In thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the ideal partial pressure in an accurate computation of chemical equilibrium. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy (chemical potential) as the real gas.

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 and

<math display="block">d\mu = dG_\mathrm{m} = -S_\mathrm{m} dT + V_\mathrm{m} dP,</math>

where and are the temperature and pressure, is the volume per mole and is the entropy per mole. 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.

The ideal gas law can still be used to describe the behavior of a real gas if the pressure is replaced by a fugacity , 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 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.

Numerical example: Nitrogen gas (N<sub>2</sub>) at 0&nbsp;°C and a pressure of atmospheres (atm) has a fugacity of &nbsp;atm.

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).

Gases

In a mixture of gases, the fugacity of each component has a similar definition, with partial molar quantities instead of molar quantities (e.g., instead of and instead of ): By Henry's law, the fugacity of the solute is proportional to its concentration. The constant of proportionality (a measured Henry's constant) depends on whether the concentration is represented by the mole fraction, molality or molarity.

Measurement

The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. In that case,

<math display="block"> \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.

For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is

<math display="block">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 based on the molar volume. Since the pressure and the molar volume are related through the equation of state; a typical procedure would be to choose a volume, calculate the corresponding pressure, and then evaluate the right-hand side of the equation.

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. The "escaping tendency" referred to the flow of matter between phases and played a similar role to that of temperature in heat flow.

At the time of Lewis' 1901 paper, physical chemistry was primarily based on inexact gas pressure laws, involving vapor pressures and partial pressures. By replacing these gas pressures with fugacity, Lewis preserved these laws’ mathematical simplicity while accounting for real-world deviations. Historians argue that Lewis, a young scientist aware of Josiah Willard Gibbs’ rigorous but abstract thermodynamics (including chemical potential), sought to reform the prevailing experimental framework with exact equations that retained practical utility. Although Lewis’ 1901 paper omitted Gibbs’ formalism entirely, his later works explicitly linked fugacity to chemical potential.

See also

  • Electrochemical potential
  • Excess chemical potential
  • Fugacity capacity
  • Multimedia fugacity model
  • Thermodynamic equilibrium

References

Further reading

Video lectures

  • Thermodynamics, University of Colorado-Boulder, 2011
  • Introduction to fugacity: Where did it come from?
  • What is fugacity?
  • What is fugacity in mixtures?