association constants and <a href="Dissociation_constant" title="Dissociation constant">dissociation constants</a> are all types of equilibrium constants.

<div class="mw-heading mw-heading2"><h2 id="Basic_definitions_and_properties">Basic definitions and properties</h2></div>

For a system undergoing a <a href="Reversible_reaction" title="Reversible reaction">reversible reaction</a> described by the general <a href="Chemical_equation" title="Chemical equation">chemical equation</a>

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<annotation encoding="application/x-tex">{\displaystyle \alpha \,\mathrm {A} +\beta \,\mathrm {B} +\cdots \rightleftharpoons \rho \,\mathrm {R} +\sigma \,\mathrm {S} +\cdots }</annotation>

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</math><img src="./bc489a976b5875e8540df14669be3bcb7556c202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.714ex; height:2.676ex;" alt="{\displaystyle \alpha \,\mathrm {A} +\beta \,\mathrm {B} +\cdots \rightleftharpoons \rho \,\mathrm {R} +\sigma \,\mathrm {S} +\cdots }" loading="lazy"></dd></dl>

a thermodynamic equilibrium constant, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\ominus }}">

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<annotation encoding="application/x-tex">{\displaystyle K^{\ominus }}</annotation>

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</math><img src="./1f45734886910651b2143da8310f2b6a094327a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.509ex;" alt="{\displaystyle K^{\ominus }}" loading="lazy">, is defined to be the value of the <a href="Reaction_quotient" title="Reaction quotient">reaction quotient</a> Qt when forward and reverse reactions occur at the same rate. At <a href="Chemical_equilibrium" title="Chemical equilibrium">chemical equilibrium</a>, the chemical composition of the mixture does not change with time, and the <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> change <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta G}</annotation>

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</math><img src="./c2c9bc9950e25c28607799cdd9cb0af1373721fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.763ex; height:2.176ex;" alt="{\displaystyle \Delta G}" loading="lazy"> for the reaction is zero. If the composition of a mixture at equilibrium is changed by addition of some reagent, a new equilibrium position will be reached, given enough time. An equilibrium constant is related to the composition of the mixture at equilibrium by

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<annotation encoding="application/x-tex">{\displaystyle K^{\ominus }={\frac {\mathrm {\{R\}} ^{\rho }\mathrm {\{S\}} ^{\sigma }...}{\mathrm {\{A\}} ^{\alpha }\mathrm {\{B\}} ^{\beta }...}}={\frac {{[\mathrm {R} ]}^{\rho }{[\mathrm {S} ]}^{\sigma }...}{{[\mathrm {A} ]}^{\alpha }{[\mathrm {B} ]}^{\beta }...}}\times \Gamma ,}</annotation>

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</math><img src="./97bbd89a5f9a7b10aca1ae7191ba1228f4b5b112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.774ex; height:7.009ex;" alt="{\displaystyle K^{\ominus }={\frac {\mathrm {\{R\}} ^{\rho }\mathrm {\{S\}} ^{\sigma }...}{\mathrm {\{A\}} ^{\alpha }\mathrm {\{B\}} ^{\beta }...}}={\frac {{[\mathrm {R} ]}^{\rho }{[\mathrm {S} ]}^{\sigma }...}{{[\mathrm {A} ]}^{\alpha }{[\mathrm {B} ]}^{\beta }...}}\times \Gamma ,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma ={\frac {\gamma _{R}^{\rho }\gamma _{S}^{\sigma }...}{\gamma _{A}^{\alpha }\gamma _{B}^{\beta }...}},}</annotation>

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</math><img src="./64645d9f5dd2d5092932d2d1497285c001fd4f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.102ex; height:7.509ex;" alt="{\displaystyle \Gamma ={\frac {\gamma _{R}^{\rho }\gamma _{S}^{\sigma }...}{\gamma _{A}^{\alpha }\gamma _{B}^{\beta }...}},}" loading="lazy"></dd></dl>

where {X} denotes the <a href="Thermodynamic_activity" title="Thermodynamic activity">thermodynamic activity</a> of reagent X at equilibrium, [X] the numerical value <a href="#cite_note-Atkins7th-3">[3]</a> of the corresponding <a href="Molar_concentration" title="Molar concentration">concentration in moles per liter</a>, and γ the corresponding <a href="Activity_coefficient" title="Activity coefficient">activity coefficient</a>. If X is a gas, instead of [X] the numerical value of the <a href="Partial_pressure" title="Partial pressure">partial pressure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{X}}</annotation>

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</math><img src="./8348dd8ce7e6f7f4778ee01fa5bdc7b828afd98c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.125ex; height:2.509ex;" alt="{\displaystyle P_{X}}" loading="lazy"> in bar is used.<a href="#cite_note-Atkins7th-3">[3]</a> If it can be assumed that the quotient of activity coefficients, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">, is constant over a range of experimental conditions, such as pH, then an equilibrium constant can be derived as a quotient of concentrations.

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<annotation encoding="application/x-tex">{\displaystyle K_{c}=K^{\ominus }/\Gamma ={\frac {[\mathrm {R} ]^{\rho }[\mathrm {S} ]^{\sigma }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}.}</annotation>

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</math><img src="./6379f8a8565fe07860036bdbd906553fe1e2a298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.966ex; height:6.509ex;" alt="{\displaystyle K_{c}=K^{\ominus }/\Gamma ={\frac {[\mathrm {R} ]^{\rho }[\mathrm {S} ]^{\sigma }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}.}" loading="lazy"></dd></dl>

An equilibrium constant is related to the standard <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> change of reaction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G^{\ominus }}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }}</annotation>

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</math><img src="./783df781bd2cde06d7956470e952c27226a4e203.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.509ex;" alt="{\displaystyle \Delta G^{\ominus }}" loading="lazy"> by

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<annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }=-RT\ln K^{\ominus },}</annotation>

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</math><img src="./dc7208315cd87e2a06bc93b32afce0667066ba84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.545ex; height:2.843ex;" alt="{\displaystyle \Delta G^{\ominus }=-RT\ln K^{\ominus },}" loading="lazy"></dd></dl>

where R is the <a href="Gas_constant" title="Gas constant">universal gas constant</a>, T is the <a href="Thermodynamic_temperature" title="Thermodynamic temperature">absolute temperature</a> (in <a href="Kelvin" title="Kelvin">kelvins</a>), and ln is the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>. This expression implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\ominus }}">

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<annotation encoding="application/x-tex">{\displaystyle K^{\ominus }}</annotation>

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</math><img src="./1f45734886910651b2143da8310f2b6a094327a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.509ex;" alt="{\displaystyle K^{\ominus }}" loading="lazy"> must be a pure number and cannot have a dimension, since <a href="Logarithm" title="Logarithm">logarithms</a> can only be taken of pure numbers. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{c}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{c}}</annotation>

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</math><img src="./86e37b50e18c03bf957b06bf74cf6ed0c1a45939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.917ex; height:2.509ex;" alt="{\displaystyle K_{c}}" loading="lazy"> must also be a pure number. On the other hand, the <a href="Reaction_quotient" title="Reaction quotient">reaction quotient</a> at equilibrium

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<annotation encoding="application/x-tex">{\displaystyle {\frac {[\mathrm {R} ]^{\rho }[\mathrm {S} ]^{\sigma }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}\ {\text{(eq)}}}</annotation>

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</math><img src="./a03c186e5533939d87dd2bb9497ae9db83ad78b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.635ex; height:6.509ex;" alt="{\displaystyle {\frac {[\mathrm {R} ]^{\rho }[\mathrm {S} ]^{\sigma }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}\ {\text{(eq)}}}" loading="lazy"></dd></dl>

does have the dimension of concentration raised to some power (see <a href="#Dimensionality">§ Dimensionality</a>, below). Such reaction quotients are often referred to, in the biochemical literature, as equilibrium constants.

For an equilibrium mixture of gases, an equilibrium constant can be defined in terms of <a href="Partial_pressure" title="Partial pressure">partial pressure</a> or <a href="Fugacity" title="Fugacity">fugacity</a>.

An equilibrium constant is related to the forward and backward <a href="Reaction_rate_constant" title="Reaction rate constant">rate constants</a>, kf and kr of the reactions involved in reaching equilibrium:

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<annotation encoding="application/x-tex">{\displaystyle K^{\ominus }={\frac {k_{\text{f}}}{k_{\text{r}}}}.}</annotation>

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</math><img src="./798c681d02f978e62fc05ef1c6e19bbb242dc885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.274ex; height:5.843ex;" alt="{\displaystyle K^{\ominus }={\frac {k_{\text{f}}}{k_{\text{r}}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Types_of_equilibrium_constants">Types of equilibrium constants</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Cumulative_and_stepwise_formation_constants">Cumulative and stepwise formation constants</h3></div>

A cumulative or overall constant, given the symbol β, is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML2 is given by

The stepwise constant, K, for the formation of the same complex from ML and L is given by

It follows that

A cumulative constant can always be expressed as the product of stepwise constants. There is no agreed notation for stepwise constants, though a symbol such as KL ML is sometimes found in the literature. It is best always to define each stability constant by reference to an equilibrium expression.

<div class="mw-heading mw-heading4"><h4 id="Competition_method">Competition method</h4></div>

A particular use of a stepwise constant is in the determination of stability constant values outside the normal range for a given method. For example, <a href="EDTA" class="mw-redirect" title="EDTA">EDTA</a> complexes of many metals are outside the range for the potentiometric method. The stability constants for those complexes were determined by competition with a weaker ligand.

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<annotation encoding="application/x-tex">{\displaystyle [\mathrm {ML} ']=K{\frac {[\mathrm {ML} ][\mathrm {L} ']}{[\mathrm {L} ]}}=K{\frac {\beta _{\mathrm {ML} }[\mathrm {M} ][\mathrm {L} ][\mathrm {L} ']}{[\mathrm {L} ]}}=K\beta _{\mathrm {ML} }[\mathrm {M} ][\mathrm {L} '];\quad \beta _{\mathrm {ML} '}=K\beta _{\mathrm {ML} }}</annotation>

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</math><img src="./3d05c614f8ba11ecb1406d6e9da06a9ba14c52c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.875ex; height:6.509ex;" alt="{\displaystyle [\mathrm {ML} ']=K{\frac {[\mathrm {ML} ][\mathrm {L} ']}{[\mathrm {L} ]}}=K{\frac {\beta _{\mathrm {ML} }[\mathrm {M} ][\mathrm {L} ][\mathrm {L} ']}{[\mathrm {L} ]}}=K\beta _{\mathrm {ML} }[\mathrm {M} ][\mathrm {L} '];\quad \beta _{\mathrm {ML} '}=K\beta _{\mathrm {ML} }}" loading="lazy"></dd></dl>

The formation constant of <a href="Palladium(II)_cyanide" class="mw-redirect" title="Palladium(II) cyanide">[Pd(CN)4]2−</a> was determined by the competition method.

<div class="mw-heading mw-heading3"><h3 id="Association_and_dissociation_constants">Association and dissociation constants</h3></div>

In organic chemistry and biochemistry it is customary to use pKa values for <a href="Acid_dissociation_constant" title="Acid dissociation constant">acid dissociation</a> equilibria.

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {p} K_{\mathrm {a} }=-\log K_{\mathrm {diss} }=\log \left({\frac {1}{K_{\mathrm {diss} }}}\right)\,}</annotation>

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</math><img src="./be519de20716b0f6ebaaf7354f5682ae1fa3b287.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.432ex; height:6.176ex;" alt="{\displaystyle \mathrm {p} K_{\mathrm {a} }=-\log K_{\mathrm {diss} }=\log \left({\frac {1}{K_{\mathrm {diss} }}}\right)\,}" loading="lazy"></dd></dl>

where log denotes a logarithm to base 10 or <a href="Common_logarithm" title="Common logarithm">common logarithm</a>, and Kdiss is a stepwise <a href="Acid_dissociation_constant" title="Acid dissociation constant">acid dissociation constant</a>. For bases, the <a href="Acid_dissociation_constant#Bases_and_basicity" title="Acid dissociation constant">base association constant</a>, pKb is used. For any given acid or base the two constants are related by pKa + pKb = pKw, so pKa can always be used in calculations.

On the other hand, stability constants for <a href="Coordination_chemistry" class="mw-redirect" title="Coordination chemistry">metal complexes</a>, and binding constants for <a href="Host%E2%80%93guest_chemistry" title="Host–guest chemistry">host–guest</a> complexes are generally expressed as association constants. When considering equilibria such as

it is customary to use association constants for both ML and HL. Also, in generalized computer programs dealing with equilibrium constants it is general practice to use cumulative constants rather than stepwise constants and to omit ionic charges from equilibrium expressions. For example, if NTA, <a href="Nitrilotriacetic_acid" title="Nitrilotriacetic acid">nitrilotriacetic acid</a>, N(CH2CO2H)3 is designated as H3L and forms complexes ML and MHL with a metal ion M, the following expressions would apply for the dissociation constants.

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ll}{\ce {H3L <=> {H2L}+ H}};&{\ce {p}}K_{1}=-\log \left({\frac {[{\ce {H2L}}][{\ce {H}}]}{[{\ce {H3L}}]}}\right)\\{\ce {H2L <=> {HL}+ H}};&{\ce {p}}K_{2}=-\log \left({\frac {[{\ce {HL}}][{\ce {H}}]}{[{\ce {H2L}}]}}\right)\\{\ce {HL <=> {L}+ H}};&{\ce {p}}K_{3}=-\log \left({\frac {[{\ce {L}}][{\ce {H}}]}{[{\ce {HL}}]}}\right)\end{array}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}{\ce {H3L <=> {H2L}+ H}};&{\ce {p}}K_{1}=-\log \left({\frac {[{\ce {H2L}}][{\ce {H}}]}{[{\ce {H3L}}]}}\right)\\{\ce {H2L <=> {HL}+ H}};&{\ce {p}}K_{2}=-\log \left({\frac {[{\ce {HL}}][{\ce {H}}]}{[{\ce {H2L}}]}}\right)\\{\ce {HL <=> {L}+ H}};&{\ce {p}}K_{3}=-\log \left({\frac {[{\ce {L}}][{\ce {H}}]}{[{\ce {HL}}]}}\right)\end{array}}}</annotation>

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</math><img src="./2fe45058f5eb3d3c3bb1958a3e9aabdb1cad2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:43.803ex; height:15.509ex;" alt="{\displaystyle {\begin{array}{ll}{\ce {H3L <=> {H2L}+ H}};&{\ce {p}}K_{1}=-\log \left({\frac {[{\ce {H2L}}][{\ce {H}}]}{[{\ce {H3L}}]}}\right)\\{\ce {H2L <=> {HL}+ H}};&{\ce {p}}K_{2}=-\log \left({\frac {[{\ce {HL}}][{\ce {H}}]}{[{\ce {H2L}}]}}\right)\\{\ce {HL <=> {L}+ H}};&{\ce {p}}K_{3}=-\log \left({\frac {[{\ce {L}}][{\ce {H}}]}{[{\ce {HL}}]}}\right)\end{array}}}" loading="lazy"></dd></dl>

The cumulative association constants can be expressed as

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ll}{\ce {{L}+ H <=> HL}};&\log \beta _{011}=\log \left({\frac {[{\ce {HL}}]}{[{\ce {L}}][{\ce {H}}]}}\right)={\ce {p}}K_{3}\\{\ce {{L}+ 2H <=> H2L}};&\log \beta _{012}=\log \left({\frac {[{\ce {H2L}}]}{[{\ce {L}}][{\ce {H}}]^{2}}}\right)={\ce {p}}K_{3}+{\ce {p}}K_{2}\\{\ce {{L}+ 3H <=> H3L}};&\log \beta _{013}=\log \left({\frac {[{\ce {H3L}}]}{[{\ce {L}}][{\ce {H}}]^{3}}}\right)={\ce {p}}K_{3}+{\ce {p}}K_{2}+{\ce {p}}K_{1}\\{\ce {{M}+ L <=> ML}};&\log \beta _{110}=\log \left({\frac {[{\ce {ML}}]}{[{\ce {M}}][{\ce {L}}]}}\right)\\{\ce {{M}+ {L}+ H <=> MLH}};&\log \beta _{111}=\log \left({\frac {[{\ce {MLH}}]}{[{\ce {M}}][{\ce {L}}][{\ce {H}}]}}\right)\end{array}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}{\ce {{L}+ H <=> HL}};&\log \beta _{011}=\log \left({\frac {[{\ce {HL}}]}{[{\ce {L}}][{\ce {H}}]}}\right)={\ce {p}}K_{3}\\{\ce {{L}+ 2H <=> H2L}};&\log \beta _{012}=\log \left({\frac {[{\ce {H2L}}]}{[{\ce {L}}][{\ce {H}}]^{2}}}\right)={\ce {p}}K_{3}+{\ce {p}}K_{2}\\{\ce {{L}+ 3H <=> H3L}};&\log \beta _{013}=\log \left({\frac {[{\ce {H3L}}]}{[{\ce {L}}][{\ce {H}}]^{3}}}\right)={\ce {p}}K_{3}+{\ce {p}}K_{2}+{\ce {p}}K_{1}\\{\ce {{M}+ L <=> ML}};&\log \beta _{110}=\log \left({\frac {[{\ce {ML}}]}{[{\ce {M}}][{\ce {L}}]}}\right)\\{\ce {{M}+ {L}+ H <=> MLH}};&\log \beta _{111}=\log \left({\frac {[{\ce {MLH}}]}{[{\ce {M}}][{\ce {L}}][{\ce {H}}]}}\right)\end{array}}}</annotation>

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</math><img src="./87e8eda75f4bbf3d8d3a13d285a53c5f29022213.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.338ex; width:69.063ex; height:27.843ex;" alt="{\displaystyle {\begin{array}{ll}{\ce {{L}+ H <=> HL}};&\log \beta _{011}=\log \left({\frac {[{\ce {HL}}]}{[{\ce {L}}][{\ce {H}}]}}\right)={\ce {p}}K_{3}\\{\ce {{L}+ 2H <=> H2L}};&\log \beta _{012}=\log \left({\frac {[{\ce {H2L}}]}{[{\ce {L}}][{\ce {H}}]^{2}}}\right)={\ce {p}}K_{3}+{\ce {p}}K_{2}\\{\ce {{L}+ 3H <=> H3L}};&\log \beta _{013}=\log \left({\frac {[{\ce {H3L}}]}{[{\ce {L}}][{\ce {H}}]^{3}}}\right)={\ce {p}}K_{3}+{\ce {p}}K_{2}+{\ce {p}}K_{1}\\{\ce {{M}+ L <=> ML}};&\log \beta _{110}=\log \left({\frac {[{\ce {ML}}]}{[{\ce {M}}][{\ce {L}}]}}\right)\\{\ce {{M}+ {L}+ H <=> MLH}};&\log \beta _{111}=\log \left({\frac {[{\ce {MLH}}]}{[{\ce {M}}][{\ce {L}}][{\ce {H}}]}}\right)\end{array}}}" loading="lazy"></dd></dl>

Note how the subscripts define the stoichiometry of the equilibrium product.

<div class="mw-heading mw-heading3"><h3 id="Micro-constants">Micro-constants</h3></div>

When two or more sites in an asymmetrical molecule may be involved in an equilibrium reaction there are more than one possible equilibrium constants. For example, the molecule <a href="Levodopa" title="Levodopa">L-DOPA</a> has two non-equivalent hydroxyl groups which may be deprotonated. Denoting L-DOPA as LH2, the following diagram shows all the species that may be formed (X = CH 2CH(NH 2)CO 2H).

The concentration of the species LH is equal to the sum of the concentrations of the two micro-species with the same chemical formula, labelled L1H and L2H. The constant K2 is for a reaction with these two micro-species as products, so that [LH] = [L1H] + [L2H] appears in the numerator, and it follows that this macro-constant is equal to the sum of the two micro-constants for the component reactions.

However, the constant K1 is for a reaction with these two micro-species as reactants, and [LH] = [L1H] + [L2H] in the denominator, so that in this case<a href="#cite_note-4">[4]</a>

and therefore K1 =k11 k12 / (k11 + k12).

Thus, in this example there are four micro-constants whose values are subject to two constraints; in consequence, only the two macro-constant values, for K1 and K2 can be derived from experimental data.

Micro-constant values can, in principle, be determined using a spectroscopic technique, such as <a href="Infrared_spectroscopy" title="Infrared spectroscopy">infrared spectroscopy</a>, where each micro-species gives a different signal. Methods which have been used to estimate micro-constant values include

<ul><li>Chemical: blocking one of the sites, for example by methylation of a hydroxyl group, followed by determination of the equilibrium constant of the related molecule, from which the micro-constant value for the "parent" molecule may be estimated.</li>

<li>Mathematical: applying numerical procedures to 13C NMR data.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a></li></ul>

Although the value of a micro-constant cannot be determined from experimental data, site occupancy, which is proportional to the micro-constant value, can be very important for biological activity. Therefore, various methods have been developed for estimating micro-constant values. For example, the isomerization constant for L-DOPA has been estimated to have a value of 0.9, so the micro-species L1H and L2H have almost equal concentrations at all <a href="PH" title="PH">pH</a> values.

<div class="mw-heading mw-heading3"><h3 id="pH_considerations_(Brønsted_constants)">pH considerations (Brønsted constants)</h3></div>

<a href="PH" title="PH">pH</a> is defined in terms of the <a href="Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a> of the hydrogen ion

In the approximation of ideal behaviour, activity is replaced by concentration. pH is measured by means of a glass electrode, a mixed equilibrium constant, also known as a Brønsted constant, may result.

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {p} K=-\log \left({\frac {[\mathrm {L} ]\{\mathrm {H} \}}{[\mathrm {HL} ]}}\right)}</annotation>

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</math><img src="./d2e6c9a676961a7d6434161b8c39b0881dc7411a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.696ex; height:6.509ex;" alt="{\displaystyle \mathrm {p} K=-\log \left({\frac {[\mathrm {L} ]\{\mathrm {H} \}}{[\mathrm {HL} ]}}\right)}" loading="lazy"></dd></dl>

It all depends on whether the electrode is calibrated by reference to solutions of known activity or known concentration. In the latter case the equilibrium constant would be a concentration quotient. If the electrode is calibrated in terms of known hydrogen ion concentrations it would be better to write p[H] rather than pH, but this suggestion is not generally adopted.

<div class="mw-heading mw-heading3"><h3 id="Hydrolysis_constants">Hydrolysis constants</h3></div>

In aqueous solution the concentration of the hydroxide ion is related to the concentration of the hydrogen ion by

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<annotation encoding="application/x-tex">{\displaystyle {\ce {{\mathit {K}}_{W}=[H][OH]}}}</annotation>

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</math><img src="./9faddc62b25f1a8557d5d72c17cb8ac46b3095d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.398ex; height:3.009ex;" alt="{\displaystyle {\ce {{\mathit {K}}_{W}=[H][OH]}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\ce {[OH]={\mathit {K}}_{W}[H]^{-1}}}}</annotation>

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</math><img src="./45481f84a02758d3c3f2b96a4b132bce158d96c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.731ex; height:3.509ex;" alt="{\displaystyle {\ce {[OH]={\mathit {K}}_{W}[H]^{-1}}}}" loading="lazy"></dd></dl>

The first step in metal ion <a href="Hydrolysis" title="Hydrolysis">hydrolysis</a><a href="#cite_note-BM-7">[7]</a> can be expressed in two different ways

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\ce {M(H2O) <=> {M(OH)}+ H}};&[{\ce {M(OH)}}]=\beta ^{*}[{\ce {M}}][{\ce {H}}]^{-1}\\{\ce {{M}+ OH <=> M(OH)}};&[{\ce {M(OH)}}]=K[{\ce {M}}][{\ce {OH}}]=KK_{{\ce {W}}}[{\ce {M}}][{\ce {H}}]^{-1}\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\ce {M(H2O) <=> {M(OH)}+ H}};&[{\ce {M(OH)}}]=\beta ^{*}[{\ce {M}}][{\ce {H}}]^{-1}\\{\ce {{M}+ OH <=> M(OH)}};&[{\ce {M(OH)}}]=K[{\ce {M}}][{\ce {OH}}]=KK_{{\ce {W}}}[{\ce {M}}][{\ce {H}}]^{-1}\end{cases}}}</annotation>

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</math><img src="./d902ebd08f7a075b8e7f1d7aebe20dd0fd86dc6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:70.805ex; height:6.509ex;" alt="{\displaystyle {\begin{cases}{\ce {M(H2O) <=> {M(OH)}+ H}};&[{\ce {M(OH)}}]=\beta ^{*}[{\ce {M}}][{\ce {H}}]^{-1}\\{\ce {{M}+ OH <=> M(OH)}};&[{\ce {M(OH)}}]=K[{\ce {M}}][{\ce {OH}}]=KK_{{\ce {W}}}[{\ce {M}}][{\ce {H}}]^{-1}\end{cases}}}" loading="lazy"></dd></dl>

It follows that β* = KKW. Hydrolysis constants are usually reported in the β* form and therefore often have values much less than 1. For example, if log K = 4 and log KW = −14, log β* = 4 + (−14) = −10 so that β* = 10−10. In general when the hydrolysis product contains n hydroxide groups log β* = log K + n log KW

<div class="mw-heading mw-heading3"><h3 id="Conditional_constants">Conditional constants</h3></div>

Conditional constants, also known as apparent constants, are concentration quotients which are not true equilibrium constants but can be derived from them.<a href="#cite_note-8">[8]</a> A very common instance is where pH is fixed at a particular value. For example, in the case of <a href="Iron(III)" class="mw-redirect" title="Iron(III)">iron(III)</a> interacting with <a href="EDTA" class="mw-redirect" title="EDTA">EDTA</a>, a conditional constant could be defined by

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<mtext>Total Fe bound to EDTA</mtext>

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<mtext>Total Fe not bound to EDTA</mtext>

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<annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {cond} }={\frac {[{\mbox{Total Fe bound to EDTA}}]}{[{\mbox{Total Fe not bound to EDTA}}]\times [{\mbox{Total EDTA not bound to Fe}}]}}}</annotation>

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</math><img src="./437377d744f8797db06771b9161cbb4d281db54d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:74.041ex; height:6.509ex;" alt="{\displaystyle K_{\mathrm {cond} }={\frac {[{\mbox{Total Fe bound to EDTA}}]}{[{\mbox{Total Fe not bound to EDTA}}]\times [{\mbox{Total EDTA not bound to Fe}}]}}}" loading="lazy"></dd></dl>

This conditional constant will vary with pH. It has a maximum at a certain pH. That is the pH where the ligand sequesters the metal most effectively.

In biochemistry equilibrium constants are often measured at a pH fixed by means of a <a href="Buffer_solution" title="Buffer solution">buffer solution</a>. Such constants are, by definition, conditional and different values may be obtained when using different buffers.

<div class="mw-heading mw-heading3"><h3 id="Gas-phase_equilibria">Gas-phase equilibria</h3></div>

For equilibria in a <a href="Gas_phase" class="mw-redirect" title="Gas phase">gas phase</a>, <a href="Fugacity" title="Fugacity">fugacity</a>, f, is used in place of activity. However, fugacity has the <a href="Dimension" title="Dimension">dimension</a> of <a href="Pressure" title="Pressure">pressure</a>, so it must be divided by a standard pressure, usually 1 bar, in order to produce a dimensionless quantity, <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠f/p<s>o</s>⁠. An equilibrium constant is expressed in terms of the dimensionless quantity. For example, for the equilibrium 2NO2 ⇌ N2O4,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {f_{\mathrm {N_{2}O_{4}} }}{p^{\ominus }}}=K\left({\frac {f_{\mathrm {NO_{2}} }}{p^{\ominus }}}\right)^{2}}</annotation>

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</math><img src="./9c338b80a3a7aa4f96cec85dc4cac50a76b6e7a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.572ex; height:6.676ex;" alt="{\displaystyle {\frac {f_{\mathrm {N_{2}O_{4}} }}{p^{\ominus }}}=K\left({\frac {f_{\mathrm {NO_{2}} }}{p^{\ominus }}}\right)^{2}}" loading="lazy"></dd></dl>

Fugacity is related to <a href="Partial_pressure" title="Partial pressure">partial pressure</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{X}}</annotation>

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</math><img src="./4bc900e2770ed796e420eb5aa0852193a1919ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="{\displaystyle p_{X}}" loading="lazy">, by a dimensionless fugacity coefficient ϕ: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}=\phi _{X}p_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{X}=\phi _{X}p_{X}}</annotation>

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</math><img src="./fcf0858310d492251dc46d5478fd5438acd41cfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.69ex; height:2.509ex;" alt="{\displaystyle f_{X}=\phi _{X}p_{X}}" loading="lazy">. Thus, for the example,

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {\phi _{\mathrm {N_{2}O_{4}} }p_{\mathrm {N_{2}O_{4}} }/{p^{\ominus }}}{\left(\phi _{\mathrm {NO_{2}} }p_{\mathrm {NO_{2}} }/{p^{\ominus }}\right)^{2}}}}</annotation>

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</math><img src="./2300297f8fca7ccf11eb1d23a331c90729b8e5cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.411ex; height:7.176ex;" alt="{\displaystyle K={\frac {\phi _{\mathrm {N_{2}O_{4}} }p_{\mathrm {N_{2}O_{4}} }/{p^{\ominus }}}{\left(\phi _{\mathrm {NO_{2}} }p_{\mathrm {NO_{2}} }/{p^{\ominus }}\right)^{2}}}}" loading="lazy"></dd></dl>

Usually the standard pressure is omitted from such expressions. Expressions for equilibrium constants in the gas phase then resemble the expression for solution equilibria with fugacity coefficient in place of activity coefficient and partial pressure in place of concentration.

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {\phi _{\mathrm {N_{2}O_{4}} }p_{\mathrm {N_{2}O_{4}} }}{\left(\phi _{\mathrm {NO_{2}} }p_{\mathrm {NO_{2}} }\right)^{2}}}}</annotation>

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</math><img src="./7f227aa266a8217a4ccab87792557342d6a8e1ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.569ex; height:7.009ex;" alt="{\displaystyle K={\frac {\phi _{\mathrm {N_{2}O_{4}} }p_{\mathrm {N_{2}O_{4}} }}{\left(\phi _{\mathrm {NO_{2}} }p_{\mathrm {NO_{2}} }\right)^{2}}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Thermodynamic_basis_for_equilibrium_constant_expressions">Thermodynamic basis for equilibrium constant expressions</h2></div>

<a href="Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a> is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant temperature and pressure the <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> is minimum.<a href="#cite_note-9">[9]</a> The slope of the reaction free energy with respect to the <a href="Extent_of_reaction" title="Extent of reaction">extent of reaction</a>, ξ, is zero when the free energy is at its minimum value.

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial G}{\partial \xi }}\right)_{T,P}=0}</annotation>

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</math><img src="./2efcb28e8d54d845e7ad90aae372d33fe0c45b80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.744ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial G}{\partial \xi }}\right)_{T,P}=0}" loading="lazy"></dd></dl>

The free energy change, dGr, can be expressed as a weighted sum of change in amount times the <a href="Chemical_potential" title="Chemical potential">chemical potential</a>, the partial molar free energy of the species. The chemical potential, μi, of the ith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, Ni

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<annotation encoding="application/x-tex">{\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P}}</annotation>

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</math><img src="./9c995be7d49a3bb55b5f0948247714376981617b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.622ex; height:6.509ex;" alt="{\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P}}" loading="lazy"></dd></dl>

A general chemical equilibrium can be written as

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<annotation encoding="application/x-tex">{\displaystyle \sum _{j}n_{j}\mathrm {Reactant} _{j}\rightleftharpoons \sum _{k}m_{k}\mathrm {Product} _{k}}</annotation>

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</math><img src="./5fa1f6de7b7f3ac1b48418959b77e5173fd406b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:35.911ex; height:5.843ex;" alt="{\displaystyle \sum _{j}n_{j}\mathrm {Reactant} _{j}\rightleftharpoons \sum _{k}m_{k}\mathrm {Product} _{k}}" loading="lazy">      </dd></dl>

where nj are the <a href="Stoichiometric_coefficient" class="mw-redirect" title="Stoichiometric coefficient">stoichiometric coefficients</a> of the reactants in the equilibrium equation, and mj are the coefficients of the products. At equilibrium

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k}m_{k}\mu _{k}=\sum _{j}n_{j}\mu _{j}}</annotation>

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</math><img src="./f36cee7abe4b03994366968fbe031f2210cc4261.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.818ex; height:5.843ex;" alt="{\displaystyle \sum _{k}m_{k}\mu _{k}=\sum _{j}n_{j}\mu _{j}}" loading="lazy"></dd></dl>

The chemical potential, μi, of the ith species can be calculated in terms of its <a href="Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a>, ai.

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<annotation encoding="application/x-tex">{\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln a_{i}}</annotation>

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</math><img src="./aa099859e0ef3ac187325e038b5ddee76fd4e55f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.196ex; height:3.176ex;" alt="{\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln a_{i}}" loading="lazy"></dd></dl>

μ<s>o</s> i is the standard chemical potential of the species, R is the <a href="Gas_constant" title="Gas constant">gas constant</a> and T is the temperature. Setting the sum for the reactants j to be equal to the sum for the products, k, so that δGr(Eq) = 0

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<annotation encoding="application/x-tex">{\displaystyle \sum _{j}n_{j}(\mu _{j}^{\ominus }+RT\ln a_{j})=\sum _{k}m_{k}(\mu _{k}^{\ominus }+RT\ln a_{k})}</annotation>

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</math><img src="./0f1cdfb255fd488ee9b808daeb010de592de883a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.826ex; height:5.843ex;" alt="{\displaystyle \sum _{j}n_{j}(\mu _{j}^{\ominus }+RT\ln a_{j})=\sum _{k}m_{k}(\mu _{k}^{\ominus }+RT\ln a_{k})}" loading="lazy"></dd></dl>

Rearranging the terms,

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }=-RT\left(\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}}\right)}</annotation>

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</math><img src="./4c6df8812a48339142f4ffd04408a8310cb24db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.861ex; height:7.676ex;" alt="{\displaystyle \sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }=-RT\left(\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}}\right)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }=-RT\ln K.}</annotation>

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</math><img src="./8c3673ccfb4184c15bc5029efd4f403c90831d5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.007ex; height:2.676ex;" alt="{\displaystyle \Delta G^{\ominus }=-RT\ln K.}" loading="lazy"></dd></dl>

This relates the <a href="Standard_state" title="Standard state">standard</a> Gibbs free energy change, ΔG<s>o</s> to an equilibrium constant, K, the <a href="Reaction_quotient" title="Reaction quotient">reaction quotient</a> of activity values at equilibrium.

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<annotation encoding="application/x-tex">{\displaystyle \Delta G^{\ominus }=\sum _{k}m_{k}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ln K=\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}};K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}\equiv {\frac {{\{\mathrm {R} \}}^{\rho }{\{\mathrm {S} \}}^{\sigma }...}{{\{\mathrm {A} \}}^{\alpha }{\{\mathrm {B} \}}^{\beta }...}}}</annotation>

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</math><img src="./22c7a52b82f330febaa240f82f82594f579dd6bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:64.097ex; height:7.176ex;" alt="{\displaystyle \ln K=\sum _{k}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}};K={\frac {\prod _{k}{a_{k}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}\equiv {\frac {{\{\mathrm {R} \}}^{\rho }{\{\mathrm {S} \}}^{\sigma }...}{{\{\mathrm {A} \}}^{\alpha }{\{\mathrm {B} \}}^{\beta }...}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Equivalence_of_thermodynamic_and_kinetic_expressions_for_equilibrium_constants">Equivalence of thermodynamic and kinetic expressions for equilibrium constants</h3></div>

At equilibrium the rate of the forward reaction is equal to the backward reaction rate. A simple reaction, such as <a href="Ester_hydrolysis" title="Ester hydrolysis">ester hydrolysis</a>

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<annotation encoding="application/x-tex">{\displaystyle {\ce {AB + H2O <=> AH + B(OH)}}}</annotation>

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</math><img src="./2afc770913f4b79fd931eb76ab0381960b4730a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.815ex; height:3.343ex;" alt="{\displaystyle {\ce {AB + H2O <=> AH + B(OH)}}}" loading="lazy"></dd></dl>

has reaction rates given by expressions

<mtext>forward rate</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{forward rate}}=k_{f}{\ce {[AB][H2O]}}}</annotation>

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</math><img src="./f1b21d8da528047145d1da206a4789ad3b2b6119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.45ex; height:3.009ex;" alt="{\displaystyle {\text{forward rate}}=k_{f}{\ce {[AB][H2O]}}}" loading="lazy"></dd>

<mtext>backward rate</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{backward rate}}=k_{b}{\ce {[AH][B(OH)]}}}</annotation>

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</math><img src="./1029a06a2c6d73da37516605438d64daf0248ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.678ex; height:2.843ex;" alt="{\displaystyle {\text{backward rate}}=k_{b}{\ce {[AH][B(OH)]}}}" loading="lazy"></dd></dl>

According to <a href="Cato_Maximilian_Guldberg" title="Cato Maximilian Guldberg">Guldberg</a> and <a href="Peter_Waage" title="Peter Waage">Waage</a>, equilibrium is attained when the forward and backward reaction rates are equal to each other. In these circumstances, an equilibrium constant is defined to be equal to the ratio of the forward and backward reaction rate constants

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {k_{f}}{k_{b}}}={\frac {{\ce {[AH][B(OH)]}}}{{\ce {[AB][H2O]}}}}}</annotation>

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</math><img src="./29ed8c668a2ca8b82e92d079faeeea81f42c693b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.362ex; height:6.676ex;" alt="{\displaystyle K={\frac {k_{f}}{k_{b}}}={\frac {{\ce {[AH][B(OH)]}}}{{\ce {[AB][H2O]}}}}}" loading="lazy">.</dd></dl>

The concentration of water may be taken to be constant, resulting in the simpler expression

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<annotation encoding="application/x-tex">{\displaystyle K^{c}={\frac {{\ce {[AH][B(OH)]}}}{{\ce {[AB]}}}}}</annotation>

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</math><img src="./88bdd5add8ce02f4f3f13912ca0e8024708ec4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.052ex; height:6.509ex;" alt="{\displaystyle K^{c}={\frac {{\ce {[AH][B(OH)]}}}{{\ce {[AB]}}}}}" loading="lazy">.</dd></dl>

This particular concentration quotient, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{c}}">

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<annotation encoding="application/x-tex">{\displaystyle K^{c}}</annotation>

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</math><img src="./78c515d97aba924991c647596862c67aea7ec112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.038ex; height:2.343ex;" alt="{\displaystyle K^{c}}" loading="lazy">, has the dimension of concentration, but the thermodynamic equilibrium constant, K, is always dimensionless.

<div class="mw-heading mw-heading2"><h2 id="Unknown_activity_coefficient_values">Unknown activity coefficient values</h2></div>

It is very rare for activity coefficient values to have been determined experimentally for a system at equilibrium. There are three options for dealing with the situation where activity coefficient values are not known from experimental measurements.

<ol><li>Use calculated activity coefficients, together with concentrations of reactants. For equilibria in solution estimates of the activity coefficients of charged species can be obtained using <a href="Debye%E2%80%93H%C3%BCckel_theory" title="Debye–Hückel theory">Debye–Hückel theory</a>, an extended version, or <a href="SIT_theory" class="mw-redirect" title="SIT theory">SIT theory</a>. For uncharged species, the activity coefficient γ0 mostly follows a "salting-out" model: log10 γ0 = bI where I stands for <a href="Ionic_strength" title="Ionic strength">ionic strength</a>.<a href="#cite_note-Butler-10">[10]</a></li>

<li>Assume that the activity coefficients are all equal to 1. This is acceptable when all concentrations are very low.</li>

<li>For equilibria in solution use a medium of high ionic strength. In effect this redefines the <a href="Standard_state" title="Standard state">standard state</a> as referring to the medium. Activity coefficients in the standard state are, by definition, equal to 1. The value of an equilibrium constant determined in this manner is dependent on the ionic strength. When published constants refer to an ionic strength other than the one required for a particular application, they may be adjusted by means of specific ion theory (SIT) and other theories.<a href="#cite_note-11">[11]</a></li></ol>

<div class="mw-heading mw-heading2"><h2 id="Dimensionality">Dimensionality</h2></div>

An equilibrium constant is related to the standard <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> of reaction change, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{R}G^{\ominus }}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{R}G^{\ominus }}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{R}G^{\ominus }=\left({\frac {\partial G}{\partial \xi }}\right)_{P,T}=-RT\ln K.}</annotation>

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</math><img src="./831957d75fb54354ee43b5f0e4c6f11844965221.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.068ex; height:6.509ex;" alt="{\displaystyle \Delta _{R}G^{\ominus }=\left({\frac {\partial G}{\partial \xi }}\right)_{P,T}=-RT\ln K.}" loading="lazy"></dd></dl>

Therefore, K, must be a <a href="Dimensionless_quantity" title="Dimensionless quantity">dimensionless number</a> from which a logarithm can be derived. In the case of a simple equilibrium

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<annotation encoding="application/x-tex">{\displaystyle {\ce {A + B <=> AB,}}}</annotation>

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</math><img src="./5dd41f94af59b59060064019b4eb5c0fc8222483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.913ex; height:3.009ex;" alt="{\displaystyle {\ce {A + B <=> AB,}}}" loading="lazy"></dd></dl>

the thermodynamic equilibrium constant is defined in terms of the <a href="Thermodynamic_activity" title="Thermodynamic activity">activities</a>, {AB}, {A} and {B}, of the species in equilibrium with each other:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {\{AB\}}{\{A\}\{B\}}}.}</annotation>

</semantics>

</math><img src="./611b4868f7a3bcb6a0e982e6d530472da5dc3554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.804ex; height:6.509ex;" alt="{\displaystyle K={\frac {\{AB\}}{\{A\}\{B\}}}.}" loading="lazy"></dd></dl>

Now, each activity term can be expressed as a product of a concentration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X]}</annotation>

</semantics>

</math><img src="./fadb3aef0836cb1d004479f470703a45972bf8fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.274ex; height:2.843ex;" alt="{\displaystyle [X]}" loading="lazy"> and a corresponding <a href="Activity_coefficient" title="Activity coefficient">activity coefficient</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (X)}</annotation>

</semantics>

</math><img src="./5183c2b567264ef8efc7da9741db9ac33d414fa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.052ex; height:2.843ex;" alt="{\displaystyle \gamma (X)}" loading="lazy">. Therefore,

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {[AB]}{[A][B]}}\times {\frac {\gamma (AB)}{\gamma (A)\gamma (B)}}={\frac {[AB]}{[A][B]}}\times \Gamma .}</annotation>

</semantics>

</math><img src="./7095de604e4ad6b49e38ce5d9544212c2694618d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.391ex; height:6.509ex;" alt="{\displaystyle K={\frac {[AB]}{[A][B]}}\times {\frac {\gamma (AB)}{\gamma (A)\gamma (B)}}={\frac {[AB]}{[A][B]}}\times \Gamma .}" loading="lazy"></dd></dl>

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">, the quotient of activity coefficients, is set equal to 1, we get

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {[AB]}{[A][B]}}.}</annotation>

</semantics>

</math><img src="./1798b4ffcfefe15308f2f9e849eaeeec245fa31f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.742ex; height:6.509ex;" alt="{\displaystyle K={\frac {[AB]}{[A][B]}}.}" loading="lazy"></dd></dl>

K then appears to have the dimension of 1/concentration. This is what usually happens in practice when an equilibrium constant is calculated as a quotient of concentration values. This can be avoided by dividing each concentration by its standard-state value (usually mol/L or bar), which is standard practice in chemistry.<a href="#cite_note-Atkins7th-3">[3]</a>

The assumption underlying this practice is that the quotient of activities is constant under the conditions in which the equilibrium constant value is determined. These conditions are usually achieved by keeping the reaction temperature constant and by using a medium of relatively high <a href="Ionic_strength" title="Ionic strength">ionic strength</a> as the solvent. It is not unusual, particularly in texts relating to biochemical equilibria, to see an equilibrium constant value quoted with a dimension. The justification for this practice is that the concentration scale used may be either mol dm−3 or mmol dm−3, so that the concentration unit has to be stated in order to avoid there being any ambiguity.

Note. When the concentration values are measured on the <a href="Mole_fraction" title="Mole fraction">mole fraction</a> scale all concentrations and activity coefficients are dimensionless quantities.

In general equilibria between two reagents can be expressed as

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {{{\mathit {p}}A}+{\mathit {q}}B<=>A_{\mathit {p}}B_{\mathit {q}},}}}">

</mrow>

</mrow>

</mrow>

<mover>

</mrow>

</mrow>

</mpadded>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\ce {{{\mathit {p}}A}+{\mathit {q}}B<=>A_{\mathit {p}}B_{\mathit {q}},}}}</annotation>

</semantics>

</math><img src="./93bdeee800dfb3bd59a83df88a9429ce134b4f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.231ex; height:3.176ex;" alt="{\displaystyle {\ce {{{\mathit {p}}A}+{\mathit {q}}B<=>A_{\mathit {p}}B_{\mathit {q}},}}}" loading="lazy"></dd></dl>

in which case the equilibrium constant is defined, in terms of numerical concentration values, as

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {[{\ce {A}}_{p}{\ce {B}}_{q}]}{[{\ce {A}}]^{p}[{\ce {B}}]^{q}}}.}</annotation>

</semantics>

</math><img src="./c52d11c0f5b77a87a5aaca226626552ef1e79c85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.671ex; height:6.509ex;" alt="{\displaystyle K={\frac {[{\ce {A}}_{p}{\ce {B}}_{q}]}{[{\ce {A}}]^{p}[{\ce {B}}]^{q}}}.}" loading="lazy"></dd></dl>

The apparent dimension of this K value is concentration1−p−q; this may be written as M(1−p−q) or mM(1−p−q), where the symbol M signifies a <a href="Molar_concentration" title="Molar concentration">molar concentration</a> (1M = 1 mol dm−3). The apparent dimension of a <a href="Dissociation_constant" title="Dissociation constant">dissociation constant</a> is the reciprocal of the apparent dimension of the corresponding <a href="Association_constant" class="mw-redirect" title="Association constant">association constant</a>, and vice versa.

When discussing the <a href="Chemical_thermodynamics" title="Chemical thermodynamics">thermodynamics</a> of chemical equilibria it is necessary to take dimensionality into account. There are two possible approaches.

<ol><li>Set the dimension of Γ to be the reciprocal of the dimension of the concentration quotient. This is almost universal practice in the field of stability constant determinations. The "equilibrium constant" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {K}{\Gamma }}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {K}{\Gamma }}}</annotation>

</semantics>

</math><img src="./2709aedf2ea66284eb53d9aad1e805fc54560c4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.902ex; height:5.176ex;" alt="{\displaystyle {\frac {K}{\Gamma }}}" loading="lazy">, is dimensionless. It will be a function of the ionic strength of the medium used for the determination. Setting the numerical value of Γ to be 1 is equivalent to re-defining the <a href="Standard_state" title="Standard state">standard states</a>.</li>

<li>Replace each concentration term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X]}</annotation>

</semantics>

</math><img src="./fadb3aef0836cb1d004479f470703a45972bf8fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.274ex; height:2.843ex;" alt="{\displaystyle [X]}" loading="lazy"> by the dimensionless quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {[X]}{[X^{0}]}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {[X]}{[X^{0}]}}}</annotation>

</semantics>

</math><img src="./6927ad27387845e0bd8ffbe6ec6dd4e2a4c3c51a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:5.181ex; height:6.509ex;" alt="{\displaystyle {\frac {[X]}{[X^{0}]}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X^{0}]}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X^{0}]}</annotation>

</semantics>

</math><img src="./f9218daa96faf3c4ac4d9a1e6c9fb3c14577e72e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.345ex; height:3.176ex;" alt="{\displaystyle [X^{0}]}" loading="lazy"> is the concentration of reagent X in its standard state (usually 1 mol/L or 1 bar).<a href="#cite_note-Atkins7th-3">[3]</a> By definition the numerical value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (X^{0})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (X^{0})}</annotation>

</semantics>

</math><img src="./02107b156ccdb31dc24fde2af4ce8d94d9a10b14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.123ex; height:3.176ex;" alt="{\displaystyle \gamma (X^{0})}" loading="lazy"> is 1, so Γ also has a numerical value of 1.</li></ol>

In both approaches the numerical value of the stability constant is unchanged. The first is more useful for practical purposes; in fact, the unit of the concentration quotient is often attached to a published stability constant value in the biochemical literature. The second approach is consistent with the standard exposition of <a href="Debye%E2%80%93H%C3%BCckel_theory" title="Debye–Hückel theory">Debye–Hückel theory</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (AB)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (AB)}</annotation>

</semantics>

</math><img src="./95a4eae545381d4a560396c267ad3081a3c12e55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.579ex; height:2.843ex;" alt="{\displaystyle \gamma (AB)}" loading="lazy">, etc. are taken to be pure numbers.

<div class="mw-heading mw-heading2"><h2 id="Water_as_both_reactant_and_solvent">Water as both reactant and solvent</h2></div>

For reactions in aqueous solution, such as an acid dissociation reaction

the concentration of water may be taken as being constant and the formation of the <a href="Hydronium_ion" class="mw-redirect" title="Hydronium ion">hydronium ion</a> is implicit.

Water concentration is omitted from expressions defining equilibrium constants, except when solutions are very concentrated.

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {[A][H]}{[AH]}}}</annotation>

</semantics>

</math><img src="./da5bec204ec1316df7b4dc4d9b598adbaf17caed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.395ex; height:6.509ex;" alt="{\displaystyle K={\frac {[A][H]}{[AH]}}}" loading="lazy"> (K defined as a dissociation constant)</dd></dl>

Similar considerations apply to <a href="#Hydrolysis_constants">metal ion hydrolysis reactions</a>.

<div class="mw-heading mw-heading2"><h2 id="Enthalpy_and_entropy:_temperature_dependence">Enthalpy and entropy: temperature dependence</h2></div>

If both the equilibrium constant, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> and the <a href="Standard_enthalpy_of_reaction" title="Standard enthalpy of reaction">standard enthalpy change</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H^{\ominus }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta H^{\ominus }}</annotation>

</semantics>

</math><img src="./e7ca2ac7d655e35bf2076604abc115a3fe696542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.55ex; height:2.509ex;" alt="{\displaystyle \Delta H^{\ominus }}" loading="lazy">, for a reaction have been determined experimentally, the <a href="Standard_molar_entropy#Chemistry" title="Standard molar entropy">standard entropy change</a> for the reaction is easily derived. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G=\Delta H-T\Delta S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta G=\Delta H-T\Delta S}</annotation>

</semantics>

</math><img src="./062377921bd620d44f679e76b7fc687d07ce2c79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.772ex; height:2.343ex;" alt="{\displaystyle \Delta G=\Delta H-T\Delta S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta G=-RT\ln K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta G=-RT\ln K}</annotation>

</semantics>

</math><img src="./7b9e5ff562d6c5f44406b558c9cc8186884866ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.849ex; height:2.343ex;" alt="{\displaystyle \Delta G=-RT\ln K}" loading="lazy">

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta S^{\ominus }={\frac {\Delta H^{\ominus }+RT\ln K}{T}}}</annotation>

</semantics>

</math><img src="./1ce0f9c1f7340b202d2eddc21fcf876db345f65d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.473ex; height:5.509ex;" alt="{\displaystyle \Delta S^{\ominus }={\frac {\Delta H^{\ominus }+RT\ln K}{T}}}" loading="lazy"></dd></dl>

To a first approximation the standard enthalpy change is independent of temperature. Using this approximation, <a href="Definite_integral" class="mw-redirect" title="Definite integral">definite integration</a> of the <a href="Van_'t_Hoff_equation" title="Van 't Hoff equation">van 't Hoff equation</a>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

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<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta H^{\ominus }=-R{\frac {d\ln K}{d(1/T)}}\ }</annotation>

</semantics>

</math><img src="./b599033b035abf21c1038cdb170b3689938faecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.624ex; height:6.176ex;" alt="{\displaystyle \Delta H^{\ominus }=-R{\frac {d\ln K}{d(1/T)}}\ }" loading="lazy"></dd></dl>

gives<a href="#cite_note-13">[13]</a>

<msub>

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<msub>

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</msub>

<mfrac>

<mrow>

<msup>

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</msup>

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</mrow>

<mrow>

<mrow>

<mfrac>

<msub>

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</msub>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln K_{2}=\ln K_{1}-{\frac {\Delta H^{\ominus }}{R}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right)}</annotation>

</semantics>

</math><img src="./ad8a8ab05b7b2e5db0fd0ba1b284e067d0020d91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.177ex; height:6.176ex;" alt="{\displaystyle \ln K_{2}=\ln K_{1}-{\frac {\Delta H^{\ominus }}{R}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right)}" loading="lazy"></dd></dl>

This equation can be used to calculate the value of log K at a temperature, T2, knowing the value at temperature T1.

The van 't Hoff equation also shows that, for an <a href="Exothermic" class="mw-redirect" title="Exothermic">exothermic</a> reaction (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H<0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta H<0}</annotation>

</semantics>

</math><img src="./e57201aba3f79014dbf0d4b2aa6e40bbbc61dfd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.26ex; height:2.176ex;" alt="{\displaystyle \Delta H<0}" loading="lazy">), when temperature increases K decreases and when temperature decreases K increases, in accordance with <a href="Le_Chatelier's_principle" title="Le Chatelier's principle">Le Chatelier's principle</a>. The reverse applies when the reaction is <a href="Endothermic" class="mw-redirect" title="Endothermic">endothermic</a>.

When K has been determined at more than two temperatures, a straight line fitting procedure may be applied to a plot of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln K}</annotation>

</semantics>

</math><img src="./370a4b66f5b61a1016e3899f22928b313e8742c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.392ex; height:2.176ex;" alt="{\displaystyle \ln K}" loading="lazy"> against <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/T}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation>

</semantics>

</math><img src="./ee06bfe8f48b840ea1c11f78977a90f661f2375e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" loading="lazy"> to obtain a value for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H^{\ominus }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta H^{\ominus }}</annotation>

</semantics>

</math><img src="./e7ca2ac7d655e35bf2076604abc115a3fe696542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.55ex; height:2.509ex;" alt="{\displaystyle \Delta H^{\ominus }}" loading="lazy">. <a href="Van_'t_Hoff_equation#Error_propagation" title="Van 't Hoff equation">Error propagation</a> theory can be used to show that, with this procedure, the error on the calculated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H^{\ominus }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta H^{\ominus }}</annotation>

</semantics>

</math><img src="./e7ca2ac7d655e35bf2076604abc115a3fe696542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.55ex; height:2.509ex;" alt="{\displaystyle \Delta H^{\ominus }}" loading="lazy"> value is much greater than the error on individual log K values. Consequently, K needs to be determined to high precision when using this method. For example, with a silver <a href="Ion-selective_electrode" title="Ion-selective electrode">ion-selective electrode</a> each log K value was determined with a precision of ca. 0.001 and the method was applied successfully.<a href="#cite_note-14">[14]</a>

Standard thermodynamic arguments can be used to show that, more generally, enthalpy will change with temperature.

<msub>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial H}{\partial T}}\right)_{p}=C_{p}}</annotation>

</semantics>

</math><img src="./c9be61f92b22b5f9151d19894854744fe19060f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.518ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial H}{\partial T}}\right)_{p}=C_{p}}" loading="lazy"></dd></dl>

where Cp is the <a href="Specific_heat_capacity" title="Specific heat capacity">heat capacity</a> at constant pressure.

<div class="mw-heading mw-heading3"><h3 id="A_more_complex_formulation">A more complex formulation</h3></div>

The calculation of K at a particular temperature from a known K at another given temperature can be approached as follows if standard thermodynamic properties are available. The effect of temperature on equilibrium constant is equivalent to the effect of temperature on Gibbs energy because:

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<annotation encoding="application/x-tex">{\displaystyle \ln K={{-\Delta _{\mathrm {r} }G^{\ominus }} \over {RT}}}</annotation>

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</math><img src="./b342087351d7c3a30e895907c4720516fea79c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.285ex; height:5.676ex;" alt="{\displaystyle \ln K={{-\Delta _{\mathrm {r} }G^{\ominus }} \over {RT}}}" loading="lazy"></dd></dl>

where ΔrG<s>o</s> is the reaction standard Gibbs energy, which is the sum of the standard Gibbs energies of the reaction products minus the sum of standard Gibbs energies of reactants.

Here, the term "standard" denotes the ideal behaviour (i.e., an infinite dilution) and a hypothetical standard concentration (typically 1 mol/kg). It does not imply any particular temperature or pressure because, although contrary to IUPAC recommendation, it is more convenient when describing aqueous systems over wide temperature and pressure ranges.<a href="#cite_note-15">[15]</a>

The standard Gibbs energy (for each species or for the entire reaction) can be represented (from the basic definitions) as:

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<annotation encoding="application/x-tex">{\displaystyle G_{T_{2}}^{\ominus }=G_{T_{1}}^{\ominus }-S_{T_{1}}^{\ominus }(T_{2}-T_{1})-T_{2}\int _{T_{1}}^{T_{2}}{{C_{p}^{\ominus }} \over {T}}\,dT+\int _{T_{1}}^{T_{2}}C_{p}^{\ominus }\,dT}</annotation>

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</math><img src="./23235dba782f830ce107fabaaddb464954a6b6d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:58.723ex; height:6.676ex;" alt="{\displaystyle G_{T_{2}}^{\ominus }=G_{T_{1}}^{\ominus }-S_{T_{1}}^{\ominus }(T_{2}-T_{1})-T_{2}\int _{T_{1}}^{T_{2}}{{C_{p}^{\ominus }} \over {T}}\,dT+\int _{T_{1}}^{T_{2}}C_{p}^{\ominus }\,dT}" loading="lazy"></dd></dl>

In the above equation, the effect of temperature on Gibbs energy (and thus on the equilibrium constant) is ascribed entirely to heat capacity. To evaluate the integrals in this equation, the form of the dependence of heat capacity on temperature needs to be known.

If the standard molar heat capacity C<s>o</s> p can be approximated by some analytic function of temperature (e.g. the Shomate equation), then the integrals involved in calculating other parameters may be solved to yield analytic expressions for them. For example, using approximations of the following forms:<a href="#cite_note-Roberge-16">[16]</a>

<ul><li>For pure substances (solids, gas, liquid): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}^{\ominus }\approx A+BT+CT^{-2}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{p}^{\ominus }\approx A+BT+CT^{-2}}</annotation>

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<li>For ionic species at T < 200 °C: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}^{\ominus }\approx (4.186a+b{\breve {S}}_{T_{1}}^{\ominus }){{(T_{2}-T_{1})} \over {\ln \left({\frac {T_{2}}{T_{1}}}\right)}}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{p}^{\ominus }\approx (4.186a+b{\breve {S}}_{T_{1}}^{\ominus }){{(T_{2}-T_{1})} \over {\ln \left({\frac {T_{2}}{T_{1}}}\right)}}}</annotation>

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then the integrals can be evaluated and the following final form is obtained:

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<annotation encoding="application/x-tex">{\displaystyle G_{T_{2}}^{\ominus }\approx G_{T_{1}}^{\ominus }+(C_{p}^{\ominus }-S_{T_{1}}^{\ominus })(T_{2}-T_{1})-T_{2}\ln \left({\frac {T_{2}}{T_{1}}}\right)C_{p}^{\ominus }}</annotation>

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</math><img src="./52b0ca6ceb4cb5db0044025df72a07caac4e13ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.076ex; height:6.176ex;" alt="{\displaystyle G_{T_{2}}^{\ominus }\approx G_{T_{1}}^{\ominus }+(C_{p}^{\ominus }-S_{T_{1}}^{\ominus })(T_{2}-T_{1})-T_{2}\ln \left({\frac {T_{2}}{T_{1}}}\right)C_{p}^{\ominus }}" loading="lazy"></dd></dl>

The constants A, B, C, a, b and the absolute entropy, S̆  <s>o</s> 298 K, required for evaluation of C<s>o</s> p(T), as well as the values of G298 K and S298 K for many species are tabulated in the literature.

<div class="mw-heading mw-heading2"><h2 id="Pressure_dependence">Pressure dependence</h2></div>

The pressure dependence of the equilibrium constant is usually weak in the range of pressures normally encountered in industry, and therefore, it is usually neglected in practice. This is true for <a href="Condensed_matter_physics" title="Condensed matter physics">condensed</a> reactant/products (i.e., when reactants and products are solids or liquid) as well as gaseous ones.

For a gaseous-reaction example, one may consider the well-studied reaction of hydrogen with nitrogen to produce ammonia:

If the pressure is increased by the addition of an inert gas, then neither the composition at equilibrium nor the equilibrium constant are appreciably affected (because the partial pressures remain constant, assuming an ideal-gas behaviour of all gases involved). However, the composition at equilibrium will depend appreciably on pressure when:

<ul><li>the pressure is changed by compression or expansion of the gaseous reacting system, and</li>

<li>the reaction results in the change of the number of moles of gas in the system.</li></ul>

In the example reaction above, the number of moles changes from 4 to 2, and an increase of pressure by system compression will result in appreciably more ammonia in the equilibrium mixture. In the general case of a gaseous reaction:

the change of mixture composition with pressure can be quantified using:

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<annotation encoding="application/x-tex">{\displaystyle K_{p}={\frac {{p_{\mathrm {S} }}^{\sigma }{p_{\mathrm {T} }}^{\tau }}{{p_{\mathrm {A} }}^{\alpha }{p_{\mathrm {B} }}^{\beta }}}={\frac {{X_{\mathrm {S} }}^{\sigma }{X_{\mathrm {T} }}^{\tau }}{{X_{\mathrm {A} }}^{\alpha }{X_{\mathrm {B} }}^{\beta }}}P^{\sigma +\tau -\alpha -\beta }=K_{X}P^{\sigma +\tau -\alpha -\beta }}</annotation>

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</math><img src="./ca2fd6f701196942f1116545972c7542fa82097b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:53.777ex; height:6.343ex;" alt="{\displaystyle K_{p}={\frac {{p_{\mathrm {S} }}^{\sigma }{p_{\mathrm {T} }}^{\tau }}{{p_{\mathrm {A} }}^{\alpha }{p_{\mathrm {B} }}^{\beta }}}={\frac {{X_{\mathrm {S} }}^{\sigma }{X_{\mathrm {T} }}^{\tau }}{{X_{\mathrm {A} }}^{\alpha }{X_{\mathrm {B} }}^{\beta }}}P^{\sigma +\tau -\alpha -\beta }=K_{X}P^{\sigma +\tau -\alpha -\beta }}" loading="lazy"></dd></dl>

where p denote the partial pressures and X the mole fractions of the components, P is the total system pressure, Kp is the equilibrium constant expressed in terms of partial pressures and KX is the equilibrium constant expressed in terms of mole fractions.

The above change in composition is in accordance with <a href="Le_Chatelier's_principle" title="Le Chatelier's principle">Le Chatelier's principle</a> and does not involve any change of the equilibrium constant with the total system pressure. Indeed, for ideal-gas reactions Kp is independent of pressure.<a href="#cite_note-17">[17]</a>

In a condensed phase, the pressure dependence of the equilibrium constant is associated with the reaction volume.<a href="#cite_note-18">[18]</a> For reaction:

the reaction volume is:

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<annotation encoding="application/x-tex">{\displaystyle \Delta {\bar {V}}=\sigma {\bar {V}}_{\mathrm {S} }+\tau {\bar {V}}_{\mathrm {T} }-\alpha {\bar {V}}_{\mathrm {A} }-\beta {\bar {V}}_{\mathrm {B} }}</annotation>

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</math><img src="./e0f76297dbd9b6b37c4b1414349939318bc1d8f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.269ex; height:2.843ex;" alt="{\displaystyle \Delta {\bar {V}}=\sigma {\bar {V}}_{\mathrm {S} }+\tau {\bar {V}}_{\mathrm {T} }-\alpha {\bar {V}}_{\mathrm {A} }-\beta {\bar {V}}_{\mathrm {B} }}" loading="lazy"></dd></dl>

where V̄ denotes a <a href="Partial_molar_volume" class="mw-redirect" title="Partial molar volume">partial molar volume</a> of a reactant or a product.

For the above reaction, one can expect the change of the reaction equilibrium constant (based either on mole-fraction or molal-concentration scale) with pressure at constant temperature to be:

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \ln K_{X}}{\partial P}}\right)_{T}={\frac {-\Delta {\bar {V}}}{RT}}.}</annotation>

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</math><img src="./f4124a57af1be19f6cd17230dc7eeaa4dd798a42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.396ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\partial \ln K_{X}}{\partial P}}\right)_{T}={\frac {-\Delta {\bar {V}}}{RT}}.}" loading="lazy"></dd></dl>

The matter is complicated as partial molar volume is itself dependent on pressure.

<div class="mw-heading mw-heading2"><h2 id="Effect_of_isotopic_substitution">Effect of isotopic substitution</h2></div>

<a href="Isotope" title="Isotope">Isotopic</a> substitution can lead to changes in the values of equilibrium constants, especially if hydrogen is replaced by <a href="Deuterium" title="Deuterium">deuterium</a> (or <a href="Tritium" title="Tritium">tritium</a>).<a href="#cite_note-Laidler-19">[19]</a> This equilibrium isotope effect is analogous to the <a href="Kinetic_isotope_effect" title="Kinetic isotope effect">kinetic isotope effect</a> on rate constants, and is primarily due to the change in <a href="Quantum_harmonic_oscillator#Hamiltonian_and_energy_eigenstates" title="Quantum harmonic oscillator">zero-point vibrational energy</a> of H–X bonds due to the change in mass upon isotopic substitution.<a href="#cite_note-Laidler-19">[19]</a> The zero-point energy is inversely proportional to the square root of the mass of the vibrating hydrogen atom, and will therefore be smaller for a D–X bond that for an H–X bond.

An example is a <a href="Hydrogen_atom_abstraction" title="Hydrogen atom abstraction">hydrogen atom abstraction</a> reaction R' + H–R ⇌ R'–H + R with equilibrium constant KH, where R' and R are organic radicals such that R' forms a stronger bond to hydrogen than does R. The decrease in zero-point energy due to deuterium substitution will then be more important for R'–H than for R–H, and R'–D will be stabilized more than R–D, so that the equilibrium constant KD for R' + D–R ⇌ R'–D + R is greater than KH. This is summarized in the rule the heavier atom favors the stronger bond.<a href="#cite_note-Laidler-19">[19]</a>

Similar effects occur in solution for <a href="Acid_dissociation_constant" title="Acid dissociation constant">acid dissociation constants</a> (Ka) which describe the transfer of H+ or D+ from a weak aqueous acid to a solvent molecule: HA + H2O = H3O+ + A− or DA + D2O ⇌ D3O+ + A−. The deuterated acid is studied in <a href="Heavy_water" title="Heavy water">heavy water</a>, since if it were dissolved in ordinary water the deuterium would rapidly exchange with hydrogen in the solvent.<a href="#cite_note-Laidler-19">[19]</a>

The product species H3O+ (or D3O+) is a stronger acid than the solute acid, so that it dissociates more easily, and its H–O (or D–O) bond is weaker than the H–A (or D–A) bond of the solute acid. The decrease in zero-point energy due to isotopic substitution is therefore less important in D3O+ than in DA so that KD < KH, and the deuterated acid in D2O is weaker than the non-deuterated acid in H2O. In many cases the difference of logarithmic constants pKD – pKH is about 0.6,<a href="#cite_note-Laidler-19">[19]</a> so that the pD corresponding to 50% dissociation of the deuterated acid is about 0.6 units higher than the pH for 50% dissociation of the non-deuterated acid.

For similar reasons the <a href="Self-ionization_of_water#Isotope_effects" title="Self-ionization of water">self-ionization of heavy water</a> is less than that of ordinary water at the same temperature.

<ul><li><a href="Determination_of_equilibrium_constants" title="Determination of equilibrium constants">Determination of equilibrium constants</a></li>

<li><a href="Stability_constants_of_complexes" title="Stability constants of complexes">Stability constants of complexes</a></li>

<li><a href="Equilibrium_fractionation" title="Equilibrium fractionation">Equilibrium fractionation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a rel="nofollow" class="external text" href="https://goldbook.iupac.org/html/S/S05915.html">IUPAC Gold Book</a>.

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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<li id="cite_note-Atkins7th-3">^ <a href="#cite_ref-Atkins7th_3-0">a</a> <a href="#cite_ref-Atkins7th_3-1">b</a> <a href="#cite_ref-Atkins7th_3-2">c</a> <a href="#cite_ref-Atkins7th_3-3">d</a> Atkins, P.; Jones, L.; Laverman, L. (2016).Chemical Principles, 7th edition, pp. 399 & 461. Freeman. ISBN 978-1-4641-8395-9

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFSplittgerberChinander1988" class="citation journal cs1">Splittgerber, A. G.; Chinander, L.L. (1 February 1988). "The spectrum of a dissociation intermediate of cysteine: a biophysical chemistry experiment". Journal of Chemical Education. 65 (2): 167. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988JChEd..65..167S">1988JChEd..65..167S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fed065p167">10.1021/ed065p167</a>.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFHagueMoreton1994" class="citation journal cs1">Hague, David N.; Moreton, Anthony D. (1994). "Protonation sequence of linear aliphatic polyamines by 13C NMR spectroscopy". J. Chem. Soc., Perkin Trans. 2 (2): 265–70. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FP29940000265">10.1039/P29940000265</a>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBorkovecKoper2000" class="citation journal cs1">Borkovec, Michal; Koper, Ger J. M. (2000). "A Cluster Expansion Method for the Complete Resolution of Microscopic Ionization Equilibria from NMR Titrations". Anal. Chem. 72 (14): 3272–9. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fac991494p">10.1021/ac991494p</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10939399">10939399</a>.</cite>

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<li id="cite_note-BM-7"><a href="#cite_ref-BM_7-0">^</a> <cite id="CITEREFBaesMesmer1976" class="citation book cs1">Baes, C. F.; Mesmer, R. E. (1976). "Chapter 18. Survey of Hydrolysis Behaviour". The Hydrolysis of Cations. Wiley. pp. 397–430.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFSchwarzenbachFlaschka1969" class="citation book cs1">Schwarzenbach, G.; Flaschka, H. (1969). Complexometric titrations. Methuen.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFDenbigh1981" class="citation book cs1">Denbigh, K. (1981). "Chapter 4". The principles of chemical equilibrium (4th ed.). Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-28150-8</bdi>.</cite>

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<li id="cite_note-Butler-10"><a href="#cite_ref-Butler_10-0">^</a> <cite id="CITEREFButler1998" class="citation book cs1">Butler, J. N. (1998). Ionic Equilibrium. John Wiley and Sons.</cite>

</li>

<cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081029193538/http://www.iupac.org/web/ins/2000-003-1-500">"Project: Ionic Strength Corrections for Stability Constants"</a>. International Union of Pure and Applied Chemistry. Archived from <a rel="nofollow" class="external text" href="http://www.iupac.org/web/ins/2000-003-1-500">the original</a> on 29 October 2008. Retrieved 2008-11-23.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a rel="nofollow" class="external text" href="https://iupac.org/wp-content/uploads/2019/05/IUPAC-GB3-2012-2ndPrinting-PDFsearchable.pdf.">Green Book (IUPAC), Quantities, Units and Symbols in Physical Chemistry, page 61, édition 2007.</a>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFAtkinsde_Paula2006" class="citation book cs1">Atkins, Peter; de Paula, Julio (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/atkinsphysicalch00atki">Physical Chemistry</a>. Oxford. p. <a rel="nofollow" class="external text" href="https://archive.org/details/atkinsphysicalch00atki/page/n246">214</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0198700722</bdi>.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFBarnesFordPettitSherringham1971" class="citation journal cs1">Barnes, D.S.; Ford, G.J; Pettit, L.D.; Sherringham, C. (1971). "Ligands containing elements of group VIB. Part V. Thermodynamics of silver complex formation of some saturated and unsaturated (alkyl-thio)acetic and (alkylseleno)acetic acids". J. Chem. Soc. A: 2883–2887. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FJ19710002883">10.1039/J19710002883</a>.</cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFMajerSedelbauerWood2004" class="citation book cs1">Majer, V.; Sedelbauer, J.; Wood (2004). "Calculations of standard thermodynamic properties of aqueous electrolytes and nonelectrolytes". In Palmer, D. A.; Fernandez-Prini, R.; Harvey, A. (eds.). Aqueous Systems at Elevated Temperatures and Pressures: Physical Chemistry of Water, Steam and Hydrothermal Solutions. Elsevier.</cite>

</li>

<li id="cite_note-Roberge-16"><a href="#cite_ref-Roberge_16-0">^</a> <cite id="CITEREFRoberge2011" class="citation book cs1">Roberge, P. R. (November 2011). "Appendix F". Handbook of Corrosion Engineering. McGraw-Hill. p. 1037ff.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFAtkins1978" class="citation book cs1">Atkins, P. W. (1978). Physical Chemistry (6th ed.). Oxford University Press. p. 210.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFVan_EldikAsanoLe_Noble1989" class="citation journal cs1">Van Eldik, R.; Asano, T.; Le Noble, W. J. (1989). "Activation and reaction volumes in solution. 2". Chem. Rev. 89 (3): 549–688. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fcr00093a005">10.1021/cr00093a005</a>.</cite>

</li>

<li id="cite_note-Laidler-19">^ <a href="#cite_ref-Laidler_19-0">a</a> <a href="#cite_ref-Laidler_19-1">b</a> <a href="#cite_ref-Laidler_19-2">c</a> <a href="#cite_ref-Laidler_19-3">d</a> <a href="#cite_ref-Laidler_19-4">e</a> <a href="Keith_J._Laidler" title="Keith J. Laidler">Laidler K.J.</a> Chemical Kinetics (3rd ed., Harper & Row 1987), p.428–433 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-06-043862-2</bdi>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Data_sources">Data sources</h3></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.acadsoft.co.uk/scdbase/scdbase.htm">IUPAC SC-Database</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170619235720/http://www.acadsoft.co.uk/scdbase/scdbase.htm">Archived</a> 2017-06-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> A comprehensive database of published data on equilibrium constants of metal complexes and ligands</li>

<li><a rel="nofollow" class="external text" href="https://www.nist.gov/ts/msd/srd/nist46.cfm">NIST Standard Reference Database 46</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100705175236/http://www.nist.gov/ts/msd/srd/nist46.cfm">Archived</a> 2010-07-05 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>: Critically selected stability constants of metal complexes</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081009060809/http://www.chem.wisc.edu/areas/reich/pkatable/">Inorganic and organic acids and bases</a> pKa data in water and <a href="Dimethyl_sulfoxide" title="Dimethyl sulfoxide">DMSO</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060928231137/http://www.grc.nasa.gov/WWW/CEAWeb/ceaHome.htm">NASA Glenn Thermodynamic Database webpage with links to (self-consistent) temperature-dependent specific heat, enthalpy, and entropy for elements and molecules</a></li></ul>

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</style><div id="Chemical_equilibria112" style="font-size:114%;margin:0 4em"><a href="Chemical_equilibrium" title="Chemical equilibrium">Chemical equilibria</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemical_stability" title="Chemical stability">Chemical stability</a></li>

<li><a href="Chelation" title="Chelation">Chelation</a></li>

<li><a href="Dynamic_equilibrium" class="mw-redirect" title="Dynamic equilibrium">Dynamic equilibrium</a></li>

<li><a href="Equilibrium_chemistry" title="Equilibrium chemistry">Equilibrium chemistry</a></li>

<li><a href="Equilibrium_stage" class="mw-redirect" title="Equilibrium stage">Equilibrium stage</a></li>

<li><a href="Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a>

<ul><li><a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs</a></li>

<li><a href="Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz</a></li></ul></li>

<li><a href="Le_Chatelier's_principle" title="Le Chatelier's principle">Le Chatelier's principle</a></li>

<li><a href="Phase_separation" title="Phase separation">Phase separation</a></li>

<li><a href="Reversible_reaction" title="Reversible reaction">Reversible reaction</a></li>

<li><a href="Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_model" title="Mathematical model">Models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Determination_of_equilibrium_constants" title="Determination of equilibrium constants">determination</a></li></ul></li>

<li><a href="Phase_diagram" title="Phase diagram">Phase diagram</a></li>

<li><a href="Predominance_diagram" title="Predominance diagram">Predominance diagram</a></li>

<li><a href="Phase_rule" title="Phase rule">Phase rule</a></li>

<li><a href="Reaction_quotient" title="Reaction quotient">Reaction quotient</a></li>

<li><a href="Thermodynamic_activity" title="Thermodynamic activity">Thermodynamic activity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Buffer_solution" title="Buffer solution">Buffer solution</a></li>

<li><a href="Equilibrium_unfolding" title="Equilibrium unfolding">Equilibrium unfolding</a></li>

<li><a href="Liquid%E2%80%93liquid_extraction" title="Liquid–liquid extraction">Liquid–liquid extraction</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Specific equilibria</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Acid_dissociation_constant" title="Acid dissociation constant">Acid dissociation</a>

<ul><li><a href="Hammett_acidity_function" title="Hammett acidity function">Hammett acidity function</a></li></ul></li>

<li><a href="Binding_constant" title="Binding constant">Binding constant</a></li>

<li><a href="Binding_selectivity" title="Binding selectivity">Binding selectivity</a></li>

<li><a href="Stability_constants_of_complexes" title="Stability constants of complexes">Coordination complexes</a>

<ul><li><a href="Macrocyclic_effect" class="mw-redirect" title="Macrocyclic effect">Macrocyclic effect</a></li></ul></li>

<li><a href="Dissociation_constant" title="Dissociation constant">Dissociation constant</a></li>

<li><a href="Hydrolysis_constant" title="Hydrolysis constant">Hydrolysis</a></li>

<li><a href="Molecular_autoionization" title="Molecular autoionization">Self-ionization</a>

<ul><li><a href="Self-ionization_of_water" title="Self-ionization of water">of water</a></li></ul></li>

<li><a href="Partition_equilibrium" title="Partition equilibrium">Partition</a>

<ul><li><a href="Partition_coefficient" title="Partition coefficient">Distribution coefficient</a></li></ul></li>

<li><a href="Solubility_equilibrium" title="Solubility equilibrium">Solubility</a>

<ul><li><a href="Common-ion_effect" title="Common-ion effect">Common-ion effect</a></li></ul></li>

<li><a href="Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">Vapor–liquid</a>

<ul><li><a href="Henry's_law" title="Henry's law">Henry's law</a></li></ul></li></ul>

</div></td></tr></tbody></table></div>

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