Fugacity

Fugacity is a quantity in thermodynamics that expresses a substance’s chemical potential in a form with the dimensions of pressure. For an ideal gas, fugacity equals pressure; for a component of an ideal gas mixture, it equals partial pressure. For nonideal substances, it preserves the logarithmic relationship between chemical potential and pressure while accounting for departures from ideal behavior. Fugacity can also be assigned to substances in liquids and solids, making it useful in phase-equilibrium calculations. (goldbook.iupac.org)

Definition and thermodynamic meaning

For a chemical species $i$, fugacity $f_i$ may be defined through

where $\mu_i$ is its molar chemical potential, $T$ is absolute temperature, $R$ is the gas constant, and $\mathbf{x}$ represents composition. The reference quantity $\mu_i^{\circ,\mathrm{ig}}$ is the chemical potential of the pure species in its hypothetical ideal-gas standard state at pressure $p^\circ$. The ratio inside the logarithm is dimensionless. The conventional standard pressure is $1$ bar, or $10^5$ pascals; specifying a standard state does not itself specify a temperature. (publications.iupac.org)

The gas-phase normalization is

at fixed temperature and composition, where $y_i$ is the gas-phase mole fraction. For a pure gas, this becomes $\lim_{p\to0}f/p=1$. Thus, the scale of fugacity is fixed by the dilute ideal-gas limit, rather than by an arbitrary multiplicative constant. (goldbook.iupac.org)

At a common temperature, two states of the same species satisfy

Fugacity therefore contains the same information about differences in chemical potential as chemical potential itself. It is not a mechanical pressure measured directly by a pressure gauge. (cpb-us-e1.wpmucdn.com)

Fugacity coefficient and nonideality

The dimensionless fugacity coefficient $\phi_i$ of a gas-mixture component is

For a pure substance, $\phi=f/p$. An ideal gas has $\phi=1$. Values below or above unity indicate that fugacity is respectively smaller or larger than the ideal-gas partial pressure at the same temperature, pressure, and composition. (goldbook.iupac.org)

The coefficient is related to the residual chemical potential by

where the ideal-gas comparison is made at the same temperature, pressure, and composition. This links fugacity to the residual partial molar Gibbs free energy, not merely to a correction in the pressure–volume relation. (cpb-us-e1.wpmucdn.com)

Fugacity coefficients and activity coefficients describe nonideality relative to different reference states. The former use an ideal-gas reference; the latter may use, for example, a pure-liquid or infinitely dilute solution reference. They are therefore not generally interchangeable. (ocw.mit.edu)

Calculation from an equation of state

An equation of state allows fugacity to be calculated from volumetric properties. For a mixture component at fixed temperature and composition,

where $\overline V_i$ is its partial molar volume. The subtraction removes the ideal-gas contribution, and the low-pressure limit supplies the integration constant. The integral requires a thermodynamically consistent description of the relevant phase. (sites.utexas.edu)

For a pure gas, introducing the compressibility factor

gives the equivalent expression

These relations show why a fugacity coefficient depends on behavior over a pressure range, rather than only on the compressibility factor at the final state. (sites.utexas.edu)

Phase equilibrium and condensed phases

When phases exchange a species at thermodynamic equilibrium, that species has the same chemical potential in each phase. With a common fugacity convention, the corresponding condition is

Equality applies to the same component across phases, not to different components within one phase. Equal fugacities also do not imply equal concentrations. (cpb-us-e1.wpmucdn.com)

For a liquid mixture, a common representation is

where $x_i$ is liquid mole fraction, $\gamma_i$ is an activity coefficient, and $f_i^{L,*}$ is the fugacity of the pure-liquid reference at the system temperature and pressure. Combining this with a vapor-phase expression gives

This formulation separates vapor nonideality from liquid-mixture nonideality. (ocw.mit.edu)

The pressure dependence of the liquid reference is often included through a Poynting correction, obtained by integrating liquid molar volume with respect to pressure. Treating liquid fugacity as simply equal to saturation vapor pressure neglects both this correction and any nonideality of the saturated vapor. (sites.utexas.edu)

Activity and chemical equilibrium

Thermodynamic activity is a dimensionless quantity related to fugacity by

when the reference fugacity $f_i^\circ$ corresponds to the chosen standard state. For an ideal-gas standard state, $f_i^\circ=p^\circ$, so $a_i=f_i/p^\circ$. Other standard-state conventions produce different activity scales without changing the underlying equilibrium condition. (publications.iupac.org)

For a chemical reaction with signed stoichiometric coefficients $\nu_i$,

At chemical equilibrium, $Q$ equals the equilibrium constant $K$. Gas-phase equilibrium calculations therefore use fugacity ratios rather than uncorrected pressures when ideal-gas behavior is inadequate. Dimensionless ratios are essential: using dimensional fugacities alone would make the numerical result depend on the pressure units. (publications.iupac.org)

Historical development, measurement, and applications

Gilbert Newton Lewis introduced fugacity in 1901. Its experimental determination can exploit selective membranes: a membrane permeable to one component establishes equilibrium between a mixture and a reference fluid, allowing the component’s fugacity to be inferred. A documented implementation used a palladium–silver membrane for hydrogen-containing mixtures. Such measurements provide tests of calculated fugacity coefficients. (nvlpubs.nist.gov)

In chemical engineering, fugacity is used in calculating vapor–liquid and other phase equilibria, gas solubility, and reaction equilibria. These calculations support separation processes such as distillation. Thermophysical-property software may provide fugacity, fugacity coefficients, and chemical potential alongside other fluid properties. (ocw.mit.edu)

The thermodynamic definition is exact, but numerical predictions depend on the property model and its parameters. Fugacity supplies an equilibrium criterion; it does not itself determine the rate at which equilibrium is reached. Models must therefore distinguish equilibrium properties from transport and reaction-rate descriptions. (ocw.mit.edu)

A distinct usage in statistical mechanics

In statistical mechanics, “fugacity” commonly denotes the dimensionless quantity

where $\mu$ is chemical potential per particle and $k_B$ is the Boltzmann constant. In the grand canonical ensemble, it weights terms with different particle numbers:

where $Z_N$ is the canonical partition function. (itp.uni-frankfurt.de)

This statistical fugacity is related to the thermodynamic fugacity through their exponential dependence on chemical potential, but it is not numerically identical to a pressure-dimensioned fugacity. Their conversion requires the appropriate reference-state and normalization factors; the symbol and units must therefore be checked when moving between the two conventions. (itp.uni-frankfurt.de)

References

1. Chemical Engineering Thermodynamics — Themis Matsoukascpb-us-e1.wpmucdn.com
2. Liquid-Vapor Equilibria in Mixtures; Ideal and Excess Chemical Potentials — MIT Lecture 16ocw.mit.edu
3. Posey Dissertation (1996)sites.utexas.edu
4. An apparatus for direct fugacity measurements on mixtures containing hydrogennvlpubs.nist.gov
5. Chemical Engineering Thermodynamics — Readings, MIT OpenCourseWareocw.mit.edu
6. REFPROP — NISTnist.gov
7. Grand Canonical Ensemble, Section 10.3: Fugacityitp.uni-frankfurt.de