Fugacity
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 chemical equilibrium. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas.[1]
Fugacities are determined experimentally or estimated from various models such as a
For an ideal gas, fugacity and pressure are equal, and so 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 RT ln φ.
The fugacity is closely related to the
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 μ. In a pure substance, μ is equal to the Gibbs energy Gm for a mole of the substance,[2]: 207 and
Gas
For an ideal gas the equation of state can be written as
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.[3] 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.[4]
The ideal gas law can still be used to describe the behavior of a real gas if the pressure is replaced by a fugacity f, defined so that
If a reference state is denoted by a zero superscript, then integrating the equation for the chemical potential gives
Numerical example:
The contribution of nonideality to the molar Gibbs energy of a real gas is equal to RT ln φ. For nitrogen at 100 atm, Gm = Gm,id + RT ln 0.9703, which is less than the ideal value Gm,id because of intermolecular attractive forces. Finally, the activity is just 97.03 without units.
Condensed phase
The fugacity of a condensed phase (liquid or solid) is defined the same way as for a gas:
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 Psat to P is
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).[8]: 141
Gases
In a mixture of gases, the fugacity of each component i has a similar definition, with partial molar quantities instead of molar quantities (e.g., Gi instead of Gm and Vi instead of Vm):[2]: 262
i is the fugacity that component 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.[2]: 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 the Lewis-Randall rule:
i is the fugacity of the pure liquid phase. This is a good approximation when the component molecules have similar size, shape and polarity.[2]: 264, 269–270
In a dilute solution with two components, the component with the larger molar fraction (the
Temperature and pressure dependence
The pressure dependence of fugacity (at constant temperature) is given by[2]: 260
The temperature dependence at constant pressure is
Measurement
The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. In that case,
The integral can be recast in an alternative form using the compressibility factor
For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is
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.[13] The "escaping tendency" referred to the flow of matter between phases and played a similar role to that of temperature in heat flow.[14][15]: 177
See also
References
- ^ ISBN 9780716787594.
- ^ ISBN 9780080500980.
- ISBN 9780840065322.
- ISBN 9780199146338.
- ISBN 9780521005777.
- ISBN 9780132693066.
- ISBN 9780132440509.
- ISBN 9781139443173.
- ISBN 9780716735397.
- ISBN 9781107069756. Note that Equations 9.24 and 9.25 left out p0 in substituting from Equation 9.6. This error is corrected in the above equation.
- ISBN 9780195345094.
- ^ David, Carl W. (2015). "Fugacity Examples 2: The fugacity of a "hard-sphere" semi-ideal gas and the van der Waals gas". Chemistry Education Materials. 91.
- JSTOR 20021635. ; the term "fugacity" is coined on p. 54.
- 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.
- ISBN 9781107175211.
Further reading
- Anderson, Greg M.; Crerar, David A. (1993). Thermodynamics in Geochemistry: The Equilibrium Model. Oxford University Press. ISBN 9780195345094.
- Mackay, Don (2011). "The fugacity approach to mass transport and MTCs". In Thibodeaux, Louis J.; Mackay, Donald (eds.). Handbook of chemical mass transport in the environment. Boca Raton, FL: CRC Press. pp. 43–50. ISBN 9781420047561.
- Mackay, Don; Arnot, Jon A. (14 April 2011). "The Application of Fugacity and Activity to Simulating the Environmental Fate of Organic Contaminants". Journal of Chemical & Engineering Data. 56 (4): 1348–1355. hdl:2027.42/143615.
External links
Video lectures
- Thermodynamics, University of Colorado-Boulder, 2011