Fugacity
Notation and definitions are described in Nomenclature.
The core material below follows Chapter 2 of Prausnitz et al. (1998). Fugacity coefficients follow Spycher and Reed (1988) as well as Appendix B of Xu et al. (2014). Section 3.1.3.3 of Bethke (2007) offers some further equations, but I find them difficult to undrestand.
The chemical potential of a gas species, which may be part of a gas mixture, is Here
[J.mol] is the chemical potential
[J.mol] is a constant
[bar] is the gas fugacity at 1atm (or it might be but I think the fugacity is not so precisely known that this small difference matters) and the temperature of interest
J.K.mol is the gas constant
[K] is temperature
is the natural logarithm
[bar] is the gas-species fugacity.
The gas-species fugacity is Here
[dimensionless] is the fugacity coefficient for the gas species
[bar] is the gas-species partial pressure
[dimensionless] is the mole fraction of the gas species within the gas mixture
[bar] is the total gas-mixture pressure
For ideal gases with ideal mixing, . Hence, for a pure gas (that is, ), , so , where is 1atm and is the chemical potential at 1atm.
For non-ideal gases, with ideal mixing (apparently a good approximation) This is the Spycher-Reed formula. [bars] is the total gas-mixture pressure, [K] is temperature, and is the natural logarithm. The quantities, , , , are given in the database.
Only the Spycher-Reed fugacity formula is used in the geochemistry module.
I believe that, by convention, the mass-action equilibrium constant for a reaction involving a gas involves just the fugacity of the gas, and includes contributions from and into the equilibrium constant.
For instance, in the reaction involving one secondary species, and a number of gas species, , mass-action equilibrium is Here is given in the database and is the fugacity of the gas species in the mixture, computed using the above formulae. Athough there is a dimensional mismatch between the left and right sides of this equation, I believe it is ignored, as and have been lumped into . All that is required is to consistently measure in bars and in Kelvin.
References
- Craig M. Bethke.
Geochemical and Biogeochemical Reaction Modeling.
Cambridge University Press, 2 edition, 2007.
doi:10.1017/CBO9780511619670.[BibTeX]
- John M. Prausnitz, Rudiger N. Lichtenthaler, and Edmundo Gomes de Azevedo.
Molecular Thermodynamics of Fluid-Phase Equilibria.
Pearson, 3 edition, 1998.
ISBN 9780139777455.[BibTeX]
- Nicolas F. Spycher and Mark H. Reed.
Fugacity coefficients of H2, CO2, CH4, H2 and of H2–CO2–CH4 mixutres: A virial equation treatment for moderate pressures and temperatures applicable to calculations of hydrothermal boiling.
Geochemica et Cosmochimica Acta, 52:739–749, 1988.
URL: http://www.sciencedirect.com/science/article/pii/0016703788903341, doi:10.1016/0016-7037(88)90334-1.[BibTeX]
- Tianfu Xu, Eric Sonnenthal, Nicolas Spycher, Liang Zhen, Norman Millar, and Karsten Pruess.
TOUGHREACT V3.0-OMP Reference Manual: A Parallel Simulation Program for Non-Isothermal Multiphase Geochemical Reactive Transport.
Technical Report, Lawrence Berkeley National Laboratory, 2014.
URL: https://tough.lbl.gov/assets/docs/TOUGHREACT_V3-OMP_RefManual.pdf.[BibTeX]