Kinetics of Water-Rock Interaction

Kinetics of Water-RockInteraction

Edited by

Susan L. Brantley

James D. Kubicki

Art F. White

123

Editors:Susan L. Brantley James D. KubickiThe Pennsylvania State University The Pennsylvania State UniversityEarth and Environmental Systems Institute Department of Geosciences and the Earth and2217 Earth-Engineering Science Building Environmental Systems InstituteUniversity Park, PA 16802 335 Deike BuildingUSA University Park, PA 16802e-mail: brantley@essc.psu.edu USA

e-mail: kubicki@geosc.psu.edu

Art F. WhiteU.S. Geological SurveyMS 420, 345 Middlefield Rd.Menlo Park, CA 94025USAe-mail: afwhite@usgs.gov

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, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission 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 inconnection 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 arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

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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 metastablemineral phases at the Earth’s surface. In addition, these energy fluxes are harvestedand transformed by living organisms. As long as these processes persist, chemicaldisequilibrium 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 contributeto chemical disequilibrium at the Earth surface. We therefore find it necessary topredict how the surface regolith will change in response to anthropogenic processesas well as long-term climatic and tectonic forcings. To address these questions, wemust understand the rates at which reactions occur and the chemical feedbacks thatrelate 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 needto 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 analysisand develops treatments for more geochemically complex systems. The focus ofthe book is low-temperature, but the treatments in the chapters lay the foundationfor 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 orchemxseer.ist.psu.edu).

v

vi Preface

The first half of the book deals with basic is-sues of chemical kinetics. In Chapter 1, Brantleyand Conrad address how geochemists define andmeasure reaction rates in the laboratory (Ch. 1,Analysis of Rates of Geochemical Reactions). InChapter 2, Kubicki discusses transition state the-ory and how molecular orbital calculations canbe used to understand or investigate geochemi-cal reaction mechanisms (Transition State The-ory and Molecular Orbital Calculations Appliedto Rates and Reaction Mechanisms in Geochem-ical Kinetics). The next chapter discusses prob-lems and approaches toward understanding howto investigate, analyze, and model the mineralsurface (Chapter 3, The Mineral-Water Interface,by Luttge and Arvidson). Important aspects ofsorption-desorption reactions on mineral surfacesand the role of organic matter in soils are de-scribed in Chapter 4, written by Chorover andBrusseau (Kinetics of Sorption-Desorption). Ap-proaches toward an integrated understanding ofmineral 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 Techniqueswith Applications to Mineral Dissolution Kinet-ics). Importantly, Chapter 6 provides the mathe-matical background for fitting rate measurementsthat are compiled in the Appendix (Bandstra et al.,Compilation of Mineral Dissolution Rates). Thelast chapter of this first half of the book reviewsthe models for assessing nucleation and growthof crystals. These models are necessary for eval-uating the stable and metastable phases that formduring mineral reaction (Benning and Waychunas,Chapter 7, Nucleation, Growth, and Aggregationof 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 (MicrobiologicalControls 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 ofMetal Sulfide Minerals and Future Prospects). Many of the ideas in the first ninechapters 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 systemssuch as soils or aquifers requires understanding of both chemical kinetics and trans-port processes, and approaches to modeling such reactive transport problems aretherefore 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 Toolfor Deciphering Kinetics of Water-Rock Interaction). Gaillardet also treats systemsat both small and global scales. In the final chapter of the book, Lerman and Wuexpand upon this treatment to discuss approaches toward modeling weathering andelemental cycling at the global scale (Chapter 13, Kinetics of Global GeochemicalCycles).

We hope the book will be useful for readers who are both experts and thoseperipherally 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 wecompiled data and models since the last such compilation edited by SLB and AFWin 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 sameforces are still destroying, by chemical decomposition or mechanical violence, even thehardest 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 ofthese and other authors interested in kinetics of the Earth.

S. L. Brantley, J. KubickiUniversity Park, PA

A.F. WhiteMenlo Park, CA

Acknowledgements

Many of the authors of this book are associated with the Center for EnvironmentalKinetics 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. Otherspecific sources of funding are acknowledged in each chapter. SLB and JK alsoacknowledge the students of the Penn State class, Geosciences 560 Kinetics ofGeochemical Systems, who reviewed chapters in an earlier form. We furthermoreacknowledge 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, alongwith her continual good humor in the face of the painstaking process of compilingthis book.

ix

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

1 Analysis of Rates of Geochemical Reactions . . . . . . . . . . . . . . . . . . . . . 1Susan L. Brantley and Christine F. Conrad1.1 Kinetics and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Rates of Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Extent of Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Rate of Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Rate Order and Rate Constant . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Reaction Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Elementary Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Heterogeneous Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Catalysis and Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Analysis of Kinetic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.1 Differential Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.2 Integral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Half Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Complex Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7.1 Opposing Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.7.2 Sequential Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7.3 Parallel Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.7.4 Chain Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.8 Temperature Dependence of Reaction Rates . . . . . . . . . . . . . . . . . . . 26

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1.9 Chemical Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.9.1 Batch Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9.2 Flow-Through Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Transition State Theory and MolecularOrbital Calculations Applied to Ratesand Reaction Mechanisms in Geochemical Kinetics . . . . . . . . . . . . . . . 39James D. Kubicki2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1 Why are Mechanisms Important? . . . . . . . . . . . . . . . . . . . . 392.1.2 Why are Reaction Mechanisms Hard to Determine? . . . . . 41

2.2 Methods for Determining Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 442.2.1 Rate Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2.2 Activation Energies—Estimate of Bond-Breaking

Energy in Rate-Determining Steps . . . . . . . . . . . . . . . . . . . 452.2.3 Isotopic Exchange—Isotopic Tracers Can Identify

Atom Types in a Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2.4 Spectroscopy—Identification of

Reactive Intermediates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2.5 Molecular Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3 Transition State Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3.1 Equilibrium Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3.2 Determining Reaction Pathways and

Transition States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.3 Calculating Activation Energies and Rate Constants . . . . . 54

2.4 Quantum Mechanical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.1 Choice of Basis Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4.2 Choice of Electron Correlation . . . . . . . . . . . . . . . . . . . . . . 582.4.3 Choice of Model System . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.5.1 O-isotope Exchange in H4SiO4(aq) . . . . . . . . . . . . . . . . . . . 602.5.2 Ligand Exchange in Aqueous Solutions . . . . . . . . . . . . . . . 622.5.3 Hydrolysis of Si-O-Si and Si-O-Al . . . . . . . . . . . . . . . . . . . 63

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 The Mineral–Water Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A. Luttge and R. S. Arvidson3.1 Introduction: Definitions and Preliminary Concepts . . . . . . . . . . . . . 73

3.1.1 Mineral–Water Interfaces are Everywhere . . . . . . . . . . . . . 733.1.2 The Mineral–Water Interface: An

Integrated Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1.3 The Relationship of the Interface to the Bulk Solid . . . . . . 763.1.4 The Fundamental Importance of Scale . . . . . . . . . . . . . . . . 77

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3.1.5 A Schematic View of the Surface Structure: TheImportance 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 UnderstandingMineral–Water Interface Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.2.1 The Concept and Quantification of Surface Area . . . . . . . 83

3.3 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.3.1 Quantification of Surface Topography . . . . . . . . . . . . . . . . 883.3.2 Quantification of Surface Chemistry and Structure . . . . . . 923.3.3 Integrated Quantitative Studies . . . . . . . . . . . . . . . . . . . . . . 94

3.4 Approaches to Modeling the Mineral–Water Interface . . . . . . . . . . . 963.4.1 Ab Initio and Density Functional Theory Calculations:

A Prerequisite for Monte Carlo Simulations . . . . . . . . . . . 973.4.2 Monte Carlo Simulations of Surface Topography

and Interface Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4 Kinetics of Sorption–Desorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Jon Chorover and Mark L. Brusseau4.1 Sorption–Desorption Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1.1 Adsorption at the Solid–Liquid Interface . . . . . . . . . . . . . . 1094.1.2 Surface Excess is the Quantitative Measure

of Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.2 Rate Limiting Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2.1 Transport Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2.2 Surface Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.2.3 Transport and Surface Reaction Control of

Sorption Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3 Sorption Mechanisms and Kinetics for Inorganic Solutes . . . . . . . . 117

4.3.1 Surface Complexes and the Diffuse Ion Swarm . . . . . . . . . 1174.3.2 Surface Complexation Kinetics for Metal Cations . . . . . . 1194.3.3 Cation Exchange on Layer Silicate Clays . . . . . . . . . . . . . . 1204.3.4 Surface Complexation Kinetics for Oxyanions . . . . . . . . . 1234.3.5 Multinuclear Surface Complexes, Surface Polymers

and Surface Precipitates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.3.6 Effects of Residence Time on Desorption Kinetics . . . . . . 130

4.4 Sorption Mechanisms and Kinetics for Organic Solutes . . . . . . . . . 1334.4.1 Polar Organic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.4.2 Hydrophobic Organic Compounds (HOCs) . . . . . . . . . . . . 1354.4.3 Vapor-Phase Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.5 Sorbent Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.6 Modeling Sorption/Desorption Kinetics . . . . . . . . . . . . . . . . . . . . . . . 142References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

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5 Kinetics of Mineral Dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Susan L. Brantley5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.1.1 Importance of Dissolution Reactions . . . . . . . . . . . . . . . . . 1515.1.2 Steady-State Dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.1.3 Stoichiometry of Dissolution . . . . . . . . . . . . . . . . . . . . . . . . 154

5.2 Mechanisms of Dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.2.1 Interface Versus Transport Control . . . . . . . . . . . . . . . . . . . 1565.2.2 Silicate and Oxide Dissolution Mechanisms . . . . . . . . . . . 161

5.3 Rate Constants as a Function of Mineral Composition . . . . . . . . . . . 1755.3.1 Silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.3.2 Feldspar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.3.3 Non-Framework Silicates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.3.4 Carbonates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.4 Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.4.1 Activation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.4.2 Solution Chemistry and Temperature Dependence . . . . . . 184

5.5 Chemical Affinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.5.1 Linear Rate Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.5.2 Non-Linear Rate Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.7 Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6 Data Fitting Techniques with Applicationsto Mineral Dissolution Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211Joel Z. Bandstra and Susan L. Brantley6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.2 Rate Law Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.2.1 Identifying Key Features . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.2.2 Modeling Key Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2226.3.1 Error Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2226.3.2 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2246.3.3 Non-Linear Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.3.4 Linear Fitting of Log Transformed Data

versus Non-linear Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.3.5 Variance Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2326.3.6 Multiple Independent Variables . . . . . . . . . . . . . . . . . . . . . . 2346.3.7 Multiple Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . 2376.3.8 Global Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

6.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2396.4.1 Graphical Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2396.4.2 Quantitative Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

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6.5 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2436.5.1 Composition of Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.5.2 Approximations from the Covariance Matrix . . . . . . . . . . . 2466.5.3 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2476.5.4 Bootstrap Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.6 Commonly Encountered Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

7 Nucleation, Growth, and Aggregation of Mineral Phases:Mechanisms and Kinetic Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Liane G. Benning and Glenn A. Waychunas7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

7.2.1 Classical Nucleation Theory (CNT) . . . . . . . . . . . . . . . . . . 2617.2.2 Kinetic Nucleation Theory (KNT) . . . . . . . . . . . . . . . . . . . . 267

7.3 Growth Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2737.3.1 Classical Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2737.3.2 (Nucleation and) Growth Far from Equilibrium. . . . . . . . . 283

7.4 Aggregation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2867.4.1 Aggregation Regimes: DLCA and RLCA . . . . . . . . . . . . . . 2867.4.2 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2897.4.3 Ostwald Ripening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2927.4.4 Example: Silica Aggregation . . . . . . . . . . . . . . . . . . . . . . . . 294

7.5 Process Quantification: Direct versus Indirect Methods . . . . . . . . . . 2967.5.1 Imaging Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2977.5.2 SAXS/WAXS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3007.5.3 XAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

7.6 Synthesis and the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

8 Microbiological Controls on Geochemical Kinetics 1: Fundamentalsand Case Study on Microbial Fe(III) Oxide Reduction . . . . . . . . . . . . 335Eric E. Roden8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3358.2 Overview of the Role of Microorganisms in Water-Rock

Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3378.2.1 Mechanisms and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3378.2.2 Key Characteristics of Microorganisms . . . . . . . . . . . . . . . 339

8.3 Kinetic Models in Microbial Geochemistry . . . . . . . . . . . . . . . . . . . . 3448.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3448.3.2 Zero-Order Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3458.3.3 First-Order Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3458.3.4 Hyperbolic Kinetics: Enzyme Activity and

Microbial Growth/Metabolism . . . . . . . . . . . . . . . . . . . . . . 347

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8.3.5 Microbial Population Dynamics and Competition . . . . . . . 3588.3.6 Kinetic Versus Thermodynamic Control

of Microbial Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . 3658.4 Case Study #1 – Microbial Fe(III) Oxide Reduction . . . . . . . . . . . . 368

8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3688.4.2 Mechanisms of Enzymatic Fe(III) Oxide Reduction . . . . . 3688.4.3 Fe(III) Oxide Mineralogy and

Microbial Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3718.4.4 Kinetics of Amorphous Fe(III) Oxide Reduction

in Sediments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3728.4.5 Pure Culture Studies of Fe(III) Oxide

Reduction Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

9 Microbiological Controls on Geochemical Kinetics 2: Case Studyon Microbial Oxidation of Metal Sulfide Minerals and FutureProspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Eric E. Roden9.1 Case Study #2 – Microbial Oxidation of Metal

Sulfide Minerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4179.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4179.1.2 Influence of Sulfide Mineral Electronic Configuration

on Dissolution/Oxidation Pathway . . . . . . . . . . . . . . . . . . . 4189.1.3 Microbial Participation in Sulfide Mineral Oxidation . . . . 4209.1.4 Kinetics of Coupled Aqueous and Solid-Phase

Oxidation Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.2 Summary and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

9.2.1 Near-Term Advances in Modeling CoupledMicrobial-Geochemical Reaction Systems . . . . . . . . . . . . . 450

9.2.2 The Genomics Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . 451References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

10 Quantitative Approaches to Characterizing Natural ChemicalWeathering Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469Art F. White10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46910.2 Scales of Chemical Weathering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47010.3 Weathering Calculations that Consider the Solid State . . . . . . . . . . . 470

10.3.1 Weathering Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47110.3.2 Case Study: Basalt Weathering Indexes . . . . . . . . . . . . . . . 47310.3.3 Case Study: Granite Weathering Indexes . . . . . . . . . . . . . . 47410.3.4 Solid Mass Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47610.3.5 Case Study: Element Mobilities During

Granite Weathering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47810.4 Weathering Calculations that Consider Solute Distributions . . . . . . 481

10.4.1 Solute Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

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10.4.2 Mineral Contributions to Solute Fluxes . . . . . . . . . . . . . . . 48210.4.3 Solute Weathering Fluxes in Soils . . . . . . . . . . . . . . . . . . . . 48310.4.4 Case Study: Solute Weathering Fluxes in a

Tropical Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48310.4.5 Solute Weathering Fluxes in Groundwater Systems . . . . . 48810.4.6 Case Study: Spring Discharge . . . . . . . . . . . . . . . . . . . . . . . 48910.4.7 Weathering along Groundwater Flow Paths . . . . . . . . . . . . 49010.4.8 Case Study: Weathering in an Unconfined Aquifer . . . . . . 49210.4.9 Weathering Fluxes in Surface Waters . . . . . . . . . . . . . . . . . 49410.4.10 Case Study: Weathering Inputs from a Small Stream . . . . 49710.4.11 Case Study: Weathering Contributions in a

Large River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49910.5 Comparison of Contemporary and Long Term Chemical

Weathering Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50110.6 Mineral Weathering Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

10.6.1 Weathering Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50410.6.2 Mineral Surface Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50510.6.3 Case Study: Weathering in a Soil Chronosequence . . . . . . 50610.6.4 Comparing Mineral Weathering Rates . . . . . . . . . . . . . . . . 507

10.7 Factors Controlling Rates of Chemical Weathering . . . . . . . . . . . . . 51210.7.1 Intrinsic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51210.7.2 Extrinsic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51410.7.3 Influences of Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51910.7.4 Chemical Weathering Under Physically

Eroding Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52310.7.5 Case Study: Steady State Denudation in a

Tropical Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52410.7.6 Influence of Erosion, Topography and Tectonics . . . . . . . . 52610.7.7 Role of Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

11 Geochemical Kinetics and Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 545Carl I. Steefel11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54511.2 Transport Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

11.2.1 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54811.2.2 Molecular Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54911.2.3 Hydrodynamic Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 558

11.3 Advection-Dispersion-Reaction Equation . . . . . . . . . . . . . . . . . . . . . 56311.3.1 Non-Dimensional Form of the Advection-Dispersion-

Reaction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56411.3.2 Equilibration Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . 56811.3.3 Reaction Fronts in Natural Systems . . . . . . . . . . . . . . . . . . 56911.3.4 Transport versus Surface Reaction Control . . . . . . . . . . . . 571

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11.3.5 Propagation of Reaction Fronts . . . . . . . . . . . . . . . . . . . . . . 57211.4 Rates of Water–Rock Interaction in Heterogeneous Systems . . . . . . 573

11.4.1 Residence Time Distributions . . . . . . . . . . . . . . . . . . . . . . . 57411.4.2 Upscaling Reaction Rates in Heterogeneous Media . . . . . 576

11.5 Determining Rates of Water–Rock InteractionAffected by Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57811.5.1 Rates from Aqueous Concentration Profiles . . . . . . . . . . . . 57811.5.2 Rates from Mineral Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 580

11.6 Feedback between Transport and Kinetics . . . . . . . . . . . . . . . . . . . . . 58111.6.1 Reactive-Infiltration Instability . . . . . . . . . . . . . . . . . . . . . . 58211.6.2 Liesegang Banding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

11.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

12 Isotope Geochemistry as a Tool for Deciphering Kinetics ofWater-Rock Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591Jerome Gaillardet12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59112.2 Isotopes as a Fingerprint of Water-Rock

Interaction Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59212.2.1 Isotopic Doping Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 59312.2.2 Experimental Mineral Dissolution Sequences . . . . . . . . . . 59712.2.3 Natural Weathering Sequences of Granitic Rocks . . . . . . . 60012.2.4 Evolution of Isotopes Along Flowpaths . . . . . . . . . . . . . . . 60212.2.5 Isotopic Tracing of Global Kinetics . . . . . . . . . . . . . . . . . . 610

12.3 The Use of Radioactive Decay to Constrain Timescales ofWater-Rock Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61512.3.1 Crystal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61512.3.2 Uranium and Thorium Series Nuclides . . . . . . . . . . . . . . . . 61712.3.3 Cosmogenic Isotopes and the Determination

of Denudation Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62612.4 Fractionation of Isotopes as a Kinetic Process . . . . . . . . . . . . . . . . . . 629

12.4.1 Equilibrium and Kinetic Fractionationof Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

12.4.2 Kinetics of Isotopic Exchange . . . . . . . . . . . . . . . . . . . . . . . 63112.4.3 Rate-Dependent Isotopic Effects . . . . . . . . . . . . . . . . . . . . . 635

12.5 Conclusion and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

13 Kinetics of Global Geochemical Cycles . . . . . . . . . . . . . . . . . . . . . . . . . 655Abraham Lerman and Lingling Wu13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65513.2 Historical Development of Geochemical Cycles . . . . . . . . . . . . . . . . 65613.3 The Rock Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65913.4 Essentials of Cycle Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

13.4.1 Calcium Carbonate and Silicate Cycle . . . . . . . . . . . . . . . . 661

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13.4.2 A Simple Cycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66213.4.3 Residence and Mixing Times . . . . . . . . . . . . . . . . . . . . . . . . 66513.4.4 Connections to Geochemical Cycles . . . . . . . . . . . . . . . . . . 667

13.5 Global Phosphorus Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66813.5.1 Phosphorus Cycle Structure . . . . . . . . . . . . . . . . . . . . . . . . . 66813.5.2 Dynamics of Mineral and Organic P Weathering . . . . . . . . 66913.5.3 Experimental and Observational Evidence . . . . . . . . . . . . . 672

13.6 Water Cycle and Physical Denudation . . . . . . . . . . . . . . . . . . . . . . . . 67313.6.1 Geographic Variation of Transport from Land to

the Oceans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67313.6.2 Land and Soil Erosion Rates . . . . . . . . . . . . . . . . . . . . . . . . 67613.6.3 Physical Denudation Rate and Residence Time . . . . . . . . . 678

13.7 Chemical Denudation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67913.7.1 Sedimentary and Crystalline Lithosphere . . . . . . . . . . . . . . 67913.7.2 Mineral Dissolution Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 68313.7.3 Chemical Denudation of Sediments . . . . . . . . . . . . . . . . . . 68413.7.4 Chemical Denudation of Continental Crust . . . . . . . . . . . . 69313.7.5 Weathering Layer Thickness . . . . . . . . . . . . . . . . . . . . . . . . 694

13.8 Mineral-CO2 Reactions in Weathering . . . . . . . . . . . . . . . . . . . . . . . . 69613.8.1 CO2 Reactions with Carbonates and Silicates . . . . . . . . . . 69613.8.2 CO2 Consumption and HCO3

− Production . . . . . . . . . . . . 69813.8.3 CO2 Consumption from Mineral-Precipitation Model . . . 70113.8.4 Mineral Dissolution Model . . . . . . . . . . . . . . . . . . . . . . . . . 706

13.9 Environmental Acid Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71113.10 CO2 in the Global Carbon Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713

13.10.1 Cycle Structure and Imbalances . . . . . . . . . . . . . . . . . . . . . . 71313.10.2 Changes in CO2 Uptake in Weathering . . . . . . . . . . . . . . . . 71413.10.3 CO2 Weathering Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . 71813.10.4 Further Ties between Carbonate and Sulfate . . . . . . . . . . . 719

13.11 Summary and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

Appendix: Compilation of Mineral Dissolution Rates . . . . . . . . . . . . . . . . . 737Joel 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738Andesine/Labradorite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742Anorthite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747Apatite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750Basalt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754Biotite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760Bytownite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762Hornblende . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764Kaolinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

xx Contents

K-feldspar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782Oligoclase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787Olivine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790Pyroxene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811Quartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825

List of Contributors

Rolf S. ArvidsonRice University, Department of Earth Science MS-126, 6100 South Main Street,Houston TX 77005, e-mail: Rolf.S.Arvidson@rice.edu

Joel Z. BandstraThe Pennsylvania State University, Center for Environmental Kinetics Analysis,2217 Earth-Engineering Science Building, University Park, PA 16802,e-mail: jxb88@psu.edu

Liane G. BenningUniversity of Leeds, Earth and Biosphere Institute, School of Earth andEnvironment, Leeds LS2 9JT, UK, e-mail: liane@earth.leeds.ac.uk

Susan L. BrantleyThe Pennsylvania State University, Earth and Environmental Systems Institute,2217 Earth-Engineering Science Building, University Park, PA 16802,e-mail: brantley@essc.psu.edu

Mark L. BrusseauUniversity of Arizona, Department of Soil, Water and Environmental Science,Shantz 429, Building #38, Tucson, AZ 85721, e-mail: Brusseau@ag.arizona.edu

Heather L. BussThe Pennsylvania State University, Department of Geosciences and the Earth andEnvironmental Systems Institute, 2217 Earth-Engineering Science Building,University Park, PA 16802, e-mail: hlbuss@usgs.gov

Richard Kramer CampenThe Pennsylvania State University, Department of Geosciences and the Center forEnvironmental Kinetics Analysis, 2217 Earth-Engineering Science Building,University Park, PA 16802, e-mail: k.campen@amolf.nl

xxi

xxii List of Contributors

Jon ChoroverUniversity of Arizona, Department of Soil, Water and Environmental Science,Shantz 429, Building #38, Tucson, AZ 85721, e-mail: Chorover@cals.arizona.edu

Christine F. ConradThe Pennsylvania State University, Center for Environmental Kinetics Analysis,2217 Earth-Engineering Science Building, University Park, PA 16802,e-mail: cconrad@wegnet.com

Jerome GaillardetInstitut de Physique du Globe de Paris, Universite Paris 7 – CNRS, 4 Place Jussieu,75252 PARIS cedex 05, France, e-mail: gaillardet@ipgp.jussieu.fr

Elisabeth M. HausrathThe Pennsylvania State University, Department of Geosciences and the Earth andEnvironmental Systems Institute, 2217 Earth-Engineering Science Building,University Park, PA 16802, e-mail: Elisabeth.M.Hausrath@nasa.gov

Je-Hun JangThe Pennsylvania State University, Department of Civil and EnvironmentalEngineering, 212 Sackett Building, University Park, PA 16802, e-mail:jehun@psu.edu

James D. KubickiThe Pennsylvania State University, Department of Geosciences and the Earth andEnvironmental Systems Institute, 335 Deike Building, University Park, PA 16802,e-mail: kubicki@geosc.psu.edu

Abraham LermanNorthwestern University, Department of Earth and Planetary Sciences, Locy Hall,1850 Campus Drive, Evanston, IL 60208, e-mail: alerman@northwestern.edu

Laura J. LiermannThe Pennsylvania State University, Department of Geosciences, 503 DeikeBuilding, University Park, PA 16802, e-mail: ljl8@psu.edu

Andreas LuttgeRice University, Department of Earth Science, Department of Chemistry, andCenter for Biological and Environmental Nanotechnology, 6100 Main Street,Houston, TX 77005, e-mail: aluttge@rice.edu

Joel MooreThe Pennsylvania State University, Department of Geosciences and the Center forEnvironmental Kinetics Analysis, 2217 Earth-Engineering Science Building,University Park, PA 16802, e-mail: joelmoore@psu.edu

Alexis K. Navarre-SitchlerThe Pennsylvania State University, Department of Geosciences and the Center forEnvironmental Kinetics Analysis, 2217 Earth-Engineering Science Building,University Park, PA 16802, e-mail: anavarre@geosc.psu.edu

List of Contributors xxiii

Eric E. RodenUniversity of Wisconsin, Department of Geology and Geophysics, 1215 W. DaytonStreet, Madison, WI 53706, e-mail: eroden@geology.wisc.edu

Carl I. SteefelLawrence Berkeley National Laboratory, Earth Sciences Division, 1 CyclotronRoad, Mail Stop 90-1116, Berkeley CA 94720, USA, e-mail: CISteefel@lbl.gov

Glenn A. WaychunasLawrence Berkeley National Laboratory, Earth Sciences Division, MS 70-108B,One Cyclotron Road, Berkeley CA 94720, e-mail: GAWaychunas@lbl.gov

Art F. WhiteU.S. Geological Survey, MS 420, 345 Middlefield Rd, Menlo Park, CA 94025,e-mail: afwhite@usgs.gov

Lingling WuNorthwestern University, Department of Earth and Planetary Sciences, 1850Campus Drive, Evanston, Il 60208, e-mail: lingling@earth.northwestern.edu

Chapter 1Analysis 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 mantleof the Earth have been exposed at the surface and have reacted to move towards anew equilibrium with the atmosphere and hydrosphere. At the same time that min-erals, liquids, and gases react abiotically and progress toward chemical equilibriumat the Earth’s surface, biological processes harvest solar energy and use it to storeelectrons 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 theEarth’s pedosphere, hydrosphere, and atmosphere. To safeguard these mineral andfluid reservoirs so that they may continue to nurture ecosystems, we must understandthe rates of chemical reactions as driven by tectonic, climatic, and anthropogenicforcings.

Chemists approach the understanding of the natural world by defining partsof the world as systems of study. The mechanically separable parts of thesystem—the crystalline and amorphous solids, liquids and gases—are known asphases. All the phases that are not inside the system are defined as the environmentsurrounding the system. By definition, a system at equilibrium will be characterizedby phases with uniform composition that exist at uniform temperature and pressure.To be precise, equilibrium is defined as that state where the chemical potential ofevery component in every phase is equal throughout the system.

To chemically understand a system, the chemical species within the system mustbe identified and characterized: the minimum number of species needed to define asystem at equilibrium comprises the set of components of that system. Likewise, thethermodynamic state of any system is completely defined by specifying the values

1 The Pennsylvania State University, Center for Environmental Kinetics Analysis, Earth andEnvironmental Systems Institute, brantley@essc.psu.edu2 The Pennsylvania State University, Center for Environmental Kinetics Analysis, Earth andEnvironmental Systems Institute, cconrad@wegnet.com

1

2 Susan L. Brantley and Christine F. Conrad

of a critical number of properties. For example, the Gibbs phase rule states that thenumber 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 H2O 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 waterwith 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 equilibriumstate 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, thermodynamicscannot define the rate at which the system evolves.

The field of irreversible thermodynamics treats systems that are removed fromequilibrium by modeling how the entropy of the system changes with time as equi-librium is approached (Prigogine, 1967). Irreversible thermodynamics defines thechange 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. Thesecond law of thermodynamics states diS ≥ 0. For a reversible process, dSi = 0, andfor an irreversible process, this term is always positive, dSi > 0. Furthermore, for aclosed system at constant temperature and pressure, it can be shown that this termis 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, theentropy produced inside the system is related to the Gibbs free energy change of thesystem.

In the case of the system with one reaction, the differentiation of Eq. (1.2) overtime and introduction of ξ , the extent of reaction (see Eq. (1.9)), results in the ex-pression

TdSi

dt= A

dξdt

, (1.3)

where the entropy production diSdt ≥ 0 and A = −∆Greaction, the chemical affinity of

the reaction. The chemical affinity, introduced by T. DeDonder, is the driving forceof the reaction. For a reaction that occurs spontaneously as written (e.g., reactantson the left and products on the right of the reaction), A > 0 and ∆Greaction < 0. Atequilibrium, the chemical affinity (−∆Greaction) is equal to 0. The negative drivingforce of reaction can be shown to be equal to a simple expression for any reactiondefined by an equilibrium constant Keq and a reaction activity quotient, Q:

∆Greaction = −A = RT ln

(Q

Keq

)(1.4)

1 Analysis of Rates of Geochemical Reactions 3

If the driving force of reaction is positive, the reaction should proceed spontaneouslyas 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 andH2O can each be set equal to unity:

NaAlSi3O8(s) +4H2O(aq) +4H+(aq) → Na+

(aq) +Al3+(aq) +3H4SiO4(aq)

(1.5)Q =

aNa+aAl3+a3H4SiO0

4

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 andthe 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 ofcrystalline 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 rateof 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 thatoccurs as written must be equal to the ratio of the rate constants for the forward andreverse 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 concentrationsare measured. Consider the reaction,

A+3B → Z (1.6)

which begins with a mixture of A and B without Z. We assume a system whereno change in volume occurs as the reaction proceeds. At any time, t, the rate ofconsumption of A, rA, is defined as the negative slope of the tangent to the plot ofconcentration of A, [A], versus time:

rA = −d[A]dt

(1.7)

The rate of formation of Z can be determined in the same manner:

rZ =d[Z]dt

(1.8)

4 Susan L. Brantley and Christine F. Conrad

For the reaction given in Eq. (1.6), rA = rZ . Note however, that the stoichiometry ofthe reaction requires that the negative of the rate of consumption of B must differby a factor of 3 from the value of rZ . Thus, if we define rates according to equationssuch as Eqs. (1.7) and (1.8), a given reaction may be characterized by differentvalues for the rate at any given time.

A useful concept, the extent of reaction, was therefore introduced by T. de Donderin 1922 to correct for the stoichiometry of reaction. The extent of reaction is definedby the following,

ξ =ni −n0

i

νi, (1.9)

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 isspecified.

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 dividedby the volume

r =1V

dξdt

(1.10)

For an individual species, i, the time derivative is given by

ξ =1νi

dni

dt(1.11)

where νi is the stoichiometric coefficient for species i. The rate of reaction thenbecomes

r =1

νiVdni

dt(1.12)

Thus, for the reaction given in Eq. (1.6)

r = − 1V

dnA

dt= − 1

3VdnB

dt=

1V

dnZ

dt(1.13)

If the volume does not change during the course of the reaction, the term dni/V inEq. (1.12) may be replaced by the change in concentration yielding

r = − 1vA

d[A]dt

= − 1vB

d[B]dt

=1vZ

d[Z]dt

. (1.14)

1 Analysis of Rates of Geochemical Reactions 5

Time (hrs)

0 200 400 600 800 10000

5

10

15

20

25

30

nepheline glassjadeite glassalbite glass

2

8

Fig. 1.1 Normalized moles of glass cm−2 released into solution as a function of time during dis-solution of nepheline (Na6Al6Si6O24), jadeite (Na4Al4Si8O24), and albite (Na3Al3Si9O24) glasspowder in batch experiments at pH 2 (Hamilton et al., 2001). QnormSi = number of moles of Sireleased into solution per cm2 glass divided by the number of moles of Si in a mole of glass (basedon 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 inrelation to the equation for the given reaction. Therefore, the reaction stoichiometrymust 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 iscommon 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 rateof release of Si to solution. If albite dissolves stoichiometrically (Eqn. 1.5) then therelease rate of Na (rNa) or Al (rAl) should be 1/3 the release rate of Si (rSi). Evenwhen the Si release is used to calculate the rate, the rate is often reported in units asmolmineral m−2 s−1. Note that the reported release rate of albite written in units ofmol albite per unit area per unit time will differ depending upon whether the formulaunit of albite is written as Na3Al3Si9O24(ralbite(24)) or as NaAlSi3O8(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 rateof dissolution of three Na-Al-Si glasses as a function of time as determined byrelease of Si to solution. In this figure, the moles of Si released are normalized bythe number of Si atoms per formula unit of glass (24 oxygen atoms per unit). Theglass with the lowest Al/Si ratio (albite) dissolves the slowest. However, Fig. 1.2demonstrates that Na and Al are preferentially released during initial albite glass

6 Susan L. Brantley and Christine F. Conrad

Time (hrs)

100 200 300 400 500 600 700 800 900

Nor

mal

ized

mol

es g

lass

/cm

2 (Q

norm

) (x

10−8

)

0

2

4

6

8

10

12

14

16

18

SiAlNa

Fig. 1.2 Normalized moles of glass cm−2 released into solution as a function of time during dis-solution of albite glass (Na3Al3Si9O24) 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 ofthese release rates indicate non-stoichiometric dissolution in which a Si-rich layer is forming onthe albite glass surface.

dissolution (rSi �= 3rNa �= 3rAl) leaving behind a silica-rich leached layer (Hamiltonet 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 levelwhere the rates of reactions are measured as a function of solution, solid, and gascomposition. 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 reactantversus time to the concentrations of species in a chemical reaction. For reactionsamong solutes, solids, and gas phases, concentrations may be denoted in a varietyof 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 betweenor 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 ofreaction. 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, thekinetics 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. Ingeneral, 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 phenomenologicallyis the oxidation or pyrite and production of sulfuric acid. This reaction is largelyresponsible for decreased pH values in sulfide mine spoils. The rate of change inthe concentration of FeS2(s) due to oxidation in pyrite-containing soils has beenobserved to be a function of the concentration of FeS2(s) in the soil, [FeS2(s)], andcan 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 isproportional to the second power of one species or, alternately, proportional to theproduct 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 thepresence of solid phase manganese dioxide, a process used to remove the toxic formof inorganic As(III) from drinking water. It has been determined that this reactionfollows 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 offerrous minerals studied by Perez et al. (2005). Batch reactors were utilized to assessthe ability of naturally occurring ferrous silicate minerals to act as an oxygen bufferin 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 themineral (sites m−2).

Some reactions cannot be defined to have a reaction order. An example of such areaction is an enzyme-catalyzed reaction that is described by the Michaelis-Mentenequation (see Chap. 8). The rate law for a reaction of this type is given by

r =Vmax[A]Km +[A]

(1.20)

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 ahyperbolic rate equation. Another rate law that is hyperbolic in form is derived ifone assumes that the rate of a reaction is proportional to the surface site density ofsorbed 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 arate that can be modeled by a hyperbolic equation such as Eq. (1.20), no true rateorder 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 therate constant depend on the order of the reaction. A first-order reaction, such asEq. (1.17) where the rate is described in units of mol L−1 s−1 is described by arate 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 bedetermined from

mol L−1s−1

(mol L−1)2 = mol−1 L s−1 (1.21)

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 canbe broken down into a series of reactions—elementary reactions—that occur ex-actly as written. Elementary reactions are the building blocks of a complex reactionand cannot be broken down further into simpler reactions. In addition, elementaryreactions have the desirable property that they exhibit the same rate regardless ofthe system: they can thus be extrapolated from one system to another at a giventemperature and pressure.

The rate equation for an elementary reaction can be written for the reaction apriori because the reaction occurs exactly as written. Thus, the rate equations forthe 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 threespecies to collide in the geometry conducive for reaction, and thus reaction ordersfor elementary reactions are seldom larger than 2, or at most, 3. The reaction orderof 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 individualchemical 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 twomolecules 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 etal. (2005) measured the rate of disappearance of aqueous silica, H4SiO4(aq), fromsolutions supersaturated with respect to amorphous silica and observed that the ratedata was not well fit by first-, second-, or third-order rate laws, but was well fit by afourth-order rate law of the form

r = −d[H4SiO4(aq)]/dt = k4[H4SiO4(aq)]4 (1.22)

The high order of reaction observed in this work suggested that this rate equationdescribed a complex reaction mechanism and not an elementary reaction. The au-thors thus derived a reaction mechanism that is consistent with a fourth-order ratelaw. They proposed the following elementary reactions to describe the mechanismfor the polymerization of monomeric to tetrameric silica in aqueous solutions:

H4SiO4(aq) +H4SiO4(aq) → H6Si2O7(aq) +H2O (1.23)

H6Si2O7(aq) +H4SiO4(aq) → H8Si3O10(aq) +H2O (1.24)

H8Si3O10(aq) +H4SiO4(aq) → H8Si4O12(aq) +2H2O (1.25)

According to their model, the polymerization of monomeric silica (H4SiO4) intotetrameric silica (H8Si4O12) 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 theformation of a cyclic compound. For this series, a composite or overall reaction canbe written as the sum over the entire mechanism:

4H4SiO4(aq) → H8Si4O12(aq) +4H2O (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 couldachieve equilibrium (with equilibrium constants K1 and K2, respectively) while thethird reaction could control the overall rate. Such a slow step that controls the rateis called the rate-controlling (or –determining or –limiting) step. If Eq. (1.25) is therate-controlling step and it is an elementary reaction then the rate constant k4 thatdescribes the rate of reaction given in Eq. (1.26) and in Eq. (1.22) can be expressedas follows:

k4 = k3 f (y)K1K2

a2H2O

(1.27)

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 forrate constants for composite reactions to contain activity coefficients, equilibriumconstants 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 isfound 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 toassume 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 ahypothetical 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 thekinetic 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 impossibleto 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, whilereactions that occur at interfaces between phases are heterogeneous reactions. Mod-eling heterogeneous reactions such as water-rock reactions tends to be difficult inthat a term must be included in the rate equation that describes the reacting speciesat the mineral interface (Chaps. 3 and 4). Often it is assumed that reactions occurat 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, andthe specific surface area (m2 g−1) is determined from this sorbed gas using theBrunauer-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 biotiteand other sheet silicates, edge sites dissolve faster than sites on the basal surfaceand often control the overall dissolution rate under acid conditions (Kalinowski andSchweda, 1996). In dissolution or precipitation reactions, a further complexity arisesbecause the mineral surface area changes with time. To date, very few attempts havebeen 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 consumedduring the course of the reaction. Therefore, they do not modify the standard Gibbsfree energy change of the reaction nor do they appear in the equilibrium constantexpression for the reaction. For example, a reaction containing a catalyst may bewritten as follows (after Laidler, 1987),

A+B+ catalyst → Y +Z + catalyst, (1.28)