Editing Activity coefficient (section)

== 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.