View source for Fugacity

{{short description|Effective partial pressure}}
In [[chemical thermodynamics]], the '''fugacity''' of a [[real gas]] is an effective [[partial pressure]] which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an [[ideal gas]] which has the same temperature and molar [[Gibbs free energy]] as the real gas.<ref name=AtkinsDePaula>{{cite book|first1=Peter |last1=Atkins |first2=Julio |last2=De Paula |title=Atkins' Physical Chemistry |url=https://archive.org/details/atkinsphysicalch00pwat |url-access=registration |edition=8th |publisher=W. H. Freeman |date=2006 |pages=[https://archive.org/details/atkinsphysicalch00pwat/page/111 111–2]}}</ref>

Fugacities are determined experimentally or estimated from various models such as a [[Van der Waals gas]] that are closer to reality than an ideal gas.  The real gas pressure and fugacity are related through the dimensionless '''fugacity coefficient''' {{math|''φ''}}.<ref name=AtkinsDePaula/>

:<math> \varphi = \frac{f}{P} \,</math>
For an ideal gas, fugacity and pressure are equal and so {{math|''φ'' {{=}} 1}}. Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to {{math|''RT'' ln ''φ''}}.

The fugacity is closely related to the [[activity (chemistry)|thermodynamic activity]]. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the [[standard state]] and normally chosen as 1 [[atmosphere (unit)|atmosphere]] or 1 [[bar (unit)|bar]].

Accurate calculations of [[chemical equilibrium]] for real gases should use the fugacity rather than the pressure. The thermodynamic condition for chemical equilibrium is that the total chemical potential of reactants is equal to that of products. If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar [[reaction quotient]] form (or [[law of mass action]]) except that the pressures are replaced by fugacities.

For a condensed phase (liquid or solid) in equilibrium with its vapor phase, the chemical potential is equal to that of the vapor, and therefore the fugacity is equal to the fugacity of the vapor. This fugacity is approximately equal to the [[vapor pressure]] when the vapor pressure is not too high.

==Pure substance==
Fugacity is closely related to the [[chemical potential]] {{math|''μ''}}. In a pure substance, {{math|''μ''}} is equal to the [[Gibbs free energy|Gibbs energy]] {{math|''G''<sub>m</sub>}} for a [[mole (unit)|mole]] of the substance,<ref name=Ott>{{cite book|last1=Ott|first1=J. Bevan|last2=Boerio-Goates|first2=Juliana|title=Chemical thermodynamics: Principles and applications|date=2000|publisher=Academic Press|location=London, UK|isbn=9780080500980}}</ref>{{rp|207}} and
:<math>d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP</math>,
where {{math|''T''}} and {{math|''P''}} are the temperature and pressure, {{math|''V''<sub>m</sub>}} is the [[molar volume|volume per mole]] and {{math|''S''<sub>m</sub>}} is the [[entropy]] per mole.<ref name=Ott/>{{rp|248}}

===Gas===
An [[ideal gas]] has the [[equation of state]]
:<math>V_\mathrm{m}^\mathrm{ideal} = \frac{RT}{P}</math>,
where {{math|''R''}} is the [[ideal gas constant]]. At constant temperature ({{math|d''T'' {{=}} 0}}),
:<math>d\mu = R T d \ln P.</math>

This equation is very useful but is only accurate when gas molecules are small compared to the distances between them and rebound elastically off each other. Gases behave most like ideal gases at low pressures and high temperatures.<ref>{{cite book|last1=Zumdahl|first1=Steven S.|last2=Zumdahl|first2=Susan A|title=Chemistry : an atoms first approach|date=2012|publisher=Brooks/Cole, CENGAGE Learning|location=Bellmont, CA|isbn=9780840065322|page=309}}</ref> At moderately high pressures, attractive interactions between molecules reduce the pressure compared to the ideal gas law; and at very  high pressures, the sizes of the molecules are no longer negligible and repulsive forces between molecules increases the pressure. At low temperatures, molecules are more likely to stick together instead of rebounding elastically.<ref>{{cite book|last1=Clugston|first1=Michael|last2=Flemming|first2=Rosalind|title=Advanced chemistry|date=2000|publisher=Univ. Press|location=Oxford|isbn=9780199146338|page=122}}</ref>

The ideal gas law can still be used to describe the behavior of a [[real gas]] if the pressure is replaced by a ''fugacity'' {{math|''f''}}, defined so that
:<math>d\mu = R T d \ln f</math>
and
:<math> \lim_{P\to 0} \frac{f}{P} = 1.</math>
That is, at low pressures {{math|''f''}} is the same as the pressure, so it has the same units as pressure. The ratio
:<math> \varphi = \frac{f}{P}</math>
is called the ''fugacity coefficient''.<ref name=Ott/>{{rp|248–249}}

If a reference state is denoted by a zero superscript, then integrating the equation for the chemical potential gives
:<math>\mu - \mu^0 = R T \ln a,</math>
where {{math|''a''}}, a dimensionless quantity, is called the ''[[Thermodynamic activity|activity]]''.<ref>{{cite book|last1=Zhu|first1=Chen|last2=Anderson|first2=Greg|title=Environmental applications of geochemical modeling|date=2002|publisher=Cambridge Univ. Press|location=Cambridge|isbn=9780521005777}}</ref>{{rp|37}}

'''Numerical example:''' [[Nitrogen]] gas (N<sub>2</sub>) at 0&nbsp;°C and a pressure of {{math|''P'' {{=}} 100}} [[Atmosphere (unit)|atmospheres]] (atm) has a fugacity of {{math|''f'' {{=}} 97.03}}&nbsp;atm.<ref name=AtkinsDePaula/> This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at {{math|97.03 atm}}. The fugacity coefficient is {{math|97.03 atm / 100 atm {{=}} 0.9703}}.

The contribution of nonideality to the molar Gibbs energy of a real gas is equal to {{math|''RT'' ln ''φ''}}. For nitrogen at 100&nbsp;atm, {{math|''G''<sub>m</sub> {{=}} ''G''<sub>m,id</sub> + ''RT'' ln 0.9703}}, which is less than the ideal value {{math|''G''<sub>m,id</sub>}} because of intermolecular attractive forces. Finally, the activity is just {{math|97.03}} without units.

=== Condensed phase ===
{{See also|Vapor–liquid equilibrium}}
The fugacity of a condensed phase (liquid or solid) is defined the same way as for a gas:
:<math>d\mu_\mathrm{c} = R T d \ln f_\mathrm{c}</math>
and
:<math> \lim_{P\to 0} \frac{f_\mathrm{c}}{P} = 1.</math>
It is difficult to measure fugacity in a condensed phase directly; but if the condensed phase is ''saturated'' (in equilibrium with the vapor phase), the chemical potentials of the two phases are equal ({{math|''&mu;''<sub>c</sub> {{=}} ''&mu;''<sub>g</sub>}}). Combined with the above definition, this implies that 
:<math>f_\mathrm{c} = f_\mathrm{g}.</math>

When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. At constant temperature, the change in fugacity as the pressure goes from the saturation press {{math|''P''<sub>sat</sub>}} to {{math|''P''}} is
:<math>\ln\frac{f}{f_\mathrm{sat}} = \frac{V_\mathrm{m}}{R T}\int_{P_\mathrm{sat}}^P dp = \frac{V\left(P-P_\mathrm{sat}\right)}{R T}.</math>
This fraction is known as the [[Poynting effect|Poynting factor]]. Using {{math|''f''<sub>sat</sub> {{=}} ''φ''<sub>sat</sub>''P''<sub>sat</sub>}}, where {{math|''φ''<sub>sat</sub>}} is the fugacity coefficient,
:<math>f = \varphi_\mathrm{sat}P_\mathrm{sat}\exp\left(\frac{V\left(P-P_\mathrm{sat}\right)}{R T}\right).</math>
This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1.<ref name= Matsoukas>{{cite book|last1=Matsoukas|first1=Themis|title=Fundamentals of chemical engineering thermodynamics : with applications to chemical processes|date=2013|publisher=Prentice Hall|location=Upper Saddle River, NJ|isbn=9780132693066}}</ref>{{rp|345–346}}<ref>{{cite book|last1=Prausnitz|first1=John M.|last2=Lichtenthaler|first2=Rudiger N.|last3=Azevedo|first3=Edmundo Gomes de|title=Molecular Thermodynamics of Fluid-Phase Equilibria|pages=40–43|isbn=9780132440509|date=1998-10-22}}</ref>

Unless pressures are very high, the Poynting factor is usually small and the exponential term is near 1. Frequently, the fugacity of the pure liquid is used as a reference state when defining and using mixture activity coefficients.

==Mixture==
The fugacity is most useful in mixtures. It does not add any new information compared to the chemical potential, but it has computational advantages. As the molar fraction of a component goes to zero, the chemical potential diverges but the fugacity goes to zero. In addition, there are natural reference states for fugacity (for example, an ideal gas makes a natural reference state for gas mixtures since the fugacity and pressure converge at low pressure).<ref name=O'Connell>{{cite book|last1=O'Connell|first1=J. P.|last2=Haile|first2=J. M.|title=Thermodynamics: Fundamentals for Applications|date=2005|publisher=Cambridge University Press|isbn=9781139443173}}</ref>{{rp|141}}

===Gases===
In a mixture of gases, the fugacity of each component {{math|''i''}} has a similar definition, with [[Partial molar property|partial molar quantities]] instead of molar quantities (e.g., {{math|''G''<sub>''i''</sub>}} instead of {{math|''G''<sub>m</sub>}} and {{math|''V''<sub>''i''</sub>}} instead of {{math|''V''<sub>m</sub>}}):<ref name=Ott/>{{rp|262}}
:<math>dG_i = R T d \ln f_i</math>
and
:<math> \lim_{P\to 0} \frac{f_i}{P_i} = 1,</math>
where {{math|''P<sub>i</sub>''}} is the [[partial pressure]] of component {{math|''i''}}. The partial pressures obey [[Dalton's law]]:
:<math>P_i = y_iP,</math>
where {{math|''P''}} is the total pressure and {{math|''y<sub>i</sub>''}} is the mole fraction of the component (so the partial pressures add up to the total pressure). The fugacities commonly obey a similar law called the Lewis and Randall rule:
:<math>f_i = y_if^*_i,</math>
where {{math|''f''<sup>*</sup><sub>''i''</sub>}} is the fugacity that component {{math|''i''}} would have if the entire gas had that composition at the same temperature and pressure. Both laws are expressions of an assumption that the gases behave independently.<ref name=Ott/>{{rp|264–265}}

===Liquids===
In a liquid mixture, the fugacity of each component is equal to that of a vapor component in equilibrium with the liquid. In an [[ideal solution]], the fugacities obey [[Raoult's law]]:
:<math>f_i = x_if^*_i,</math>
where {{math|''x<sub>i</sub>''}} is the mole fraction in the liquid and {{math|''f''<sup>*</sup><sub>''i''</sub>}} is the fugacity of the pure vapor phase. This is a good approximation when the component molecules have similar size, shape and polarity.<ref name=Ott/>{{rp|264,269–270}}

In a dilute solution with two components, the component with the larger molar fraction (the [[solvent]]) may still obey Raoult's law even if the other component (the [[solute]]) has different properties. That is because its molecules experience essentially the same environment that they do in the absence of the solute. By contrast, each solute molecule is surrounded by solvent molecules, so it obeys a different law known as [[Henry's law]].<ref>{{cite book|last1=Atkins|first1=Peter|last2=Paula|first2=Julio de|title=Physical chemistry|date=2002|publisher=W.H. Freeman|location=New York|isbn=9780716735397|edition=7th|url-access=registration|url=https://archive.org/details/physicalchemistr0000atki}}</ref>{{rp|171}} By Henry's law, the fugacity of the solute is proportional to its concentration. The constant of proportionality depends on whether the concentration is represented by the mole fraction, [[molality]] or [[molarity]].<ref name=Ott/>{{rp|274}}

==Temperature and pressure dependence==
The pressure dependence of fugacity (at constant temperature) is given by<ref name=Ott/>{{rp|260}}
:<math> \left(\frac{\partial \ln f}{\partial P}\right)_T = \frac{V_\mathrm{m}}{R T}</math>
and is always positive.<ref name=Ott/>{{rp|260}}

The temperature dependence at constant pressure is
:<math>\left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2},</math>
where {{math|&Delta; ''H''<sub>m</sub>}} is the change in molar [[enthalpy]] as the gas expands, liquid vaporizes, or solid sublimates into a vacuum.<ref name=Ott/>{{rp|262}} Also, if the pressure is {{math|''p''<sup>0</sup>}}, then
:<math>\left(\frac{\partial \left(T \ln (f/p^0)\right)}{\partial T}\right)_P = -\frac{S_\mathrm{m}}{R} <  0.</math>
Since the temperature and entropy are positive, {{math|ln (''f''/''p''<sup>0</sup>)}} decreases with increasing temperature.<ref name=Franses>{{cite book|last1=Franses|first1=Elias I.|title=Thermodynamics with chemical engineering applications|date=2014|publisher=Cambridge University Press|isbn=9781107069756|page=248}} Note that Equations 9.24 and 9.25 left out {{math|''p''<sup>0</sup>}} in substituting from Equation 9.6. This error is corrected in the above equation.</ref>

==Relationship to entropy==
Fugacity, activity, and [[chemical potential]] are very closely related to entropy.  Specifically, the fugacity of a species (divided by the reference pressure) is the partial [[multiplicative calculus#Multiplicative derivatives|geometric derivative]] of the dimensionless [[entropy]] of the system with respect to exchanging elementary entities of that component for those of a pure ideal gas at reference pressure.{{cn|date=April 2019}}

==Measurement==
The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. In that case,
:<math> \ln\varphi = \frac{1}{R T}\int_0^P \left(V_m - V_\mathrm{m}^\mathrm{ideal}\right) d P.</math>
This integral can also be calculated using an equation of state.<ref name=Ott/>{{rp|251–252}}

The integral can be recast in an alternative form using the [[compressibility factor]]
:<math>Z = \frac{P V_\mathrm{m}}{R T}.</math>
Then
:<math> \ln\varphi = \int_0^P \left(\frac{Z-1}{P}\right) d P.</math>
This is useful because of the [[theorem of corresponding states]]: If the pressure and temperature at the [[Critical point (thermodynamics)|critical point]] of the gas are {{math|''P''<sub>c</sub>}} and {{math|''T''<sub>c</sub>}}, we can define [[reduced properties]] {{math|''P''<sub>r</sub> {{=}} ''P'' /''P''<sub>c</sub>}} and {{math|''T''<sub>r</sub> {{=}} ''T'' /''T''<sub>c</sub>}}. Then, to a good approximation, most gases have the same value of {{math|''Z''}} for the same reduced temperature and pressure. However, in [[geochemistry|geochemical]] applications, this principle ceases to be accurate at pressures where [[metamorphism]] occurs.<ref name=Anderson>{{cite book|last1=Anderson|first1=Greg M.|last2=Crerar|first2=David A.|title=Thermodynamics in Geochemistry: The Equilibrium Model|date=1993|publisher=Oxford University Press|isbn=9780195345094}}</ref>{{rp|247}}
  
For a gas obeying the [[van der Waals equation]], the explicit formula for the fugacity coefficient is
:<math>RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right )</math>
This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation.<ref>{{cite journal|last=David |first=Carl W. |title=Fugacity Examples 2: The fugacity of a "hard-sphere" semi-ideal gas and the van der Waals gas |date=2015 |journal=Chemistry Education Materials |volume=91|url=http://digitalcommons.uconn.edu/chem_educ/91}}</ref>

==History==
The word ''fugacity'' is derived from the Latin ''fugere'', to flee. In the sense of an "escaping tendency", it was introduced to thermodynamics in 1901 by the American chemist [[Gilbert N. Lewis]] and popularized in an influential textbook by Lewis and [[Merle Randall]], ''Thermodynamics and the Free Energy of Chemical Substances'', in 1923.<ref>{{cite journal |last1=Lewis |first1=Gilbert Newton |title=The law of physico-chemical change |journal=Proceedings of the American Academy of Arts and Sciences |date=June 1901 |volume=37 |issue=3 |pages=49–69 |url=https://babel.hathitrust.org/cgi/pt?id=njp.32101050586062;view=1up;seq=57 |doi=10.2307/20021635 |jstor=20021635 }} ; the term "fugacity" is coined on p. 54.</ref> The "escaping tendency" referred to the flow of matter between phases and played a similar role to that of temperature in heat flow.<ref>{{cite journal |last1=Lewis |first1=Gilbert Newton |title=A new conception of thermal pressure and a theory of solutions |journal=Proceedings of the American Academy of Arts and Sciences |date=1900 |volume=36 |issue=9 |pages=145–168 |url=https://babel.hathitrust.org/cgi/pt?id=njp.32101050586054;view=1up;seq=165|doi=10.2307/20020988 |jstor=20020988 }} The term "escaping tendency" is introduced on p. 148, where it is represented by the Greek letter ''ψ'' ; ''ψ'' is defined for ideal gases on p. 156.</ref><ref name=Anderson2017>{{cite book|last1=Anderson|first1=Greg|title=Thermodynamics of Natural Systems: Theory and Applications in Geochemistry and Environmental Science|date=2017|publisher=Cambridge University Press|isbn=9781107175211}}</ref>{{rp|177}}

==See also==
{{div col}}
*[[Electrochemical potential]]
*[[Excess chemical potential]]
*[[Fugacity capacity]]
*[[Multimedia fugacity model]]
*[[Thermodynamic equilibrium]]
{{div col end}}

==References==
{{Reflist}}

==Further reading==
{{Refbegin}}
*{{cite book|last1=Anderson|first1=Greg M.|last2=Crerar|first2=David A.|title=Thermodynamics in Geochemistry: The Equilibrium Model|date=1993|publisher=Oxford University Press|isbn=9780195345094}}
*{{cite book|last1=Mackay |first1=Don |chapter=The fugacity approach to mass transport and MTCs  |editor-last1=Thibodeaux |editor-first1=Louis J. |editor-last2=Mackay |editor-first2=Donald |title=Handbook of chemical mass transport in the environment |date=2011 |publisher=CRC Press |location=Boca Raton, FL |isbn=9781420047561 |pages=43&ndash;50}}
*{{cite journal|last1=Mackay |first1=Don |last2=Arnot |first2=Jon A. |title=The Application of Fugacity and Activity to Simulating the Environmental Fate of Organic Contaminants |journal=Journal of Chemical & Engineering Data |date=14 April 2011 |volume=56 |issue=4 |pages=1348–1355 |doi=10.1021/je101158y}}
{{Refend}}

==External links==

===Video lectures===

*Thermodynamics, University of Colorado-Boulder, 2011
**[https://www.youtube.com/watch?v=fkqmf_QR4Yg Introduction to fugacity: Where did it come from?]
**[https://www.youtube.com/watch?v=AMBoLiQJMFQ What is fugacity?]
**[https://www.youtube.com/watch?v=OZFB48wVtjw What is fugacity in mixtures?]

[[Category:Physical chemistry]]
[[Category:Chemical thermodynamics]]
[[Category:Thermodynamic properties]]
[[Category:State functions]]