Editing Thermodynamic activity

{{short description|Measure of the effective concentration of a species in a mixture}}
In [[chemical thermodynamics|thermodynamics]], '''activity''' (symbol {{mvar|a}}) is a measure of the "effective concentration" of a [[chemical species|species]] in a mixture, in the sense that the species' [[chemical potential]] depends on the activity of a real solution in the same way that it would depend on [[concentration]] for an [[ideal solution]].  The term "activity" in this sense was coined by the American chemist [[Gilbert N. Lewis]] in 1907.<ref>{{cite journal |last1=Lewis |first1=Gilbert Newton |title=Outlines of a new system of thermodynamic chemistry |journal=Proceedings of the American Academy of Arts and Sciences |date=1907 |volume=43 |issue=7 |pages=259–293 |doi=10.2307/20022322 |jstor=20022322 |url=https://babel.hathitrust.org/cgi/pt?id=njp.32101050586120;view=1up;seq=275|url-access=subscription }} ; the term "activity" is defined on p. 262.</ref>

By convention, activity is treated as a [[dimensionless quantity]], although its value depends on customary choices of [[standard state]] for the species. The activity of pure substances in condensed phases (solids and liquids) is taken as {{mvar|a}} = 1.<ref>{{cite book |last1=Petrucci |first1=Ralph H. |last2=Harwood |first2=William S. |last3=Herring |first3=F. Geoffrey |title=General Chemistry |date=2002 |publisher=Prentice Hall |isbn=0-13-014329-4 |page=804 |edition=8th}}</ref> Activity depends on temperature, pressure and composition of the mixture, among other things. For gases, the activity is the effective partial pressure, and is usually referred to as [[fugacity]].

The difference between activity and other measures of concentration arises because the interactions between different types of [[molecule]]s in non-ideal [[gas]]es or [[Solution (chemistry)|solution]]s are different from interactions between the same types of molecules. The activity of an [[ion]] is particularly influenced by its surroundings.

[[Equilibrium constant]]s should be defined by activities but, in practice, are often defined by [[concentration]]s instead. The same is often true of equations for [[reaction rate]]s. However, there are circumstances where the activity and the concentration are ''significantly different'' and, as such, it is not valid to approximate with concentrations where activities are required. Two examples serve to illustrate this point:
*In a solution of [[potassium hydrogen iodate]] KH(IO<sub>3</sub>)<sub>2</sub> at 0.02&nbsp;[[molar concentration|M]] the activity is 40% lower than the calculated hydrogen ion concentration, resulting in a much higher [[pH]] than expected. 
*When a 0.1&nbsp;M [[hydrochloric acid]] solution containing [[methyl green]] [[pH indicator|indicator]] is added to a 5&nbsp;M solution of [[magnesium chloride]], the color of the indicator changes from green to yellow—indicating increasing acidity—when in fact the acid has been diluted. Although at low ionic strength (<&nbsp;0.1&nbsp;M) the [[activity coefficient]] approaches unity, this coefficient can actually increase with ionic strength in a high ionic strength regime. For hydrochloric acid solutions, the minimum is around 0.4&nbsp;M.<ref name="McCarty2006">{{citation | title = pH Paradoxes: Demonstrating that it is not true that pH ≡ −log[H<sup>+</sup>] | first1 = Christopher G. | last1 = McCarty | first2 = Ed | last2 = Vitz | journal = [[Journal of Chemical Education|J. Chem. Educ.]] | volume = 83 | pages = 752 | year = 2006 | doi = 10.1021/ed083p752 | issue = 5|bibcode = 2006JChEd..83..752M }}</ref>

==Definition==
The relative activity of a species {{mvar|i}}, denoted {{mvar|a<sub>i</sub>}}, is defined<ref name="GoldBook">{{GoldBookRef|title=activity (relative activity), ''a''|file=A00115}}</ref><ref name="GreenBook">{{GreenBookRef2nd|pages=49–50}}</ref> as:
:<math>a_i = e^{\frac{\mu_i - \mu^{\ominus}_i}{RT}}</math>
where {{mvar|μ<sub>i</sub>}} is the (molar) [[chemical potential]] of the species {{mvar|i}} under the conditions of interest, {{math|''μ''{{su|p=<s>o</s>|b=''i''}}}} is the (molar) chemical potential of that species under some defined set of standard conditions, {{mvar|R}} is the [[gas constant]], {{mvar|T}} is the [[thermodynamic temperature]] and {{mvar|e}} is [[e (mathematical constant)|the exponential constant]].

Alternatively, this equation can be written as:
:<math>\mu_i = \mu_i^{\ominus} + RT\ln{a_i}</math>

In general, the activity depends on any factor that alters the chemical potential. Such factors may include: concentration, temperature, pressure, interactions between chemical species, electric fields, etc. Depending on the circumstances, some of these factors, in particular concentration and interactions, may be more important than others.

The activity depends on the choice of standard state such that changing the standard state will also change the activity. This means that activity is a relative term that describes how "active" a compound is compared to when it is under the standard state conditions. In principle, the choice of standard state is arbitrary; however, it is often chosen out of mathematical or experimental convenience. Alternatively, it is also possible to define an "absolute activity", {{mvar|λ}}, which is written as:

:<math>\lambda_i = e^{\frac{\mu_i}{RT}}\,</math>

Note that this definition corresponds to setting as standard state the solution of <math>\mu_i = 0</math>, if the latter exists.

===Activity coefficient===
{{Main|Activity coefficient}}
The activity coefficient {{mvar|γ}}, which is also a dimensionless quantity, relates the activity to a measured [[mole fraction]] {{mvar|x<sub>i</sub>}} (or {{mvar|y<sub>i</sub>}} in the gas phase), [[molality]] {{mvar|b<sub>i</sub>}}, [[mass fraction (chemistry)|mass fraction]] {{mvar|w<sub>i</sub>}},  [[molar concentration]] (molarity) {{mvar|c<sub>i</sub>}} or [[mass concentration (chemistry)|mass concentration]] {{mvar|ρ<sub>i</sub>}}:<ref name="McQuarrie&Simon"> McQuarrie, D. A.; Simon, J. D. ''Physical Chemistry – A Molecular Approach'', p. 990 & p. 1015 (Table 25.1), University Science Books, 1997. </ref> 
:<math>a_{ix} = \gamma_{x,i} x_i,\ a_{ib} = \gamma_{b,i}\frac {b_i} {b^{\ominus}},\, a_{iw}=\gamma_{w,i} w_i,\ a_{ic} = \gamma_{c,i} \frac{c_i}{c^{\ominus}},\, a_{ir} = \gamma_{\rho,i} \frac{\rho_i}{\rho^{\ominus}}\, </math>

The division by the standard molality {{math|''b''<sup><s>o</s></sup>}} (usually 1 mol/kg) or the standard molar concentration {{math|''c''<sup><s>o</s></sup>}} (usually 1 mol/L) is necessary to ensure that both the activity and the activity coefficient are dimensionless, as is conventional.<ref name="GreenBook"/>

The activity depends on the chosen standard state and composition scale;<ref name="McQuarrie&Simon"/> for instance, in the dilute limit it approaches the mole fraction, mass fraction, or numerical value of molarity, all of which are different. However, the activity coefficients are similar.{{citation needed|date=February 2022}}

When the activity coefficient is close to 1, the substance shows almost ideal behaviour according to [[Henry's law]] (but not necessarily in the sense of an [[ideal solution]]). In these cases, the activity can be substituted with the appropriate dimensionless measure of composition {{mvar|x<sub>i</sub>}}, {{math|{{sfrac|''b<sub>i</sub>''|''b''<sup><s>o</s></sup>}}}} or {{math|{{sfrac|''c<sub>i</sub>''|''c''<sup><s>o</s></sup>}}}}. It is also possible to define an activity coefficient in terms of [[Raoult's law]]: the [[International Union of Pure and Applied Chemistry]] (IUPAC) recommends the symbol {{mvar|f}} for this activity coefficient,<ref name="GreenBook"/> although this should not be confused with [[fugacity]].
:<math>a_i = f_i x_i\,</math>

==Standard states==
{{see also|Standard state}}

===Gases===
In most laboratory situations, the difference in behaviour between a real gas and an ideal gas is dependent only on the pressure and the temperature, not on the presence of any other gases. At a given temperature, the "effective" pressure of a gas {{mvar|i}} is given by its [[fugacity]] {{mvar|f<sub>i</sub>}}: this may be higher or lower than its mechanical pressure. By historical convention, fugacities have the dimension of pressure, so the dimensionless activity is given by:
:<math>a_i = \frac{f_i}{p^{\ominus}} = \varphi_i y_i \frac{p}{p^{\ominus}}</math>
where {{mvar|φ<sub>i</sub>}} is the dimensionless fugacity coefficient of the species, {{mvar|y<sub>i</sub>}} is its [[mole fraction]] in the gaseous mixture ({{math|''y'' {{=}} 1}} for a pure gas) and {{mvar|p}} is the total pressure. The value {{math|''p''<sup><s>o</s></sup>}} is the standard pressure: it may be equal to 1&nbsp;[[Atmosphere (unit)|atm]] (101.325&nbsp;[[Pascal (unit)|kPa)]] or 1&nbsp;[[Bar (unit)|bar]] (100&nbsp;kPa) depending on the source of data, and should always be quoted.

===Mixtures in general===
The most convenient way of expressing the composition of a generic mixture is by using the [[mole fraction]]s {{mvar|x{{sub|i}}}} (written {{mvar|y{{sub|i}}}} in the gas phase) of the different components (or chemical species: atoms or molecules) present in the system, where

: <math>x_i = \frac{n_i}{n}\,, \qquad n =\sum_i n_i\,, \qquad \sum_i x_i = 1\,</math>

: with {{mvar|n{{sub|i}}}}, the number of moles of the component ''i'', and {{mvar|n}}, the total number of moles of all the different components present in the mixture. 

The standard state of each component in the mixture is taken to be the pure substance, ''i.e.'' the pure substance has an activity of one. When activity coefficients are used, they are usually defined in terms of [[Raoult's law]],
: <math>a_i = f_i x_i\,</math>
where {{mvar|f<sub>i</sub>}} is the Raoult's law activity coefficient: an activity coefficient of one indicates ideal behaviour according to Raoult's law.

===Dilute solutions (non-ionic)===
A solute in dilute solution usually follows [[Henry's law]] rather than Raoult's law, and it is more usual to express the composition of the solution in terms of the molar concentration {{mvar|c}} (in mol/L) or the molality {{mvar|b}} (in mol/kg) of the solute rather than in mole fractions. The standard state of a dilute solution is a hypothetical solution of concentration {{math|''c''<sup><s>o</s></sup>}}&nbsp;= 1&nbsp;mol/L (or molality {{math|''b''<sup><s>o</s></sup>}}&nbsp;= 1&nbsp;mol/kg) which shows ideal behaviour (also referred to as "infinite-dilution" behaviour). The standard state, and hence the activity, depends on which measure of composition is used. Molalities are often preferred as the volumes of non-ideal mixtures are not strictly additive and are also temperature-dependent: molalities do not depend on volume, whereas molar concentrations do.<ref name="Kaufman2002">{{Citation | first = Myron | last = Kaufman | title = Principles of Thermodynamics | page = 213 | publisher = CRC Press | year = 2002 | isbn = 978-0-8247-0692-0}}</ref>

The activity of the solute is given by:
:<math>\begin{align}
a_{c,i} &= \gamma_{c,i}\, \frac{c_i}{c^{\ominus}} \\[6px]
a_{b,i} &= \gamma_{b,i}\, \frac{b_i}{b^{\ominus}}
\end{align}</math>

===Ionic solutions===
When the solute undergoes ionic dissociation in solution (for example a salt), the system becomes decidedly non-ideal and we need to take the dissociation process into consideration. One can define activities for the cations and anions separately ({{math|''a''<sub>+</sub>}} and {{math|''a''<sub>–</sub>}}).

In a liquid solution the activity coefficient of a given [[ion]] (e.g. Ca<sup>2+</sup>) isn't measurable because it is experimentally impossible to independently measure the electrochemical potential of an ion in solution.  (One cannot add cations without putting in anions at the same time). Therefore, one introduces the notions of

;mean ionic activity
:{{math|''a''{{su|b=±|p=''ν''}} {{=}} ''a''{{su|b=+|p=''ν''<sub>+</sub>}}''a''{{su|b=−|p=''ν''<sub>−</sub>}}}}
;mean ionic molality
:{{math|''b''{{su|b=±|p=''ν''}} {{=}} ''b''{{su|b=+|p=''ν''<sub>+</sub>}}''b''{{su|b=−|p=''ν''<sub>−</sub>}}}}
;mean ionic activity coefficient
:{{math|''γ''{{su|b=±|p=''ν''}} {{=}} ''γ''{{su|b=+|p=''ν''<sub>+</sub>}}''γ''{{su|b=−|p=''ν''<sub>−</sub>}}}}
where {{math|''ν'' {{=}} ''ν''<sub>+</sub> + ''ν''<sub>–</sub>}} represent the stoichiometric coefficients involved in the ionic dissociation process

Even though {{math|''γ''<sub>+</sub>}} and {{math|''γ''<sub>–</sub>}} cannot be determined separately, {{math|''γ''<sub>±</sub>}} is a measurable quantity that can also be predicted for sufficiently dilute systems using [[Debye–Hückel theory]]. For electrolyte solutions at higher concentrations, Debye–Hückel theory needs to be extended and replaced, e.g., by a [[Pitzer equations|Pitzer]] electrolyte solution model (see [[#External_links|external links]] below for examples). For the activity of a strong ionic solute (complete dissociation) we can write:
:{{math|''a''<sub>2</sub> {{=}} ''a''{{su|b=±|p=''ν''}} {{=}} ''γ''{{su|b=±|p=''ν''}}''m''{{su|b=±|p=''ν''}}}}

==Measurement==
The most direct way of measuring the activity of a volatile species is to measure its equilibrium partial [[vapor pressure]]. For water as solvent, the [[water activity]] ''a<sub>w</sub>'' is the equilibrated [[Humidity|relative humidity]]. For non-volatile components, such as [[sucrose]] or [[sodium chloride]], this approach will not work since they do not have measurable vapor pressures at most temperatures. However, in such cases it is possible to measure the vapor pressure of the ''solvent'' instead. Using the [[Gibbs–Duhem relation]] it is possible to translate the change in solvent vapor pressures with concentration into activities for the solute.

The simplest way of determining how the activity of a component depends on pressure is by measurement of densities of solution, knowing that real solutions have deviations from the additivity of (molar) volumes of pure components compared to the (molar) volume of the solution. This involves the use of [[partial molar volume]]s, which measure the change in chemical potential with respect to pressure.

Another way to determine the activity of a species is through the manipulation of [[colligative properties]], specifically [[freezing point depression]]. Using freezing point depression techniques, it is possible to calculate the activity of a weak acid from the relation,
:<math>b^{\prime} = b(1 + a)\,</math>
where {{mvar|b′}} is the total equilibrium molality of solute determined by any colligative property measurement (in this case {{math|Δ''T''<sub>fus</sub>}}), {{mvar|b}} is the nominal molality obtained from titration and {{mvar|a}} is the activity of the species.

There are also electrochemical methods that allow the determination of activity and its coefficient.

The value of the mean ionic activity coefficient {{math|''γ''<sub>±</sub>}} of [[ion]]s in solution can also be estimated with the [[Debye–Hückel equation]], the [[Davies equation]] or the [[Pitzer equations]].

===Single ion activity measurability revisited===
The prevailing view that single ion activities are unmeasurable, or perhaps even physically meaningless, has its roots in the work of [[Edward A. Guggenheim]] in the late 1920s.<ref>{{cite journal|first=E. A. |last=Guggenheim |title=The Conceptions of Electrical Potential Difference between Two Phases and the Individual Activities of Ions |journal=[[J. Phys. Chem.]] |volume=33 |issue=6 |date=1929 |pages=842–849 |doi=10.1021/j150300a003}}</ref> However, chemists have not given up the idea of single ion activities. For example, [[pH]] is defined as the negative logarithm of the hydrogen ion activity. By implication, if the prevailing view on the physical meaning and measurability of single ion activities is correct it relegates pH to the category of thermodynamically unmeasurable quantities. For this reason the [[International Union of Pure and Applied Chemistry]] (IUPAC) states that the activity-based definition of pH is a notional definition only and further states that the establishment of primary pH standards requires the application of the concept of 'primary method of measurement' tied to the Harned cell.<ref>{{GoldBookRef|file=P04524|title=pH}}</ref>  Nevertheless, the concept of single ion activities continues to be discussed in the literature, and at least one author purports to define single ion activities in terms of purely thermodynamic quantities. The same author also proposes a method of measuring single ion activity coefficients based on purely thermodynamic processes.<ref>{{cite journal|first=A.L. |last=Rockwood |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 |date=2015 |pages=1978–1991 |doi=10.1002/cphc.201500044 |pmid=25919971 |pmc=4501315 }}</ref> A different approach <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> has a similar objective.

==Use==
Chemical activities should be used to define [[chemical potential]]s, where the chemical potential depends on the [[temperature]] {{mvar|T}}, [[pressure]] {{mvar|p}} and the activity {{mvar|a<sub>i</sub>}} according to the [[formula]]:
:<math>\mu_i = \mu_i^{\ominus} + RT\ln{a_i}</math>
where {{mvar|R}} is the [[gas constant]] and {{math|''μ''{{su|b=''i''|p=<s>o</s>}}}} is the value of {{mvar|μ<sub>i</sub>}} under standard conditions. Note that the choice of concentration scale affects both the activity and the standard state chemical potential, which is especially important when the reference state is the infinite dilution of a solute in a solvent. Chemical potential has units of joules per mole (J/mol), or energy per amount of matter. Chemical potential can be used to characterize the specific [[Gibbs free energy]] changes occurring in chemical reactions or other transformations.

Formulae involving activities can be simplified by considering that:
* For a chemical solution:
** the [[solvent]] has an activity of unity (only a valid approximation for rather dilute solutions)
** At a low concentration, the activity of a solute can be approximated to the ratio of its concentration over the standard concentration: <math display="block">a_i = \frac{c_i}{c^{\ominus}}</math>
Therefore, it is approximately equal to its concentration.

* For a mix of [[gas]] at low pressure, the activity is equal to the ratio of the [[partial pressure]] of the gas over the standard pressure: <math display="block">a_i = \frac{p_i}{p^{\ominus}}</math> Therefore, it is equal to the partial pressure in atmospheres (or bars), compared to a standard pressure of 1 atmosphere (or 1 bar).
* For a solid body, a uniform, single species solid has an activity of unity at standard conditions. The same thing holds for a pure liquid.

The latter follows from any definition based on Raoult's law, because if we let the solute concentration {{math|''x''<sub>1</sub>}} go to zero, the vapor pressure of the solvent {{mvar|p}} will go to {{mvar|p*}}. Thus its activity {{math|1=''a'' = {{sfrac|''p''|''p''*}}}} will go to unity. This means that if during a reaction in dilute solution more solvent is generated (the reaction produces water for example) we can typically set its activity to unity.

Solid and liquid activities do not depend very strongly on pressure because their molar volumes are typically small. [[Graphite]] at 100&nbsp;bars has an activity of only 1.01 if we choose {{math|1=''p''<sup><s>o</s></sup> = 1 bar}} as standard state. Only at very high pressures do we need to worry about such changes. Activity expressed in terms of pressure is called [[fugacity]].

==Example values==
Example values of activity coefficients of [[sodium chloride]] in aqueous solution are given in the table.<ref name="Cohen1988">{{citation | first = Paul | last = Cohen | title = The ASME Handbook on Water Technology for Thermal Systems | publisher = American Society of Mechanical Engineers | year = 1988 | page = 567 | isbn = 978-0-7918-0300-4}}</ref> In an ideal solution, these values would all be unity. The deviations ''tend'' to become larger with increasing molality and temperature, but with some exceptions.
{| class="wikitable sortable"
|+Activity coefficients of sodium chloride in aqueous solution
|-
![[Molality]] (mol/kg)
! 25&nbsp;°C
! 50&nbsp;°C
! 100&nbsp;°C
! 200&nbsp;°C
! 300&nbsp;°C
! 350&nbsp;°C
|-
| 0.05
| 0.820
| 0.814
| 0.794
| 0.725 
| 0.592 
| 0.473
|-
| 0.50
| 0.680
| 0.675
| 0.644
| 0.619 
| 0.322 
| 0.182
|-
|-
| 2.00
| 0.669
| 0.675
| 0.641
| 0.450 
| 0.212 
| 0.074
|-
| 5.00
| 0.873
| 0.886
| 0.803
| 0.466 
| 0.167 
| 0.044
|-
|}

==See also==
{{Portal|Chemistry}}
*[[Fugacity]], the equivalent of activity for [[partial pressure]]
*[[Chemical equilibrium]]
*[[Electrochemical potential]]
*[[Excess chemical potential]]
*[[Partial molar property]]
*[[Thermodynamic equilibrium]]
*[[Thermal expansion]]
*[[Virial expansion]]
*[[Water activity]]
* [[Non-random two-liquid model]] (NRTL model) – phase equilibrium calculations
* [[UNIQUAC]] model – phase equilibrium calculations

==References==
{{Reflist}}

==External links==
* [http://phasediagram.dk/chemical_potentials.htm Equivalences among different forms of activity coefficients and chemical potentials]
* [http://www.aim.env.uea.ac.uk/aim/aim.php Calculate activity coefficients of common inorganic electrolytes and their mixtures]
* [https://aiomfac.lab.mcgill.ca/ AIOMFAC online-model]: calculator for activity coefficients of inorganic ions, water, and organic compounds in aqueous solutions and multicomponent mixtures with organic compounds.
{{Chemical equilibria}}{{Authority control}}

[[Category:Dimensionless numbers of chemistry]]
[[Category:Physical chemistry]]
[[Category:Thermodynamic properties]]