Editing Chemical equilibrium (section)

==Thermodynamics==

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

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

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

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

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

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

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

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

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

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

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

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

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

By substituting the chemical potentials:

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

the relationship becomes:

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

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

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

leading to:

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

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

[[File:Diag eq.svg|thumb|350px|right]]

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

*If activity of a reagent ''i'' increases <math display="block">Q_\mathrm{r} = \frac{\prod (a_j)^{\nu_j}}{\prod(a_i)^{\nu_i}}~,</math> the reaction quotient decreases. Then <math display="block">Q_\mathrm{r} < K_\mathrm{eq}~</math> and <math display="block">\left(\frac {dG}{d\xi}\right)_{T,p} < 0~</math> The reaction will shift to the right (i.e. in the forward direction, and thus more products will form).
*If activity of a product ''j'' increases, then <math display="block">Q_\mathrm{r} > K_\mathrm{eq}~</math> and <math display="block">\left(\frac {dG}{d\xi}\right)_{T,p} >0~</math> The reaction will shift to the left (i.e. in the reverse direction, and thus less products will form).

Note that activities and equilibrium constants are dimensionless numbers.

===Treatment of activity===

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

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

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

===Concentration quotients===

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

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

===Metastable mixtures===

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

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

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