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Modern geochemistry studies the distribution and amounts of the chemical elements in minerals, ores, rocks, soils, waters, and the atmosphere, and the circulation of the elements in nature, on the basis of the properties of their atoms and ions.
The distribution and circulation of the chemical elements in and on the Earth is influenced by a myriad of chemical and physical factors, many of which have changed over geological time. Understanding the role of these factors in geological processes requires us to condense information about elemental abundances and distributions into models. This book is about geochemical models for situations where time plays a key role. Geoscientists have always appreciated the importance of time in fashioning the Earth. Many geological processes require time spans that are far too long for human observation, but we can use models to extrapolate rates based on short-term observations to predict geochemistry in deep time. Equally important are models that forecast the future behavior of geochemical systems because those models are needed for environmental management and resources recovery projects.
Some of the models described in this book were developed by geochemists but many others come from applied sciences and engineering. Because of this diverse provenance, the models in their original form used a confusing mix of units, terminology, and notation. This book attempts to remedy that problem by recasting the models using internally consistent notation, units, and terminology familiar to geochemists. Furthermore, whenever possible the models are developed from fundamental theory showing a sufficient number of intermediate steps to allow the reader to follow the derivations.
Before any model is ready for use it must be verified. The verification step is greatly simplified if the model is constructed using conventional computational methods, notation, and units. This chapter reviews some procedures and conventions that are recommended for geochemical model construction.
Balancing chemical reactions
The first step in building a geochemical model is to write balanced equations that describe the governing chemical reactions. It is often possible to recognize these reactions based on past experiences, but when experience is lacking a general strategy is needed to identify these key reactions. The strategy should recognize that, with few exceptions, the key reactions involve the most abundant phases and chemical species. Creating a mineral inventory listing the possible hosts for the elements of interest is a first step toward selecting the solid phases to include in the model. Similarly, a chemical analysis of the aqueous phase can be used along with an aqueous speciation model to identify important aqueous species. The reactions among these mineral and aqueous species are expressed as balanced chemical reactions and these reactions become the basis for the subsequent model.
This well-organised, comprehensive reference and textbook describes rate models developed from fundamental kinetic theory and presents models using consistent terminology and notation. Major topics include rate equations, reactor theory, transition state theory, surface reactivity, advective and diffusive transport, aggregation kinetics, nucleation kinetics and solid-solid transformation rates. The theoretical basis and mathematical derivation of each model is presented in detail and illustrated with worked examples from real-world applications to geochemical problems. The book is also supported by online resources: self-study problems put students' new learning into practice, and spreadsheets provide the full data used in figures and examples, enabling students to manipulate the data for themselves. This is an ideal overview for graduate students, providing a solid understanding of geochemical kinetics. It will also provide researchers and professional geochemists with a valuable reference for solving scientific and engineering problems.
Under near-equilibrium conditions, solids form from aqueous solutions via the addition of monomer growth units. The terrace, ledge, and kink (TLK) model and the Burton–Cabrera–Frank (BCF) theory give reasonable descriptions of this process. At higher degrees of supersaturation, monomers addition is joined by the accretion of polymer and larger growth units, up to and including stable crystallites. The solids formed by accretion of these larger units are often metastable and eventually transform to more stable forms. This means that the formation of a stable crystalline solid from a very supersaturated solution involves several steps. This chapter presents models that describe the formation and accretion of larger growth units and transformation of the resulting metastable solids.
Polymers such as DNA, proteins, polysaccharides, and polyphenols are essential components of organisms; and synthetic polymers such as Nylon, polyethylene, styrene, and Teflon are basic raw materials for modern technology, so it is not surprising that there is a rich literature about the formation of polymers. Most of that literature focuses on the formation of biopolymers and plastic but the polymerization models (Dotson et al., 1996) for those cases are potentially useful to geochemists.
Rate equations are quantitative models of the time course of chemical reactions. Although rate equations are based on macroscopic observations, they reflect processes that occur at the molecular scale. This chapter reviews some of the important models that link these two scales. These models are especially useful because they constrain the mathematical form of rate equations and they provide a conceptual basis for thinking about the reactions. Because water is so important in geochemical systems, this chapter focuses on models for reaction rates in the aqueous phase.
Chemical reactions break and reform bonds within and between molecules so that the bonding arrangement of the reactants gives way to the bonding arrangement of the products. For a reaction to occur: (1) the molecules must “collide” with each other to form a cluster; (2) the atoms in that cluster must be configured in the approximate geometry of the products; and (3) the cluster must contain sufficient energy to allow the rearrangement of the electrons from the breaking bonds to the developing bonds.
Many important reactions happen at interfaces. Because reactions between aqueous species and mineral surfaces are so important in low temperature geochemistry, this chapter focuses on reactions at solid/solution interfaces.
Reaction rates at solid/solution interfaces are controlled by the area of the interface as well as by the chemical and physical conditions that occur there. Surface reactions are approximately confined to a two-dimensional region, so their rates are expressed in terms of how fast species are created per unit of surface area, and this means that the rates have units of flux (J, mol/m2sec). The flux notation (J) and terminology is used throughout this book. The environment at the solid/solution interface is a hybrid of the bulk solid and bulk solution, so models of the chemical and physical conditions controlling the reaction rates must account for this transitional character. Equilibrium thermodynamics provides a powerful starting point for constraining the surface conditions. At equilibrium the chemical potential of each component must be the same throughout the system, so the chemical potential of the components in the surface are equal to their chemical potentials in the solid and solution phases. At low temperatures, the slow rate of equilibration between the bulk solid and the surface may void this requirement for the solid but it should apply for the components in the bulk solution. Also, at equilibrium the principle of detailed balance requires that the rates of forward reactions in the interface must equal the rates of the reverse reactions. In addition, the forward and reverse reaction steps must be the same. Models of reaction rates at equilibrium are well constrained by these principles but as the system departs from equilibrium these requirements fall away and we must search for other principles to model interfacial reaction rates.
So far this book has offered appetizers. This chapter deals with the preparation of the main course by introducing concepts for linking the simple models discussed in this book to understand the complex processes that lead to pattern formation in geological settings. Geoscientists spend much of their time and effort identifying and explaining naturally occurring spatial and temporal patterns with the goal of interpreting those patterns to understand the processes and conditions that formed them. They are challenged by the need to identify meaningful patterns in situations that often appear to be chaotic (Crutchfield, 2012). Meaningful patterns are frequently subtle and recognizing them often requires clues provided by process models. Most pattern-forming systems are too complex to interpret in a holistic way so our strategy is to first parse them into simple processes, which can be accurately modeled using methods like those explained in this book. The resulting models of discrete processes are then linked to simulate the overall pattern-forming scenario. This strategy is widely used in science and technology. For example, engineers design processing plants by dividing the overall process into unit operations, each of which is responsible for a single chemical or physical transformation of a feedstock (Gupta and Yan, 2006; Hendricks, 2006; McCabe et al., 1993). These unit operations are modeled separately and the models are combined to simulate the entire processing plant. This strategy is especially effective when the unit operations occur in a linear array of steps so that the product of one step is the feed for the next. Natural processes are often more complicated because they can switch from one path to another in a stochastic manner or because there is one or more feedback loops in the overall process. Regardless of the complexity of the situation, the divide and analyze strategy is the most effective way to understand how observed patterns are related to unit processes. The challenge of interpreting pattern-forming processes is a very exciting scientific frontier (Ball, 1999; Nicolis and Prigogine, 1989).
This book is intended for use as a textbook for an upper-level undergraduate or a first-year graduate course and as a handy reference for professionals who need to refresh their memories about a particular rate model. It focuses on rate models that are well understood, widely used, and pertinent to geochemical processes. Its scope is limited to an amount of material that can be covered in a one-semester course and the depth of presentation is restricted to the principal aspects of each model. The reference list directs the reader to sources that contain the additional details needed to produce more sophisticated models.
As the scientific and technological enterprise grows larger, scientists cope with the onslaught of new knowledge by retreating to more narrowly defined sub-disciplines. Increasing specialization tends to make each sub-discipline less relevant to the overall scientific and technological enterprise. This tempts scientists to pursue questions of trivial importance relative to the larger picture. Furthermore, the terminology that grows up in these enclaves becomes abstruse and incomprehensible. Geochemistry is challenged to avoid this trap by aggressively integrating knowledge from cognate disciplines in order to remain vibrant and meaningful. This is my justification for choosing many of the topics in this book from scientific and technological areas that might not be considered as traditional geochemistry. Hopefully this intellectual cross-fertilization will improve geochemistry’s vigor. My choice of topics was predicated on the question: What should everyone know about rates of geochemical processes? I hope that this book answers that question. If I have done my job properly, this book will provide an intellectual foundation for anyone, including scientists and technologists from cognate disciplines, who deal with geochemical processes. I have tried to meld these topics by using a standard notation and terminology that is consistent with that already used in geochemistry.
Quasi-kinetic models deal with processes that are controlled by mass transfer rates rather than by chemical reaction rates. These models assume nearly instantaneous attainment of equilibrium within the region of interest, so changes in the species distribution are controlled by the rate of transfer of substances into or out of that region. These models are constrained by continuity equations making them similar to the chemical reactors models in Chapter 4.
Local equilibrium assumption
Most of the models considered in this chapter rely on the local equilibrium assumption. This assumption requires that the rates of chemical reaction and local mass transfer within the model’s spatial domain are fast relative to the residence time of a slug of solution within that domain. Knapp (1989) and Bahr and Rubin (1987) have evaluated conditions where the local equilibrium assumption is valid and the Knapp treatment is summarized by Zhu and Anderson (2002).
For the local equilibrium assumption to be valid, both the mineral and solution reaction rates and the transport rate to and from the minerals’ surfaces must be fast. These constraints are best tested using the i rst Damköhler number , DaI, and the Péclet number , Pe. DaI compares the rate of consumption (or production) of a species by chemical reaction to the rate of delivery (or removal) of that species by advection.
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