Editing Fugacity (section)

==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|p=207}} and
<math display="block">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===
For an [[ideal gas]] the [[equation of state]] can be written as 
<math display="block">V_\mathrm{m}^\text{ideal} = \frac{RT}{P},</math>
where {{math|''R''}} is the [[ideal gas constant]]. The differential change of the chemical potential between two states of slightly different pressures but equal temperature (i.e., {{math|1=d''T'' = 0}}) is given by
<math display="block">d\mu = V_\mathrm{m}dP = RT \,  \frac{dP}{P} = R T\, d \ln P,</math>where '''ln p''' is the [[natural logarithm]] of p.

For real gases the equation of state will depart from the simpler one, and the result above derived for an ideal gas will only  be a good approximation provided that (a) the typical size of the molecule is negligible compared to the average distance between the individual molecules, and (b) 
the short range behavior of the inter-molecular potential can be neglected, i.e., when the molecules can be considered to rebound elastically off each other during molecular collisions. In other words, real gases behave 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| url=https://archive.org/details/chemistryatomsfi00zumd|url-access=limited|date=2012|publisher=Brooks/Cole, CENGAGE Learning| location=Bellmont, CA|isbn=9780840065322|page=[https://archive.org/details/chemistryatomsfi00zumd/page/n332 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 display="block">d\mu = R T \,d \ln f</math>
and
<math display="block"> \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 display="block"> \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 display="block">\mu - \mu^0 = R T \,\ln \frac{f}{P^0},</math>
Note this can also be expressed with <math>a = f/P^0</math>, a dimensionless quantity, 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|1=''P'' = 100}} [[Atmosphere (unit)|atmospheres]] (atm) has a fugacity of {{math|1=''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 {{nowrap|97.03 atm}}. The fugacity coefficient is {{nowrap|1={{sfrac|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|1=''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 display="block">d\mu_\mathrm{c} = R T \,d \ln f_\mathrm{c}</math>
and
<math display="block"> \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|1=''μ''<sub>c</sub> = ''μ''<sub>g</sub>}}). Combined with the above definition, this implies that 
<math display="block">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 {{mvar|P}} is
<math display="block">\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|1=''f''<sub>sat</sub> = ''φ''<sub>sat</sub>&thinsp;''p''<sub>sat</sub>}}, where {{math|''φ''<sub>sat</sub>}} is the fugacity coefficient,
<math display="block">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|publisher=Pearson Education }}</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.