Editing Activity coefficient (section)

== 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>