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Among the most vexing tasks for geochemical modelers, especially when they work with concentrated solutions, is estimating values for the activity coefficients of electrolyte species. To understand in a qualitative sense why activity coefficients in electrolyte solutions vary, we can imagine how solution concentration affects species activities. In the solution, electrical attraction draws anions around cations and cations around anions. We might think of a dilute solution as an imperfect crystal of loosely packed, hydrated ions that, within a matrix of solvent water, is constantly rearranging itself by Brownian motion. A solution of uncharged, nonpolar species, by contrast, would be nearly random in structure. The electrolyte solution is lower in free energy G than it would be if the species did not interact electrically because of the energy liberated by moving ions of opposite charge together while separating those of like charge. The chemical potentials of the species, for the same reason, are smaller than they would be in the absence of electrostatic forces.

Aqueous geochemists work daily with equations that describe the equilibrium points of chemical reactions among dissolved species, minerals, and gases. To study an individual reaction, a geochemist writes the familiar expression, known as the mass action equation, relating species’ activities to the reaction’s equilibrium constant. In this chapter we carry this type of analysis a step farther by developing expressions that describe the conditions under which not just one but all of the possible reactions in a geochemical system are at equilibrium. We consider a geochemical system comprising at least an aqueous solution in which the species of many elements are dissolved. We generally have some information about the fluid’s bulk composition, perhaps directly because we have analyzed it in the laboratory. The system may include one or more minerals, up to the limit imposed by the phase rule (see Section 3.4), that coexist with and are in equilibrium with the aqueous fluid. The fluid's composition might also be buffered by equilibrium with a gas reservoir (perhaps the atmosphere) that contains one or more gases. The gas buffer is large enough that its composition remains essentially unchanged if gas exsolves from or dissolves into the fluid. How can we express the equilibrium state of such a system? A direct approach would be to write each reaction that could occur among the system’s species, minerals, and gases. To solve for the equilibrium state, we would determine a set of concentrations that simultaneously satisfy the mass action equation corresponding to each possible reaction. The concentrations would also have to add up, together with the mole numbers of any minerals in the system, to give the system’s bulk composition. In other words, the concentrations would also need to satisfy a set of mass balance equations. Such an approach, however, is unnecessarily difficult to carry out. Dissolving even a few elements in water produces many tens of species that need be considered, and complex solutions contain many hundreds of species. Each species represents an independent variable, namely its concentration, in our scheme. For any but the simplest of chemical systems, the problem would contain too many unknown values to be solved conveniently.

Diagenesis is the set of processes by which sediments evolve after they are deposited and begin to be buried. Diagenesis includes physical effects such as compaction and the deformation of grains in the sediment (or sedimentary rock), as well as chemical reactions such as the dissolution of grains and the precipitation of minerals to form cements in the sediment's pore space. The chemical aspects of diagenesis are of special interest here. Formerly, geologists considered chemical diagenesis to be a process by which the minerals and pore fluid in a sediment reacted with each other in response to changes in temperature, pressure, and stress. As early as the 1960s and especially since the 1970s, however, geologists have recognized that many diagenetic reactions occur in systems open to groundwater flow and mass transfer. The reactions proceed in response to a supply of reactants introduced into the sediments by flowing groundwater, which also serves to remove reaction products. Hay (1963, 1966), in studies of the origin of diagenetic zeolite, was perhaps the first to emphasize the effects of mass transport on sediment diagenesis. He showed that sediments open to groundwater flow followed reaction pathways different from those observed in sediments through which flow was restricted. Sibley and Blatt (1976) used cathodoluminescence microscopy to observe the Tuscarora orthoquartzite of the Appalachian basin. The almost nonporous Tuscarora had previously been taken as a classic example of pressure welding, but the microscopy demonstrated that the rock is not especially well compacted but, instead, tightly cemented. The rock consists of as much as 40% quartz (SiO2) cement that was apparently deposited by advecting groundwater. By the end of the decade, Hayes (1979) and Surdam and Boles (1979) argued forcefully that the extent to which diagenesis has altered sediments in sedimentary basins can be explained only by recognition of the role of groundwater flow in transporting dissolved mass. This view has become largely accepted among geoscientists, although it is clear that the scale of groundwater flow might range from the regional (e.g., Bethke and Marshak, 1990) to circulation cells perhaps as small as tens of meters (e.g., Bjorlykke and Egeberg, 1993; Aplin and Warren, 1994).

As geochemists, we frequently need to describe the chemical states of natural waters, including how dissolved mass is distributed among aqueous species, and to understand how such waters will react with minerals, gases, and fluids of the Earth's crust and hydrosphere. We can readily undertake such tasks when they involve simple chemical systems, in which the relatively few reactions likely to occur can be anticipated through experience and evaluated by hand calculation. As we encounter more complex problems, we must rely increasingly on quantitative models of solution chemistry and irreversible reaction to find solutions. The field of geochemical modeling has grown rapidly since the early 1960s, when the first attempt was made to predict by hand calculation the concentrations of dissolved species in seawater. Today's challenges might be addressed by using computer programs to trace many thousands of reactions in order, for example, to predict the solubility and mobility of forty or more elements in buried radioactive waste. Geochemists now use quantitative models to understand sediment diagenesis and hydrothermal alteration, explore for ore deposits, determine which contaminants will migrate from mine tailings and toxic waste sites, predict scaling in geothermal wells and the outcome of steam-flooding oil reservoirs, solve kinetic rate equations, manage injection wells, evaluate laboratory experiments, and study acid rain, among many examples. Teachers let their students use these models to learn about geochemistry by experiment and experience. Many hundreds of scholarly articles have been written on the modeling of geochemical systems, giving mathematical, geochemical, mineralogical, and practical perspectives on modeling techniques. Dozens of computer programs, each with its own special abilities and prejudices, have been developed (and laboriously debugged) to analyze various classes of geochemical problems. In this book, I attempt to treat geochemical modeling as an integrated subject, progressing from the theoretical foundations and computational concerns to the ways in which models can be applied in practice. In doing so, I hope to convey, by principle and by example, the nature of modeling and the results and uncertainties that can be expected. Hollywood may never make a movie about geochemical modeling, but the field has its roots in top-secret efforts to formulate rocket fuels in the 1940s and 1950s.

The purpose of the first chapter was to give an overview of the book’s contents: the topics covered, results gained, limitations, and so on. The next seven chapters, starting with this one, give the groundwork on which the main conclusions are based. The intention is to assemble the needed ideas, taking advantage of the fact that an extensive literature exists in which the ideas are established, discussed, restricted, etc. What follows is thus an extract of selected essentials from other documents, rather than being a free-standing and self-contained development of the ideas. The reader is asked to relate the ideas as summarized here to the longer discussions in which they appear elsewhere. The total energy in a portion of material can be split in either of two ways: . . . Total energy = internal energy + external energy, or . . . . . . Total energy = free energy + bound energy . . . In symbols, . . . U + PV = total = G + TS . . . where U = internal energy of the portion; G = free energy of the portion, specifically the Gibbs free energy or enthalpy; P = pressure; V = volume of the portion; T = temperature; S = entropy of the portion. All the terms except the free energy, G, have independent definitions, so the equations just given define that quantity: . . . G = U + PV – TS (2.1) . . . The equation relates to whatever portion of material one has in view. We now suppose that the material has n components and that, in the portion considered, the masses of each are m1, m2, . . . , m_n. Then we imagine increasing m₁ by a small amount δm1 while keeping P, T, and m2, m3, . . , ,m_n constant. Let the consequent change in G be δG: then the limit of the ratio δG/δm1 as δm1 → 0 is the quantity of interest, henceforth written μ1; it is the chemical potential of component 1 in the material at its current pressure, temperature, and composition.

As in Chapter 2, so again here the intention is to review ideas that are already familiar, rather than to introduce the unfamiliar; to build a springboard, but not yet to leap off into space. The familiar idea is of flow down a gradient—water running downhill. Parallels are electric current in a wire, salt diffusing inland from the sea, heat flowing from the fevered brow into the cool windowpane, and helium diffusing through the membrane of a helium balloon. For any of these, we can imagine a linear relation: . . . Flow rate across a unit area = (conductivity) x (driving gradient) . . . where the conductivity retains a constant value, and if the other two quantities change, they do so in a strictly proportional way. Real life is not always so simple, but this relation serves to introduce the right quantities, some suitable units and some orders of magnitude. For present purposes, the second and fourth of the examples listed are the most relevant. To make comparison easier we imagine a barrier through which salt can diffuse and through which water can percolate, but we imagine circumstances such that only one process occurs at a time. Specifically, imagine a lagoon separated from the ocean by a manmade dike of gravel and sand 4 m thick, as in Figure 3.1. If the lagoon is full of seawater but the water levels on the two sides of the dike are unequal, water will percolate through the dike, whereas if the levels are the same and the dike is saturated but the lagoon is fresh water, salt will diffuse through but there will be no bulk flow of water. (More correctly, because seawater and fresh water have different densities, and because of other complications, the condition of no net water flow would be achieved in circumstances a little different from what was just stated. For present purposes all we need is the idea that conditions exist where water does not percolate but salt does diffuse.) For flow of water driven by a pressure gradient, suitable units are shown in the upper part of Table 3.1 and for diffusion of salt driven by a concentration gradient, suitable units are shown in the lower part.