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{{short description|Measure of the effective concentration of a species in a mixture}}
In [[chemical 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}} ; 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 (
The difference between activity and other measures of
*In a solution of [[potassium hydrogen iodate]] KH(IO<sub>3</sub>)<sub>2</sub> at 0.02 [[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 M [[hydrochloric acid]] solution containing [[methyl green]] [[pH indicator|indicator]] is added to a 5 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 (< 0.1 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 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>
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:<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.
:<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 [[
:<math>
The division by the standard molality {{math|''b''<sup><s>o</s></sup>}} (usually 1 mol/kg) or the standard
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>
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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 ({{
===Mixtures in general===
The most convenient way of expressing the composition of a generic mixture is by using the
: <math>x_i = \frac{n_i}{n}\,, \qquad n =\sum_i n_i\,, \qquad \sum_i x_i = 1\,</math>
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>▼
: 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
The activity of the solute is given by:
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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
:{{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
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.
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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.
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===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
==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|''
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
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''
▲* For a solid body, a uniform, single species solid at one bar has an activity of unity. The same thing holds for a pure liquid.
Solid and liquid activities do not depend very strongly on pressure because their molar volumes are typically small. [[Graphite]] at 100 bars has an activity of only 1.01 if we choose {{math|1=''p''<sup><s>o</s></sup>
▲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|''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 e.g.) 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 bars has an activity of only 1.01 if we choose {{math|''p''<sup><s>o</s></sup>}} = 1 bar as standard state. Only at very high pressures do we need to worry about such changes.
==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.
|+Activity coefficients of sodium chloride in aqueous solution
▲:{| cellpadding="10" class="wikitable sortable"
|-
![[Molality]] (mol/kg)
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* [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]
* [
{{Chemical equilibria}}{{Authority control}}▼
▲{{Authority control}}
[[Category:Dimensionless numbers of chemistry]]
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