Thermodynamic activity: Difference between revisions

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 (solidsolids orand liquids) is normally taken as [[Unity{{mvar|a}} = 1.<ref>{{cite book (mathematics)|unity]]last1=Petrucci (the|first1=Ralph numberH. 1)|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 compositionconcentration arises because the interactions between different types of [[molecule]]s in non-ideal [[gas]]es or [[Solution (chemistry)|solution]]s interactare withdifferent eachfrom other,interactions eitherbetween to attract orthe tosame repeltypes eachof othermolecules. The activity of an [[ion]] is particularly influenced by its surroundings.

Activities ''should'' be used to define [[equilibriumEquilibrium constant]]s should be defined by activities but, in practice, are often defined by [[concentration]]s are often used 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 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" (i.e., the [[fugacity]] in statistical mechanics), {{mvar|λ}}, which is written as:

The activity coefficient {{mvar|γ}}, which is also a dimensionless quantity, relates the activity to a measured [[amountmole fraction]] {{mvar|x_i}} (or {{mvar|y_i}} in the gas phase), [[molality]] {{mvar|b_i}}, [[mass fraction (chemistry)|mass fraction]] {{mvar|w_i}}, [[amountmolar concentration]] (molarity) {{mvar|c_i}} or [[mass concentration (chemistry)|mass concentration]] {{mvar|ρ_i}}:<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_ia_{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''^<s>o</s>}} (usually 1 mol/kg) or the standard amountmolar concentration {{math|''c''^<s>o</s>}} (usually 1 mol/L) is necessary to ensure that both the activity and the activity coefficient are dimensionless, as is conventional.<ref name="GreenBook"/>

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_i}}, {{math|{{sfrac|''b_i''|''b''^<s>o</s>}}}} or {{math|{{sfrac|''c_i''|''c''^<s>o</s>}}}}. 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]].

where {{mvar|φ_i}} is the dimensionless fugacity coefficient of the species, {{mvar|y_i}} 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''^<s>o</s>}} 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.

The most convenient way of expressing the composition of a generic mixture is by using the amount[[mole fractionsfraction]]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>

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 amountmolar concentration {{mvar|c}} (in mol/L) or the molality {{mvar|b}} (in mol/kg) of the solute rather than in amountmole fractions. The standard state of a dilute solution is a hypothetical solution of concentration {{math|''c''^<s>o</s>}}&nbsp;= 1&nbsp;mol/L (or molality {{math|''b''^<s>o</s>}}&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 amountmolar 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>

Even though {{math|''γ''₊}} and {{math|''γ''_–}} cannot be determined separately, {{math|''γ''_±}} 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:

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_w'' 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.

where {{mvar|b′}} is the total equilibrium molality of solute determined by any colligative property measurement (in this case {{math|Δ''T''_fus}}), {{mvar|b}} is the nominal molality obtained from titration and {{mvar|a}} is the activity of the species.

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 never been able tonot givegiven 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>

where {{mvar|R}} is the [[gas constant]] and {{math|''μ''{{su|b=''i''|p=<s>o</s>}}}} is the value of {{mvar|μ_i}} 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.

** 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>

* 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).