2008_Brantley_Book_Kinetics Of Water-Rock Interaction.pdf

Kinetics of Water-Rock Interaction Kinetics of Water-Rock Interaction Edited by Susan L. Brantley James D. Kubicki Art F. White 123 Editors: Susan L. Brantley James D. Kubicki The Pennsylvania State University The Pennsylvania State University Earth and Environmental Systems Institute Department of Geosciences and the Earth and 2217 Earth-Engineering Science Building Environmental Systems Institute University Park, PA 16802 335 Deike Building USA University Park, PA 16802 e-mail: [email protected] USA e-mail: [email protected] Art F. White U.S. Geological Survey MS 420, 345 Middleﬁeld Rd. Menlo Park, CA 94025 USA e-mail: [email protected] Cover photograph © Brady McTigue, courtesy of Brady McTigue Photography and Design. ISBN 978-0-387-73562-7 e-ISBN 978-0-387-73563-1 Library of Congress Control Number: 2007937090 © 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover illustration: Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com Preface Systems at the surface of the Earth are continually responding to energy inputs de- rived ultimately from radiation from the Sun or from the radiogenic heat in the in- terior. These energy inputs drive plate movements and erosion, exposing metastable mineral phases at the Earth’s surface. In addition, these energy ﬂuxes are harvested and transformed by living organisms. As long as these processes persist, chemical disequilibrium at the Earth’s surface will be perpetuated. In addition, as human populations grow, the need to produce food, extract wa- ter, and extract energy resources increases. These processes continually contribute to chemical disequilibrium at the Earth surface. We therefore ﬁnd it necessary to predict how the surface regolith will change in response to anthropogenic processes as well as long-term climatic and tectonic forcings. To address these questions, we must understand the rates at which reactions occur and the chemical feedbacks that relate these reactions across extreme temporal and spatial scales. Scientists and en- gineers who work on soil fertility, nuclear waste disposal, hydrocarbon production, and contaminant and CO2 sequestration are among the many researchers who need to understand geochemical kinetics. Fundamental questions concerning the long- term geological, climatic and biological evolution of the planet also rely on geo- kinetic information. In this book, we summarize approaches toward measuring and predicting the ki- netics of water-rock interactions which contribute to the processes mentioned above. In our treatment, we transect multiple length and time scales to integrate molecu- lar and macroscopic viewpoints of processes that shape our world. The treatment, as discussed below, begins at a chemical level with fundamental kinetic analysis and develops treatments for more geochemically complex systems. The focus of the book is low-temperature, but the treatments in the chapters lay the foundation for discussions of geochemical kinetics regardless of temperature and some high- temperature systems are treated in Ch. 12. An Appendix of data for mineral disso- lution reaction rates is included that will be expanded online (see www.czen.org or chemxseer.ist.psu.edu). v vi Preface The ﬁrst half of the book deals with basic is- sues of chemical kinetics. In Chapter 1, Brantley and Conrad address how geochemists deﬁne and measure reaction rates in the laboratory (Ch. 1, Analysis of Rates of Geochemical Reactions). In Chapter 2, Kubicki discusses transition state the- ory and how molecular orbital calculations can be used to understand or investigate geochemi- cal reaction mechanisms (Transition State The- ory and Molecular Orbital Calculations Applied to Rates and Reaction Mechanisms in Geochem- ical Kinetics). The next chapter discusses prob- lems and approaches toward understanding how to investigate, analyze, and model the mineral surface (Chapter 3, The Mineral-Water Interface, by Lüttge and Arvidson). Important aspects of sorption-desorption reactions on mineral surfaces and the role of organic matter in soils are de- scribed in Chapter 4, written by Chorover and Brusseau (Kinetics of Sorption-Desorption). Ap- proaches toward an integrated understanding of mineral dissolution and rigorous ﬁtting of min- eral dissolution data are discussed in Chapters 5 (Brantley, Kinetics of Mineral Dissolution) and 6 (Bandstra and Brantley, Data Fitting Techniques with Applications to Mineral Dissolution Kinet- ics). Importantly, Chapter 6 provides the mathe- matical background for ﬁtting rate measurements that are compiled in the Appendix (Bandstra et al., Compilation of Mineral Dissolution Rates). The last chapter of this ﬁrst half of the book reviews the models for assessing nucleation and growth of crystals. These models are necessary for eval- uating the stable and metastable phases that form during mineral reaction (Benning and Waychunas, Chapter 7, Nucleation, Growth, and Aggregation of Mineral Phases: Mechanisms and Kinetic Con- trols). In the last six chapters of the book, kinetic the- ory is applied to complex environmental systems. Roden introduces concepts of kinetic theory as ap- plied to reductive ferric oxide dissolution medi- ated by microbes in Chapter 8 (Microbiological Controls on Geochemical Kinetics 1: Fundamen- tals and Case Study on Microbial Fe(III) Oxide Preface vii Reduction) and to microbiological metal sulﬁde oxidation in Chapter 9 (Microbio- logical Controls on Geochemical Kinetics 2: Case Study on Microbial Oxidation of Metal Sulﬁde Minerals and Future Prospects). Many of the ideas in the ﬁrst nine chapters are utilized in discussions of chemical weathering in Chapter 10 by White (Quantitative Approaches to Characterizing Natural Chemical Weathering Rates). In this chapter, the problems associated with extrapolating kinetics from the labo- ratory to the ﬁeld are also discussed. Modelling kinetics in environmental systems such as soils or aquifers requires understanding of both chemical kinetics and trans- port processes, and approaches to modeling such reactive transport problems are therefore discussed by Steefel in Chapter 11 (Geochemical Kinetics and Transport). In Chapter 12, Gaillardet explores the utility of isotopic techniques, including sev- eral new isotopic systems that are under development, to unravel complex environ- mental systems at both low and high temperature (Isotope Geochemistry as a Tool for Deciphering Kinetics of Water-Rock Interaction). Gaillardet also treats systems at both small and global scales. In the ﬁnal chapter of the book, Lerman and Wu expand upon this treatment to discuss approaches toward modeling weathering and elemental cycling at the global scale (Chapter 13, Kinetics of Global Geochemical Cycles). We hope the book will be useful for readers who are both experts and those peripherally involved in geochemical kinetics. We aimed the book at graduate stu- dents as well as professional earth and environmental scientists. From its inception, the book was to be used as a teaching tool for graduate students; however, as we compiled data and models since the last such compilation edited by SLB and AFW in 1995 (Chemical Weathering Rates of Silicate Minerals), we realized that our in- troductory text could also be used as a professional guidebook for geochemical ki- netics. Harkening back to that earlier volume, perhaps it is ﬁtting to quote (again) from James Hutton who wrote, The ruins of an older world are visible in the present structure of our planet... The same forces are still destroying, by chemical decomposition or mechanical violence, even the hardest rocks and transporting the materials to the sea. or even to quote again from Al Hibbler in 1955, Time goes by so slowly, but time can do so much. We hope this text will contribute at least in a small way to the collected wisdom of these and other authors interested in kinetics of the Earth. S. L. Brantley, J. Kubicki University Park, PA A.F. White Menlo Park, CA Acknowledgements Many of the authors of this book are associated with the Center for Environmental Kinetics Analysis at Penn State. Funding for this center from both the National Sci- ence Foundation (Environmental Molecular Sciences Institute Grant CHE-0431328) and from the U.S. Department of Energy, Biological and Environmental Research, was instrumental in most aspects of writing the book and compiling the data. Other speciﬁc sources of funding are acknowledged in each chapter. SLB and JK also acknowledge the students of the Penn State class, Geosciences 560 Kinetics of Geochemical Systems, who reviewed chapters in an earlier form. We furthermore acknowledge advice from Ken Howell (Springer), and help from Lee Carpenter, Debbie Lambert, and Sue Rockey (Penn State). Finally, Denise Kowalski (Penn State) is acknowledged for her inﬁnite patience and attention to detail, along with her continual good humor in the face of the painstaking process of compiling this book. ix Contents Preface........................................................ v Acknowledgements.............................................. ix List of Contributors............................................. xxi 1 Analysis of Rates of Geochemical Reactions..................... 1 Susan L. Brantley and Christine F. Conrad 1.1 Kinetics and Thermodynamics.............................. 1 1.2 Rates of Reactions........................................ 3 1.2.1 Extent of Reaction................................. 3 1.2.2 Rate of Reaction.................................. 4 1.3 Rate Equations........................................... 6 1.3.1 Rate Order and Rate Constant....................... 6 1.4 Reaction Mechanisms..................................... 8 1.4.1 Elementary Reactions.............................. 8 1.4.2 Heterogeneous Reactions........................... 10 1.4.3 Catalysis and Inhibition............................ 10 1.5 Analysis of Kinetic Results................................. 12 1.5.1 Differential Method................................ 12 1.5.2 Integral Method................................... 13 1.6 Half Life................................................ 17 1.7 Complex Reactions........................................ 20 1.7.1 Opposing Reactions............................... 20 1.7.2 Sequential Reactions............................... 22 1.7.3 Parallel Reactions................................. 25 1.7.4 Chain Reactions................................... 25 1.8 Temperature Dependence of Reaction Rates................... 26 xi xii Contents 1.9 Chemical Reactors........................................ 28 1.9.1 Batch Reactors.................................... 29 1.9.2 Flow-Through Reactors............................ 30 1.10 Conclusions.............................................. 34 References..................................................... 35 2 Transition State Theory and Molecular Orbital Calculations Applied to Rates and Reaction Mechanisms in Geochemical Kinetics............... 39 James D. Kubicki 2.1 Introduction.............................................. 39 2.1.1 Why are Mechanisms Important?.................... 39 2.1.2 Why are Reaction Mechanisms Hard to Determine?..... 41 2.2 Methods for Determining Mechanisms....................... 44 2.2.1 Rate Laws........................................ 44 2.2.2 Activation Energies—Estimate of Bond-Breaking Energy in Rate-Determining Steps................... 45 2.2.3 Isotopic Exchange—Isotopic Tracers Can Identify Atom Types in a Reaction.......................... 45 2.2.4 Spectroscopy—Identiﬁcation of Reactive Intermediates............................. 45 2.2.5 Molecular Modeling............................... 46 2.3 Transition State Theory.................................... 48 2.3.1 Equilibrium Assumption........................... 48 2.3.2 Determining Reaction Pathways and Transition States.................................. 51 2.3.3 Calculating Activation Energies and Rate Constants..... 54 2.4 Quantum Mechanical Calculations........................... 56 2.4.1 Choice of Basis Set................................ 57 2.4.2 Choice of Electron Correlation...................... 58 2.4.3 Choice of Model System........................... 60 2.5 Examples................................................ 60 2.5.1 O-isotope Exchange in H4 SiO4(aq)................... 60 2.5.2 Ligand Exchange in Aqueous Solutions............... 62 2.5.3 Hydrolysis of Si-O-Si and Si-O-Al................... 63 2.6 Summary................................................ 65 References..................................................... 66 3 The Mineral–Water Interface................................. 73 A. Lüttge and R. S. Arvidson 3.1 Introduction: Deﬁnitions and Preliminary Concepts............. 73 3.1.1 Mineral–Water Interfaces are Everywhere............. 73 3.1.2 The Mineral–Water Interface: An Integrated Approach............................... 75 3.1.3 The Relationship of the Interface to the Bulk Solid...... 76 3.1.4 The Fundamental Importance of Scale................ 77 Contents xiii 3.1.5 A Schematic View of the Surface Structure: The Importance of Defects and Dislocations............... 80 3.1.6 Introduction to the Processes of Adsorption, Dissolution, Nucleation, and Growth................. 82 3.2 Quantiﬁcation: The Key to Understanding Mineral–Water Interface Processes........................... 83 3.2.1 The Concept and Quantiﬁcation of Surface Area....... 83 3.3 Analytical Methods....................................... 88 3.3.1 Quantiﬁcation of Surface Topography................ 88 3.3.2 Quantiﬁcation of Surface Chemistry and Structure...... 92 3.3.3 Integrated Quantitative Studies...................... 94 3.4 Approaches to Modeling the Mineral–Water Interface........... 96 3.4.1 Ab Initio and Density Functional Theory Calculations: A Prerequisite for Monte Carlo Simulations........... 97 3.4.2 Monte Carlo Simulations of Surface Topography and Interface Processes............................. 99 3.5 Summary and Outlook..................................... 101 References..................................................... 102 4 Kinetics of Sorption–Desorption............................... 109 Jon Chorover and Mark L. Brusseau 4.1 Sorption–Desorption Reactions.............................. 109 4.1.1 Adsorption at the Solid–Liquid Interface.............. 109 4.1.2 Surface Excess is the Quantitative Measure of Adsorption..................................... 110 4.2 Rate Limiting Steps....................................... 111 4.2.1 Transport Processes................................ 112 4.2.2 Surface Reactions................................. 114 4.2.3 Transport and Surface Reaction Control of Sorption Kinetics.................................. 116 4.3 Sorption Mechanisms and Kinetics for Inorganic Solutes........ 117 4.3.1 Surface Complexes and the Diffuse Ion Swarm......... 117 4.3.2 Surface Complexation Kinetics for Metal Cations...... 119 4.3.3 Cation Exchange on Layer Silicate Clays.............. 120 4.3.4 Surface Complexation Kinetics for Oxyanions......... 123 4.3.5 Multinuclear Surface Complexes, Surface Polymers and Surface Precipitates............................ 125 4.3.6 Effects of Residence Time on Desorption Kinetics...... 130 4.4 Sorption Mechanisms and Kinetics for Organic Solutes......... 133 4.4.1 Polar Organic Compounds.......................... 133 4.4.2 Hydrophobic Organic Compounds (HOCs)............ 135 4.4.3 Vapor-Phase Processes............................. 140 4.5 Sorbent Heterogeneity..................................... 141 4.6 Modeling Sorption/Desorption Kinetics....................... 142 References..................................................... 145 xiv Contents 5 Kinetics of Mineral Dissolution................................ 151 Susan L. Brantley 5.1 Introduction.............................................. 151 5.1.1 Importance of Dissolution Reactions................. 151 5.1.2 Steady-State Dissolution........................... 151 5.1.3 Stoichiometry of Dissolution........................ 154 5.2 Mechanisms of Dissolution................................. 156 5.2.1 Interface Versus Transport Control................... 156 5.2.2 Silicate and Oxide Dissolution Mechanisms........... 161 5.3 Rate Constants as a Function of Mineral Composition........... 175 5.3.1 Silica............................................ 175 5.3.2 Feldspar......................................... 176 5.3.3 Non-Framework Silicates........................... 177 5.3.4 Carbonates....................................... 180 5.4 Temperature Dependence................................... 183 5.4.1 Activation Energy................................. 183 5.4.2 Solution Chemistry and Temperature Dependence...... 184 5.5 Chemical Afﬁnity......................................... 185 5.5.1 Linear Rate Laws................................. 185 5.5.2 Non-Linear Rate Laws............................. 187 5.6 Conclusion............................................... 193 5.7 Glossary of Symbols...................................... 194 References..................................................... 196 6 Data Fitting Techniques with Applications to Mineral Dissolution Kinetics................................ 211 Joel Z. Bandstra and Susan L. Brantley 6.1 Introduction.............................................. 211 6.2 Rate Law Selection........................................ 212 6.2.1 Identifying Key Features........................... 212 6.2.2 Modeling Key Features............................. 216 6.3 Parameter Estimation...................................... 222 6.3.1 Error Minimization................................ 222 6.3.2 Linear Regression................................. 224 6.3.3 Non-Linear Fitting................................ 226 6.3.4 Linear Fitting of Log Transformed Data versus Non-linear Fitting........................... 229 6.3.5 Variance Heterogeneity............................. 232 6.3.6 Multiple Independent Variables...................... 234 6.3.7 Multiple Dependent Variables....................... 237 6.3.8 Global Analysis................................... 238 6.4 Error Analysis............................................ 239 6.4.1 Graphical Diagnostics.............................. 239 6.4.2 Quantitative Diagnostics............................ 242 Contents xv 6.5 Uncertainty Quantiﬁcation.................................. 243 6.5.1 Composition of Errors............................. 245 6.5.2 Approximations from the Covariance Matrix........... 246 6.5.3 Monte Carlo Methods.............................. 247 6.5.4 Bootstrap Methods................................ 248 6.6 Commonly Encountered Problems........................... 249 6.7 Conclusions.............................................. 251 References..................................................... 252 7 Nucleation, Growth, and Aggregation of Mineral Phases: Mechanisms and Kinetic Controls............................. 259 Liane G. Benning and Glenn A. Waychunas 7.1 Introduction.............................................. 259 7.2 Nucleation............................................... 261 7.2.1 Classical Nucleation Theory (CNT).................. 261 7.2.2 Kinetic Nucleation Theory (KNT).................... 267 7.3 Growth Processes......................................... 273 7.3.1 Classical Growth Theory........................... 273 7.3.2 (Nucleation and) Growth Far from Equilibrium......... 283 7.4 Aggregation Processes..................................... 286 7.4.1 Aggregation Regimes: DLCA and RLCA.............. 286 7.4.2 Fractals.......................................... 289 7.4.3 Ostwald Ripening................................. 292 7.4.4 Example: Silica Aggregation........................ 294 7.5 Process Quantiﬁcation: Direct versus Indirect Methods.......... 296 7.5.1 Imaging Techniques............................... 297 7.5.2 SAXS/WAXS..................................... 300 7.5.3 XAS............................................ 310 7.6 Synthesis and the Future................................... 319 References..................................................... 323 8 Microbiological Controls on Geochemical Kinetics 1: Fundamentals and Case Study on Microbial Fe(III) Oxide Reduction............ 335 Eric E. Roden 8.1 Introduction.............................................. 335 8.2 Overview of the Role of Microorganisms in Water-Rock Interactions.............................................. 337 8.2.1 Mechanisms and Deﬁnitions........................ 337 8.2.2 Key Characteristics of Microorganisms............... 339 8.3 Kinetic Models in Microbial Geochemistry.................... 344 8.3.1 Introduction...................................... 344 8.3.2 Zero-Order Kinetics............................... 345 8.3.3 First-Order Kinetics............................... 345 8.3.4 Hyperbolic Kinetics: Enzyme Activity and Microbial Growth/Metabolism...................... 347 xvi Contents 8.3.5 Microbial Population Dynamics and Competition....... 358 8.3.6 Kinetic Versus Thermodynamic Control of Microbial Reaction Rates......................... 365 8.4 Case Study #1 – Microbial Fe(III) Oxide Reduction............ 368 8.4.1 Introduction...................................... 368 8.4.2 Mechanisms of Enzymatic Fe(III) Oxide Reduction..... 368 8.4.3 Fe(III) Oxide Mineralogy and Microbial Reducibility............................. 371 8.4.4 Kinetics of Amorphous Fe(III) Oxide Reduction in Sediments..................................... 372 8.4.5 Pure Culture Studies of Fe(III) Oxide Reduction Kinetics................................ 380 References..................................................... 400 9 Microbiological Controls on Geochemical Kinetics 2: Case Study on Microbial Oxidation of Metal Sulﬁde Minerals and Future Prospects.................................................. 417 Eric E. Roden 9.1 Case Study #2 – Microbial Oxidation of Metal Sulﬁde Minerals.......................................... 417 9.1.1 Introduction...................................... 417 9.1.2 Inﬂuence of Sulﬁde Mineral Electronic Conﬁguration on Dissolution/Oxidation Pathway................... 418 9.1.3 Microbial Participation in Sulﬁde Mineral Oxidation.... 420 9.1.4 Kinetics of Coupled Aqueous and Solid-Phase Oxidation Reactions............................... 431 9.2 Summary and Prospects.................................... 450 9.2.1 Near-Term Advances in Modeling Coupled Microbial-Geochemical Reaction Systems............. 450 9.2.2 The Genomics Revolution.......................... 451 References..................................................... 455 10 Quantitative Approaches to Characterizing Natural Chemical Weathering Rates........................................... 469 Art F. White 10.1 Introduction.............................................. 469 10.2 Scales of Chemical Weathering.............................. 470 10.3 Weathering Calculations that Consider the Solid State........... 470 10.3.1 Weathering Indexes................................ 471 10.3.2 Case Study: Basalt Weathering Indexes............... 473 10.3.3 Case Study: Granite Weathering Indexes.............. 474 10.3.4 Solid Mass Transfers............................... 476 10.3.5 Case Study: Element Mobilities During Granite Weathering................................ 478 10.4 Weathering Calculations that Consider Solute Distributions...... 481 10.4.1 Solute Fluxes..................................... 481 Contents xvii 10.4.2 Mineral Contributions to Solute Fluxes............... 482 10.4.3 Solute Weathering Fluxes in Soils.................... 483 10.4.4 Case Study: Solute Weathering Fluxes in a Tropical Soil..................................... 483 10.4.5 Solute Weathering Fluxes in Groundwater Systems..... 488 10.4.6 Case Study: Spring Discharge....................... 489 10.4.7 Weathering along Groundwater Flow Paths............ 490 10.4.8 Case Study: Weathering in an Unconﬁned Aquifer...... 492 10.4.9 Weathering Fluxes in Surface Waters................. 494 10.4.10 Case Study: Weathering Inputs from a Small Stream.... 497 10.4.11 Case Study: Weathering Contributions in a Large River...................................... 499 10.5 Comparison of Contemporary and Long Term Chemical Weathering Fluxes........................................ 501 10.6 Mineral Weathering Rates.................................. 503 10.6.1 Weathering Gradients.............................. 504 10.6.2 Mineral Surface Areas............................. 505 10.6.3 Case Study: Weathering in a Soil Chronosequence...... 506 10.6.4 Comparing Mineral Weathering Rates................ 507 10.7 Factors Controlling Rates of Chemical Weathering............. 512 10.7.1 Intrinsic Effects................................... 512 10.7.2 Extrinsic Effects.................................. 514 10.7.3 Inﬂuences of Climate.............................. 519 10.7.4 Chemical Weathering Under Physically Eroding Conditions................................ 523 10.7.5 Case Study: Steady State Denudation in a Tropical Soil..................................... 524 10.7.6 Inﬂuence of Erosion, Topography and Tectonics........ 526 10.7.7 Role of Biology................................... 527 10.8 Summary................................................ 531 References..................................................... 532 11 Geochemical Kinetics and Transport........................... 545 Carl I. Steefel 11.1 Introduction.............................................. 545 11.2 Transport Processes....................................... 548 11.2.1 Advection........................................ 548 11.2.2 Molecular Diffusion............................... 549 11.2.3 Hydrodynamic Dispersion.......................... 558 11.3 Advection-Dispersion-Reaction Equation..................... 563 11.3.1 Non-Dimensional Form of the Advection-Dispersion- Reaction Equation................................. 564 11.3.2 Equilibration Length Scales......................... 568 11.3.3 Reaction Fronts in Natural Systems.................. 569 11.3.4 Transport versus Surface Reaction Control............ 571 xviii Contents 11.3.5 Propagation of Reaction Fronts...................... 572 11.4 Rates of Water–Rock Interaction in Heterogeneous Systems...... 573 11.4.1 Residence Time Distributions....................... 574 11.4.2 Upscaling Reaction Rates in Heterogeneous Media..... 576 11.5 Determining Rates of Water–Rock Interaction Affected by Transport...................................... 578 11.5.1 Rates from Aqueous Concentration Proﬁles............ 578 11.5.2 Rates from Mineral Proﬁles......................... 580 11.6 Feedback between Transport and Kinetics..................... 581 11.6.1 Reactive-Inﬁltration Instability...................... 582 11.6.2 Liesegang Banding................................ 583 11.7 Concluding Remarks...................................... 584 References..................................................... 585 12 Isotope Geochemistry as a Tool for Deciphering Kinetics of Water-Rock Interaction...................................... 591 Jérôme Gaillardet 12.1 Introduction.............................................. 591 12.2 Isotopes as a Fingerprint of Water-Rock Interaction Pathways...................................... 592 12.2.1 Isotopic Doping Techniques......................... 593 12.2.2 Experimental Mineral Dissolution Sequences.......... 597 12.2.3 Natural Weathering Sequences of Granitic Rocks....... 600 12.2.4 Evolution of Isotopes Along Flowpaths............... 602 12.2.5 Isotopic Tracing of Global Kinetics.................. 610 12.3 The Use of Radioactive Decay to Constrain Timescales of Water-Rock Interactions.................................... 615 12.3.1 Crystal Growth................................... 615 12.3.2 Uranium and Thorium Series Nuclides................ 617 12.3.3 Cosmogenic Isotopes and the Determination of Denudation Rates............................... 626 12.4 Fractionation of Isotopes as a Kinetic Process.................. 629 12.4.1 Equilibrium and Kinetic Fractionation of Isotopes....................................... 630 12.4.2 Kinetics of Isotopic Exchange....................... 631 12.4.3 Rate-Dependent Isotopic Effects..................... 635 12.5 Conclusion and Perspectives................................ 643 References..................................................... 645 13 Kinetics of Global Geochemical Cycles......................... 655 Abraham Lerman and Lingling Wu 13.1 Introduction.............................................. 655 13.2 Historical Development of Geochemical Cycles................ 656 13.3 The Rock Cycle.......................................... 659 13.4 Essentials of Cycle Modeling............................... 661 13.4.1 Calcium Carbonate and Silicate Cycle................ 661 Contents xix 13.4.2 A Simple Cycle Model............................. 662 13.4.3 Residence and Mixing Times........................ 665 13.4.4 Connections to Geochemical Cycles.................. 667 13.5 Global Phosphorus Cycle................................... 668 13.5.1 Phosphorus Cycle Structure......................... 668 13.5.2 Dynamics of Mineral and Organic P Weathering........ 669 13.5.3 Experimental and Observational Evidence............. 672 13.6 Water Cycle and Physical Denudation........................ 673 13.6.1 Geographic Variation of Transport from Land to the Oceans....................................... 673 13.6.2 Land and Soil Erosion Rates........................ 676 13.6.3 Physical Denudation Rate and Residence Time......... 678 13.7 Chemical Denudation...................................... 679 13.7.1 Sedimentary and Crystalline Lithosphere.............. 679 13.7.2 Mineral Dissolution Rates.......................... 683 13.7.3 Chemical Denudation of Sediments.................. 684 13.7.4 Chemical Denudation of Continental Crust............ 693 13.7.5 Weathering Layer Thickness........................ 694 13.8 Mineral-CO2 Reactions in Weathering........................ 696 13.8.1 CO2 Reactions with Carbonates and Silicates.......... 696 13.8.2 CO2 Consumption and HCO3 − Production............ 698 13.8.3 CO2 Consumption from Mineral-Precipitation Model... 701 13.8.4 Mineral Dissolution Model......................... 706 13.9 Environmental Acid Forcing................................ 711 13.10 CO2 in the Global Carbon Cycle............................. 713 13.10.1 Cycle Structure and Imbalances...................... 713 13.10.2 Changes in CO2 Uptake in Weathering................ 714 13.10.3 CO2 Weathering Pathways.......................... 718 13.10.4 Further Ties between Carbonate and Sulfate........... 719 13.11 Summary and Overview.................................... 720 References..................................................... 723 Appendix: Compilation of Mineral Dissolution Rates................. 737 Joel Z. Bandstra, Heather L. Buss, Richard K. Campen, Laura J. Liermann, Joel Moore, Elisabeth M. Hausrath, Alexis K. Navarre-Sitchler, Je-Hun Jang, Susan L. Brantley Albite......................................................... 738 Andesine/Labradorite............................................ 742 Anorthite...................................................... 747 Apatite........................................................ 750 Basalt......................................................... 754 Biotite........................................................ 760 Bytownite..................................................... 762 Hornblende.................................................... 764 Kaolinite...................................................... 771 xx Contents K-feldspar..................................................... 782 Oligoclase..................................................... 787 Olivine........................................................ 790 Pyroxene...................................................... 811 Quartz......................................................... 818 Index............................................................. 825 List of Contributors Rolf S. Arvidson Rice University, Department of Earth Science MS-126, 6100 South Main Street, Houston TX 77005, e-mail: [email protected] Joel Z. Bandstra The Pennsylvania State University, Center for Environmental Kinetics Analysis, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] Liane G. Benning University of Leeds, Earth and Biosphere Institute, School of Earth and Environment, Leeds LS2 9JT, UK, e-mail: [email protected] Susan L. Brantley The Pennsylvania State University, Earth and Environmental Systems Institute, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] Mark L. Brusseau University of Arizona, Department of Soil, Water and Environmental Science, Shantz 429, Building #38, Tucson, AZ 85721, e-mail: [email protected] Heather L. Buss The Pennsylvania State University, Department of Geosciences and the Earth and Environmental Systems Institute, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] Richard Kramer Campen The Pennsylvania State University, Department of Geosciences and the Center for Environmental Kinetics Analysis, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] xxi xxii List of Contributors Jon Chorover University of Arizona, Department of Soil, Water and Environmental Science, Shantz 429, Building #38, Tucson, AZ 85721, e-mail: [email protected] Christine F. Conrad The Pennsylvania State University, Center for Environmental Kinetics Analysis, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] Jérôme Gaillardet Institut de Physique du Globe de Paris, Université Paris 7 – CNRS, 4 Place Jussieu, 75252 PARIS cedex 05, France, e-mail: [email protected] Elisabeth M. Hausrath The Pennsylvania State University, Department of Geosciences and the Earth and Environmental Systems Institute, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] Je-Hun Jang The Pennsylvania State University, Department of Civil and Environmental Engineering, 212 Sackett Building, University Park, PA 16802, e-mail: [email protected] James D. Kubicki The Pennsylvania State University, Department of Geosciences and the Earth and Environmental Systems Institute, 335 Deike Building, University Park, PA 16802, e-mail: [email protected] Abraham Lerman Northwestern University, Department of Earth and Planetary Sciences, Locy Hall, 1850 Campus Drive, Evanston, IL 60208, e-mail: [email protected] Laura J. Liermann The Pennsylvania State University, Department of Geosciences, 503 Deike Building, University Park, PA 16802, e-mail: [email protected] Andreas Lüttge Rice University, Department of Earth Science, Department of Chemistry, and Center for Biological and Environmental Nanotechnology, 6100 Main Street, Houston, TX 77005, e-mail: [email protected] Joel Moore The Pennsylvania State University, Department of Geosciences and the Center for Environmental Kinetics Analysis, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] Alexis K. Navarre-Sitchler The Pennsylvania State University, Department of Geosciences and the Center for Environmental Kinetics Analysis, 2217 Earth-Engineering Science Building, University Park, PA 16802, e-mail: [email protected] List of Contributors xxiii Eric E. Roden University of Wisconsin, Department of Geology and Geophysics, 1215 W. Dayton Street, Madison, WI 53706, e-mail: [email protected] Carl I. Steefel Lawrence Berkeley National Laboratory, Earth Sciences Division, 1 Cyclotron Road, Mail Stop 90-1116, Berkeley CA 94720, USA, e-mail: [email protected] Glenn A. Waychunas Lawrence Berkeley National Laboratory, Earth Sciences Division, MS 70-108B, One Cyclotron Road, Berkeley CA 94720, e-mail: [email protected] Art F. White U.S. Geological Survey, MS 420, 345 Middleﬁeld Rd, Menlo Park, CA 94025, e-mail: [email protected] Lingling Wu Northwestern University, Department of Earth and Planetary Sciences, 1850 Campus Drive, Evanston, Il 60208, e-mail: [email protected] Chapter 1 Analysis of Rates of Geochemical Reactions Susan L. Brantley1 and Christine F. Conrad2 1.1 Kinetics and Thermodynamics Over the last several billion years, rocks formed at equilibrium within the mantle of the Earth have been exposed at the surface and have reacted to move towards a new equilibrium with the atmosphere and hydrosphere. At the same time that min- erals, liquids, and gases react abiotically and progress toward chemical equilibrium at the Earth’s surface, biological processes harvest solar energy and use it to store electrons in reservoirs which are vastly out of equilibrium with the Earth’s other sur- face reservoirs. In addition to these processes, over the last several thousand years, humans have produced and disseminated non-equilibrated chemical phases into the Earth’s pedosphere, hydrosphere, and atmosphere. To safeguard these mineral and ﬂuid reservoirs so that they may continue to nurture ecosystems, we must understand the rates of chemical reactions as driven by tectonic, climatic, and anthropogenic forcings. Chemists approach the understanding of the natural world by deﬁning parts of the world as systems of study. The mechanically separable parts of the system—the crystalline and amorphous solids, liquids and gases—are known as phases. All the phases that are not inside the system are deﬁned as the environment surrounding the system. By deﬁnition, a system at equilibrium will be characterized by phases with uniform composition that exist at uniform temperature and pressure. To be precise, equilibrium is deﬁned as that state where the chemical potential of every component in every phase is equal throughout the system. To chemically understand a system, the chemical species within the system must be identiﬁed and characterized: the minimum number of species needed to deﬁne a system at equilibrium comprises the set of components of that system. Likewise, the thermodynamic state of any system is completely deﬁned by specifying the values 1 The Pennsylvania State University, Center for Environmental Kinetics Analysis, Earth and Environmental Systems Institute, [email protected] 2 The Pennsylvania State University, Center for Environmental Kinetics Analysis, Earth and Environmental Systems Institute, [email protected] 1 2 Susan L. Brantley and Christine F. Conrad of a critical number of properties. For example, the Gibbs phase rule states that the number of properties that must be deﬁned to completely describe a system (the de- grees of freedom, F) is dependent upon the number of phases, P, and components, C: F =C−P+2 (1.1) So, for example, to completely deﬁne the one-component one-phase pure H2 O sys- tem, we must only deﬁne the temperature, T , and pressure, P, of the system (F = 2). If one adds sufﬁcient NaCl as a second component so as to supersaturate this water with respect to halite and then isolates the system, the second law of thermodynam- ics states that the properties of this isolated system will evolve until the equilibrium state is reached. Indeed, the degrees of freedom of the ﬁnal two-component, two- phase (NaCl-saturated water and solid NaCl) system must also equal two: in effect, the state of this system is deﬁned solely by the temperature and pressure. Thermody- namics completely deﬁnes the ﬁnal state of the system: however, thermodynamics cannot deﬁne the rate at which the system evolves. The ﬁeld of irreversible thermodynamics treats systems that are removed from equilibrium by modeling how the entropy of the system changes with time as equi- librium is approached (Prigogine, 1967). Irreversible thermodynamics deﬁnes the change in entropy of the system, dS, as the sum of the entropy supplied to the sys- tem by its surroundings, dSe , and the entropy produced inside the system, dSi. The second law of thermodynamics states di S ≥ 0. For a reversible process, dSi = 0, and for an irreversible process, this term is always positive, dSi > 0. Furthermore, for a closed system at constant temperature and pressure, it can be shown that this term is related to the change in Gibbs free energy of the system, dGsys : T dSi = −dGsys (1.2) Therefore, for spontaneous reactions in closed systems at constant T and P, the entropy produced inside the system is related to the Gibbs free energy change of the system. In the case of the system with one reaction, the differentiation of Eq. (1.2) over time and introduction of ξ , the extent of reaction (see Eq. (1.9)), results in the ex- pression dSi dξ T =A , (1.3) dt dt where the entropy production ddti S ≥ 0 and A = −∆Greaction , the chemical afﬁnity of the reaction. The chemical afﬁnity, introduced by T. DeDonder, is the driving force of the reaction. For a reaction that occurs spontaneously as written (e.g., reactants on the left and products on the right of the reaction), A > 0 and ∆ Greaction < 0. At equilibrium, the chemical afﬁnity (−∆ Greaction ) is equal to 0. The negative driving force of reaction can be shown to be equal to a simple expression for any reaction deﬁned by an equilibrium constant Keq and a reaction activity quotient, Q: Q ∆Greaction = −A = RT ln (1.4) Keq 1 Analysis of Rates of Geochemical Reactions 3 If the driving force of reaction is positive, the reaction should proceed spontaneously as written. For example, for the reaction of albite with water, the reaction and ac- tivity quotient can be written by inspection assuming that the activity of albite and H2 O can each be set equal to unity: + NaAlSi3 O8(s) + 4H2 O(aq) + 4H(aq) → Na+ (aq) + Al(aq) + 3H4 SiO4(aq) 3+ aNa+ aAl 3+ a3H 0 (1.5) 4 SiO4 Q = a4H + The value of log Keq equals 4.70 at standard temperature and pressure (Drever, 1997). For a soil porewater in contact with albite where Q < 104.70 , thermodynam- ics predicts that albite should dissolve spontaneously as ∆Greaction is negative and the reaction should proceed as written. In contrast, if Q > 104.70 , albite should pre- cipitate. Of course, under ambient conditions, it is observed that precipitation of crystalline albite does not occur at a measurable rate; therefore, the kinetics of pre- cipitation are extremely slow, even if albite is supersaturated. Although thermodynamics does not allow the prediction of rates of chemical re- actions, it does place a constraint on kinetics. In particular, at equilibrium, the rate of the forward reaction must equal the rate of the reverse reaction. This constraint, known as microscopic reversibility, is discussed further in Chap. 2. Microscopic re- versibility leads to the conclusion that the equilibrium constant for a reaction that occurs as written must be equal to the ratio of the rate constants for the forward and reverse reactions (see Eq. (1.81)). 1.2 Rates of Reactions 1.2.1 Extent of Reaction In kinetic experiments, the rates of change of reactant and product concentrations are measured. Consider the reaction, A + 3B → Z (1.6) which begins with a mixture of A and B without Z. We assume a system where no change in volume occurs as the reaction proceeds. At any time, t, the rate of consumption of A, rA , is deﬁned as the negative slope of the tangent to the plot of concentration of A, [A], versus time: d[A] rA = − (1.7) dt The rate of formation of Z can be determined in the same manner: d[Z] rZ = (1.8) dt 4 Susan L. Brantley and Christine F. Conrad For the reaction given in Eq. (1.6), rA = rZ. Note however, that the stoichiometry of the reaction requires that the negative of the rate of consumption of B must differ by a factor of 3 from the value of rZ. Thus, if we deﬁne rates according to equations such as Eqs. (1.7) and (1.8), a given reaction may be characterized by different values for the rate at any given time. A useful concept, the extent of reaction, was therefore introduced by T. de Donder in 1922 to correct for the stoichiometry of reaction. The extent of reaction is deﬁned by the following, ni − n0i ξ= , (1.9) νi where n0i is the initial number of moles of a reactant or product, ni is the moles at time t, and νi is the stoichiometric coefﬁcient for that species in the written reaction. The extent of reaction can only be determined unequivocally for reactions with time- independent stoichiometries where the stoichiometric equation for the reaction is speciﬁed. 1.2.2 Rate of Reaction For a reaction in a system where reaction stoichiometry does not change with time, rate of reaction is deﬁned as the derivative of the extent of reaction with time divided by the volume 1 dξ r= (1.10) V dt For an individual species, i, the time derivative is given by 1 dni ξ= (1.11) νi dt where νi is the stoichiometric coefﬁcient for species i. The rate of reaction then becomes 1 dni r= (1.12) νiV dt Thus, for the reaction given in Eq. (1.6) 1 dnA 1 dnB 1 dnZ r=− =− = (1.13) V dt 3V dt V dt If the volume does not change during the course of the reaction, the term dni /V in Eq. (1.12) may be replaced by the change in concentration yielding 1 d[A] 1 d[B] 1 d[Z] r=− =− =. (1.14) vA dt vB dt vZ dt 1 Analysis of Rates of Geochemical Reactions 5 30 nepheline glass 25 jadeite glass 2 albite glass 20 8 15 10 5 0 0 200 400 600 800 1000 Time (hrs) Fig. 1.1 Normalized moles of glass cm−2 released into solution as a function of time during dis- solution of nepheline (Na6 Al6 Si6 O24 ), jadeite (Na4 Al4 Si8 O24 ), and albite (Na3 Al3 Si9 O24 ) glass powder in batch experiments at pH 2 (Hamilton et al., 2001). QnormSi = number of moles of Si released into solution per cm2 glass divided by the number of moles of Si in a mole of glass (based on a 24 oxygen formula unit). The slopes of these lines are the rates of dissolution of each glass. The extent of reaction and stoichiometric coefﬁcients has no meaning except in relation to the equation for the given reaction. Therefore, the reaction stoichiometry must be speciﬁed when referencing the rate of reaction. For the reaction given in Eq. (1.6), the rate of reaction could be analyzed by mea- suring [A], [B], or [Z] versus time. Similarly, for mineral dissolution kinetics, it is common to measure multiple products of the reaction. For example, for albite disso- lution, either Na, Al, or Si concentrations could be monitored versus time; however, it has become common to calculate silicate dissolution rates solely based on the rate of release of Si to solution. If albite dissolves stoichiometrically (Eqn. 1.5) then the release rate of Na (rNa ) or Al (rAl ) should be 1/3 the release rate of Si (rSi ). Even when the Si release is used to calculate the rate, the rate is often reported in units as molmineral m−2 s−1. Note that the reported release rate of albite written in units of mol albite per unit area per unit time will differ depending upon whether the formula unit of albite is written as Na3 Al3 Si9 O24 (ralbite(24) ) or as NaAlSi3 O8 (ralbite(8) ): rSi = 3rNa = 3rAl = 3ralbite(8) = 9ralbite(24) (1.15) These ideas are demonstrated in Figs. 1.1 and 1.2. The ﬁrst ﬁgure shows the rate of dissolution of three Na-Al-Si glasses as a function of time as determined by release of Si to solution. In this ﬁgure, the moles of Si released are normalized by the number of Si atoms per formula unit of glass (24 oxygen atoms per unit). The glass with the lowest Al/Si ratio (albite) dissolves the slowest. However, Fig. 1.2 demonstrates that Na and Al are preferentially released during initial albite glass 6 Susan L. Brantley and Christine F. Conrad 18 Normalized moles glass/cm2 (Qnorm) 16 Si Al 14 Na 12 (x 10−8) 10 8 6 4 2 0 100 200 300 400 500 600 700 800 900 Time (hrs) Fig. 1.2 Normalized moles of glass cm−2 released into solution as a function of time during dis- solution of albite glass (Na3 Al3 Si9 O24 ) at pH 2 in a batch experiment based on Si, Al, or Na con- centrations (Hamilton et al., 2001). See Fig. 1.1 for deﬁnition of Qnorm. According to Eq. (1.15), the rate of release of Si should be three times faster than the release of Na and Al. The ratios of these release rates indicate non-stoichiometric dissolution in which a Si-rich layer is forming on the albite glass surface. dissolution (rSi = 3rNa = 3rAl ) leaving behind a silica-rich leached layer (Hamilton et al., 2001). 1.3 Rate Equations 1.3.1 Rate Order and Rate Constant Analysis of reaction rates is usually ﬁrst attempted at a phenomenological level where the rates of reactions are measured as a function of solution, solid, and gas composition. At this phenomenological level, a rate equation or rate law is a math- ematical expression that relates the change in concentration of a product or reactant versus time to the concentrations of species in a chemical reaction. For reactions among solutes, solids, and gas phases, concentrations may be denoted in a variety of units such as moles L−1 for aqueous or solid phase species, m2 L−1 or sites L−1 for solid phase species, or partial pressures for gases. For the reaction given in Eq. (1.6), the rate equation might be written as r = k[A]α [B]β [Z]σ (1.16) where ideally, k, α , β , and σ are constants independent of concentration and time. Notice that rate laws are written in units of concentration rather than activities. In 1 Analysis of Rates of Geochemical Reactions 7 contrast, in thermodynamic equations, activity coefﬁcients are used to calculate ac- tivities from concentrations. However, in kinetics it is the spatial concentration (e.g., moles cm−3 ) of the species that determines the rate of molecular collisions between or among reactants, and the rate of collision partially controls the rate of reaction (Lasaga, 1981). The exponents α , β , and σ are known as the partial orders of reaction with re- spect to A, B, and Z and the sum of all of the partial orders is the overall order of reaction. For phenomenological treatments of reaction kinetics, these orders are em- pirical and need not be integral values. Note also that no simple relationship need ex- ist between the stoichiometry of an equation and the order of the reaction. In fact, the kinetics of many geochemical systems are only treated with a phenomenological ap- proach where equations such as Eq. (1.16) are treated simply as ﬁtting equations. In general, this is also the level of analysis that is ﬁrst utilized to understand a system. An example of a ﬁrst-order reaction that has been treated phenomenologically is the oxidation or pyrite and production of sulfuric acid. This reaction is largely responsible for decreased pH values in sulﬁde mine spoils. The rate of change in the concentration of FeS2(s) due to oxidation in pyrite-containing soils has been observed to be a function of the concentration of FeS2(s) in the soil, [FeS2(s) ], and can be written as (Hossner and Doolittle, 2003) r = −d[FeS2(s) ]/dt = k[FeS2(s) ] (1.17) A second-order reaction may refer to either the case where the rate of reaction is proportional to the second power of one species or, alternately, proportional to the product of the concentrations of two species each raised to the ﬁrst power. An ex- ample of the ﬁrst case can be seen in the rate of oxidation of dissolved As(III) in the presence of solid phase manganese dioxide, a process used to remove the toxic form of inorganic As(III) from drinking water. It has been determined that this reaction follows second-order kinetics (Driehaus et al., 1995): r = −d[As(III)(aq) ]/dt = k[As(III)(aq) ]2. (1.18) The second type of second-order kinetics is exhibited by the oxidation kinetics of ferrous minerals studied by Perez et al. (2005). Batch reactors were utilized to assess the ability of naturally occurring ferrous silicate minerals to act as an oxygen buffer in a nuclear fuel repository. The experimental oxidation data were ﬁt by a second- order rate law of the form, r = −d[O2(aq) ]/dt = k[O2(aq) ][Fe(II)(s) ] (1.19) where [Fe(II)(s) ] refers to the concentration of ferrous sites on the surface of the mineral (sites m−2 ). Some reactions cannot be deﬁned to have a reaction order. An example of such a reaction is an enzyme-catalyzed reaction that is described by the Michaelis-Menten equation (see Chap. 8). The rate law for a reaction of this type is given by Vmax [A] r= (1.20) Km + [A] 8 Susan L. Brantley and Christine F. Conrad where Vmax and Km are constants and [A] is the concentration of the substrate re- acting with the enzyme. The Michaelis-Menten rate equation is an example of a hyperbolic rate equation. Another rate law that is hyperbolic in form is derived if one assumes that the rate of a reaction is proportional to the surface site density of sorbed species on a solid where this sorbate concentration is modeled with a Lang- muir adsorption isotherm. Regardless of the derivation of the rate equation, for a rate that can be modeled by a hyperbolic equation such as Eq. (1.20), no true rate order exists: for low [A], the rate is observed to be ﬁrst order in A, while for high [A], the rate is observed to be zeroth order. The constant k in a rate equation is called the rate constant. The units of the rate constant depend on the order of the reaction. A ﬁrst-order reaction, such as Eq. (1.17) where the rate is described in units of mol L−1 s−1 is described by a rate constant with units of s−1. The units for a second-order reaction rate constant (e.g., Eq. (1.19)) where the concentration terms are expressed in mol L−1 can be determined from mol L−1 s−1 = mol −1 L s−1 (1.21) (mol L−1 )2 Whenever a value for a rate constant is cited, to interpret this rate constant the re- searcher must know both how the reaction rate and rate equation have been deﬁned. 1.4 Reaction Mechanisms 1.4.1 Elementary Reactions Many reactions take place in a series of steps and involve the formation of interme- diate species. This set of steps is called the mechanism. Complex mechanisms can be broken down into a series of reactions—elementary reactions—that occur ex- actly as written. Elementary reactions are the building blocks of a complex reaction and cannot be broken down further into simpler reactions. In addition, elementary reactions have the desirable property that they exhibit the same rate regardless of the system: they can thus be extrapolated from one system to another at a given temperature and pressure. The rate equation for an elementary reaction can be written for the reaction a priori because the reaction occurs exactly as written. Thus, the rate equations for the elementary reactions, A → B, 2A → B, and A + B → C, are written r = k[A], r = k[A]2 , and r = k[A][B], respectively. It is rare for more than two or at most three species to collide in the geometry conducive for reaction, and thus reaction orders for elementary reactions are seldom larger than 2, or at most, 3. The reaction order of an elementary reaction is also related to the molecularity—the number of reac- tant particles (e.g., atoms, molecules, or free radicals) involved in each individual chemical event. For example, the reaction A → Z involves only one molecule, A, and is therefore said to be unimolecular. The reaction A + B → Z involves two molecules and is said to be bimolecular. 1 Analysis of Rates of Geochemical Reactions 9 Characteristically, a kineticist measures a reaction rate as a function of the con- centration of reactants or products, and then proposes a rate equation such as Eq. (1.18) that describes the observations. In the next step of analysis, a rate mecha- nism that is consistent with the rate equation is proposed. For example, Icopini et al. (2005) measured the rate of disappearance of aqueous silica, H4 SiO4(aq) , from solutions supersaturated with respect to amorphous silica and observed that the rate data was not well ﬁt by ﬁrst-, second-, or third-order rate laws, but was well ﬁt by a fourth-order rate law of the form r = −d[H4 SiO4(aq) ]/dt = k4 [H4 SiO4(aq) ]4 (1.22) The high order of reaction observed in this work suggested that this rate equation described a complex reaction mechanism and not an elementary reaction. The au- thors thus derived a reaction mechanism that is consistent with a fourth-order rate law. They proposed the following elementary reactions to describe the mechanism for the polymerization of monomeric to tetrameric silica in aqueous solutions: H4 SiO4(aq) + H4 SiO4(aq) → H6 Si2 O7(aq) + H2 O (1.23) H6 Si2 O7(aq) + H4 SiO4(aq) → H8 Si3 O10(aq) + H2 O (1.24) H8 Si3 O10(aq) + H4 SiO4(aq) → H8 Si4 O12(aq) + 2H2 O (1.25) According to their model, the polymerization of monomeric silica (H4 SiO4 ) into tetrameric silica (H8 Si4 O12 ) is hypothesized to occur via monomer addition (Eqs. (1.23)–(1.25)). In the ﬁnal reaction (Eq. (1.25)) the extra water released is due to the formation of a cyclic compound. For this series, a composite or overall reaction can be written as the sum over the entire mechanism: 4H4 SiO4(aq) → H8 Si4 O12(aq) + 4H2 O (1.26) If the reactions given in Eq. (1.23) (rate constant k1 ) and Eq. (1.24) (k2 ) are signiﬁ- cantly faster than the reaction given in Eq. (1.25) (k3 ), then these two reactions could achieve equilibrium (with equilibrium constants K1 and K2 , respectively) while the third reaction could control the overall rate. Such a slow step that controls the rate is called the rate-controlling (or –determining or –limiting) step. If Eq. (1.25) is the rate-controlling step and it is an elementary reaction then the rate constant k4 that describes the rate of reaction given in Eq. (1.26) and in Eq. (1.22) can be expressed as follows: K1 K2 k4 = k3 f (y) 2 (1.27) aH2 O In this equation, f (y) represents a term incorporating activity coefﬁcients for the sil- ica species, and aH2O is the activity of water. Note that, because the overall rate equa- tion is written in terms of concentration of species rather than activities of species, activity correction terms are incorporated into the rate constant k4. It is common for rate constants for composite reactions to contain activity coefﬁcients, equilibrium constants and rate constants for elementary reactions as shown by this example. 10 Susan L. Brantley and Christine F. Conrad Another example of simplifying complex functions with a single rate constant is found in the discussion of calcite precipitation in Chap. 5. It is common to hypothesize a mechanism such as Eqs. (1.23)–(1.25) and then to assume that the individual reactions are elementary reactions for which rate equa- tions can be written a priori. Such an approach allows the kineticist to propose a hypothetical mechanism to test experimentally. If the data and rate equations de- rived from a mechanism agree, the mechanism can be said to be consistent with the kinetic evidence. If they do not agree, a new mechanism can be derived and tested. In general, without detailed spectroscopy to determine a mechanism, it is impossible to prove that a mechanism occurs as written, especially for geochemical reactions. 1.4.2 Heterogeneous Reactions Reactions that occur in one phase are referred to as homogenous reactions, while reactions that occur at interfaces between phases are heterogeneous reactions. Mod- eling heterogeneous reactions such as water-rock reactions tends to be difﬁcult in that a term must be included in the rate equation that describes the reacting species at the mineral interface (Chaps. 3 and 4). Often it is assumed that reactions occur at all sites on the mineral surface or at some constant fraction of the surface sites. For such a case, the concentration of reactant sites may be included in the rate equa- tion as the total mineral-water interfacial area. Generally, to assess such an area, the mineral surface is measured by adsorption of an inert gas to the surface, and the speciﬁc surface area (m2 g−1 ) is determined from this sorbed gas using the Brunauer-Emmet-Taylor (BET) isotherm (Brantley and Mellott, 2000). However, since surface sites do not react identically, most reactions are proportional to the re- active surface area rather than the total surface area (Ch 3). For example, for biotite and other sheet silicates, edge sites dissolve faster than sites on the basal surface and often control the overall dissolution rate under acid conditions (Kalinowski and Schweda, 1996). In dissolution or precipitation reactions, a further complexity arises because the mineral surface area changes with time. To date, very few attempts have been made to incorporate the change in surface area with time into mineral dissolu- tion/precipitation models. 1.4.3 Catalysis and Inhibition Catalysts are substances that increase the rate of reaction but are not consumed during the course of the reaction. Therefore, they do not modify the standard Gibbs free energy change of the reaction nor do they appear in the equilibrium constant expression for the reaction. For example, a reaction containing a catalyst may be written as follows (after Laidler, 1987), A + B + catalyst → Y + Z + catalyst, (1.28) 1 Analysis of Rates of Geochemical Reactions 11 and the rate of the reaction proceeding from left to right, r1 , can be written as r1 = k1 [A][B][catalyst] (1.29) Accordingly, the rate of the reaction occurring from right to left, r−1 , can be written r−1 = k−1 [Y ][Z][catalyst] (1.30) Using the condition that r1 = r−1 at equilibrium, the catalyst concentration cancels out in the expression for the equilibrium constant, K, for the reaction k1 [Y ][Z] K= = (1.31) k−1 [A][B] eq Catalysts can be classed as either homogeneous when only one phase is involved, or they can be heterogeneous where the reaction occurs at an interface between phases. Enzymes are examples of catalysts that can be either homogeneous or heteroge- neous: some enzymes are soluble and catalyze soluble reactants homogeneously, but other enzymes are embedded in membranes that catalyze soluble reactants het- erogeneously. In addition, enzymes can be produced intracellularly, thus capable only of catalysis of reactions inside the cell, or they can be secreted extracellularly, and can diffuse to environmental substrates. Some examples of important biogeo- chemical reactions catalyzed by enzymes are summarized in Chaps. 8 and 9. Many authors have also documented that mineral surfaces such as Mn oxides can act as catalysts for the reactions of organic molecules. Autocatalysts are catalysts that are products of a reaction. Autocatalysts that build up in concentration in a solution cause the rate of a reaction to increase with time. An example in geochemistry of an autocatalyst is aqueous Fe(III) that forms during oxidation of pyrite, and which, once formed, acts as an oxidant for the continued ox- idation of pyrite. The oxidation of pyrite is one of the main processes contributing to the acidiﬁcation of lakes, streams and rivers by acid mine and acid rock drainage. The mechanism of the oxidation of pyrite is extremely complex and can vary ac- cording to species in solution, the nature of the intermediates, and the mechanism by which these intermediates produce the ﬁnal product (Williamson and Rimstidt, 1994). The reaction for the oxidation of pyrite by ferric iron can be written as + FeS2 + 14Fe3+ + 8H2 O → 15Fe2+ + 2SO2− 4 + 16H (1.32) The measured rate of oxidation of pyrite in the presence of dissolved oxygen in- creases with increasing concentration of Fe(III): thus, the ferric ion acts as an auto- catalyst. Williamson and Rimstidt (1994) proposed the following rate law for pyrite oxidation in the presence of dissolved oxygen: 0.93 mFe r = 10−6.07 3+ 0.40 (1.33) mFe 2+ where r is the rate of pyrite destruction in mol m−2 s−1 and m is the concentration of Fe3+ or Fe2+. 12 Susan L. Brantley and Christine F. Conrad Catalysts should not be confused with species that simply accelerate reactions: such species, known as accelerators, appear in the equilibrium constant expression. For example, many have observed that organic ligands accelerate the rate of disso- lution of Fe-containing minerals (Kraemer, 2004). In a similar vein, inhibitors are species that reduce the rate of a reaction generally by affecting the ∆ G of reaction. Inhibitors do not behave in the same manner as catalysts as they are often consumed during reaction. Inhibitors can sometimes decrease the rate of a heterogeneous re- action by physically blocking reactive sites. The degree of inhibition, εi , is deﬁned as the rate of reaction without the inhibitor, r0 , minus the rate with the inhibitor, r, divided by r0 (after Laidler, 1987): r0 − r r εi = = 1− (1.34) r0 r0 Inhibition and catalysis or acceleration are often observed to be important in control- ling the overall morphology and growth rate of crystals (Nagy, 1995). For example, Al-hydroxide crystallization rates have been shown to be slowed considerably by the presence of Si at concentrations as low as 2 mg L−1 (Hem et al., 1973). In contrast, the presence of aqueous Li+ ions greatly accelerates the growth rate, suggesting that a Li-aluminate species acts to nucleate bayerite (Van Straten et al., 1985). Addition- ally, in weathering environments, organic components have been shown to acceler- ate the rates of formation of Al-hydroxides and kaolinite, promoting the formation of clays in organic-rich environments (Nagy, 1995 and references therein). 1.5 Analysis of Kinetic Results The ﬁrst step in analysis of geochemical kinetics is to measure the rates of change of reactants or products. The next step is generally to see how this rate varies as a function of the concentrations of species in solution, partial pressure of gases, or area of mineral surfaces in order to propose a rate equation. In general, two methods for determining rate equations are used: the differential method or the method of integration. The methods are described and exempliﬁed below largely following the notation and methods of Laidler (1987). 1.5.1 Differential Method The differential method is implemented by calculating the slope from a plot of con- centrations of reactants or products versus time (e.g., Figs. 1.1 and 1.2). If these slopes are measured for experiments with different initial concentrations of reac- tants, then the logarithm of the rates of reaction can be plotted versus the logarithms of concentration to calculate the reaction order. For example, Hamilton et al. (2001) measured the dissolution rates of albite glass as a function of H+ activity in solution 1 Analysis of Rates of Geochemical Reactions 13 1.6 −15.3 log Normalized Dissolution Rate Glass Normalized moles glass/cm2 (QnormSi ) (a) (b) 1.4 −15.4 pH 1 1.2 pH 2 −15.5 (mole glass/cm2/s) pH 4 1.0 −15.6 (x10−8) 0.8 −15.7 0.6 −15.8 0.4 −15.9 0.2 −16.0 0.0 −16.1 0 200 400 600 800 1000 0 1 2 3 4 5 Time (hrs) pH Fig. 1.3 Use of the differential method to determine the order of reaction with respect to H+ for silica release from albite glass. (a) Rate of dissolution is determined under acidic pH conditions (pH = 1–4). Log rate is then plotted versus pH (b) to determine the order of reaction with respect to H+ (≈0.2, after Hamilton et al., 2001) (pH 1–4). The dissolution rates were determined from the observed moles of Si re- leased divided by the number of Si atoms per glass formula unit (on a 24 oxygen unit basis) as a function of time. To determine the order of reaction with respect to H+ , the logarithm of the dissolution rates were plotted against pH, and the order was found to be ∼0.2 (Fig. 1.3). For a reaction that is dependent upon concentrations of more than one reactant, the experimentalist can measure the rate as a function of one reactant by varying that reactant concentration at the same time that all other concentrations are held in excess. By using this so-called isolation method for each reactant, the partial order with respect to each reactant can be determined. If one reactant is held at constant excess concentration and the rate is observed to vary as the ﬁrst order of the concentration of a second reactant, such behavior is described as pseudo-ﬁrst order. 1.5.2 Integral Method In using this method, a researcher assumes that a reaction proceeds according to a particular rate equation and then derives the integrated rate expression for that as- sumed rate law. This assumption is then tested by using the measured observations of rate versus concentrations of reactants or products to calculate and plot integrated rate expressions. When the experimental data matches the characteristic plots, the data are said to be consistent with the corresponding rate equation. Below, we ex- plicitly demonstrate examples of the method of integration applied to zero-order, ﬁrst-order, second-order and nth -order reactions (after Laidler, 1987). The results are tabulated in Table 1.1. 14 Susan L. Brantley and Christine F. Conrad Table 1.1 Analytical solutions for integrated rate laws of nth order reactions (after Laidler, 1987) Differential Integrated Rate constant Order Stoichiometry form form units dx x 0 A→Z = kA kA = mol L−1 s−1 dt t dx 1 a0 1 A→Z = k(a0 − x) kA = ln s−1 dt t a0 − x dx 1 x 2 2A → Z, = k(a0 − x)2 kA = L mol−1 s−1 dt t a0 (a0 − x) A+B → Z dx 1 1 3 A + 2B → Z = k(a0 − x) kA = dt t a0 − b0 b0 (a0 − x) × (a0 − 2x) ln L mol−1 s−1 a0 (b0 − x) dx 1 n A → Z, = k(a0 − x)n kA = Ln−1 mol1−n s−1 dt t(n − 1) 1 1 A+B+··· → Z × − (a0 − x)n−1 an−10 For conditions of one reactant or reactants at equal initial concentrations (mol L−1 ). 1.5.2.1 Zero-Order Reactions A zero-order rate law for a reaction describes a rate that is independent of the concentration of reactants. These rate laws are uncommon because most reactions increase in rate as the collision frequency of reactants increases; thus, the rate increases with the concentration of reactants. Zeroth-order reaction kinetics are, however, observed for some complex reaction mechanisms. Consider a zero-order reaction of the form A → Z in a system where the starting concentration of Z is zero and A is a0. The rate of the reaction expressed as the change of A with time is given by dx r= = kA (1.35) dt where x represents the concentration of Z, or amount of A that has been consumed, at time t. Separating variables, integrating both sides of the equation, and applying the boundary condition that x = 0 when t = 0 yields the integrated rate law for a zero-order equation: x = kAt (1.36) Plotting the measured values of the product concentration versus time yields a straight line with slope equal to the apparent zero-order rate constant. The data shown in Figs. 1.1–1.3 can be interpreted as zeroth-order kinetics for dissolution of albite, jadeite, and nepheline glasses. However, these are really pseudo-zeroth-order because rates of dissolution of these glasses vary with concen- tration of protons in solution (among other variables). 1 Analysis of Rates of Geochemical Reactions 15 1.5.2.2 First-Order Reactions A ﬁrst-order elementary reaction must be a reaction such as the following, A → Z, A → 2Z, A → B +C, (1.37) although other more complex reactions can also show apparent ﬁrst-order kinetics (Bandstra and Tratnyek, 2005). Following Laidler (1987), if the initial concentra- tions of A and Z can be deﬁned as a0 and zero respectively, then after time t the concentration of Z equals x and the concentration of A is a0 − x. Thus, the rate of change in the concentration of Z can be written as dx r= = kA (a0 − x) (1.38) dt where kA is the ﬁrst order rate constant that relates to the consumption of A. Sepa- ration of variables yields dx = kA dt (1.39) a0 − x Integration of Eq. (1.39) yields − ln(a0 − x) = kAt − I (1.40) where I is the constant of integration. Evaluating I using the boundary condition that x = 0 when t = 0 yields a value for I (= − ln a0 ), and Eq. (1.40) can be rewritten: a0 kAt = ln (1.41) a0 − x Here the rate constant, kA , has units of s−1. This equation is also commonly written as a0 − x = a0 e−kA t (1.42) For a reaction that is ﬁrst order in A, a plot of ln[A] versus t is linear with slope = −kA. Many geochemical systems are described with ﬁrst-order kinetics: perhaps the best known examples of such kinetics known by all geologists are the decay of radioactive elements whose ﬁrst-order kinetics are often described by their half-lives (see Sect. 1.6). 1.5.2.3 Second-Order Reactions Second-order reactions can be treated in much the same way as ﬁrst-order reactions (again, we follow the treatment of Laidler, 1987). However, with second-order el- ementary reactions, the rate of reaction may be proportional to the square of the concentration of one reactant (Eq. (1.43)) or to the product of two concentrations of two reactants (Eq. (1.44)) 2A → Z (1.43) A+B → Z (1.44) 16 Susan L. Brantley and Christine F. Conrad If analysis of the second reaction were to be measured with the same value of start- ing concentration of A and B, the rate could be expressed as dx r= = kA (a0 − x)2 (1.45) dt where x is the amount of A that has reacted in unit volume at time t and a0 is again the initial concentration of A at t = 0. Again the variables are separated to yield dx = kA dt (1.46) (a0 − x)2 Integration of this equation leads to 1 = kAt + I (1.47) a0 − x Applying the boundary condition that x = 0 when t = 0 yields 1 I= (1.48) a0 Substituting this into Eq. (1.47) leads to the integrated form of the second-order rate law shown below x kAt = (1.49) a0 (a0 − x) Here, the rate constant has units of L mol−1 s−1. Note that the variation of x with t is no longer exponential. A characteristic kinetic plot for a second-order reaction can be obtained by plotting 1/[A] versus time. If the rate is proportional to the product of the concentrations of two different sub- stances, and the concentrations are not initially equal, the procedure for integration follows a different approach (following notation and treatment of Laidler, 1987). For example, we assume a reaction stoichiometry described by νA A + νB B → Z, where νA and νB are the stoichiometric coefﬁcients for A and B, respectively, and the initial concentrations are a0 and b0. The rate after x moles of A have reacted per unit volume is dx r= = kA (a0 − νA x)(b0 − νB x) (1.50) dt For simplicity, we assume stoichiometric coefﬁcients are unity. Therefore, integra- tion of Eq. (1.50) and application of the boundary condition that x = 0 at t = 0 yields 1 b0 (a0 − x) ln = kAt (1.51) a0 − b0 a0 (b0 − x) Plotting the left hand side of this equation versus t yields a straight line for a second- order reaction whose slope is kA. 1 Analysis of Rates of Geochemical Reactions 17 1.5.2.4 nth -Order Reactions Higher order reactions are uncommon due to the extremely low probability of simul- taneous multiple body collisions. However, many rate laws incorporate multiple ele- mentary reactions, resulting in a composite rate law with order >2 (see for example, the treatment of silica oligomerization in Sect. 1.4.1). While such treatments are largely phenomenological rather than molecular, the quantiﬁcation of such a rate law followed by the proposal and testing of a hypothesized mechanism can lead to molecular insights. Furthermore, some phenomenological rate laws are successfully used to extrapolate kinetics. To calculate the integrated rate law for an nth -order rate, we consider a reaction of the nth -order involving a single reactant A with an initial concentration a0 (Laidler, 1987). As seen previously, the concentration of A remaining after time t is a0 − x. Thus, following the previous examples, the rate of consumption of A is dx r= = kA (a0 − x)n (1.52) dt Applying the boundary condition that x = 0 at t = 0 and assuming n = 1 leads to 1 1 1 kA = − (1.53) t(n − 1) (a0 − x)n−1 an−1 0 We can employ the integral method to calculate integrated expressions for nth order rate laws (Fig. 1.4) for the previous example of oligomerization of silica in aqueous solutions (Eq. (1.26)). For example, for the fourth-order rate law we derive 1 1 1 k4t = − (1.54) 3 [H4 SiO4 ]3 [H4 SiO4 ]30 where t is the elapsed time, [H4 SiO4 ] is the concentration of monomeric silica at time t, and [H4 SiO4 ]0 is the initial concentration of monomeric silica. The rate constant is deﬁned as the slope of a linear ﬁt of the right-hand side of Eq. (1.54) plotted versus time (Fig. 1.4c). The linear relationship in the 4th -order plot (Fig. 1.4c) indicates that the rate of change of monomeric silica can be described by Eq. (1.53) with n = 4. We cannot say that we have proven the mechanism to be correct, however, but only that the proposed mechanism is consistent with the data. 1.6 Half Life It is often difﬁcult to compare the relative rates of reactions between two processes because of differences in units of rate constants and varying orders of reaction. A very useful approach for evaluating relative rates of reaction is to determine the half-life, or t1/2 , for each reaction of interest. The half-life is deﬁned as the amount 18 Susan L. Brantley and Christine F. Conrad Fig. 1.4 Use of the integral 2.0 method to determine the order

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