# Fugacity

**Fugacity** is a pressure-like thermodynamic quantity, denoted \( f \) or \( \tilde{p} \), that replaces pressure in the ideal-gas expression for chemical potential so that the same logarithmic form remains valid for real gases and mixtures. The [International Union of Pure and Applied Chemistry](https://www.edgepedia.org/international-union-of-pure-and-applied-chemistry) defines the fugacity of a substance B in a gaseous mixture by

\[ f_{\mathrm{B}} = \lambda_{\mathrm{B}} \lim_{p \to 0} \frac{p_{\mathrm{B}}}{\lambda_{\mathrm{B}}} \]

where \( p_{\mathrm{B}} \) is the partial pressure of B and \( \lambda_{\mathrm{B}} \) its absolute activity.<sup>[1](https://goldbook.iupac.org/terms/view/F02543)</sup> In practical terms, fugacity is the pressure that a hypothetical ideal gas would need, at the same temperature, to have the same chemical potential as the real substance of interest.<sup>[2](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%253A_Pure_Substances_in_Single_Phases/7.08%253A_Chemical_Potential_and_Fugacity)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( f_{\mathrm{B}} = \lambda_{\mathrm{B}} \lim_{p \to 0} p_{\mathrm{B}}/\lambda_{\mathrm{B}} \), with \( p_{\mathrm{B}} \) the partial pressure and \( \lambda_{\mathrm{B}} \) the absolute activity of B<sup>[1](https://goldbook.iupac.org/terms/view/F02543)</sup> |
| Origin | Introduced by Gilbert N. Lewis in 1905 to extend \( dG = RT\,d\ln P \) from ideal gases to all substances<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> |
| Ideal-gas limit | For a pure ideal gas \( f = P \); as \( P \to 0 \), \( f \to P \), and for a component in a mixture \( f_i \to y_i P \)<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> |
| Chemical potential link | \( d\ln f = d\mu/(RT) \) at constant temperature, so fugacity is chemical potential expressed in units of pressure<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> |
| Fugacity coefficient | \( \varphi = f/P \) for a pure gas and \( \varphi_i = f_i/(y_i P) \) in a mixture; \( \varphi = 1 \) for ideal-gas behavior<sup>[4](https://goldbook.iupac.org/terms/view/F02544/pdf)</sup><sup> • </sup><sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> |
| Equilibrium criterion | At phase equilibrium the fugacity of each component is equal in every phase, \( f_i^{(l)} = f_i^{(v)} \), equivalent to equality of chemical potentials<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> |
| Calculation | Fugacity is a function of pressure, temperature, and volume; any equation of state \( f = f(P,V,T) \) yields it through \( \ln \varphi_i = \int_0^P (\bar{Z}_i - 1)\,dP/P \)<sup>[5](https://courses.ems.psu.edu/png520/node/1998)</sup> |

## Why fugacity is needed

For an ideal gas the chemical potential has the exact form

\[ \mu = \mu^{\circ} + RT \ln\left( \frac{p}{p^{\circ}} \right) \]

which is useful because it relates the chemical potential to a measurable quantity, the pressure. The derivation assumes ideal-gas behavior, however, and at sufficiently high pressures real gases deviate from the ideal gas law as molecular attractions and repulsions become important; pressure alone no longer describes the thermodynamic behavior.<sup>[6](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap07/fugacity.htm)</sup> Lewis's 1905 proposal was to define a new property, the fugacity, such that

\[ \mu = \mu^{\circ} + RT \ln\left( \frac{f}{p^{\circ}} \right) \]

has exactly the same form as the ideal-gas expression, with fugacity replacing pressure.<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup><sup> • </sup><sup>[6](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap07/fugacity.htm)</sup> Fugacity is related to pressure but contains the deviations from ideality within it.<sup>[7](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%3A_Chemical_Equilibrium/26.10%3A_Real_Gases_Are_Expressed_in_Terms_of_Partial_Fugacities)</sup>

**Interpretation.** Fugacity is the pressure of the hypothetical ideal gas, with intermolecular forces "turned off", that would have the same chemical potential as the real gas at the given temperature.<sup>[2](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%253A_Pure_Substances_in_Single_Phases/7.08%253A_Chemical_Potential_and_Fugacity)</sup> It is a measure of the difference between the chemical potential of the substance in the system of interest and its chemical potential in the hypothetical ideal-gas standard state at the same temperature.<sup>[8](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_%28Ellgen%29/15%3A_Chemical_Potential_Fugacity_Activity_and_Equilibrium/15.01%3A_The_Chemical_Potential_and_Fugacity_of_a_Gas)</sup> Because the chemical potential is central to thermodynamics, fugacity inherits that central role: the correction term \( RT \ln f \) adjusts the tabulated standard free energy of a gas, usually given at one bar, to the pressure of interest.<sup>[9](https://www.geol.umd.edu/~piccoli/lmdr/fugacity.html)</sup>

## The fugacity coefficient

The fugacity coefficient \( \varphi \) is defined by IUPAC as the ratio of fugacity to the partial pressure of a gaseous constituent.<sup>[4](https://goldbook.iupac.org/terms/view/F02544/pdf)</sup> For a pure substance \( \varphi = f/P \), and for a component in a mixture \( \varphi_i = f_i/(y_i P) \), where \( y_i \) is the mole fraction.<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> The coefficient equals unity when the substance behaves as an ideal gas, so it serves as a measure of non-ideality: the closer \( \varphi \) is to one, the closer the gas is to ideal behavior.<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> All gases approach \( \varphi = 1 \) as the pressure tends to zero, since all gases behave ideally in that limit.<sup>[7](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%3A_Chemical_Equilibrium/26.10%3A_Real_Gases_Are_Expressed_in_Terms_of_Partial_Fugacities)</sup>

The coefficient follows from the compressibility factor \( Z = pV/(RT) \). Differentiating the fugacity form of the chemical potential and comparing with \( (\partial \mu/\partial p)_T = V \) gives \( [\partial \ln f/\partial p]_T = Z/p \), and integration from zero pressure yields

\[ \ln \varphi = \int_0^{p} \left( \frac{Z - 1}{p} \right)_T dp \]

so the fugacity coefficient is obtained directly from any measured or modeled \( pVT \) behavior.<sup>[7](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%3A_Chemical_Equilibrium/26.10%3A_Real_Gases_Are_Expressed_in_Terms_of_Partial_Fugacities)</sup> For a component in a multicomponent mixture the corresponding relation is \( \ln \varphi_i = \int_0^P (\bar{Z}_i - 1)\,dP/P \), where \( \bar{Z}_i \) is the partial compressibility factor of component i.<sup>[5](https://courses.ems.psu.edu/png520/node/1998)</sup>

## Fugacity in mixtures and at equilibrium

For a component i of a mixture, the fugacity \( f_i \) is defined through the partial molar Gibbs energy, \( d\bar{G}_i = RT\,d\ln f_i \) at constant temperature, and in the ideal-gas limit \( f_i \) reduces to the partial pressure \( y_i P \).<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup> The fugacity of a component in a real-gas mixture depends on pressure, temperature, and composition.<sup>[8](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_%28Ellgen%29/15%3A_Chemical_Potential_Fugacity_Activity_and_Equilibrium/15.01%3A_The_Chemical_Potential_and_Fugacity_of_a_Gas)</sup>

**Phase equilibrium.** The equilibrium condition stated in chemical potentials, \( \mu_i^{(l)} = \mu_i^{(v)} \), can be restated exactly in fugacities: integrating the defining relation between the liquid and vapor states shows that \( \ln(f_i^{(v)}/f_i^{(l)}) = 0 \), hence

\[ f_i^{(l)} = f_i^{(v)}, \quad i = 1, 2, \ldots, n_c \]

The two criteria are equivalent, and the fugacity form is generally the preferred one in practice because fugacities are calculable from equations of state.<sup>[3](https://courses.ems.psu.edu/png520/print/m16_p4.html)</sup>

## Calculation from equations of state

Because \( \ln \varphi_i \) is an integral over \( (\bar{Z}_i - 1)\,dP/P \), choosing an equation of state for a substance immediately provides a route to its fugacity.<sup>[5](https://courses.ems.psu.edu/png520/node/1998)</sup> For mixtures at low to moderate pressures the virial equation \( V/n = RT/p + B \) is accurate, and the fugacity coefficient of constituent i then takes the simple form

\[ \ln \varphi_i = \frac{B'_i\, p}{RT} \]

where \( B'_i = 2 \sum_j y_j B_{ij} - B \) is built from the second virial coefficients \( B_{ij} \) of the pure components and their cross terms, all functions of temperature only.<sup>[10](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.03%3A_Gas_Mixtures)</sup> As a worked example, for \( \mathrm{CO_2} \) gas at \( 0.00\ ^\circ\mathrm{C} \) the pressure-series virial coefficients are \( B_p = -6.67 \times 10^{-3}\ \mathrm{bar}^{-1} \) and \( C_p = -3.4 \times 10^{-5}\ \mathrm{bar}^{-2} \), values that determine the fugacity at any pressure at that temperature.<sup>[11](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.10%3A_Chapter_7_Problems)</sup>

Modern software implements the same relations: in the isochoric formalism used by the NIST-supported teqp library, the fugacity-coefficient array is obtained from the gradient of the residual Helmholtz energy density and the compressibility factor,<sup>[12](https://pages.nist.gov/teqp-docs/en/stable/_static/doxygen/html/structteqp_1_1IsochoricDerivatives.html)</sup> and the fugacity coefficient appears as a standard output of mixture equation-of-state calculations in chemical engineering thermodynamics courses.<sup>[13](https://ocw.mit.edu/courses/10-40-chemical-engineering-thermodynamics-fall-2003/pages/readings/)</sup>

## Fugacity in statistical mechanics

The same word appears in statistical mechanics with a related meaning. In the grand canonical ensemble, where systems exchange both energy and particles with a reservoir, the fugacity \( z \) weights the number of particles. For the ideal gas the grand partition function gives a [Poisson distribution](https://www.edgepedia.org/poisson-distribution) of particle number with mean \( \langle N \rangle = Vz/\Lambda^3 \), where \( \Lambda \) is the thermal wavelength, so

\[ z = \frac{\langle N \rangle}{V} \Lambda^3 = \rho_0 \Lambda^3 \]

and the chemical potential becomes \( \mu = k_B T \ln(\rho_0 \Lambda^3) \).<sup>[14](https://www.mit.edu/~jgt/Content-Web/Teaching/8333/Chapter-3.pdf)</sup> This \( z \) is dimensionless, while the thermodynamic fugacity \( f \) carries units of pressure; the two are proportional at fixed temperature, and both encode the same exponential dependence of particle exchange on chemical potential.

## Scope of the concept

Although introduced for gases, the fugacity criterion extends to condensed phases: any solid or liquid taken to zero pressure would eventually become a gas for which \( f = p \), which fixes the proportionality constant in the general definition.<sup>[15](https://www.sciencedirect.com/topics/chemistry/fugacity)</sup> In solution thermodynamics the parallel quantity is the activity, with the activity coefficient \( \gamma_i \) playing for liquids and solids the correcting role that the fugacity coefficient plays for vapors.<sup>[16](https://ocw.mit.edu/courses/3-20-materials-at-equilibrium-sma-5111-fall-2003/764ba46faf34fb6f2439c6b52eb37fb9_mixtures.pdf)</sup> For most gas mixtures near atmospheric pressure, deviations from ideality are small and the partial pressure is a good approximation to the fugacity; significant corrections arise mainly at high pressures.<sup>[16](https://ocw.mit.edu/courses/3-20-materials-at-equilibrium-sma-5111-fall-2003/764ba46faf34fb6f2439c6b52eb37fb9_mixtures.pdf)</sup>

## References

1. [Fugacity, IUPAC Compendium of Chemical Terminology (Gold Book), F02543](https://goldbook.iupac.org/terms/view/F02543)
2. [The Chemical Potential and Fugacity of a Gas, DeVoe, Thermodynamics and Chemistry, LibreTexts 7.8](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%253A_Pure_Substances_in_Single_Phases/7.08%253A_Chemical_Potential_and_Fugacity)
3. [Fugacity, PNG 520: Phase Behavior of Natural Gas and Condensate Fluids, Penn State College of Earth and Mineral Sciences](https://courses.ems.psu.edu/png520/print/m16_p4.html)
4. [Fugacity coefficient, IUPAC Compendium of Chemical Terminology (Gold Book), F02544](https://goldbook.iupac.org/terms/view/F02544/pdf)
5. [Expressions for Fugacity Calculation, PNG 520, Penn State](https://courses.ems.psu.edu/png520/node/1998)
6. [Fugacity, Chem 351 Help, California State University East Bay](https://chemistry.csueastbay.edu/~pfleming/chem/351/chap07/fugacity.htm)
7. [Real Gases Are Expressed in Terms of Partial Fugacities, LibreTexts 26.10](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_%28LibreTexts%29/26%3A_Chemical_Equilibrium/26.10%3A_Real_Gases_Are_Expressed_in_Terms_of_Partial_Fugacities)
8. [The Chemical Potential and Fugacity of a Gas, Ellgen, Thermodynamics and Chemical Equilibrium, LibreTexts 15.1](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_%28Ellgen%29/15%3A_Chemical_Potential_Fugacity_Activity_and_Equilibrium/15.01%3A_The_Chemical_Potential_and_Fugacity_of_a_Gas)
9. [Fugacity, Department of Geology, University of Maryland](https://www.geol.umd.edu/~piccoli/lmdr/fugacity.html)
10. [Gas Mixtures, DeVoe, Thermodynamics and Chemistry, LibreTexts 9.3](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.03%3A_Gas_Mixtures)
11. [Chapter 7 Problems, DeVoe, Thermodynamics and Chemistry, LibreTexts 7.10](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.10%3A_Chapter_7_Problems)
12. [teqp::IsochoricDerivatives, NIST teqp documentation](https://pages.nist.gov/teqp-docs/en/stable/_static/doxygen/html/structteqp_1_1IsochoricDerivatives.html)
13. [Readings, 10.40 Chemical Engineering Thermodynamics, MIT OpenCourseWare](https://ocw.mit.edu/courses/10-40-chemical-engineering-thermodynamics-fall-2003/pages/readings/)
14. [Thermodynamics, Chapter 3, MIT 8.333 course notes](https://www.mit.edu/~jgt/Content-Web/Teaching/8333/Chapter-3.pdf)
15. [Fugacity, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/chemistry/fugacity)
16. [Mixtures, SMA 5101 Thermodynamics of Materials, MIT OpenCourseWare (Ceder, 2001)](https://ocw.mit.edu/courses/3-20-materials-at-equilibrium-sma-5111-fall-2003/764ba46faf34fb6f2439c6b52eb37fb9_mixtures.pdf)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Laws, states, and potentials › Equilibrium and state functions › Equations of state › Real-gas and virial equations*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license