in terms of magnetite oxidation. The first reaction is simpler, of course, and hence preferred, but it may sometimes be necessary to consider the proportions of the actual gas species present.
If we can regard magnetite and hematite as pure phases, then their activities are equal to one and the equilibrium constant for reaction 3.122 is the inverse of the oxygen fugacity:
(3.124)
We can rewrite eqn. 3.86 as:
(3.125)
and taking the standard state as 1000 K and 1 bar, we can write:
Thus, oxygen fugacity can be calculated directly from the difference in the free energy of formation of magnetite and hematite at the appropriate T and P. Substituting appropriate values into this equation yields a value for log
It is important to understand that the oxygen fugacity is fixed at this level (though the exact level at which it is fixed is still disputed because of uncertainties in the thermodynamic data) simply by the equilibrium coexistence of magnetite and hematite. The oxygen fugacity does not depend on the proportion of these minerals. For this reason, it is appropriately called a buffer. To understand how this works, imagine some amount of magnetite, hematite and oxygen present in a magma. If the oxygen fugacity is increased by the addition of oxygen to the system, equilibrium in the reaction in eqn. 3.121 is driven to the right until the log of the oxygen fugacity returns to a value of −10.86. Only when all magnetite is converted to hematite can the oxygen fugacity rise. A drop in oxygen fugacity would be buffered in exactly the opposite way until all hematite were gone. A number of other buffers can be constructed based on reactions such as:
and
These can be used to construct the oxygen buffer curves in Figure 3.22.
3.12 SUMMARY
Natural systems often contain multiple phases, many of which are solutions of several components; in this chapter, we developed the thermodynamic tools to deal with them.
We began by defining components, phases, and species. Together, the number of components and phases in a system determine the degrees of freedom of the system:(3.2) which are the number of independent variables we need to specify to completely describe the system. We derived the Clapeyron equation, which described the boundary between two phases, such as graphite and diamond, the P−T space:Figure 3.22 Oxygen buffer curves in the system Fe−Si−O at 1 bar. QIF, IW, WM, FMQ, and MH refer to the quartz–iron–fayalite, iron–wüstite, wüstite–magnetite, fayalite–magnetite–quartz and magnetite–hematite buffers, respectively.(3.3)
We found the thermodynamic properties of solutions depend on their composition as well as T and P and to deal with this we introduced partial molar quantities, particularly the partial molar Gibbs free energy or chemical potential:(3.13)
The simplest solutions are ideal ones, where there are no energetic or volumetric effects of solution (ΔH = 0; ΔV= 0), so the enthalpy and volume of an ideal
solution are simply their sum of the partial molar quantities. There are, however, entropic effects associated with solution, so that(3.31)
In nonideal solutions, the availability of a species for reaction can differ from its concentration; to deal with this we introduced fugacity and activity; the latter is related to concentration through an activity coefficient:(3.48) The activity coefficient is related to the excess Gibbs free energy associated with nonideal behavior:(3.56a) Much of the problem with dealing with nonideal solutions is reduced to finding values for the activity coefficients.
Electrolyte solutions, of which seawater is a good example, are common nonideal solutions. We reviewed the nature of these solutions and introduced approaches for calculating activity coefficients in them, such as the Debye–Hückel extended law:(3.74) We then reviewed ways to calculate activities in ideal solid solutions.
In section 3.9, we introduced the equilibrium constant:(3.85) and found we could directly relate it to the Gibbs free energy of reaction.
In section 3.11, we introduced the electrochemical potential to deal with changing valance states of elements, that is, oxidation–reduction reactions. This too we could relate to our thermodynamic framework:(3.106) A useful way to represent redox potential in low-temperature systems is the electron activity(3.112) which we could also directly relate to electrochemical potential. Since oxygen is the most common oxidant, in high-temperature systems, the redox state of the system is more commonly represented with oxygen fugacity, ƒO2.
REFERENCES AND SUGGESTIONS FOR FURTHER READING
1 Anderson, G.M. and Crerar, D.A. 1993. Thermodynamics in Geochemistry. New York, Oxford University Press.
2 Blount, C.W. 1977. Barite solubilities and thermodynamic quantities up to 300°C and 1400 bars. American Mineralogist 62: 942–57.
3 Brookins, D.G. 1988. Eh-pH Diagrams for Geochemistry. New York, Springer Verlag.
4 Davies, C.W. 1938. The extent of dissociation of salts in water. VIII. An equation for the mean ionic activity coefficient in water, and a revision of the dissociation constant of some sulfates. Journal of the Chemical Society 2093–8.
5 Davies, C.W. 1962. Ion Association. London, Butterworths.
6 Debye, P. and Hückel, E. 1923. On the theory of electrolytes. Phys. Z. 24: 185–208.
7 Fletcher, P. 1993. Chemical Thermodynamics for Earth Scientists. Essex, Longman Scientific and Technical.
8 Garrels, R.M. and Christ, C.L. 1965. Solutions, Minerals, and Equilibria. San Francisco: Freeman Cooper.
9 Helgeson, H.C. 1967. Solution chemistry and metamorphism, in Researches in Geochemistry vol. 2 (ed. P.H. Abelson), pp. 362–402. New York, John Wiley and Sons, Ltd.
10 Helgeson, H.C. and Kirkham, D.H. 1974. Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures, Debye–Hückel
