Editing Activity coefficient (section)

=== 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.