Environmental process analysis principles and modeling

Tai Lieu Chat Luong



Environmental Process
Analysis



Environmental Process
Analysis
Principles and Modeling

Henry V. Mott, Professor Emeritus
Department of Civil and Environmental Engineering
South Dakota School of Mines and Technology
Rapid City, SD, USA


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Library of Congress Cataloging-in-Publication Data
Mott, Henry V., 1951–
Environmental process analysis : principles and modeling / Henry V. Mott, professor emeritus,
Department of Civil and Environmental Engineering, South Dakota School of Mines and Technology,
Rapid City, SD.
  pages cm
  Includes bibliographical references and index.
  ISBN 978-1-118-11501-5 (cloth)
1.  Environmental chemistry.  2.  Chemical processes.  I.  Title.
  TD193.M735 2013
 577′.14–dc23
2013016208
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1


To my deceased grandparents, Ida and Floyd Slingsby, and Ragna and Henry Mott;
to my deceased parents, Marge Marie and Henry Valentine, who raised me;

to my sisters, Jean, Judy, and Jane, with whom I shared childhood;
to my children, Harrison, Graeme, and Sarah, with whom I now share adulthood;
to my daughter-in-law, Lana, and my granddaughter, Samantha;
to Marty, my sweet bride, with whom I share a wonderful life.



Contents

Prefacexiii
Acknowledgmentsxvii
1. Introductory Remarks

1

1.1  Perspective / 1
1.2  Organization and Objectives  /  2
1.2.1 Water / 2
1.2.2 Concentration Units / 3
1.2.3 Chemical Equilibria and the Law of Mass Action  /  3
1.2.4 Henry’s Law / 4
1.2.5 Acids and Bases  /  4
1.2.6 Mixing / 5
1.2.7 Reactions in Ideal Reactors  /  5
1.2.8 Nonideal Reactors / 6
1.2.9 Acids and Bases: Advanced Principles  /  6
1.2.10  Metal Complexation and Solubility  /  7
1.2.11  Oxidation and Reduction  /  8
1.3  Approach / 8
2.Water

11

2.1  Perspective / 11
2.2  Important Properties of Water  /  12
vii


viii

Contents

3. Concentration Units for Gases, Liquids, and Solids

16

3.1  Selected Concentration Units  /  16
3.2 The Ideal Gas Law and Gas Phase
Concentration Units / 20
3.3  Aqueous Concentration Units  /  23
3.4  Applications of Volume Fraction Units  /  28
4. The Law of Mass Action and Chemical Equilibria
4.1 
4.2 
4.3 
4.4 
4.5 
4.6 
4.7 

36

Perspective / 36
The Law of Mass Action  /  37
Gas/Water Distributions / 38
Acid/Base Systems / 39
Metal Complexation Systems  /  40
Water/Solid Systems (Solubility/Dissolution)  /  41
Oxidation/Reduction Half Reactions  /  43

5. Air / Water Distribution: Henry’s Law

44

5.1  Perspective / 44
5.2  Henry’s Law Constants  /  46
5.3  Applications of Henry’s Law  /  51
6. Acid/Base Component Distributions
6.1  Perspective / 64
6.2  Proton Abundance in Aqueous Solutions: pH and
the Ion Product of Water  /  65
6.3  Acid Dissociation Constants  /  69
6.4  Mole Accounting Relations / 70
6.5  Combination of Mole Balance and Acid/Base Equilibria  /  74
6.5.1  Monoprotic Acids / 74
6.5.2  Diprotic Acids / 76
6.5.3  Triprotic and Tetraprotic Acids / 80
6.5.4  Abundance (Ionization) Fractions  /  82
6.6  Alkalinity, Acidity, and the Carbonate System  /  82
6.6.1  The Alkalinity Test: Carbonate System Abundance and

Speciation / 82
6.6.2  Acidity / 90
6.7  Applications of Acid/Base Principles in Selected
Environmental Contexts / 91
6.7.1  Monoprotic Acids / 91
6.7.2  Multiprotic Acids / 101

64


Contents

7. Mass Balance, Ideal Reactors, and Mixing

ix

119

7.1  Perspective / 119
7.2  The Mass Balance  /  120
7.3  Residence Time Distribution (RTD) Analyses  /  121
7.3.1  RTD Experimental Apparatus / 121
7.3.2  Tracers / 121
7.3.3  Tracer Input Stimuli  /  122
7.4  Exit Responses for Ideal Reactors  /  125
7.4.1  The Ideal Plug-Flow Reactor (PFR)  /  125
7.4.2  The Ideal Completely Mixed Flow Reactor (CMFR)  /  128
7.4.3  The Ideal (Completely Mixed) Batch Reactor (CMBR)  /  130
7.5  Modeling of Mixing in Ideal CMFRs  /  130
7.5.1  Zero-Volume Applications / 130

7.5.2  Time-Dependent Mixing / 137
7.6  Applications of CMFR Mixing Principles in Environmental Systems  /  144
8. Reactions in Ideal Reactors

157

8.1  Perspective / 157
8.2  Chemical Stoichiometry and Mass/Volume Relations  /  158
8.2.1  Stoichiometry and Overall Reaction Rates  /  159
8.2.2  Some Useful Mass, Volume, and Density Relations  /  160
8.2.3 Applications of Stoichiometry and Bulk
Density Relations / 162
8.3  Reactions in Ideal Reactors  /  171
8.3.1  Reaction Rate Laws  /  171
8.3.2  Reactions in Completely Mixed Batch Reactors  /  174
8.3.3  Reactions in Plug-Flow Reactors  /  176
8.3.4  Reactions in Completely Mixed Flow Reactors  /  179
8.3.5  Unsteady-State Applications of Reactions in Ideal Reactors  /  181
8.4  Applications of Reactions in Ideal Reactors  /  183
8.4.1  Batch Reactor Systems  /  184
8.4.2  Plug-Flow Reactor Systems  /  190
8.4.3  Completely Mixed Flow Reactor Systems  /  198
8.4.4  Some Context-Specific Advanced Applications  /  206
8.5  Interfacial Mass Transfer in Ideal Reactors  /  216
8.5.1  Convective and Diffusive Flux  /  217
8.5.2  Mass Transfer Coefficients / 218
8.5.3 Some Special Applications of Mass Transfer in Ideal
Reactors / 222

x

Contents

9. Reactions in Nonideal Reactors
9.1
9.2

Perspective / 265
Exit Concentration Versus Time Traces  /  266
9.2.1  Impulse Stimulus / 266
9.2.2  Positive Step Stimulus  /  267
9.3 Residence Time Distribution Density  /  267
9.3.1 E(t) Curve and Quantitation of Tracer Mass  /  268
9.3.2 E(t) and E(q) RTD Density Curves  /  269
9.4 Cumulative Residence Time Distributions  /  271
9.5 Characterization of RTD Distributions  /  272
9.5.1  Mean and Variance from RTD Density  /  272
9.5.2  Mean and Variance from Cumulative RTD  /  274
9.6 Models for Addressing Longitudinal Dispersion in Reactors  /  275
9.6.1  CMFRs (Tanks) in Series (TiS) Model  /  275
9.6.2  Plug-Flow with Dispersion (PFD) Model  /  277
9.6.3  Segregated Flow (SF) Model  /  279
9.7 Modeling Reactions in CMFRs in Series (TiS) Reactors  /  280
9.7.1 Pseudo-First-Order Reaction Rate Law in TiS
Reactors / 280
9.7.2  Saturation Reaction Rate Law with the TiS Model  /  281
9.8 Modeling Reactions with the Plug-Flow with
Dispersion Model / 282
9.8.1 Pseudo-First-Order Reaction Rate Law with

the PFD Model  /  282
9.8.2  Saturation Rate Law with the PFD Model  /  287
9.9 Modeling Reactions Using the Segregated
Flow (SF) Model  /  289
9.10  Applications of Nonideal Reactor Models  /  291
9.10.1  Translation of RTD Data for Use with
Nonideal Models / 291
9.10.2  Modeling Pseudo-First-Order Reactions  /  297
9.10.3  Modeling Saturation-Type Reactions with the
TiS and SF Models  /  302
9.11 Considerations for Analyses of Spatially
Variant Processes / 305
9.11.1  Internal Concentration Profiles in Real Reactors  /  305
9.11.2  Oxygen Consumption in PFR-Like Reactors  /  312
9.12 Modeling Utilization and Growth in PFR-Like Reactors Using
TiS and SF  /  318

265


Contents

10.  Acid-Base Advanced Principles

xi

335

10.1  Perspective / 335
10.2  Activity Coefficient / 336

10.2.1  Computing Activity Coefficients / 337
10.2.2  Activity Coefficient and Law of Mass Action  /  340
10.3  Temperature Dependence of Equilibrium Constants  /  344
10.3.1  Standard State Gibbs Energy of Reaction  /  344
10.3.2  Temperature Corrections for Equilibrium Constants  /  347
10.4  Nonideal Conjugate Acid/Conjugate Base Distributions  /  350
10.5  The Proton Balance (Proton Condition)  /  358
10.5.1  The Reference Conditions and Species  /  358
10.5.2  The Proton Balance Equation  /  359
10.5.3  The Reference and Initial Conditions for the Proton
Balance / 363
10.6 Analyses of Solutions Prepared by Addition of Acids,
Bases, and Salts to Water  /  365
10.6.1  Additions to Freshly Distilled Water (FDW)  /  365
10.6.2  Dissolution of a Weak Acid in Water  /  371
10.6.3  Dissolution of a Basic Salt in Water  /  374
10.6.4  A Few Words about the Charge Balance  /  379
10.7  Analysis of Mixed Aqueous Solutions  /  380
10.7.1  Mixing Computations with Major Ions  /  381
10.7.2  Final Solution Composition for Mixing of Two or More
Solutions / 382
10.8  Acid and Base Neutralizing Capacity  /  396
10.8.1  ANC and BNC of Closed Systems  /  396
10.8.2  ANC and BNC of Open Systems  /  403
10.8.3  ANC and BNC of Semi-Open Systems  /  408
10.9  Activity Versus Concentration for Nonelectrolytes  /  417
10.9.1  The Setschenow Equation  /  417
10.9.2  Definitions of Salt Abundance  /  419
10.9.3  Activity of Water in Salt Solutions  /  422
11.  Metal Complexation and Solubility

11.1  Perspective / 439
11.2  Hydration of Metal Ions  /  440
11.3  Cumulative Formation Constants  /  441
11.3.1  Deprotonation of Metal/Water Complexes  /  441
11.3.2  Metal Ion Hydrolysis (Formation) Reactions  /  442

439


xii

Contents

11.4 
11.5 

11.6 

11.7 

11.8 

11.3.3  Cumulative Hydrolysis (Formation) Reactions  /  443
11.3.4  The Cumulative Formation Constant for
Metal/Ligand Complexes / 446
Formation Equilibria for Solids  /  447
Speciation of Metals in Aqueous Solutions Containing Ligands  /  448
11.5.1  Metal Hydroxide Systems  /  448
11.5.2  Metals with Multiple Ligands  /  451
Metal Hydroxide Solubility  /  456

11.6.1  Solubility in Dilute Solution  /  456
11.6.2  Solubility in the Presence of Ligands other than
Hydroxide / 463
Solubility of Metal Carbonates  /  467
11.7.1  Calcium Carbonate Solubility  /  468
11.7.2  Solubility of Metal Carbonates—the Controlling Solid
Phase / 476
11.7.3  Solubility of Phosphates  /  498
Solubility of Other Metal–Ligand Solids  /  511

12.  Oxidation and Reduction

519

12.1  Perspective / 519
12.2  Redox Half Reactions  /  520
12.2.1  Assigning Oxidation States  /  521
12.2.2  Writing Half Reactions  /  523
12.2.3  Adding Half Reactions  /  526
12.2.4  Equilibrium Constants for Redox Half Reactions  /  530
12.3  The Nernst Equation  /  533
12.4  Electron Availability in Environmental Systems  /  535
12.4.1 pE–pH (EH–pH) Predominance Diagrams  /  537
12.4.2  Effect of pE on Redox Couple Speciation  /  545
12.4.3  Determining System pE / 550
12.4.4  Speciation Using Electron Availability  /  560
Appendices571
References599
Index602

Preface

This book is about mathematical and numerical modeling of processes in contexts
associated with both natural and engineered environmental systems. In its assembly,
I have relied on some very traditional but highly ubiquitous principles from natural
and engineering science—chemical equilibria, reaction kinetics, ideal (and nonideal)
reactor theory, and mass accounting. As necessary to the contexts of interest, I have
incorporated principles from fluid dynamics, soil science, mass transfer, and microbial processes.
Many texts addressing introductory environmental engineering include discussions
of these principles, but in opting to semiquantitatively address specific environmental
contexts, never really apply them. Introductory modeling efforts seldom tread quantitatively beyond situations that are solved by single, explicit relations. This approach is
fully appropriate at the entry level. Broad-based knowledge gained from an introductory course and text is essential to full appreciation of the portability of principles to
myriad environmental systems. This text is not intended to replace an introductory
environmental engineering textbook but to build on the contextual knowledge gained
through completion of an introductory environmental engineering course.
In Chapter 2, some properties of water important to the understanding and
employment of chemical equilibria are discussed. In Chapter 3, a collection of the
various units describing abundance of components in gas, liquid, and solid systems
is assembled. In Chapter 4, several specific conventions of the law of mass action,
applicable to specific chemical “systems” are detailed. Then in Chapters 5 and 6,
modeling of systems employing Henry’s law and acid/base principles is examined. In
Chapters 7 and 8, modeling of mixing and reactions in ideal reactors is addressed.
These first eight chapters constitute the “basic” portion of this text. These topics and
associated modeling work are appropriate for a third- or fourth-year undergraduate
xiii


xiv

Preface

course, beyond the introductory level. I employ MathCAD as a powerful computational tool to illustrate, in the environmental contexts considered, the power of
­modeling in process analysis. In Chapter 9, I have extended the applications of three
nonideal reactor models: completely-mixed flow reactors in series; plug-flow with
dispersion; and segregated flow, beyond the level of treatment found in current texts.
While containing good “food for thought” at the fourth-year undergraduate level,
Chapter 9 is most appropriate for the graduate level.
Traditional water or aquatic chemistry texts introduce and discuss the chemical
equilibria of acids/bases, metal complexes, solubility/dissolution, and oxidation/
reduction. Mention is made of the proton balance, but this powerful tool is most often
discarded or treated cursorily in favor of the seemingly much simpler charge balance.
In fact, for systems that are not infinitely dilute (virtually all real systems) the charge
balance most often fails at the outset. I have extended the application of the proton
balance (or condition) to provide for significant advances in understandings of the
acid- and base-neutralizing capacity of aqueous solutions and both solution–vapor
and solution–solid systems. I have also demonstrated the relative ease with which
nondilute solution principles can be incorporated into chemical equilibrium
modeling.
For modeling of systems, traditional texts most often rely heavily upon simplifying assumptions, leading to graphical or approximate solutions, or upon sophisticated chemical equilibrium modeling software for quantitative description of
chemical equilibria. Some recent texts have begun to chip away at the computational
wall separating pencil/paper/graphical solutions from those involving sophisticated
software but have not made significant headway. No other existing text known to me
addresses, in transparent detail, the process of coupling mathematics with chemical
equilibria and both mass and proton accounting for numerical modeling of chemical
equilibrium systems.
Herein, I employ the general mathematical/numerical worksheet software
MathCAD to occupy the region beyond approximate solutions and encroaching upon
that of sophisticated software. A huge assembly of mathematical capability is available in a “what you see is what you get” user interface. Key to modeling of chemical
equilibrium systems is ready capability to write user-defined functions, to program

the solution of systems of nonlinear equations, and to create structured-code-like
programs, all entirely visible in printable, portable worksheets. In fact, the vast
majority of work illustrated in examples of this text has been conveniently exported
into the manuscript as captures directly from worksheets. I make few, if any, simplifying assumptions beyond those associated with the first principles used in
the mathematical modeling. The modeling efforts described herein, associated with
the traditional water chemistry principles, are numerically as capable as those of the
sophisticated software but much more flexible. These created models can be used not
only to numerically model the equilibria but also to employ the equilibrium modeling
to assess the consequences of perturbing the systems. Coupled with Chapters 2–6,
Chapters 10–12 constitute the “advanced” portion of this text addressing chemical
equilibrium modeling.


Preface

xv

Those who will benefit from reading and studying this text are those who wish to
mathematically and numerically model environmental processes and systems and
who wish to fully understand the connections among the various factors leading to
the results. Practitioners, depending upon their level of fundamental understandings,
would benefit in a manner similar to students. No specific numerical methods skills
are necessary, beyond attention to detail and an understanding that for numerical
solution methods to work, they must be started in some vicinity of the final solution,
assigning initial guesses to all unknowns sought. Although not absolutely necessary,
it is certainly recommended that the reader obtain the MathCAD software and
­carefully follow through the worked examples. Such an approach promotes both
understandings of the principles and mathematical modeling as well as capability for
implementation of numeric solutions.
Henry V. Mott

Additional MathCAD files that accompany this text are available at booksupport.
wiley.com by entering ISBN 9781118115015.
Additionally adopters of the text can obtain the solutions manual to the text by
going to the books landing page at www.wiley.com and requesting the solutions
manual.



Acknowledgments

I offer my special thanks to four former students, Zane Green, Nathan Kutil, Ulrike
Lashley, and Teryl Stacey, who painstakingly reviewed the manuscript of this text,
freely offering their time and abilities to make this effort as useful as possible for the
students to come. I also offer my thanks to the many graduate and undergraduate
­students who sat in my classrooms, and with great enthusiasm engaged in the discussions and related efforts necessary to the development of the understandings manifest
in the many example problems included in this text. I also offer my heartfelt thanks
to my friend and colleague, Melvin Klasi, who, through my many years as a member
of the Faculty of the SD School of Mines, was always willing to assist me in my
understandings of mathematics and its implementation in modeling efforts.
I also must acknowledge some of my many teachers and mentors. Sam Ruzick
and John Willard helped me unlock my love of chemistry, although it was to remain
dormant for many of the years I studied to be and called myself a civil engineer. Hank
Trangsrud taught me to ask tough questions and then to answer them. Al Wallace
was, well, Al Wallace. My good friend Tom Nielsen and I learned much as we tackled
the tough problems and topics with which Al charged us. Don Johnstone and Harry
Gibbons were instrumental in the development of my understanding of microbes and
aquatic insects as living, breathing beings. David Yonge, Erv Hinden, and Ken Hartz
helped propel me onward by suggesting, at my MS thesis proposal presentation, that
I extend it to a PhD dissertation, although I left Washington State to pursue my PhD.
Walt Weber presented me with a challenging and relevant PhD thesis project and

solid mentoring and support for its completion. Then, Walt, Don Gray, Linda Abriola,
and Rane Curl helped me ensure that my work was top notch. I learned much from
my common struggles alongside and interactions with my peer PhD students: Yo
Chin, Lynn Katz, Domenic Grasso, Kevin Ohlmstead, Chip Kilduff, Margaret Carter,
xvii


xviii

Acknowledgments

and Ed Smith. In the classrooms of Bernie Van Wie, Linda Abriola, Rich Kapuscinski,
Jon Bulkley, Rane Curl, Ray Canale, Scott Fogler, and Bob Kadlec, I learned to
couple mathematics with physical, chemical, and biological processes. The understandings of the portability of fundamental principles among systems quite naturally
arose as an added bonus. In the classrooms of Brice Carnahan and James Wilkes,
I learned that quantitative answers need not be exact, but certainly as close as reasonably possible.
I am the primary author of this text; I have no coauthors. However, I have chosen to
employ the first person plural, we, in many of the discussions of the text. The knowledge
and understandings employed in those discussions and companion examples arise as a
consequence of the foundational work I did as assisted and guided by my many
teachers and mentors. Their collective pursuit of personal and student betterment
­certainly contributed greatly to the expertise that I now claim as my own. In this text,
when I use the term “we,” it is I and my teachers and mentors to whom I refer.


Chapter

1

Introductory Remarks

1.1  Perspective
From the outset, let us make no mistakes about the purpose and content of this textbook. The main title—Environmental Process Analysis—suggests that we will analyze processes. The targeted processes are those operative homogeneously in aqueous
solutions, involving the gas–water interface, and involving the water–solid interface.
Understandings of the behavior of environmental systems can arise from examination
of both natural or engineered processes under equilibrium or near-equilibrium conditions. The effects of perturbations on systems can be determined using the initial and
predicted final equilibrium conditions. In addition, understandings can arise from
examination of the progress of such processes under transient or near (quasi) steadystate conditions. Then, Environmental Process Analysis is the examination of the
processes operative in conjunction with perturbations of environmental systems,
either natural or engineered, arising mostly from actions of our society. Certain of
these perturbations beget negative consequences associated with actions that, while
well-intentioned, contribute to the detriment of an environmental system. Others are
intended to positively affect a compromised natural system or to implement a desired
outcome within the context of an engineered system. The subtitle—Principles and
Modeling—suggests that we will employ appropriate principles, develop models in
support of our analyses, and employ these models to predict the outcomes from
intended or unintended perturbations. Modeling has three distinct levels. Conceptual
modeling involves identifying, understanding, and interrelating processes operative
Environmental Process Analysis: Principles and Modeling, First Edition. Henry V. Mott.
© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

1


2

Introductory Remarks

within targeted systems. Mathematical modeling involves coupling of relevant
mathematical relations with processes identified by conceptual modeling efforts and

assembling those mathematical relations into overall models describing behaviors of
processes within systems. Lastly, numerical modeling involves work with the developed mathematical model to produce quantitative predictions of behavior.
We examine the scientific literature to understand processes and the means by
which they may be mathematically described and consult resources assembled by the
mathematicians to develop sets of or even single equations that might be used to
describe the behavior of the system. It is not until we have collected these relations
and devised means to use them to obtain quantitative answers that we have accomplished the process called modeling. A model can be as simple as a single linear relation or as complex as a set of coupled, higher-order, partial differential equations. The
key is that, in either case, the conceptual, mathematical, and numerical aspects are
employed. Even today, in the minds of many, numerical modeling is associated with
the writing of lines and lines of structured programs that employ numerical methods
in solution of sets of mathematical relations that defy closed-form analytic solution.
We prefer the simpler idea that numerical modeling merely involves the production
of numerical results using appropriate means to describe behaviors of processes in
systems. Fortunately, with the development of the microchip, personal computers,
and general computational software, the numerical part of modeling efforts has
become much more conveniently accomplished. Then, in this text we illustrate and
employ the modeling process to analyze effects of perturbations on both natural and
engineered systems. We also illustrate the portability of key principles and concepts
among the myriad contexts within which environmental engineering operates.
1.2  Organization and Objectives
Our prime objective with this textbook is the education of the student, interested faculty member, or practitioner in the means and methodologies to conceptually, mathematically, and numerically model processes operative in environmental systems.
We begin with very basic processes and simple systems and progress to processes
that are somewhat complex and to systems well beyond the simplistic. We have organized the text into 11 additional chapters beyond this introduction. Chapters 2–6
build upon each other in the general area of equilibrium aqueous chemistry.
Chapters 7–9 are aligned along an alternative thread addressing reactions and reactors. Then Chapters 10–12 return to the aqueous equilibrium chemistry thread to
address more advanced applications of the principles. In the following sections, we
briefly describe the focus of each of the ensuing chapters.
1.2.1  Water
Although vital to environmental systems and perhaps of greatest importance relative
to the future of the Earth and its inhabitants, water is somewhat ancillary to our

analyses herein. We are mostly concerned about constituents within water and are


Organization and Objectives

3

mostly interested in the properties of water that contribute to the behaviors of these
constituents. We have thus included a short chapter addressing the properties of water
that are important in examination of the behaviors of acids and bases, cations and
anions, and specifically hydronium and hydroxide in aqueous solutions. For those
wishing to delve more deeply into the mechanical or other properties of water, we
suggest examination of the many texts written addressing fluid properties and
physical chemistry of water.
1.2.2  Concentration Units
Each scientific and engineering discipline, and subdiscipline in many cases, has its
own means to specify the abundances of constituents in gases, liquids, and solids.
Since environmental engineering must embrace most of the natural sciences (e.g.,
chemistry, physics, biology, geology, limnology, etc.) and many of the engineering
disciplines (e.g., chemical, civil, geological, metallurgical, mining, etc.), we environmental engineers must be conversant with the preferred means to describe specie
abundances by the many disciplines. To that end, we have included Chapter 3, in
which we have assembled a database of concentration units used across these disciplines. Chapter 3 also contains a review of the means to interconvert units from one
set to another using the basic chemical concepts of molecular mass and the ideal gas
equation of state.
1.2.3  Chemical Equilibria and the Law of Mass Action
Over the past three plus centuries, the chemists have assembled a wonderful system
with which to describe chemical processes. Tendencies for processes to proceed,
rates at which they would proceed, and associated ending points (the equilibrium
conditions) are all addressed within this very logical, quantitative system. In examination of perturbations of environmental systems, herein we choose to predict the
final state of a system via close attention to the processes operative within. To that

end, we employ chemical equilibria in combination with mass or molar accounting.
Distinct styles for describing these equilibria arise from special applications of the
law of mass action. Specifically, Henry’s law, acid deprotonation, metal–ligand complex formation, solubility and dissolution, and oxidation/reduction half reactions all
have their characteristic formulations of the law of mass action. These are reviewed
in Chapter 4. For chemical equilibria, the change in standard-state Gibbs energy
associated with a reaction as written is employed to define the equilibrium constant
under standard conditions. The change in standard state enthalpy associated with a
reaction as written is used in adjusting the magnitude of the equilibrium constant
for varying temperature. We leave detailed discussions of these topics to the physical
chemists and choose to employ two important results. Use of standard-state Gibbs
energy changes to determine the magnitude of equilibrium constants is introduced in
Chapter 10 and employed in detail in Chapter 12. Use of standard-state enthalpy
changes to adjust equilibrium constants for alternative temperatures is employed
in Chapter 10.


4

Introductory Remarks

1.2.4  Henry’s Law
Chapter 5 is devoted to developing understandings of the application of Henry’s law
to distributions of nonelectrolyte species between vapor and water. We employ
Henry’s law to predict abundances in the vapor from known abundances in water, and
to predict abundances in water from known abundances in the vapor. We employ
varying discipline-specific concentration units in these analyses. We begin our
integrated modeling efforts by carrying Henry’s law with us into a number of environmental contexts addressing air/water distributions in atmospheric, terrestrial, biogeochemical, and engineered systems. We showcase its portability.
1.2.5  Acids and Bases
In Chapter 6, we introduce the concept of water as an acid and a base and examine
the interactions between water and the hydrogen ion (often simply called a proton)

to form the hydronium ion, and begin the discussion of the hydration of cations in
general, using the hydronium ion as an example. We introduce and solidify the concept that each acid has a conjugate base and that each base has a conjugate acid.
Mono- and multiprotic acids are examined. Unlike many texts which focus on the
carbonate system, the sulfur system, the nitrogen system, and the phosphorus
system, we approach acid deprotonation from the standpoint of the general behavior
of acids, employing a systematic approach to quantitate the behaviors of specific
acids in defined systems. We stress that if any specie of an acid system is present in
an aqueous solution, all must be present. We introduce the mole balance concept and
strive toward an understanding of the idea of the predominant specie or species as
dictated by the relation between the hydronium ion abundance within the system
and the acid dissociation constant of the targeted acid system. We illustrate the
connection between Henry’s law and acid deprotonation equilibria. For a system
that has attained the equilibrium condition, all equilibria must be simultaneously
satisfied. We illustrate the prediction of aqueous speciation when the abundance of
a vapor-phase specie and one critical condition of the aqueous solution are known.
Similarly, from knowledge of at least two conditions relative to an acid system
within an aqueous solution, we can predict the entire speciation within the aqueous
solution as well as the abundance of the nonelectrolyte acid specie in vapor with
which the water is in equilibrium. Employing the proton balance in the context of
conjugate bases accepting protons and conjugate acids donating protons, we seek to
develop beginning understandings of buffering capacity and the functional pro­
perties termed alkalinity and acidity. We make a beginning foray into the concepts of
acid and base neutralizing capacity. We extend our integrated modeling efforts by
carrying our understandings of acid deprotonation with us to join our understandings of Henry’s law from Chapter 5 in contextual applications, again involving the
atmospheric, terrestrial, biogeochemical, and engineered systems. In a manner similar to that employed in Chapter 5, we illustrate the portability of these principles
and concepts.


Organization and Objectives

5

1.2.6  Mixing
The mixing of two or more continuous streams is an important environmental ­process
often given but cursory treatment in environmental engineering texts. While “zero
volume mixing” is simple in concept, the nuances regarding when, how, and to what
systems we can employ this principle often smudge the understandings of its applicability. In Chapter 7, we use continuous mixing of flows to begin our examination
of the differences between transient and steady-state processes. Understandings of
mixing phenomena are employed in developing beginning understandings of ideal
reactors. The principles behind residence time distribution analyses are addressed
and used in the definitions of completely mixed flow and plug flow reactors (CMFRs
and PFRs). Impulse and step input stimuli are introduced, and exit responses for
CMFRs and PFRs are examined. We introduce the process mass balance: the rate of
accumulation within a control volume is the sum of the rates of input, output, and
generation of a targeted substance. We employ the process mass balance to model the
behavior of CMFRs receiving impulse and step input stimuli. We carry these zerovolume and transient mixing principles into environmental contexts, using them to
model responses of selected natural and engineered systems to perturbations
involving substances that are assumed to be nonreactive. In this manner, we illustrate
the portability of these principles.
1.2.7  Reactions in Ideal Reactors
Although chemical stoichiometry is examined in preuniversity courses as well as in
general chemistry courses completed by environmental engineers, the ability to
employ these principles in specific environmental applications is not assured.
Therefore, in Chapter 8 we begin with a review of the use of stoichiometry to determine reactant requirements and production of products using a number of common
environmental engineering contexts. With these we illustrate quantitatively the conversions of one substance to another, without the complication associated with examination of the rates of transformation. We include mass–volume–porosity relations so
that both the requirements for reactants and creation of products, for example, from
water treatment operations can be expressed using molar, mass, and volume units.
Mass–volume–porosity relations are also useful in quantitating rates of a process in
natural systems considered as reactors (either ideal as examined in Chapter 8 or nonideal as examined in Chapter 9).
We introduce two formulations of the reaction rate law: pseudo-first-order and

saturation (arising from enzyme-limited microbial processes). Beyond radioactive
decay, few processes rates are directly and linearly dependent only upon the abundance of the reactant. The pseudo-first-order rate law arises when certain of the reactants, aside from a target reactant, upon which the reaction rate is truly dependent, are
maintained at constant abundance. If we can quantitate the abundances of these nontarget reactants, we can mathematically treat the overall reaction as if it were a firstorder reaction, greatly simplifying the resultant mathematics. Microbial reactions are