Editing Activity coefficient

{{Short description|Value accounting for thermodynamic non-ideality of mixtures}}

In [[thermodynamics]], an '''activity coefficient''' is a factor used to account for deviation of a [[mixture]] of [[chemical substance]]s from ideal behaviour.<ref>{{GoldBookRef|title=Activity coefficient|file=A00116}}</ref> In an [[ideal mixture]], the microscopic interactions between each pair of [[chemical species]] are the same (or macroscopically equivalent, the [[enthalpy change of solution]] and volume variation in mixing is zero) and, as a result, properties of the mixtures can be expressed directly in terms of simple [[concentration]]s or [[partial pressure]]s of the substances present e.g. [[Raoult's law]]. Deviations from ideality are accommodated by modifying the concentration by an ''activity coefficient''. Analogously, expressions involving gases can be adjusted for non-ideality by scaling partial pressures by a [[fugacity]] coefficient.

The concept of activity coefficient is closely linked to that of [[activity (chemistry)|activity in chemistry]].

== Thermodynamic definition ==
[[File:Chemical potentials vs log mole fraction.svg|thumb|Chemical potentials for various hypothetical non-ideal substances in solution.]]
[[File:Activity coefficients vs log mole fraction.svg|thumb|Activity coefficients for the above figure. Activity coefficients quantify the deviation of <math>\mu</math> from an ideal curve (dashed line in above figure).]]

The [[chemical potential]], <math>\mu_\mathrm{B}</math>, of a substance B in an [[ideal mixture]] of liquids or an [[ideal solution]] is given by
:<math>\mu_\mathrm{B} = \mu_\mathrm{B}^{\ominus} + RT \ln x_\mathrm{B} \,</math>,
where ''μ''{{su|b=B|p=<s>o</s>}} is the chemical potential of a pure substance <math>\mathrm{B}</math>, and <math> x_\mathrm{B} </math> is the [[mole fraction]] of the substance in the mixture.

This is generalised to include non-ideal behavior by writing
:<math>\mu_\mathrm{B} = \mu_\mathrm{B}^{\ominus} + RT \ln a_\mathrm{B} \,</math>
when <math>a_\mathrm{B}</math> is the activity of the substance in the mixture,
:<math>a_\mathrm{B} = x_\mathrm{B} \gamma_\mathrm{B}</math>,
where <math>\gamma_\mathrm{B}</math> is the activity coefficient, which may itself depend on <math>x_\mathrm{B}</math>. As <math>\gamma_\mathrm{B}</math> approaches 1, the substance behaves as if it were ideal. For instance, if <math>\gamma_\mathrm{B}</math>&nbsp;≈&nbsp;1, then [[Raoult's law]] is accurate. For <math>\gamma_\mathrm{B}</math>&nbsp;>&nbsp;1 and <math>\gamma_\mathrm{B}</math>&nbsp;<&nbsp;1, substance B shows positive and negative deviation from Raoult's law, respectively. A positive deviation implies that substance B is more volatile.

In many cases, as <math>x_\mathrm{B}</math> goes to zero, the activity coefficient of substance B approaches a constant; this relationship is [[Henry's law]] for the solvent.  These relationships are related to each other through the [[Gibbs–Duhem equation]].<ref>{{Cite journal|last1=DeHoff|first1=Robert|title=Thermodynamics in materials science|journal=Entropy|volume=20|issue=7|isbn=9780849340659|pages=230–231|edition=2nd|bibcode=2018Entrp..20..532G|doi=10.3390/e20070532|year=2018|pmid=33265621 |pmc=7513056 |doi-access=free}}</ref>
Note that in general activity coefficients are dimensionless.

In detail: [[Raoult's law]] states that the partial pressure of component B is related to its vapor pressure (saturation pressure) and its mole fraction <math>x_\mathrm{B}</math> in the liquid phase,
:<math> p_\mathrm{B} = x_\mathrm{B} \gamma_\mathrm{B} p^{\sigma}_\mathrm{B} \;,</math>
with the convention
<math> \lim_{x_\mathrm{B} \to 1} \gamma_\mathrm{B} = 1 \;.</math>
In other words: Pure liquids represent the ideal case.

At infinite dilution, the activity coefficient approaches its limiting value, <math>\gamma_\mathrm{B}</math><sup>∞</sup>. Comparison with [[Henry's law]],
:<math> p_\mathrm{B} = K_{\mathrm{H,B}} x_\mathrm{B} \quad \text{for} \quad x_\mathrm{B} \to 0 \;,</math>
immediately gives
:<math>K_{\mathrm{H,B}} = p_\mathrm{B}^\sigma \gamma_\mathrm{B}^\infty \;.</math>
In other words: The compound shows nonideal behavior in the dilute case.

The above definition of the activity coefficient is impractical if the compound does not exist as a pure liquid. This is often the case for electrolytes or biochemical compounds. In such cases, a different definition is used that considers infinite dilution as the ideal state:
:<math>\gamma_\mathrm{B}^\dagger \equiv \gamma_\mathrm{B} / \gamma_\mathrm{B}^\infty</math>
with
<math> \lim_{x_\mathrm{B} \to 0} \gamma_\mathrm{B}^\dagger = 1 \;,</math>
and
:<math> \mu_\mathrm{B} = \underbrace{\mu_\mathrm{B}^\ominus + RT \ln \gamma_\mathrm{B}^\infty}_{\mu_\mathrm{B}^{\ominus\dagger}} + RT \ln \left(x_\mathrm{B} \gamma_\mathrm{B}^\dagger\right)</math>

The <math>^\dagger</math> symbol has been used here to distinguish between the two kinds of activity coefficients. Usually it is omitted, as it is clear from the context which kind is meant. But there are cases where both kinds of activity coefficients are needed and may even appear in the same equation, e.g., for solutions of salts in (water + alcohol) mixtures. This is sometimes a source of errors.

Modifying mole fractions or concentrations by activity coefficients gives the ''effective activities'' of the components, and hence allows expressions such as [[Raoult's law]] and [[equilibrium constant]]s to be applied to both ideal and non-ideal mixtures.

=== Ionic solutions ===
{{anchor|Mean activity coefficient}}

Knowledge of activity coefficients is particularly important in the context of [[electrochemistry]] since the behaviour of [[electrolyte]] solutions is often far from ideal, even starting at low densities due to the effects of the [[ionic atmosphere]].  Additionally, they are particularly important in the context of [[soil chemistry]] due to the low volumes of solvent and, consequently, the high concentration of [[electrolytes]].<ref>{{cite book | first1= Jorge G. |last1=Ibáñez|first2=Margarita |last2=Hernández Esparza|first3=Carmen |last3=Doría Serrano|first4=Mono Mohan |last4=Singh| title= Environmental Chemistry: Fundamentals| year= 2007| publisher= Springer| isbn= 978-0-387-26061-7}}</ref>

For solution of substances which ionize in solution the activity coefficients of the cation and anion cannot be experimentally determined independently of each other because solution properties depend on both ions. Single ion activity coefficients must be linked to the activity coefficient of the dissolved electrolyte as if undissociated. In this case a mean stoichiometric activity coefficient of the dissolved electrolyte, ''γ''<sub>±</sub>, is used. It is called stoichiometric because it expresses both the deviation from the ideality of the solution and the incomplete ionic dissociation of the ionic compound which occurs especially with the increase of its concentration.

For a 1:1 electrolyte, such as [[sodium chloride|NaCl]] it is given by the following:

:<math> \gamma_\pm=\sqrt{\gamma_+\gamma_-}</math>

where <math>\gamma_\mathrm{+}</math> and <math>\gamma_\mathrm{-}</math> are the activity coefficients of the cation and anion respectively. <!-- "This definition involves a [[tacit assumption]] of a degree of 100% ionic dissociation of the electrolyte." No it does not. the activities in the expression are activities of ions, irrespective of whether ion association is also occurring. (signed) petergans-->

More generally, the mean activity coefficient of a compound of formula <math>A_\mathrm{p} B_\mathrm{q}</math> is given by<ref>{{cite book|last1=Atkins|first1=Peter|last2=dePaula|first2=Julio|title=Physical Chemistry|date=2006|publisher=OUP|isbn=9780198700722|chapter=Section 5.9, The activities of ions in solution|edition=8th}}</ref>

:<math> \gamma_\pm=\sqrt[p+q]{\gamma_\mathrm{A}^p\gamma_\mathrm{B}^q}.</math>

The prevailing view that single ion activity coefficients are unmeasurable independently, or perhaps even physically meaningless, has its roots in the work of Guggenheim in the late 1920s.<ref name="Guggenheim1928">{{cite journal|last1=Guggenheim|first1=E. A.|title=The Conceptions of Electrical Potential Difference between Two Phases and the Individual Activities of Ions|journal=The Journal of Physical Chemistry|volume=33|issue=6|year=1928|pages=842–849|issn=0092-7325|doi=10.1021/j150300a003}}</ref> In this view, the partitioning of the physical [[electrochemical potential]]s into an activity contribution and a [[Galvani potential]] contribution is arbitrary, thus nonidealities in ion activities can be remapped to nonidealities in Galvani potential and vice versa. Nevertheless, certain products of activities (such as <math> \gamma_\pm</math>) reflect a charge-neutral stoichiometry that is anyway insensitive to this partitioning, so these products are physically meaningful even if the single-ion activities are not.<ref name="Guggenheim1928"/> However, chemists have never been able to give up the idea of single ion activities, and by implication single ion activity coefficients. For example, [[pH]] is defined as the negative logarithm of the hydrogen ion activity. If the prevailing view on the physical meaning and measurability of single ion activities is correct then defining pH as the negative logarithm of the hydrogen ion activity places the quantity squarely in the unmeasurable category. Recognizing this logical difficulty, [[International Union of Pure and Applied Chemistry]] (IUPAC) states that the activity-based definition of pH is a notional definition only.<ref>{{GoldBookRef|title=pH| file = P04524}}</ref> Despite the prevailing negative view on the measurability of single ion coefficients, the concept of single ion activities continues to be discussed in the literature.<ref name="Rockwood2015">{{cite journal|last1=Rockwood|first1=Alan L.|title=Meaning and Measurability of Single-Ion Activities, the Thermodynamic Foundations of pH, and the Gibbs Free Energy for the Transfer of Ions between Dissimilar Materials|journal=ChemPhysChem|volume=16|issue=9|year=2015|pages=1978–1991|issn=1439-4235|doi=10.1002/cphc.201500044|pmid=25919971|pmc=4501315}}</ref><ref>{{Cite journal |last=May |first=Peter M. |last2=May |first2=Eric |date=2024 |title=Ion Trios: Cause of Ion Specific Interactions in Aqueous Solutions and Path to a Better pH Definition |url=https://pubs.acs.org/doi/10.1021/acsomega.4c07525 |journal=ACS Omega |volume=9 |issue=46 |pages=46373–46386 |doi=10.1021/acsomega.4c07525|pmc=11579776 }}</ref>

== Experimental determination of activity coefficients ==
Activity coefficients may be determined experimentally by making measurements on non-ideal mixtures. Use may be made of [[Raoult's law]] or [[Henry's law]] to provide a value for an ideal mixture against which the experimental value may be compared to obtain the activity coefficient. Other [[colligative]] properties, such as [[osmotic pressure]] may also be used.

=== Radiochemical methods ===
Activity coefficients can be determined by [[radiochemistry|radiochemical]] methods.<ref>{{cite journal|title=Radiochemical Measurements of Activity Coefficients in Mixed Electrolytes|first1=R. H.|last1=Betts|first2=Agnes N.|last2=MacKenzie|journal=Canadian Journal of Chemistry|volume=30|issue=2|pages=146–162|doi=10.1139/v52-020|year = 1952}}</ref>

=== At infinite dilution ===
Activity coefficients for binary mixtures are often reported at the infinite dilution of each component. Because activity coefficient models simplify at infinite dilution, such empirical values can be used to estimate interaction energies.  Examples are given for water:

{| class="wikitable"
|+ Binary solutions with water<ref>{{cite web |title=Activity Coefficients at Infinite Dilution of 30 Important Components from Dortmund Data Bank |url=http://www.ddbst.com/en/EED/ACT/ACTindex.php |website=Dortmund Data Bank |publisher=DDBST GmbH |access-date=13 December 2018}}</ref>
|-
! X
! {{math|γ<sub>x</sub><sup>∞</sup>}} (K)
! {{math|γ<sub>W</sub><sup>∞</sup>}} (K)
|-
| [[Ethanol]] || 4.3800 (283.15) || 3.2800 (298.15)
|-
| [[Acetone]] || || 6.0200 (307.85)
|}

== Theoretical calculation of activity coefficients ==
[[File:UNIQUACRegressionChloroformMethanol.png|thumb|UNIQUAC [[Regression analysis|Regression]] of activity coefficients ([[chloroform]]/[[methanol]] mixture)]]
Activity coefficients of electrolyte solutions may be calculated theoretically, using the [[Debye–Hückel equation]] or extensions such as the [[Davies equation]],<ref name="King1964">{{cite journal|last1=King|first1=E. L.|title=Book Review: Ion Association, C. W. Davies, Butterworth, Washington, D.C., 1962 |journal=Science| volume=143| issue=3601|year=1964|page=37|issn=0036-8075|doi=10.1126/science.143.3601.37|bibcode=1964Sci...143...37D}}</ref> [[Pitzer equations]]<ref name="davies">{{cite web |first1=I. |last1=Grenthe |first2=H. |last2=Wanner |title=Guidelines for the extrapolation to zero ionic strength |url=http://www.nea.fr/html/dbtdb/guidelines/tdb2.pdf |access-date=2007-07-23 |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> or TCPC model.<ref name="GeWang2007">{{cite journal|last1=Ge|first1=Xinlei|last2=Wang|first2=Xidong|last3=Zhang|first3=Mei|last4=Seetharaman|first4=Seshadri|title=Correlation and Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model|journal=Journal of Chemical & Engineering Data|volume=52|issue=2|year=2007|pages=538–547|issn=0021-9568|doi=10.1021/je060451k}}</ref><ref name="GeZhang2008">{{cite journal|last1=Ge|first1=Xinlei|last2=Zhang|first2=Mei|last3=Guo|first3=Min|last4=Wang|first4=Xidong|title=Correlation and Prediction of Thermodynamic Properties of Nonaqueous Electrolytes by the Modified TCPC Model|journal=Journal of Chemical & Engineering Data|volume=53|issue=1|year=2008|pages=149–159|issn=0021-9568|doi=10.1021/je700446q}}</ref><ref>{{cite journal|last1=Ge|first1=Xinlei|last2=Zhang|first2=Mei|last3=Guo|first3=Min|last4=Wang|first4=Xidong|title=Correlation and Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model|journal=Journal of Chemical & Engineering Data|volume=53|issue=4|year=2008|pages=950–958|issn=0021-9568|doi=10.1021/je7006499}}</ref><ref name="GeWang2009">{{cite journal|last1=Ge|first1=Xinlei|last2=Wang|first2=Xidong|title=A Simple Two-Parameter Correlation Model for Aqueous Electrolyte Solutions across a Wide Range of Temperatures|journal=Journal of Chemical & Engineering Data|volume=54|issue=2|year=2009|pages=179–186|issn=0021-9568|doi=10.1021/je800483q}}</ref> [[Specific ion interaction theory]] (SIT)<ref>{{cite web|url=http://www.iupac.org/web/ins/2000-003-1-500 |title=Project: Ionic Strength Corrections for Stability Constants |access-date=2008-11-15 |publisher=IUPAC |archive-url=https://web.archive.org/web/20081029193538/http://www.iupac.org/web/ins/2000-003-1-500 |archive-date=29 October 2008 |url-status=dead }}</ref> may also be used.

For non-electrolyte solutions correlative methods such as [[UNIQUAC]], [[Non-random two-liquid model|NRTL]], [[MOSCED]] or [[UNIFAC]] may be employed, provided fitted component-specific or model parameters are available. COSMO-RS is a theoretical method which is less dependent on model parameters as required information is obtained from [[quantum mechanics]] calculations specific to each molecule (sigma profiles) combined with a statistical thermodynamics treatment of surface segments.<ref name="Klamt">{{cite book|last1=Klamt|first1=Andreas|title=COSMO-RS from quantum chemistry to fluid phase thermodynamics and drug design|date=2005|publisher=Elsevier|location=Amsterdam|isbn=978-0-444-51994-8|edition=1st}}</ref>

For uncharged species, the activity coefficient ''γ''<sub>0</sub> mostly follows a [[salting-out]] model:<ref name=Butler>{{cite book|last1=N. Butler|first1=James|title=Ionic equilibrium: solubility and pH calculations|date=1998|publisher=Wiley|location=New York, NY [u.a.]|isbn=9780471585268}}</ref>

:<math> \log_{10}(\gamma_{0}) = b I</math>

This simple model predicts activities of many species (dissolved undissociated gases such as CO<sub>2</sub>, H<sub>2</sub>S, NH<sub>3</sub>, undissociated acids and bases) to high [[ionic strength]]s (up to 5&nbsp;mol/kg). The value of the constant ''b'' for CO<sub>2</sub> is 0.11 at 10&nbsp;°C and 0.20 at 330&nbsp;°C.<ref name="EllisGolding1963">{{cite journal|last1=Ellis|first1=A. J.|last2=Golding|first2=R. M.|title=The solubility of carbon dioxide above 100 degrees C in water and in sodium chloride solutions|journal=American Journal of Science|volume=261|issue=1|year=1963|pages=47–60|issn=0002-9599|doi=10.2475/ajs.261.1.47|bibcode=1963AmJS..261...47E}}</ref>

For [[water]] as solvent, the activity ''a''<sub>w</sub> can be calculated using:<ref name = "Butler"/>

:<math> \ln(a_\mathrm{w}) = \frac{-\nu b}{55.51} \varphi</math>

where ''ν'' is the number of ions produced from the dissociation of one molecule of the dissolved salt, ''b'' is the molality of the salt dissolved in water, ''φ'' is the [[osmotic coefficient]] of water, and the constant 55.51 represents the [[molality]] of water. In the above equation, the activity of a solvent (here water) is represented as inversely proportional to the number of particles of salt versus that of the solvent.

===Link to ionic diameter===
The ionic activity coefficient is connected to the [[ionic radius|ionic diameter]] by the formula obtained from [[Debye–Hückel theory]] of [[electrolyte]]s:
:<math>\log (\gamma_{i}) = - \frac {A z_i^2 \sqrt {I}}{1+ B a \sqrt {I}}</math>
where ''A'' and ''B'' are constants, ''z<sub>i</sub>'' is the valence number of the ion, and ''I'' is [[ionic strength]].

=== Concentrated ionic solutions ===

Ionic activity coefficients can be calculated theoretically, for example by using the [[Debye–Hückel equation]]. The theoretical equation can be tested by combining the calculated single-ion activity coefficients to give mean values which can be compared to experimental values.

==== Stokes–Robinson model ====
For concentrated ionic solutions the hydration of ions must be taken into consideration, as done by Stokes and Robinson in their hydration model from 1948.<ref>{{Cite journal |doi = 10.1021/ja01185a065|pmid = 18861802|title = Ionic Hydration and Activity in Electrolyte Solutions|journal = Journal of the American Chemical Society|volume = 70|issue = 5|pages = 1870–1878|year = 1948|last1 = Stokes|first1 = R. H|last2 = Robinson|first2 = R. A}}</ref> The activity coefficient of the electrolyte is split into electric and statistical components by E. Glueckauf who modifies the Robinson–Stokes model.

The statistical part includes [[solvation shell|hydration index number]] {{mvar|h}}, the number of ions from the dissociation and the ratio {{mvar|r}} between the [[apparent molar property|apparent molar volume]] of the electrolyte and the molar volume of water and molality {{mvar|b}}.

Concentrated solution statistical part of the activity coefficient is:
:<math>\ln \gamma_s = \frac{h- \nu}{\nu} \ln \left (1 + \frac{br}{55.5} \right) - \frac{h}{\nu} \ln \left (1 - \frac{br}{55.5} \right) + \frac{br(r + h -\nu)}{55.5 \left (1 + \frac{br}{55.5} \right)}</math><ref name="Glueckauf1955">{{Cite journal |url=https://pubs.rsc.org/en/content/articlelanding/1955/tf/tf9555101235 |doi = 10.1039/TF9555101235|title = The influence of ionic hydration on activity coefficients in concentrated electrolyte solutions|journal = Transactions of the Faraday Society|volume = 51|pages = 1235|year = 1955|last1 = Glueckauf|first1 = E.|url-access = subscription}}</ref><ref name="Glueckauf1957">{{Cite journal |url=https://pubs.rsc.org/en/content/articlelanding/1957/TF/tf9575300305 |doi = 10.1039/TF9575300305|title = The influence of ionic hydration on activity coefficients in concentrated electrolyte solutions|journal = Transactions of the Faraday Society|volume = 53|pages = 305|year = 1957|last1 = Glueckauf|first1 = E.|url-access = subscription}}</ref><ref name="Kortüm1960">{{cite journal|last1=Kortüm|first1=G.|title=The Structure of Electrolytic Solutions |publisher=Herausgeg. von W. J. Hamer; John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd. |location=London |url=https://onlinelibrary.wiley.com/doi/10.1002/ange.19600722427 |year=1959 |journal=[[Angewandte Chemie]]|volume=72|issue=24|page=97|issn=0044-8249|doi=10.1002/ange.19600722427|author1-link=Gustav Kortüm|url-access=subscription}}</ref>

The Stokes–Robinson model has been analyzed and improved by other investigators.<ref name="Miller1956">{{Cite journal |last= Miller|first = Donald G. |url=https://pubs.acs.org/doi/pdf/10.1021/j150543a034 |doi = 10.1021/j150543a034 |title = On the Stokes-Robinson Hydration Model for Solutions |journal = The Journal of Physical Chemistry|volume = 60|issue = 9|pages = 1296–1299|year = 1956|url-access = subscription}}</ref><ref>{{Cite journal |last=Nesbitt |first=H. Wayne |doi=10.1007/BF00649040 |url=https://link.springer.com/article/10.1007/BF00649040 |title = The stokes and robinson hydration theory: A modification with application to concentrated electrolyte solutions |journal =[[Journal of Solution Chemistry]] |volume = 11 |issue = 6 |pages = 415–422 |year=1982 |s2cid= 94189765|url-access=subscription }}</ref> The problem with this widely accepted idea that electrolyte activity coefficients are driven at higher concentrations by changes in hydration is that water activities are completely dependent on the concentration of the ions themselves, as imposed by a thermodynamic relationship called the Gibbs-Duhem equation. This means that the activity coefficients and the corresponding water activities are linked together fundamentally, regardless of molecular-level hypotheses. Due to this high correlation, such hypotheses are not independent enough to be satisfactorily tested.

==== Ion trios ====

The rise in activity coefficients found with most aqueous strong electrolyte systems can be explained by increasing electrostatic repulsions between ions of the same charge which are forced together as the available space between them decreases. In this way, the initial attractions between cations and anions at the low concentrations described by Debye and Hueckel are progressively overcome. It has been proposed<ref>{{Cite journal |last=May |first=Peter M. |last2=May |first2=Eric |date=2024 |title=Ion Trios: Cause of Ion Specific Interactions in Aqueous Solutions and Path to a Better pH Definition |url=https://pubs.acs.org/doi/10.1021/acsomega.4c07525 |journal=ACS Omega |volume=9 |issue=46 |pages=46373–46386 |doi=10.1021/acsomega.4c07525|pmc=11579776 }}</ref> that these electrostatic repulsions take place predominantly through the formation of so-called ion trios in which two ions of like charge interact, on average and at distance, with the same counterion as well as with each other. This model accurately reproduces the experimental patterns of activity and osmotic coefficients exhibited by numerous 3-ion aqueous electrolyte mixtures.

==Dependence on state parameters==
The derivative of an activity coefficient with respect to temperature is related to [[Excess molar quantity|excess molar enthalpy]] by 
: <math>\bar{H}^{\mathsf{E}}_i= -RT^2 \frac{\partial}{\partial T}\ln(\gamma_i)</math>
Similarly, the derivative of an activity coefficient with respect to  pressure can be related to excess molar volume.
: <math>\bar{V}^{\mathsf{E}}_i= RT \frac{\partial}{\partial P}\ln(\gamma_i)</math>

== Application to chemical equilibrium ==
At equilibrium, the sum of the chemical potentials of the reactants is equal to the sum of the chemical potentials of the products. The [[Gibbs free energy]] change for the reactions, Δ<sub>r</sub>''G'', is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

:<math> \alpha_\mathrm{A} + \beta_\mathrm{B} = \sigma_\mathrm{S} + \tau_\mathrm{T},</math>
:<math> \Delta_\mathrm{r} G =  \sigma \mu_\mathrm{S} + \tau \mu_\mathrm{T} - (\alpha \mu_\mathrm{A} + \beta \mu_\mathrm{B}) = 0\,</math>
Substitute in the expressions for the chemical potential of each reactant:

:<math> \Delta_\mathrm{r} G = \sigma \mu_S^\ominus + \sigma RT \ln a_\mathrm{S} + \tau \mu_\mathrm{T}^\ominus + \tau RT \ln a_\mathrm{T} -(\alpha \mu_\mathrm{A}^\ominus + \alpha RT \ln a_\mathrm{A} + \beta \mu_\mathrm{B}^\ominus + \beta RT \ln a_\mathrm{B})=0</math>
Upon rearrangement this expression becomes

:<math> \Delta_\mathrm{r} G =\left(\sigma \mu_\mathrm{S}^\ominus+\tau \mu_\mathrm{T}^\ominus -\alpha \mu_\mathrm{A}^\ominus- \beta \mu_\mathrm{B}^\ominus \right) + RT \ln \frac{a_\mathrm{S}^\sigma a_\mathrm{T}^\tau} {a_\mathrm{A}^\alpha a_\mathrm{B}^\beta} =0</math>

The sum 
{{nowrap|''σμ''{{su|b=S|p=<s>o</s>}} + ''τμ''{{su|b=T|p=<s>o</s>}} − ''αμ''{{su|b=A|p=<s>o</s>}} − ''βμ''{{su|b=B|p=<s>o</s>}}}} is the standard free energy change for the reaction, <math>\Delta_\mathrm{r} G^\ominus</math>. 

Therefore,
:<math> \Delta_r G^\ominus = -RT \ln K </math>

where {{mvar|K}} is the [[equilibrium constant]]. Note that activities and equilibrium constants are dimensionless numbers.

This derivation serves two purposes. It shows the relationship between standard free energy change and equilibrium constant. It also shows that an equilibrium constant is defined as a quotient of activities. In practical terms this is inconvenient. When each activity is replaced by the product of a concentration and an activity coefficient, the equilibrium constant is defined as

:<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}</math>
where [S] denotes the [[concentration]] of S, etc. In practice equilibrium constants are [[Determination of equilibrium constants|determined]] in a medium such that the quotient of activity coefficients is constant and can be ignored, leading to the usual expression
:<math>K= \frac{[\mathrm{S}]^\sigma[\mathrm{T}]^\tau}{[\mathrm{A}]^\alpha[\mathrm{B}]^\beta}</math>
which applies under the conditions that the activity quotient has a particular (constant) value.

==References==
{{Reflist}}

== External links ==
* [https://aiomfac.lab.mcgill.ca/ AIOMFAC online-model] An interactive group-contribution model for the calculation of activity coefficients in organic&ndash;inorganic mixtures.
* [http://www.sciencedirect.com/science/article/pii/0013468676850256?np=y ''Electrochimica Acta''] Single-ion activity coefficients

{{Authority control}}

{{DEFAULTSORT:Activity Coefficient}}
[[Category:Thermodynamic models]]
[[Category:Equilibrium chemistry]]
[[Category:Dimensionless numbers of chemistry]]