Multimedia Environmental Models: The Fugacity Approach, Second Edition

Multimedia Environmental Models The Fugacity Approach Second Edition

Multimedia Environmental Models The Fugacity Approach Second Edition

Donald Mackay

LEWIS PUBLISHERS Boca Raton London New York Washington, D.C.

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Preface This book is about the behavior of organic chemicals in our multimedia environment or biosphere of air, water, soil, and sediments, and the diversity of biota that reside in these media. It is a response to the concern that we have unwisely contaminated our environment with a large number of chemicals in the mistaken belief that the environment’s enormous capacity to dilute and degrade will reduce concentrations to negligible levels. We now know that the environment has only a finite capacity to dilute and degrade. Certain chemicals have persisted and accumulated to levels that have caused adverse effects on wildlife and even humans. Some chemicals have the potential to migrate from medium to medium, reaching unexpected destinations in unexpectedly high concentrations. We need to understand these processes, not only qualitatively in the form of assertions that DDT evaporates and bioaccumulates, but quantitatively as statements that DDT in a particular region evaporates at a rate of 100 kg per year and bioaccumulates from water at a concentration of 1 ng/L to fish at levels of 1 µg/g. We have learned that chemical behavior in the complex assembly of environmental media is not a random process like leaves blowing in the wind. The chemicals behave in accordance with the laws of nature, which dictate chemical partitioning and rates of transport and transformation. Most fundamentally, the chemicals are subject to the law of conservation of mass, i.e., a mass balance exists for the chemical that is a powerful constraint on quantities, concentrations, and fluxes. By coupling the mass balance principle with expressions based on our understanding of the laws of nature, we can formulate a quantitative accounting of chemical inputs and outputs. This book is concerned with developing and applying these expressions in the form of mathematical statements or “models” of chemical fate. These accounts or models are invaluable summaries of chemical behavior. They can form the basis of remedial and proactive strategies. Such models can confirm (or deny) that we really understand chemical fate in the environment. Since many environmental calculations are complex and repetitive, they are particularly suitable for implementation on computers. Accordingly, for many of the calculations described in this book, computer programs are described and made available on the Internet with which a variety of chemicals can be readily assessed in a multitude of environmental situations. The models are formulated using the concept of fugacity, which was introduced by G.N. Lewis in 1901 as a criterion of equilibrium and has proved to be a very convenient and elegant method of calculating multimedia equilibrium partitioning. It has been widely and successfully used in chemical processing calculations. In this book, we exploit it as a convenient and elegant method of explaining and deducing the environmental fate of chemicals. Since publication of the first edition of this book ten years ago, there has been increased acceptance of the benefits of using fugacity to formulate models and interpret environmental fate. Multimedia fugacity models are now routinely used for evaluating chemicals before and after production. Much of the experience gained in these ten years is incorporated in this second edition. Mathematical simulations of chemical fate are now more accurate, compre-

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hensive, and reliable, and they have gained greater credibility as decision-support tools. No doubt this trend will continue, especially as young environmental scientists and engineers take over the reins of environmental science and continue to develop new fugacity models. This book has been written as a result of the author teaching graduate-level courses at the University of Toronto and Trent University. It is hoped that it will be suitable for other graduate courses and for practitioners of the environmental science of chemical fate in government, industry, and the private consulting sector. The simpler concepts are entirely appropriate for undergraduate courses, especially as a means of promoting sensitivity to the concept that chemicals, which provide modern society with so many benefits, must also be more carefully managed from their cradle, in the chemical synthesis plant, to their grave of ultimate destruction. At the end of most chapters is a “Concluding Example” in which a student may be asked to apply the principles discussed in that chapter to one or more chemicals of their choice. Necessary data are given in Table 3.5 in Chapter 3. I have found this useful as a method of assigning different problems to a large number of students, while allowing them to explore the properties and fate of substances of particular interest to them. We no longer regard the environment as a convenient, low-cost dumping ground for unwanted chemicals. When we discharge chemicals into the environment, it must be with a full appreciation of their ultimate fate and possible effects. We must ensure that mistakes of the past with PCBs, mercury, and DDT are not repeated. This is best guaranteed by building up a quantitative understanding of chemical fate in our total multimedia environment, how chemicals will be transported and transformed, and where, and to what extent they may accumulate. It is hoped that this book is one step toward this goal and will be of interest and use to all those who value the environment and seek its more enlightened stewardship. Donald Mackay

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Acknowledgments It is a pleasure to acknowledge the contribution of many colleagues. Much of the credit for the approaches devised in this book is due to the pioneering work by George Baughman, who saw most clearly the evolution of multimedia environmental modeling as a coherent and structured branch of environmental science amid the often frightening complexity of the environment and the formidable number of chemicals with which it is contaminated. Brock Neely, Russ Christman, and Don Crosby were instrumental in encouraging me to apply the fugacity concept to environmental calculations. I am indebted to my former colleagues at the University of Toronto, especially Wan Ying Shiu and Sally Paterson, whose collaboration has been crucial in developing the fugacity approach. I am grateful to my more recent colleagues at Trent University, and our industrial and government partners who have made the Canadian Environmental Modelling Centre a successful focus for the development, validation, and dissemination of mass balance models. This second edition was written in part when on research leave at the Department of Environmental Toxicology at U.C. Davis, where Marion Miller, Don Crosby, and their colleagues were characteristically generous and supportive. At Trent, I was greatly assisted by David Woodfine, Rajesh Seth, Merike Perem, Lynne Milford, Angela McLeod, Adrienne Holstead, Todd Gouin, Alison Fraser, Ian Cousins, Tom Cahill, Jenn Brimecombe, and Andreas Beyer. I am particularly grateful to Steve Sharpe for the figures, to Matt MacLeod and Christopher Warren for their critical review and comments, and to Eva Webster for her outstanding scientific and editorial contributions. Without the support and diligent typing of my wife, Ness, this book would not have been possible. Thank you. I dedicate this book to Ness, Neil, Ian, Julia, and Gwen, and especially to Beth, who was born as this edition neared completion. I hope it will help to ensure that her life is spent in a cleaner, more healthful environment.

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Contents Chapter 1 Introduction................................................................................................................1 Chapter 2 Some Basic Concepts.................................................................................................5 2.1 Introduction ...............................................................................................5 2.2 Units ..........................................................................................................5 2.3 The Environment as Compartments..........................................................9 2.4 Mass Balances.........................................................................................11 2.5 Eulerian and Lagrangian Coordinate Systems ........................................20 2.6 Steady State and Equilibrium..................................................................22 2.7 Diffusive and Nondiffusive Environmental Transport Processes...........25 2.8 Residence Times and Persistence............................................................26 2.9 Real and Evaluative Environments .........................................................27 2.10 Summary .................................................................................................28 Chapter 3 Environmental Chemicals and Their Properties ......................................................29 3.1 Introduction and Data Sources ................................................................29 3.2 Identifying Priority Chemicals................................................................31 3.3 Key Chemical Properties and Classes.....................................................40 3.4 Concluding Example...............................................................................54 Chapter 4 The Nature of Environmental Media .......................................................................55 4.1 Introduction .............................................................................................55 4.2 The Atmosphere ......................................................................................55 4.3 The Hydrosphere or Water......................................................................58 4.4 Bottom Sediments ...................................................................................60 4.5 Soils.........................................................................................................63 4.6 Summary .................................................................................................65 4.7 Concluding Example...............................................................................65 Chapter 5 Phase Equilibrium....................................................................................................69 5.1 Introduction .............................................................................................69 5.2 Properties of Pure Substances .................................................................75 5.3 Properties of Solutes in Solution.............................................................78 5.4 Partition Coefficients ..............................................................................83 5.5 Environmental Partition Coefficients and Z Values ...............................94 5.6 Multimedia Partitioning Calculations .....................................................99 5.7 Level I Calculations ..............................................................................111 5.8 Concluding Examples ...........................................................................113 Chapter 6 Advection and Reactions .......................................................................................117 6.1 Introduction ...........................................................................................117 6.2 Advection ..............................................................................................119

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6.3 6.4 6.5 6.6 6.7 6.8 6.9

Degrading Reactions .............................................................................125 Combined Advection and Reaction ......................................................131 Unsteady-State Calculations .................................................................133 The Nature of Environmental Reactions...............................................135 Level II Computer Calculations ............................................................141 Summary ...............................................................................................142 Concluding Example.............................................................................142

Chapter 7 Intermedia Transport..............................................................................................145 7.1 Introduction ...........................................................................................145 7.2 Diffusive and Nondiffusive Processes ..................................................145 7.3 Molecular Diffusion within a Phase......................................................148 7.4 Turbulent or Eddy Diffusion within a Phase ........................................153 7.5 Unsteady-State Diffusion ......................................................................156 7.6 Diffusion in Porous Media ....................................................................159 7.7 Diffusion between Phases: The Two-Resistance Concept....................161 7.8 Measuring Transport D Values .............................................................167 7.9 Combining Series and Parallel D Values ..............................................173 7.10 Level III Calculations............................................................................175 7.11 Level IV Calculations ...........................................................................181 7.12 Concluding Examples ...........................................................................183 Chapter 8 Applications of Fugacity Models...........................................................................185 8.1 Introduction, Scope, and Strategies.......................................................185 8.2 Level I, II, and III Models.....................................................................189 8.3 An Air-Water Exchange Model ............................................................191 8.4 A Surface Soil Model............................................................................194 8.5 A Sediment-Water Exchange Model ....................................................199 8.6 QWASI Model of Chemical Fate in a Lake..........................................201 8.7 QWASI Model of Chemical Fate in Rivers ..........................................208 8.8 QWASI Multi-segment Models ............................................................210 8.9 A Fish Bioaccumulation Model ............................................................213 8.10 Sewage Treatment Plants ......................................................................220 8.11 Indoor Air Models.................................................................................221 8.12 Uptake by Plants ...................................................................................223 8.13 Pharmacokinetic Models.......................................................................224 8.14 Human Exposure to Chemicals.............................................................226 8.15 The PBT–LRT Attributes......................................................................229 8.16 Global Models.......................................................................................230 8.17 Closure ..................................................................................................232 Appendix Fugacity Forms ......................................................................................................235 References and Bibliography ................................................................................239 Index ......................................................................................................................257

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CHAPTER 1 Introduction

Since the Second World War, and especially since the publication of Rachel Carson’s Silent Spring in 1962, there has been growing concern about contamination of the environment by “man-made” chemicals. These chemicals may be present in industrial and municipal effluents, in consumer or commercial products, in mine tailings, in petroleum products, and in gaseous emissions. Some chemicals such as pesticides may be specifically designed to kill biota present in natural or agricultural ecosystems. They may be organic, inorganic, metallic, or radioactive in nature. Many are present naturally, but usually at much lower concentrations than have been established by human activity. Most of these chemicals cause toxic effects in organisms, including humans, if applied in sufficiently large doses or exposures. They may therefore be designated as “toxic substances.” There is a common public perception and concern that when these substances are present in air, water, or food, there is a risk of adverse effects to human health. Assessment of this risk is difficult (a) because the exposure is usually (fortunately) well below levels at which lethal toxic effects and even sub-lethal effects can be measured with statistical significance against the “noise” of natural population variation, and (b) because of the simultaneous multiple toxic influences of other substances, some taken voluntarily and others involuntarily. There is a growing belief that it is prudent to ensure that the functioning of natural ecosystems is unimpaired by these chemicals, not only because ecosystems have inherent value, but because they can act as sensing sites or early indicators of possible impact on human well-being. Accordingly, there has developed a branch of environmental science concerned with describing, first qualitatively and then quantitatively, the behavior of chemicals in the environment. This science is founded on earlier scientific studies of the condition of the natural environment—meteorology, oceanography, limnology, hydrology, and geomorphology and their physical, energetic, biological, and chemical sub-sciences. This newer branch of environmental science has been variously termed environmental chemistry, environmental toxicology, or chemodynamics. 1

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MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

It is now evident that our task is to design a society in which the benefits of chemicals are enjoyed while the risk of adverse effects from them is virtually eliminated. To do this, we must exert effective and cost-effective controls over the use of such chemicals, and we must have available methods of calculating their environmental behavior in terms of concentration, persistence, reactivity, and partitioning tendencies between air, water, soils, sediments, and biota. Such calculations are useful when assessing or implementing remedial measures to treat alreadycontaminated environments. They become essential as the only available method for predicting the likely behavior of chemicals that (a) may be newly introduced into commerce or that (b) may be subject to production increases or introduction into new environments. In response to this societal need, this book develops, describes, and illustrates a framework and procedures for calculating the behavior of chemicals in the environment. It employs both conventional procedures that are based on manipulations of concentrations and procedures that use the concepts of activity and fugacity to characterize the equilibrium that exists between environmental phases such as air, water, and soil. Most of the emphasis is placed on organic chemicals, which are fortunately more susceptible to generalization than metals and other inorganic chemicals when assessing environmental behavior. The concept of fugacity, which was introduced by G.N. Lewis in 1901 as a more convenient thermodynamic equilibrium criterion than chemical potential, has been widely used in chemical process calculations. Its convenience in environmental chemical equilibrium or partitioning calculations has become apparent only in the last two decades. It transpires that fugacity is also a convenient quantity for describing mathematically the rates at which chemicals diffuse, or are transported, between phases; for example, volatilization of pesticides from soil to air. The transfer rate can be expressed as being driven by, or proportional to, the fugacity difference that exists between the source and destination phases. It is also relatively easy to transform chemical reaction, advective flow, and nondiffusive transport rate equations into fugacity expressions and build up sets of fugacity equations describing the quite complex behavior of chemicals in multiphase, nonequilibrium environments. These equations adopt a relatively simple form, which facilitates their formulation, solution, and interpretation to determine the dominant environmental phenomena. We develop these mathematical procedures from a foundation of thermodynamics, transport phenomena, and reaction kinetics. Examples are presented of chemical fate assessments in both real and evaluative multimedia environments at various levels of complexity and in more localized situations such as at the surface of a lake. These calculations of environmental fate can be tedious and repetitive, thus there is an incentive to use the computer as a calculating aid. Accordingly, computer programs are made available for many of the calculations described later in the text. It is important that the computer be viewed and used as merely a rather fast and smart adding machine and not as a substitute for understanding. The reader is encouraged to write his or her own programs and modify those provided. The author was “brought up” to write computer programs in languages such as FORTRAN, BASIC, and C. The first edition of this book was regarded as very advanced by including a diskette of programs in BASIC. Such programs have the

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INTRODUCTION

3

immense benefit that the sequence and details of calculations are totally transparent. Executable versions can be run on any computer. Unfortunately, it is not always easy to change input parameters or equations, and the output is usually printed tables. The modern trend is to use spreadsheets, such as Microsoft EXCEL®, which have improved input and output features, including the ability to draw graphs and charts. Spreadsheets have the disadvantages that calculations are less transparent, there may be problems when changing versions of the spreadsheet program, and not everyone has the same spreadsheet. Sufficient information is given on each mass balance model that readers can write their own programs using the system of their choice. Microsoft Windows® software for performing model calculations is available from the Internet site www.trentu.ca/envmodel. Older DOS-based models are also available. They are updated regularly and are subject to revision. In all cases, the equations correspond closely to those in this book (unless otherwise stated), and they are totally transparent. Some are used in a regulatory context, thus the user is prevented from changing the coding, although all code can be viewed. Preparing a second edition of this book has enabled me to update, expand, and reorganize much of the material presented in the first (1991) edition. I have benefited greatly from the efforts of those who have sought to understand environmental phenomena and who have applied the fugacity approach when deducing the fate of chemicals in the environment. There is no doubt that, as we enter the new millennium, environmental science is becoming more quantitative. It is my hope that this book will contribute to that trend.

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CHAPTER

2

Some Basic Concepts 2.1

INTRODUCTION

Much of the scientific fascination with the environment lies in its incredible complexity. It consists of a large number of phases such as air, soil, and water, which vary in properties and composition from place to place (spatially) and with time (temporally). It is very difficult to assemble a complete, detailed description of the condition (temperature, pressure, and composition) of even a small environmental system or microcosm consisting, for example, of a pond with sediment below and air above. It is thus necessary to make numerous simplifying assumptions or statements about the condition of the environment. For example, we may assume that a phase is homogeneous, or it may be in equilibrium with another phase, or it may be unchanging with time. The art of successful environmental modeling lies in the selection of the best, or “least-worst,” set of assumptions that yields a model that is not so complex as to be excessively difficult to understand yet is sufficiently detailed to be useful and faithful to reality. The excessively simple model may be misleading. The excessively detailed model is unlikely to be useful, trusted, or even understandable. The aim is to suppress the less necessary detail in favor of the important processes that control chemical fate. In this chapter, several concepts are introduced that are used when we seek to compile quantitative descriptions of chemical behavior in the environment. But first, it is essential to define the system of units and dimensions that forms the foundation of all calculations.

2.2

UNITS

The introduction of the “SI” or “Système International d’Unités” or International System of Units in 1960 has greatly simplified scientific calculations and communication. With few exceptions, we adopt the SI system. The system is particularly convenient, because it is “coherent” in that the basic units combine one-to-one to 5

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MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

give the derived units directly with no conversion factors. For example, energy (joules) is variously the product of force (newtons) and distance (metres), or pressure (pascals) and volume (cubic metres), or power (watts) and time (seconds). Thus, the foot-pound, the litre-atmosphere, and the kilowatt-hour become obsolete in favor of the single joule. Some key aspects of the SI system are discussed below. Conversion tables from obsolete or obsolescent unit systems are available in scientific handbooks. Length (metre, m) This base unit is defined as the specified number of wavelengths of a krypton light emission. Area Square metre (m2). Occasionally, the hectare (ha) (an area 100 × 100 m or 104 m2) or the square kilometre (km2) is used. For example, pesticide dosages to soils are often given in kg/ha. Volume (cubic metre, m3) The litre (L) (0.001 m3) is also used because of its convenience in analysis, but it should be avoided in environmental calculations. In the United States, the spellings “meter” and “liter” are often used. Mass (kilogram, kg) Kilogram (kg). The base unit is the kilogram (kg), but it is often more convenient to use the gram (g), especially for concentrations. For large masses, the megagram (Mg) or the equivalent metric tonne (t) may be used. Amount (mole abbreviated to mol) This unit, which is of fundamental importance in environmental chemistry, is really a number of constituent entities or particles such as atoms, ions, or molecules. It is the actual number of particles divided by Avogadro’s number (6.0 × 1023), which is defined as the number of atoms in 12 g of the carbon-12 isotope. When reactions occur, the amounts of substances reacting and forming are best expressed in moles rather than mass, since atoms or molecules combine in simple stoichiometric ratios. The need to involve atomic or molecular masses is thus avoided. Molar Mass or Molecular Mass (or Weight) (g/mol) This is the mass of 1 mole of matter and is sometimes (wrongly) referred to as molecular weight or molecular mass. Strictly, the correct unit is kg/mol, but it is often more convenient to use g/mol, which is obtained by adding the atomic masses (weights). Benzene (C6H6) is thus approximately 78 g/mol or 0.078 kg/mol.

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SOME BASIC CONCEPTS

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Time (second or hour, s or h) The standard unit of a second (s) is inconveniently short when considering environmental processes such as flows in large lakes when residence times may be many years. The use of hours, days, and years is thus acceptable. We generally use hours as a compromise. Concentration The preferred unit is the mole per cubic metre (mol/m3) or the gram per cubic metre (g/m3). Most analytical data are reported in amount or mass per litre (L), because a litre is a convenient volume for the analytical chemist to handle and measure. Complications arise if the litre is used in environmental calculations, because it is not coherent with area or length. The common mg/L, which is often ambiguously termed the “part per million,” is equivalent to g/m3. In some circumstances, the use of mass fraction, volume fraction, or mole fraction as concentrations is desirable. It is acceptable, and common, to report concentrations in units such as mol/L or mg/L but, prior to any calculation, they should be converted to a coherent unit of amount of substance per cubic metre. Concentrations such as parts per thousand (ppt), parts per million (ppm), parts per billion (ppb), and parts per trillion (also ppt) should not be used. There can be confusion between parts per thousand and per trillion. The billion is 109 in North America and 1012 in Europe. The air ppm is usually on a volume/volume basis, whereas the water ppm is usually on a mass/volume basis. The mixing ratio used for air is the ratio of numbers of molecules or volumes and is often given in ppm. Concentrations must be presented with no possible ambiguity. Density (kg/m3) This has identical units to mass concentrations, but the use of kg/m3 is preferred, water having a density of 1000 kg/m3 and air a density of approximately 1.2 kg/m3. Force (newton, N) The newton is the force that causes a mass of 1 kg to accelerate at 1 m/s2. It is 105 dynes and is approximately the gravitational force operating on a mass of 102 g at the Earth’s surface. Pressure (pascal, Pa) The pascal or newton per square metre (N/m2) is inconveniently small, since it corresponds to only 102 grams force over one square metre, but it is the standard unit, and it is used here. The atmosphere (atm) is 101325 Pa or 101.325 kPa. The torr or mm of mercury (mmHg) is 133 Pa and, although still widely used, should be regarded as obsolescent.

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Energy (joule, J) The joule, which is one N-m or Pa-m3, is also a small quantity. It replaces the obsolete units of calorie (which is 4.184 J) and Btu (1055 J). Temperature (K) The kelvin is preferred, although environmental temperatures may be expressed in degrees Celsius, °C, and not centigrade, where 0°C is 273.15 K. There is no degree symbol prior to K. Frequency (hertz, Hz) The hertz is one event per second (s–1). It is used in descriptions of acoustic and electromagnetic waves, stirring, and in nuclear decay processes where the quantity of a radioactive material may be described in becquerels (Bq), where 1 Bq corresponds to the amount that has a disintegration rate of 1 Hz. The curie (Ci), which corresponds to 3.7 × 1010 disintegrations per second (and thus 3.7 × 1010 Bq), was formerly used. Gas Constant (R) This constant, which derives from the gas law, is 8.314 J/mol K or Pa-m3/mol K. An advantage of the SI system is that R values in diverse units such as cal/mol K and cm3·atm/mol K become obsolete and a single universal value now applies. 2.2.1

Prefices

The following prefices are used: Factor

Prefix

Factor

101

deka da

10–1

deci d

102

hecto h

10–2

centi c

103

kilo k

10–3

milli m

mega M

10–6

micro µ

10

6

Prefix

109

giga G

10–9

nano n

1012

tera T

10–12

pico p

15

peta P

10

–15

femto f

1018

exa E

10–18

10

atto a

Note that these prefices precede the unit. It is inadvisable to include more than one prefix in a unit, e.g., ng/mg, although mg/kg may be acceptable, because the base unit of mass is the kg. The equivalent µg/g is clearer. The use of expressions such as an aerial pesticide spray rate of 900 g/km2 can be ambiguous, since a kilo(metre2) is not equal to a square kilometre, i.e., a (km)2. The former style is not permissible.

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Expressing the rate as 9 g/ha or 0.9 mg/m2 removes all ambiguity. The prefices deka, hecto, deci, and centi are restricted to lengths, areas, and volumes. A common (and disastrous) mistake is to confuse milli, micro, and nano. We use the convention J/mol-K meaning J mol–1 K–1. Strictly, J/(mol-K) is correct but, in the interests of brevity, the parentheses are omitted. 2.2.2

Dimensional Strategy and Consistency

When undertaking calculations of environmental fate, it is highly desirable to adopt the practice of first converting all the supplied input data, in its diversity of units, into the SI units described above and eliminate the prefices, e.g., 10 kPa should become 104 Pa. Calculations should be done using only these SI units. If necessary, the final results can then be converted to other units for the convenience of the user. When assembling quantities in expressions or equations, it is critically important that the dimensions be correct and consistent. It is always advisable to write down the units on each side of the equation, cancel where appropriate, and check that terms that add or subtract have identical units. For example, a lake may have an inflow or reaction rate of a chemical expressed as follows: A flow rate: (water flow rate G m3/h) × (concentration C g/m3) = GC g/h A reaction rate: (volume V m3) × (rate constant k h–1) × (concentration C mol/m3) = VkC mol/h Obviously, it is erroneous to express the above concentration in mol/L or the volume in cm3. When checking units it may be necessary to allow for changes in the prefices (e.g. kg to g), and for unit conversions (e.g., h to s). 2.2.3

Logarithms

The preferred logarithmic quantity is the natural logarithm to the base e or 2.7183, designated as ln. Base 10 logarithms are still used for certain quantities such as the octanol-water partition coefficient and for plotting on log-log or log-linear graph paper. The natural antilog or exponential of x is written either ex or exp(x). The base 10 log of a quantity is the natural log divided by 2.303 or ln10.

2.3

THE ENVIRONMENT AS COMPARTMENTS

It is useful to view the environment as consisting of a number of connected phases or compartments. Examples are the atmosphere, terrestrial soil, a lake, the bottom sediment under the lake, suspended sediment in the lake, and biota in soil or water. The phase may be continuous (e.g., water) or consist of a number of particles that are not in contact, but all of which reside in one phase [e.g., atmospheric

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particles (aerosols), or biota in water]. In some cases, the phases may be similar chemically but different physically, e.g., the troposphere or lower atmosphere, and the stratosphere or upper atmosphere. It may be convenient to lump all biota together as one phase or consider them as two or more classes each with a separate phase. Some compartments are in contact, thus a chemical may migrate between them (e.g., air and water), while others are not in contact, thus direct transfer is impossible (e.g., air and bottom sediment). Some phases are accessible in a short time to migrating chemicals (e.g., surface waters), but others are only accessible slowly (e.g., deep lake or ocean waters), or effectively not at all (e.g., deep soil or rock). Some confusion is possible when expressing concentrations for mixed phases such as water containing suspended solids (SS). An analysis may give a total or bulk concentration expressed as amount of chemical per m3 of mixed water and particles. Alternatively, the water may be filtered to give the concentration or amount of chemical that is dissolved in water per m3 of water. The difference between these is the amount of chemical in the SS phase per m3 of water. This is different from the concentration in the SS phase expressed as amount of chemical per m3 of particle. Concentrations in soils, sediments, and biota can be expressed on a dry or wet weight basis. Occasionally, concentrations in biota are expressed on a lipid or fat content basis. Concentrations must be expressed unambiguously. 2.3.1 Homogeneity and Heterogeneity A key modeling concept is that of phase homogeneity and heterogeneity. Well mixed phases such as shallow pond waters tend to be homogeneous, and gradients in chemical concentration or temperature are negligible. Poorly mixed phases such as soils and bottom sediments are usually heterogeneous, and concentrations vary with depth. Situations in which chemical concentrations are heterogeneous are difficult to describe mathematically, thus there is a compelling incentive to assume homogeneity wherever possible. A sediment in which a chemical is present at a concentration of 1 g/m3 at the surface, dropping linearly to zero at a depth of 10 cm, can be described approximately as a well mixed phase with a concentration of 1 g/m3 and 5 cm deep, or 0.5 g/m3 and 10 cm deep. In all three cases, the amount of chemical present is the same, namely 0.05 g per square metre of sediment horizontal area. Even if a phase is not homogeneous, it may be nearly homogeneous in one or two of the three dimensions. For example, lakes may be well mixed horizontally but not vertically, thus it is possible to describe concentrations as varying only in one dimension (the vertical). A wide, shallow river may be well mixed vertically but not horizontally in the cross-flow or down-flow directions. 2.3.2 Steady- and Unsteady-State Conditions If conditions change relatively slowly with time, there is an incentive to assume “steady-state” behavior, i.e., that properties are independent of time. A severe mathematical penalty is incurred when time dependence has to be characterized, and “unsteady-state,” dynamic, or time-varying conditions apply. We discuss this issue in more detail later.

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SOME BASIC CONCEPTS

2.3.3

11

Summary

In summary, our simplest view of the environment is that of a small number of phases, each of which is homogeneous or well mixed and unchanging with time. When this is inadequate, the number of phases may be increased; heterogeneity may be permitted in one, two, or three dimensions; and variation with time may be included. The modeler’s philosophy should be to concede each increase in complexity reluctantly, and only when necessary. Each concession results in more mathematical complexity and the need for more data in the form of kinetic or equilibrium parameters. The model becomes more difficult to understand and thus less likely to be used, especially by others. This is not a new idea. William of Occam expressed the same sentiment about 650 years ago, when he formulated his principle of parsimony or “Occam’s Razor,” stating Essentia non sunt multiplicanda praeter necessitatem which can be translated as, “What can be done with fewer (assumptions) is done in vain with more,” or more colloquially, “Don’t make models more complicated than is necessary.”

2.4

MASS BALANCES

When describing a volume of the environment, it is obviously essential to define its limits in space. This may simply be the boundaries of water in a pond or the air over a city to a height of 1000 m. The volume is presumably defined exactly, as are the areas in contact with adjoining phases. Having established this control “envelope” or “volume” or “parcel,” we can write equations describing the processes by which a mass of chemical enters and leaves this envelope. The fundamental and now axiomatic law of conservation of mass, which was first stated clearly by Antoine Lavoisier, provides the basis for all mass balance equations. Rarely do we encounter situations in which nuclear processes violate this law. Mass balance equations are so important as foundations of all environmental calculations that it is essential to define them unambiguously. Three types can be formulated and are illustrated below. We do not treat energy balances, but they are set up similarly. 2.4.1

Closed System, Steady-State Equations

This is the simplest class of equation. It describes how a given mass of chemical will partition between various phases of fixed volume. The basic equation simply expresses the obvious statement that the total amount of chemical present equals the sum of the amounts in each phase, each of these amounts usually being a product of a concentration and a volume. The system is closed or “sealed” in that no entry or exit of chemical is permitted. In environmental calculations, the concentrations are usually so low that the presence of the chemical does not affect the phase volumes.

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Worked Example 2.1 A three-phase system consists of air (100 m3), water (60 m3), and sediment (3 m ). To this is added 2 mol of a hydrocarbon such as benzene. The phase volumes are not affected by this addition, because the volume of hydrocarbon is small. Subscripting air, water, and sediment symbols with A, W, and S, respectively, and designating volume as V (m3) and concentration as C (mol/m3), we can write the mass balance equation. 3

total amount = sum of amounts in each phase mol 2 = VACA + VWCW + VSCS = 100 CA + 60 CW + 3 CS mol To proceed further, we must have information about the relationships between CA, CW , and CS. This could take the form of phase equilibrium equations such as CA/CW = 0.4 and CS/CW = 100 These ratios are usually referred to as partition coefficients or distribution coefficients and are designated KAW and KSW , respectively. We discuss them in more detail later. We can now eliminate CA and CS by substitution to give 2 = 100 (0.4 CW) + 60 CW + 3(100CW) = 400 CW mol Thus, CW = 2/400 = 0.005 mol/m3 It follows that CA = 0.4 CW = 0.002 mol/m3 CS = 100 CW = 0.5 mol/m3 The amounts in each phase (mi) mol are the VC products as follows: mW = VWCW = 0.30 mol

(15%)

mA = VACA = 0.20 mol

(10%)

mS = VSCS = 1.50 mol

(75%)

Total

2.00 mol

This simple algebraic procedure has established the concentrations and amounts in each phase using a closed system, steady-state, mass balance equation and equilibrium relationships. The essential concept is that the total amount of chemical present

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13

must equal the sum of the individual amounts in each compartment. We later refer to this as a Level I calculation. It is useful because it is not always obvious where concentrations are high, as distinct from amounts. Example 2.2 In this example, 0.04 mol of a pesticide of molar mass 200 g/mol is applied to a closed system consisting of 20 m3 of water, 90 m3 of air, 1 m3 of sediment, and 2 L of biota (fish). If the concentration ratios are air/water 0.1, sediment/water 50, and biota/water 500, what are the concentrations and amounts in each phase in both gram and mole units? Answer The fish contains 0.1 g or 0.0005 mol at a concentration of 50 g/m3 or 0.25 mol/m3. Example 2.3 A circular lake of diameter 2 km and depth 10 m contains suspended solids (SS) with a volume fraction of 10–5, i.e., 1 m3 of SS per 105 m3 water, and biota (such as fish) at a concentration of 1 mg/L. Assuming a density of biota of 1.0 g/cm3, a SS/water partition coefficient of 104, and a biota/water partition coefficient of 105. Calculate the disposition and concentrations of 1.5 kg of a PCB in this system. Answer In this case, 8.3% is present in each of SS and biota and 83% in water with a concentration in water of 39.8 µg/m3. 2.4.2

Open System, Steady-State Equations

In this class of mass balance equation, we introduce the possibility of the chemical flowing into and out of the system and possibly reacting or being formed. The conditions within the system do not change with time, i.e., its condition looks the same now as in the past and in the future. The basic mass balance assertion is that the total rate of input equals the total rate of output, these rates being expressed in moles or grams per unit time. Whereas the basic unit in the closed system balance was mol or g, it is now mol/h or g/h. Worked Example 2.4 A 104 m3 thoroughly mixed pond has a water inflow and outflow of 5 m3/h. The inflow water contains 0.01 mol/m3 of chemical. Chemical is also discharged directly into the pond at a rate of 0.1 mol/h. There is no reaction, volatilization, or other losses of the chemical; it all leaves in the outflow water.

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(i) What is the concentration (C) in the outflow water? We designate this as an unknown C mol/m3. total input rate = total output rate 5 m3/h × 0.01 mol/m3 + 0.1 mol/h = 0.15 mol/h = 5 m3/h × C mol/m3 = 5 C mol/h Thus, C = 0.03 mol/m3 The total inflow and outflow rates of chemical are 0.15 mol/h. (ii) If the chemical also reacts in a first-order manner such that the rate is VCk mol/h where V is the water volume, C is the chemical concentration in the well mixed water of the pond, and k is a first-order rate constant of 10–3 h–1, what will be the new concentration? The output by reaction is VCk or 104 × 10–3 C or 10 C mol/h, thus we rewrite the equation as: 0.05 + 0.1 = 5 C + 10 C = 15 C mol/h Thus, C = 0.01 mol/m3 The total input of 0.15 mol/h is thus equal to the total output of 0.15 mol/h, consisting of 0.05 mol/h outflow and 0.10 mol/h reaction. An inherent assumption is that the prevailing concentration in the pond is constant and equal to the outflow concentration. This is the “well mixed” or “continuously stirred tank” assumption. It may not always apply, but it greatly simplifies calculations when it does. The key step is to equate the sum of the input rates to the sum of the output rates, ensuring that the units are equivalent in all the terms. This often requires some unit-to-unit conversions. Worked Example 2.5 A lake of area (A) 106 m2 and depth 10 m (volume V 107 m3) receives an input of 400 mol/day of chemical in an effluent discharge. Chemical is also present in the inflow water of 104 m3/day at a concentration of 0.01 mol/m3. The chemical reacts with a first-order rate constant k of 10–3 h–1, and it volatilizes at a rate of (10–5 C) mol/m2s, where C is the water concentration and m2 refers to the air-water area. The outflow is 8000 m3/day, there being some loss of water by evaporation. Assuming that the lake water is well mixed, calculate the concentration and all the inputs and outputs in mol/day. Use a time unit of days in this case.

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Discharge Inflow Total input Reaction rate

15

= = = =

400 mol/day 104 m3/day × 0.01 mol/m3 = 100 mol/day 500 mol/day VCk = 107 m3 × C mol/m3 × 10–3 h–1 × 24 h/day = 24 × 104C mol/day Volatilization rate = 106 m2 × 10–5 C mol/m2 s × 3600 s/h × 24 h/day = 86.4 × 104C mol/day Outflow = 8000 m3/day × C mol/m3 = 0.8 × 104C mol/day Thus,

500 C Reaction rate Volatilization rate Outflow Total rate of loss

= = = = = =

24 × 104C + 86.4 × 104C + 0.8 × 104C = 111.2 × 104C 4.5 × 10–4 mol/m3 107.9 mol/day (i.e., 108 mol/day) 388.5 mol/day (i.e., 390 mol/day) 3.6 mol/day 500 mol/day = input rate

Until proficiency is gained in manipulating these multi-unit equations, it is best to write out all quantities and units and check that the units are consistent. Judgement should be exercised when selecting the number of significant figures to be carried through the calculation. It is preferable to carry more than is needed, then go back and truncate. Remember that environmental quantities are rarely known with better than 5% accuracy. Avoid conveying an erroneous impression of accuracy by using too many significant figures. Example 2.6 A building, 20 m wide × 25 m long × 5 m high is ventilated at a rate of 200 m3/h. The inflow air contains CO2 at a concentration of 0.6 g/m3. There is an internal source of CO2 in the building of 500 g/h. What is the mass of CO2 in the building and the exit CO2 concentration? Answer 7.75 kg and 3.1 g/m3 Example 2.7 A pesticide is applied to a 10 ha field at an average rate of 1 kg/ha every 4 weeks. The soil is regarded as being 20 cm deep and well mixed. The pesticide evaporates at a rate of 2% of the amount present per day, and it degrades microbially with a rate constant of 0.05 days–1. What is the average standing mass of pesticide present at steady state? What will be the steady-state average concentration of pesticide (g/m3), and in units of µg/g assuming a soil solids density of 2500 kg/m3?

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Answer 5.1 kg, 0.255 g/m3, 0.102 µg/g In all these examples, chemical is flowing or reacting, but observed conditions in the envelope are not changing with time, thus the steady-state condition applies. In Example 2.7, the concentration will change in a “sawtooth” manner but, over the long term, it is constant. 2.4.3

Unsteady-State Equations

Whereas the first two types of mass balances lead to simple algebraic equations, unsteady-state conditions give differential equations in time. The simplest method of setting up the equation is to write d(contents)/dt = total input rate – total output rate The input and output rates should be in units of amount/time, e.g., mol/h or g/h. The “contents” must be in consistent units, e.g., in mol or g, and dt, the time increment, in units consistent with the time unit in the input and output terms, (e.g., h). The differential equation can then be solved along with an appropriate initial or boundary condition to give an algebraic expression for concentration as a function of time. The simplest example is the first-order decay equation. Worked Example 2.8 A lake of 106 m3 with no inflow or outflow is treated with 10 mol of piscicide (a chemical that kills fish), which has a first-order reaction (degradation or decay) rate constant k of 10–2 h–1. What will the concentration be after 1 and 10 days, assuming no further input, and when will half the chemical have been degraded? The contents are VC or 106C mol. The output is only by reaction at a rate of VCk or 106 × 10–2C or 104C mol/h. There is zero input, thus, d (106C)/dt = 106dC/dt = 0 – 104 C mol/h Thus, dC/dt = –10–2C mol/h This differential equation is easily solved by separating the variables C and t to give dC/C = –10–2 dt Integrating gives lnC = –10–2t + IC

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where IC is an integration constant that is usually evaluated from an initial condition, i.e., C = Co when t = 0; thus, IC is lnCo and ln(C/Co) = –10–2 t or C = Co exp (–10–2t) Now, Co is (10 mol)/106m3 or 10–5 mol/m3 Thus, C = 10–5 exp (–10–2t) mol/m3 After 1 day (24 h), C will be 0.79 × 10–5 mol/m3, i.e., 79% remains After 10 days (240 h), C will be 0.091 × 10–5 mol/m3, i.e., 9.1% remains Half the chemical will have degraded when C/Co is 0.5; or 10–2 t is –ln 0.5 or 0.693; or t is 69.3 h Note that the half-time t is 0.693/k. This relationship, that the half-time is 0.693 divided by the rate constant, is very important and is used extensively. It is also possible to have inflow and outflow as well as reaction, as shown in the next example. Worked Example 2.9 A well mixed lake of volume V 106 m3 containing no chemical starts to receive an inflow of 10 m3/s containing chemical at a concentration of 0.2 mol/m3. The chemical reacts with a first-order rate constant of 10–2 h–1, and it also leaves with the outflow of 10 m3/s. By “first-order,” we specify that the rate is proportional to C raised to the power one. What will be the concentration of chemical in the lake one day after the start of the input of chemical? Input rate = 10 × 0.2 = 2 mol/s (we choose a time unit of seconds here) Output by reaction = (106 m3)(10–2 h–1)(1 h/3600s)C mol/m3 = 2.78 C mol/s Output by flow = 10 C mol/s Thus,

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Input – Output = d(contents)/dt 2 – 2.78C – 10C = d(106 C)/dt or dC/(2 – 12.78C) = 10–6 dt or ln(2 – 12.78C)/(–12.78) = 10–6 t + IC When t is zero, C is zero, thus, IC = –ln(2)/12.78 and ln[(2 – 12.78C)/2] = –12.78 × 10–6 t or (2 – 12.78 C) = 2 exp(–12.78 × 10–6 t) or C = (2/12.78)[1 – exp(–12.78 × 10–6 t)] Note that when t is zero, exp(0) is unity and C is zero, as dictated by the initial condition. When t is very large, the exponential group becomes zero, and C approaches (2/12.78) or 0.157 mol/m3. At such times, the input of 2 mol/s is equal to the total of the output by flow of 10 × 0.157 or 1.57 mol/s plus the output by reaction of 2.78 × 0.157 or 0.44 mol/s. This is the steady-state solution, which the lake eventually approaches after a long period of time. When t is 1 day or 86400s, C will be 0.105 mol/m3 or 67% of the way to its final value. C will be halfway to its final value when 12.78 × 10–6 t is 0.693 or t is 54200 s or 15 h. This time is largely controlled by the residence time of the water in the lake, which is (106 m3)/(10 m3/s) or 105 s or 27.8 h Worked Example 2.10 A well mixed lake of 105 m3 is initially contaminated with chemical at a concentration of 1 mol/m3. The chemical leaves by the outflow of 0.5 m3/s, and it reacts with a rate constant of 10–2 h–1. What will be the chemical concentration after 1 and 10 days, and when will 90% of the chemical have left the lake?

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Input = 0 Output by flow = 0.5C Output by reaction = VCk = 105 · C · 10–2h–1(1/3600) = 0.278C Thus, 0 – 0.5C – 0.278C = 105dC/dt dC/C = –0.778 × 10–5dt C = Co exp(–0.778 × 10–5 t) Since CO is 1.0 mol/m3, after 1 day or 864000 s, C will be 0.51 mol/m3. t C

= 10 days = 86400s; C = 0.0012 mol/m3 = 0.1 when 0.778 × 10–5 t = –ln 0.1 or 2.3 or when t is 296000 s or 3.4 days

Example 2.11 If the concentration of CO2 in Example 2.6 has reached steady state of 3.1 g/m3, and then the internal source is reduced to 100 g/h, deduce the equation expressing the time course of CO2 concentration decay and the new steady-state value. Answer New steady-state 1.1 g/m3 and C = 1.1 + 2.0 exp(–0.08 t) Example 2.12 A lake of volume 106 m3 has an outflow of 500 m3/h. It is to be treated with a piscicide, the concentration of which must be kept above 1 mg/m3. It is decided to add 3 kg, thus achieving a concentration of 3 mg/m3, and to allow the concentration to decay to 1 mg/m3 before adding another 2 kg to bring the concentration back to 3 mg/m3. If the piscicide has a degradation half-life of 693 hours (29 days), what will be the interval before the second (and subsequent) applications are required? Answer 30 days Mr. MacLeod, being economically and ecologically perceptive, claims that if he is allowed to make applications every 10 days instead of 30 days, he can maintain the concentration above 1 mg/m3 but reduce the piscicide usage by 35%. Is he correct? Answer Yes

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What is the absolute minimum piscicide usage every 30 days to maintain 1 mg/m3? Answer A total of 1.08 kg added continuously over a 30 day period These unsteady-state solutions usually contain exponential terms such as exp(–kt). The term k is a characteristic rate constant with units of reciprocal time. It is thus somewhat difficult to grasp and remember. A quantity of 0.01 h–1 does not convey an impression of rapidity. It is convenient to calculate its reciprocal 1/k or 100 h, which is a characteristic time. This is the time required for the process to move exp(–1) or to within 37% of the final value, i.e., it is 63% completed. Those working with radioisotopes prefer to use half-lives rather than k, i.e., the time for half completion. This occurs when the term exp(–kt) is 0.5 or kt is ln2 or 0.693, thus the half-time τ is 0.693/k. Another useful time is the 90% completion value, which is 2.303/k. Two common mistakes are made if rate constants are manipulated as times rather than frequencies. A rate constant of 1 day–1 is 0.042 h–1, not 24 h–1—a common mistake. If there are two first-order reactions, the total rate constant is the sum of the individual rate constants. This has the effect of giving a total half-time or halflife that is less than either individual half-time. It is a disastrous mistake to add halflives. Their reciprocals add. In some cases, the differential equation can become quite complex, and there may be several of them applying simultaneously. Setting up these equations requires practice and care. There is a common misconception that solving the equations is the difficult task. On the contrary, it is setting them up that is most difficult and requires the most skill. If the equation is difficult to solve, tables of integrals can be consulted, computer programs such as Mathematica or Matlabs can be used, or an obliging mathematician can be sought. For many differential equations, an analytical solution is not feasible, and numerical methods must be used to generate a solution. We discuss techniques for doing this later.

2.5

EULERIAN AND LAGRANGIAN COORDINATE SYSTEMS

It is usually best to define the mass balance envelope as being fixed in space. This can be called the Eulerian coordinate system. When there is appreciable flow through the envelope, it may be better to define the envelope as being around a certain amount of material and allow that envelope of material to change position. This “fix a parcel of material then follow it in time as it moves” approach is often applied to rivers when we wish to examine the changing condition of a volume of water as it flows downstream and undergoes various reactions. This can be called the Lagrangian coordinate system. It is also applied to “parcels” of air emitted from a stack and subject to wind drift. Both systems must give the same results, but it may be easier to write the equations in one system than the other. The following example is an illustration. It also demonstrates the need to convert units to the SI system.

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Worked Example 2.13 Consider a river into which the 1.8 million population of a city discharges a detergent at a rate of 1 pound per capita per year, i.e., the discharge is 1.8 million pounds per year. The aim is to calculate the concentrations at distances of 1 and 10 miles downstream from a knowledge of the degradation rate of the detergent and the constant downstream flow conditions, which are given below. This can be done in Eulerian or Lagrangian coordinate systems. The input data are first converted to SI units. 1.8 × 106 lb per year

93300 g/h

(E)

1 ft/s

1097 m/h

(U)

River depth

3 ft

0.91 m

(h)

River width

20 yards

18.3 m

(w)

Degradation half-life

0.3 days

7.2 h

(τ1/2)

Discharge rate River flow velocity

The river flow rate is Uhw, i.e., 18270 m3/h. The rate constant k is 0.693/τ1/2, i.e. 0.096 h–1. When the detergent mixes into the river, the concentration will be CO or E/(Uhw) or 5.1 g/m3. Lagrangian Solution A parcel of water that maintains its integrity, i.e., it does not diffuse or disperse, will decay according to the equation C = CO exp(–kt) where t is the time from discharge. At 1 mile (1609 m), the time t will be 1609/U or 1.47 h, and at 10 miles, it will be 14.7 h. Substituting shows that, after 1 and 10 miles, the concentrations will be 4.4 and 1.24 g/m3. The chemical will reach half its input concentration when t is 0.693/k or 7.2 h, which corresponds to 7900 m or 4.92 miles. This Lagrangian solution is straightforward, but it is valid only if conditions in the river remain constant and negligible upstream-downstream dispersion occurs. Eulerian Solution We now simulate the river as a series of connected reaches or segments or well mixed lakes, each being L or 200 m long. Each reach thus has a volume V of Lhw or 3330 m3. A steady-state mass balance on the first reach gives input rate = UhwCO = output rate = UhwC1 + VkC1 where CO and C1 are the input and output concentrations. C1 is also the concentration in the segment. It follows that

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C1 = CO/(1 + Vk/(Uhw)) = CO/(1 + kL/U) Note the consistency of the dimensions, kL/U being dimensionless. The group (1 + kL/U) has a value of 1.0175, thus C1 is 0.983CO. 1.7% of the chemical is lost in each segment. The same equation applies to the second reach, thus C2 is 0.983C1 or 0.9832CO. In general, for the nth reach, Cn is (0.983)nCO or CO/(1 + kL/U)n. One mile is reached when n is 8, and 10 miles corresponds to n of 80, thus C8 is 0.9838CO or 4.45, and C8 is 1.29. The half distance will occur when 0.983n is 0.5, i.e., when n is log 0.5/log 0.983 or 40, corresponding to 8000 m or 5 miles. The Eulerian answer is thus slightly different. It could be made closer to the Lagrangian result by carrying more significant figures or by decreasing L and increasing n. An advantage of the Eulerian system is that it is possible to have segments with different properties such as depth, width, velocity, volume, and temperature. There can be additional inputs. The general equation employing the group (1 + kL/U)n will not then apply, each segment having a specific value of this factor. The mathematical enthusiast will note that L/U is t/n, where t is the flow time to a distance nL m. The Lagrangian factor is thus also (1 + kt/n)n, which approaches exp(kt) when n is large. It is good practice to do the calculation in both systems (even approximately) and check that the results are reasonable. Some water quality models of rivers and estuaries can have several hundred segments, thus it is difficult to grasp the entirety of the results, and mistakes can go undetected.

2.6

STEADY STATE AND EQUILIBRIUM

In the previous section, we introduced the concept of “steady state” as implying unchanging with time, i.e., all time derivatives are zero. There is frequent confusion between this concept and that of “equilibrium,” which can also be regarded as a situation in which no change occurs with time. The difference is very important and, regrettably, the terms are often used synonymously. This is entirely wrong and is best illustrated by an example. Consider the vessel in Figure 2.1A, which contains 100 m3 of water and 100 m3 of air. It also contains a small amount of benzene, say 1000 g. If this is allowed to stand at constant conditions for a long time, the benzene present will equilibrate between the water and the air and will reach unchanging but different concentrations, possibly 8 g/m3 in the water and 2 g/m3 in the air, i.e., a factor of 4 difference in favor of the water. There is thus 800 g in the water and 200 g in the air. In this condition, the system is at equilibrium and at a steady state. If, somehow, the air and its benzene were quickly removed and replaced by clean air, leaving a total of 800 g in the water, and the volumes remained constant, the concentrations would adjust (some benzene transferring from water to air) to give a new equilibrium (and steady state) of 6.4 g/m3 in the water (total 640 g) and 1.6 g/m3 in the air (total 160 g), again with a factor of 4 difference. This factor is a partition coefficient or distribution coefficient or, as is discussed later, a form of Henry’s law constant. During the adjustment period (for example, immediately after removal of the air when the benzene concentration in air is near zero and the water is still near 8 g/m3),

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Figure 2.1

23

Illustration of the difference between equilibrium and steady-state conditions. Equilibrium implies that the oxygen concentrations in the air and water achieve a ratio or partition coefficient of 20. Steady state implies unchanging with time, even if flow occurs and regardless of whether equilibrium applies.

the concentrations are not at a ratio of 4, the conditions are nonequilibrium, and, since the concentrations are changing with time, they are also of unsteady-state in nature. This correspondence between equilibrium and steady state does not, however, necessarily apply when flow conditions prevail. It is possible for air and water

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containing nonequilibrium quantities of benzene to flow into and out of the tank at constant rates as shown in Figure 2.1B. But equilibrium and a steady-state condition are maintained, since the concentrations in the tank and in the outflows are at a ratio of 1:4. It is possible for near equilibrium to apply in the vessel, even when the inflow concentrations are not in equilibrium, if benzene transfer between air and water is very rapid. Figure 2.1B thus illustrates a flow, equilibrium, and steadystate conditions, whereas Figure 2.1A is a nonflow, equilibrium, and steady-state situation. In Figure 2.1C, there is a deficiency of benzene in the inflow water (or excess in the air) and, although in the time available some benzene transfers from air to water, there is insufficient time for equilibrium to be reached. Steady state applies, because all concentrations are constant with time. This is a flow, nonequilibrium, steady-state condition in which the continuous flow causes a constant displacement from equilibrium. In Figure 2.1D, the inflow water and/or air concentration or rates change with time, but there is sufficient time for the air and water to reach equilibrium in the vessel, thus equilibrium applies (the concentration ratio is always 4), but unsteadystate conditions prevail. Similar behavior could occur if the tank temperature changes with time. This represents a flow, equilibrium, and unsteady-state condition. Finally, in Figure 2.1E, the concentrations change with time, and they are not in equilibrium; thus, a flow, nonequilibrium, unsteady-state condition applies, which is obviously quite complex. The important point is that equilibrium and steady state are not synonymous; neither, either, or both can apply. Equilibrium implies that phases have concentrations (or temperatures or pressures) such that they experience no tendency for net transfer of mass. Steady state merely implies constancy with time. In the real environment, we observe a complex assembly of phases in which some are (approximately) in steady state, others in equilibrium, and still others in both steady state and equilibrium. By carefully determining which applies, we can greatly simplify the mathematics used to describe chemical fate in the environment. A couple of complications are worthy of note. Chemical reactions also tend to proceed to equilibrium but may be prevented from doing so by kinetic or activation considerations. An unlit candle seems to be in equilibrium with air, but in reality it is in a metastable equilibrium state. If lit, it proceeds toward a “burned” state. Thus, some reaction equilibria are not achieved easily, or not at all. Second, “steady state” depends on the time frame of interest. Blood circulation in a sleeping child is nearly in steady state; the flow rates are fairly constant, and no change is discernible over several hours. But, over a period of years, the child grows, and the circulation rate changes; thus, it is not a true steady state when viewed in the long term. The child is in a “pseudo” or “short-term” steady state. In many cases, it is useful to assume steady state to apply for short periods, knowing that it is not valid over long periods. Mathematically, a differential equation that truly describes the system is approximated by an algebraic equation by setting the differential or the d(contents)/dt term to zero. This can be justified by examining the relative magnitude of the input, output, and inventory change terms.

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2.7

25

DIFFUSIVE AND NONDIFFUSIVE ENVIRONMENTAL TRANSPORT PROCESSES

In the air-water example, it was argued that equilibrium occurs when the ratio of the benzene concentrations in water and air is 4. Thus, if the concentration in water is 4 mol/m3, equilibrium conditions exist when the concentration in air is 1 mol/m3. If the air concentration rises to 2 mol/m3, we expect benzene to transfer by diffusion from air to water until the concentration in air falls, concentration in water rises, and a new equilibrium is reached. This is easily calculated if the total amount of benzene is known. In a nonflow system, if the initial concentrations in air and water are CAO and CWO mol/m3, respectively, and the volumes are VA and VW , then the total amount M is, as shown earlier, M = CAOVA + CWOVW mol Here, CAO is 2, and CWO is 4 mol/m3, and since the volumes are both 100 m3, M is 600 mols. This will distribute such that CW is 4CA or M = 600 = CAVA + CWVW = CAVA + 4CAVW = CA(VA + 4VW) = CA 500 Thus, CA is 1.2 mol/m3, and CW is 4.8 mol/m3. Thus, the water concentration rises from 4.0 to 4.8, while that of the air drops from 2.0 to 1.2 mol/m3. Conversely, if the concentration in water is increased to 10 mol/m3, there will be transfer from water to air until a new equilibrium state is reached. A worrisome dilemma is, “How does the benzene in the water know the concentration in the air so that it can decide to start or stop diffusing?” In fact, it does not know or care. It diffuses regardless of the condition at the destination. Equilibrium merely implies that there is no net diffusion, the water-to-air and air-to-water diffusion rates being equal and opposite. Chemicals in the environment are always striving to reach equilibrium. They may not always achieve this goal, but it is useful to know the direction in which they are heading. Other transport mechanisms occur that are not driven by diffusion. For example, we could take 1 m3 of the water with its associated 1 mol of benzene and physically convey it into the air, forcing it to evaporate, thus causing the concentration of benzene in the air to increase. This nondiffusive, or “bulk,” or “piggyback” transfer occurs at a rate that depends on the rate of removal of the water phase and is not influenced by diffusion. Indeed, it may be in a direction opposite to that of diffusion. In the environment, it transpires that there are many diffusive and nondiffusive processes operating simultaneously. Examples of diffusive transfer processes include 1. 2. 3. 4.

Volatilization from soil to air Volatilization from water to air Absorption or adsorption by sediments from water Diffusive uptake from water by fish during respiration

Some nondiffusive processes are

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1. Fallout of chemical from air to water or soil in dustfall, rain, or snow 2. Deposition of chemical from water to sediments in association with suspended matter which deposits on the bed of sediment 3. The reverse process of resuspension 4. Ingestion and egestion of food containing chemical by biota

The mathematical expressions for these rates are quite different. For diffusion, the net rate of transfer or flux is written as the product of the departure from equilibrium and a kinetic quantity, and the net flux becomes zero when the phases are in equilibrium. We examine these diffusive processes in Chapter 7. For nondiffusive processes, the flux is the product of the volume of the phase transferred (e.g., quantity of sediment or rain) and the concentration. We treat nondiffusive processes in Chapter 6. We use the word flux as short form for transport rate. It has units such as mol/h or g/h. Purists insist that flux should have units of mol/h·m2, i.e., it should be area specific. We will apply it to both. It is erroneous to use the term flux rate since flux, like velocity, already contains the “per time” term.

2.8

RESIDENCE TIMES AND PERSISTENCE

In some environments, such as lakes, it is convenient to define a residence time or detention time. If a pond has a volume of 1000 m3 and experiences inflow and outflow of 2 m3/h, it is apparent that, on average, the water spends 500 h (20.8 days) (i.e., 1000 m3/2 m3/h) in the lake. This residence or detention time may not bear much relationship to the actual time that a particular parcel of water spends in the pond, since some water may bypass most of the pond and reside for only a short time, and some may be trapped for years. The quantity is very useful, however, because it gives immediate insight into the time required to flush out the contents. Obviously, a large lake with a long residence time will be very slow to recover from contamination. Comparison of the residence time with a chemical reaction time (e.g., a half-life) indicates whether a chemical is removed from a lake predominantly by flow or by reaction. If a well mixed lake has a volume V m3 and equal inflow and outflow rates G 3 m /h, then the flow residence time τF is V/G (h). If it is contaminated by a nonreacting (conservative) chemical at a concentration CO mol/m3 at zero time and there is no new emission, a mass balance gives, as was shown earlier, C = C0 exp(–Gt/V) = C0 exp (–t/τF) = CO exp(–kF t) The residence time is thus the reciprocal of a rate constant kF with units of h–1. The half-time for recovery occurs when t/τF or kFt is ln 2 or 0.693, i.e., when t is 0.693τ or 0.693/k. If the chemical also undergoes a reaction with a rate constant kR h–1, it can be shown that C = C0 exp[–(kF + kR)t] = C0 exp(–kTt)

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Thus, the larger (faster) rate constant dominates. The characteristic times τF and τR (i.e., 1/kF and 1/kR) combine as reciprocals to give the total time τT , as do electrical resistances in parallel, i.e., 1/τF + 1/τR = 1/τT = kT + kR Thus, the smaller (shorter) τ dominates. The term τR can be viewed as a reaction persistence. Characteristic times such as τR and τF are conceptually easy to grasp and are very convenient quantities to deduce when interpreting the relative importance of environmental processes. For example, if τF is 30 years and τR is 3 years, τT is 2.73 years; thus, reaction dominates the chemical’s fate. Ten out of every 11 molecules react, and only one leaves the lake by flow.

2.9

REAL AND EVALUATIVE ENVIRONMENTS

The environmental scientist who is attempting to describe the behavior of a pesticide in a system such as a lake soon discovers that real lakes are very complex. Considerable effort is required to measure, analyze, and describe the lake, with the result that little energy (or research money) remains with which to describe the behavior of the pesticide. This is an annoying problem, because it diverts attention from the pesticide (which is important) to the condition of the lake (which may be relatively unimportant). A related problem also arises when a new chemical is being considered. Into which lake should it be placed (hypothetically) for evaluation? A significant advance in environmental science was made in 1978, when Baughman and Lassiter (1978) proposed that chemicals may be assessed in “evaluative environments” that have fictitious but realistic properties such as volume, composition, and temperature. Evaluative environments can be decreed to consist of a few homogeneous phases of specified dimensions with constant temperature and composition. Essentially, the environmental scientist designs a “world” to desired specifications, then explores mathematically the likely behavior of chemicals in that world. No claim is made that the evaluative world is identical to any real environment, although broad similarities in chemical behavior are expected. There are good precedents for this approach. In 1824, Carnot devised an evaluative steam engine, now termed the Carnot cycle, which leads to a satisfying explanation of entropy and the second law of thermodynamics. The kinetic theory of gases uses an evaluative assumption of gas molecule behavior. The principal advantage of evaluative environments is that they act as an intellectual stepping stone when tackling the difficult task of describing both chemical behavior and an environment. The task is simplified by sidestepping the effort needed to describe a real environment. The disadvantage is that results of evaluative environment calculations cannot be validated directly, so they are suspect and possibly quite wrong. Some validation can be sought by making the evaluative environment similar to a simple real environment, such as a small pond or a laboratory microcosm. Later, we construct evaluative environments or “unit worlds” and use them to explore the likely behavior of chemicals. In doing so, we use equations that can be

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validated using real environments. A somewhat different assembly of equations proves to be convenient for real environments, but the underlying principles are the same.

2.10

SUMMARY

In this chapter, we have introduced the system of units and dimensions. A view of the environment has been presented as an assembly of phases or compartments that are (we hope) mostly homogeneous rather than heterogeneous in properties, and that vary greatly in volume and composition. We can define these phases or parts of them as “envelopes” about which we can write mass, mole, and, if necessary, energy balance equations. Steady-state conditions will yield algebraic equations, and unsteady-state conditions will yield differential equations. These equations may contain terms for discharges, flow (diffusive and nondiffusive) of material between phases, and for reaction or formation of a chemical. We have discriminated between equilibrium and steady state and introduced the concepts of residence time and persistence. Finally, the use of both real and evaluative environments has been suggested. Having established these basic concepts, or working tools, our next task is to develop the capability of quantifying the rates of the various flow, transport, and reaction processes.

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CHAPTER

3

Environmental Chemicals and Their Properties 3.1

INTRODUCTION AND DATA SOURCES

In this book, we focus on techniques for building mass balance models of chemical fate in the environment, rather than on the detailed chemistry that controls transport and transformation, as well as toxic interactions. For a fuller account of the basic chemistry, the reader is referred to the excellent texts by Crosby (1988), Tinsley (1979), Stumm and Morgan (1981), Pankow (1991), Schwarzenbach et al. (1993), Seinfeld and Pandis (1997), Findlayson-Pitts and Pitts (1986), Thibodeaux (1996), and Valsaraj (1995). There is a formidable and growing literature on the nature and properties of chemicals of environmental concern. Numerous handbooks list relevant physicalchemical and toxicological properties. Especially extensive are compilations on pesticides, chemicals of potential occupational exposure, and carcinogens. Government agencies such as the U.S. Environmental Protection Agency (EPA), Environment Canada, scientific organizations such as the Society of Environmental Toxicology and Chemistry (SETAC), industry groups, and individual authors have published numerous reports and books on specific chemicals or classes of chemicals. Conferences are regularly held and proceedings published on specific chemicals such as the “dioxins.” Computer-accessible databases are now widely available for consultation. Table 3.1 lists some of the more widely used texts and scientific journals. Most are available in good reference libraries. Most of the chemicals that we treat in this book are organic, but the mass balancing principles also apply to metals, organometallic chemicals, gases such as oxygen and freons, inorganic compounds, and ions containing elements such as phosphorus and arsenic. Metals and other inorganic compounds tend to require individual treatment, because they usually possess a unique set of properties. Organic compounds, on the other hand, tend to fall into certain well defined classes. We are often able to estimate the properties and behavior of one organic chemical from that 29

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Table 3.1 Sources of information on chemical properties and estimation methods (See Chapter 1.5 of Mackay, et al., Illustrated Handbooks of Physical Chemical Properties and Environmental Fate for Organic Chemicals, cited below, for more details) The Merck Index: An Encyclopedia of Chemicals, Drugs, and Biologicals (Annual), S. Budavarie, ed. Whitehouse Station, NJ: Merck & Co., 1996. Handbook of Chemistry and Physics, D. R. Lide, ed., 81/e. Boca Raton, FL: CRC Press. Verschueren’s Handbook of Environmental Data on Organic Chemicals. New York: John Wiley & Sons, 1997. Illustrated Handbook of Physical Chemical Properties and Environmental Fate for Organic Chemicals (in 5 volumes). D. Mackay, W. Y Shiu, and K. C. Ma. Boca Raton, FL: CRC Press, 1991–1997. Also available as a CD ROM. Handbook of Environmental Fate and Exposure Data for Organic Chemicals (several volumes), P. H. Howard, ed. Boca Raton, FL: Lewis Publications. Handbook of Environmental Degradation Rates, P. H. Howard et al. Boca Raton, FL: Lewis Publications. Lange’s Handbook of Chemistry, 15/e, J. A. Dean, ed. New York: McGraw-Hill, 1998. Dreisbach’s Physical Properties of Chemical Compounds, Vol I to III. Washington, DC, Amer. Chem. Soc. Technical Reports, European Chemical Industry Ecology and Toxicology Centre (ECETOC). Brussels, Belgium. Sax’s Dangerous Properties of Industrial Materials, 10/e. R. J. Lewis, ed. New York: John Wiley & Sons. Groundwater Chemicals Desk Reference, J. J. Montgomery. Boca Raton, FL: Lewis Publications, 1996. Genium Materials Safety Data Sheets Collection. Amsterdam, NY: Genium Publishing Corp. The Properties of Gases and Liquids, R. C. Reid, J. M. Prausnitz, and B. E. Poling. New York: McGraw-Hill, 1987. NIOSH/OSHA Occupational Health Guidelines for Chemical Hazards. Washington, DC: U.S. Government Printing Office. The Pesticide Manual, 12/e. C. D. S. Tomlin, ed. Loughborough, UK: British Crop Protection Council. The Agrochemicals Handbook, H. Kidd and D. R. James, eds. London: Royal Society of Chemistry. Agrochemicals Desk Reference, 2/e, J. H. Montgomery. Boca Raton, FL: Lewis Publications. ARS Pesticide Properties Database, R. Nash, A. Herner, and D. Wauchope. Beltsville, MD: U.S. Department of Agriculture, www.arsusda.gov/rsml/ppdb.html. Substitution Constants for Correlation Analysis in Chemistry and Biology, C. H. Hansch (currently out of print). New York: Wiley-Interscience. Handbook of Chemical Property Estimation Methods, W. J. Lyman, W. F. Reehl, D. H. Rosenblatt (currently out of print). New York: McGraw-Hill. Handbook of Property Estimation Methods for Chemicals, R. S. Boethling and D. Mackay. Boca Raton, FL: CRC Press, 2000. Chemical Property Estimation: Theory and Practice, E. J. Baum. Boca Raton, FL: Lewis Publications, 1997. Toolkit for Estimating Physiochemical Properties of Organic Compounds, M. Reinhard and A. Drefahl. New York: John Wiley & Sons, 1999. IUPAC Handbook. Research Triangle Park, NC: International Union of Pure and Applied Chemistry. Website for database and EPIWIN estimation methods, Syracuse, NY: Syracuse Research Corporation (http://www.syrres.com).

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of other, somewhat similar or homologous chemicals. An example is the series of chlorinated benzenes that vary systematically in properties from benzene to hexachlorobenzene. It is believed that some 50,000 to 80,000 chemicals are used in commerce. The number of chemicals of environmental concern runs to a few thousand. There are now numerous lists of “priority” chemicals of concern, but there is considerable variation between lists. It is not possible, or even useful, to specify an exact number of chemicals. Some inorganic chemicals ionize in contact with water and thus lose their initial identity. Some lists name PCBs (polychlorinated biphenyls) as one chemical and others as six groups of chemicals whereas, in reality, the PCBs consist of 209 possible individual congeners. Many chemicals, such as surfactants and solvents, are complex mixtures that are difficult to identify and analyze. One designation, such as naphtha, may represent 1000 chemicals. There is a multitude of pesticides, dyes, pigments, polymeric substances, drugs, and silicones that have valuable social and commercial applications. These are in addition to the numerous “natural” chemicals, many of which are very toxic. For legislative purposes, most jurisdictions have compiled lists of chemicals that are, or may be, encountered in the environment, and from these “raw” lists of chemicals of potential concern they have established smaller lists of “priority” chemicals. These chemicals, which are usually observed in the environment, are known to have the potential to cause adverse ecological and/or biological effects and are thus believed to be worthy of control and regulation. In practice, a chemical that fails to reach the “priority” list is usually ignored and receives no priority rather than less priority. These lists should be regarded as dynamic. New chemicals are being added as enthusiastic analytical chemists detect them in unexpected locations or toxicologists discover subtle new effects. Examples are brominated flame retardants, chlorinated alkanes, and certain very stable fluorinated substances (e.g., trifluoroacetic acid) that have only recently been detected and identified. In recent years, concern has grown about the presence of endocrine modulating substances such as nonylphenol, which can disrupt sex ratios and generally interfere with reproductive processes. The popular book Our Stolen Future, by Colborn et al. (1996) brought this issue to public attention. Some of these have industrial or domestic sources, but there is increasing concern about the general contamination by drugs used by humans or in agriculture. Table 3.2 lists about 200 chemicals by class and contains many of the chemicals of current concern.

3.2

IDENTIFYING PRIORITY CHEMICALS

It is a challenging task to identify from “raw lists” of chemicals a smaller, more manageable number of “priority” chemicals. Such chemicals receive intense scrutiny, analytical protocols are developed, properties and toxicity are measured, and reviews are conducted of sources, fate, and effects. This selection contains an element of judgement and is approached by different groups in different ways. A common thread among many of the selection processes is the consideration of six factors: quantity,

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Table 3.2 List of Chemicals Commonly Found on Priority Chemical Lists Volatile Halogentated Hydrocarbons Chloromethane Methylene chloride Chloroform Carbontetrachloride Chloroethane 1,1-Dichloroethane 1,2-Dichloroethane cis-1,2-Dichloroethene trans-1,2-Dichloroethene Vinyl chloride 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichloroethylene Tetrachloroethylene Hexachloroethane 1,2-Dichloropropane 1,3-Dichloropropane cis-1,3-Dichloropropylene trans-1,3-Dichloropropylene Chloroprene Bromomethane Bromoform Ethylenedibromide Chlorodibromomethane Dichlorobromomethane Dichlorodibromomethane Freons (chlorofluoro-hydrocarbons) Dichlorodifluoromethane Trichlorofluoromethane Halogenated Monoaromatics Chlorobenzene 1,2-Dichlorobenzene 1,3-Dichlorobenzene 1,4-Dichlorobenzene 1,2,3-Trichlorobenzene 1,2,4-Trichlorobenzene 1,2,3,4-Tetrachlorobenzene 1,2,3,5-Tetrachlorobenzene

Monoaromatic Hydrocarbons Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Styrene Polycyclic Aromatic Hydrocarbons Naphthalene 1-Methylnaphthalene 2-Methylnaphthalene Trimethylnaphthalene Biphenyl Acenaphthene Acenaphthylene Fluorene Anthracene Fluoranthene Phenanthrene Pyrene Chrysene Benzo(a)anthracene Dibenz(a,h)anthracene Benzo(b)fluoranthene Benzo(k)fluoranthene Benzo(a)pyrene Perylene Benzo(g,h,i)perylene Indeno(1,2,3)pyrene

Dienes 1,3-Butadiene Cyclopentadiene Hexachlorobutadiene Hexachlorocyclopentadiene Alcohols and Phenols Benzyl alcohol Phenol o-Cresol m-Cresol p-Cresol 2-Hydroxybiphenyl 4-Hydroxybiphenyl Eugenol

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Table 3.2 List of Chemicals Commonly Found on Priority Chemical Lists 1,2,4,5-Tetrachlorobenzene Pentachlorobenzene Hexachlorobenzene 2,4,5-Trichlorotoluene Octachlorostyrene Halogenated Biphenyls and Naphthalenes Polychlorinated Biphenyls (PCBs) Polybrominated Biphenyls (PBBs) 1-Chloronaphthalene 2-Chloronaphthalene Polychlorinated Naphthalenes (PCNs) Aroclor Aroclor Aroclor Aroclor Aroclor Aroclor Aroclor Aroclor

Mixtures (PCBs) 1016 1221 1232 1242 1248 1254 1260

Chlorinated Dibenzo-p-dioxins 2,3,7,8-Tetrachlorodibenzo-p-dioxin Tetrachlorinated dibenzo-p-dioxins Pentachlorinated dibenzo-p-dioxins Hexachlorinated dibenzo-p-dioxins Heptachlorinated dibenzo-p-dioxins Octachlorinated dibenzo-p-dioxin Brominated dibenzo-p-dioxins Chlorinated Dibenzofurans Tetrachlorinated dibenzofurans Pentachlorinated dibenzofurans Hexachlorinated dibenzofurans Heptachlorinated dibenzofurans Octachlorodibenzofuran

Halogenated Phenols 2-Chlorophenol 2,4-Dichlorophenol 2,6-Dichlorophenol 2,3,4-Trichlorophenol 2,3,5-Trichlorophenol 2,4,5-Trichlorophenol 2,4,6-Trichlorophenol 2,3,4,5-Tetrachlorophenol 2,3,4,6-Tetrachlorophenol 2,3,5,6-Tetrachlorophenol Pentachlorophenol 4-Chloro-3-methylphenol 2,4-Dimethylphenol 2,6-Di-t-butyl-4-methylphenol Tetrachloroguaiacol Nitrophenols, Nitrotoluenes and Related Compounds 2-Nitrophenol 4-Nitrophenol 2,4-Dinitrophenol 4,6-Dinitro-o-cresol Nitrobenzene 2,4-Dinitrotoluene 2,6-Dinitrotoluene 1-Nitronaphthalene 2-Nitronaphthalene 5-Nitroacenaphthalene Fluorinated Compounds Polyfluorinated alkanes Trifluoroacetic acid Fluoro-chloro acids Polyfluorinated chemicals

Phthalate Esters Nitrosamines and Other Nitrogen Compounds Dimethylphthalate N-Nitrosodimethylamine Diethylphthalate N-Nitrosodiethylamine Di-n-butylphthalate N-Nitrosodiphenylamine Di-n-octylphthalate N-Nitrosodi-n-propylamine Di(2-ethylhexyl) phthalate Diphenylamine Benzylbutylphthalate Indole 4-aminoazobenzene Chlorinated longer chain alkanes Pesticides, including biocides, fungicides, rodenticides, insecticides and herbicides

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persistence, bioaccumulation, potential for transport to distant locations, toxicity, and a miscellaneous group of other adverse effects. 3.2.1

Quantity

The first factor is the quantity produced, used, formed or transported, including consideration of the fraction of the chemical that may be discharged to the environment during use. Some chemicals, such as benzene, are used in very large quantities in fuels, but only a small fraction (possibly less than a fraction of a percent) is emitted into the environment through incomplete combustion or leakage during storage. Other chemicals, such as pesticides, are used in much smaller quantities but are discharged completely and directly into the environment; i.e., 100% is emitted. At the other extreme, there are chemical intermediates that may be produced in large quantities but are emitted in only minuscule amounts (except during an industrial accident). It is difficult to compare the amounts emitted from these various categories, because they are highly variable and episodic. It is essential, however, to consider this factor; many toxic chemicals have no significant adverse impact, because they enter the environment in negligible quantities. Central to the importance of quantity is the adage first stated by Paracelsus, nearly five centuries ago, that the dose makes the poison. This can be restated in the form that all chemicals are toxic if administered to the victim in sufficient quantities. A corollary is that, in sufficiently small doses, all chemicals are safe. Indeed, certain metals and vitamins are essential to survival. The general objective of environmental regulation or “management” must therefore be to ensure that the quantity of a specific substance entering the environment is not excessive. It need not be zero; indeed, it is impossible to achieve zero. Apart from cleaning up past mistakes, the most useful regulatory action is to reduce emissions to acceptable levels and thus ensure that concentrations and exposures are tolerable. Not even the EPA can reduce the toxicity of benzene. It can only reduce emissions. This implies knowing what the emissions are and where they come from. This is the focus of programs such as the Toxics Release Inventory (TRI) in the U.S.A. or the National Pollutant Release Inventory (NPRI) system in Canada. There are similar programs in Europe, Australia, and Japan. Regrettably, the data are often incomplete. A major purpose of this book is to give the reader the ability to translate emission rates into environmental concentrations so that the risk resulting from exposure to these concentrations can be assessed. When this can be done, it provides an incentive to improve release inventories. 3.2.2

Persistence

The second factor is the chemical’s environmental persistence, which may also be expressed as a lifetime, half-life, or residence time. Some chemicals, such as DDT or the PCBs, may persist in the environment for several years by virtue of their resistance to transformation by degrading processes of biological and physical origin. They may have the opportunity to migrate widely throughout the environment and reach vulnerable organisms. Their persistence results in the possibility of establishing

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relatively high concentrations. This arises because, in principle, the amount in the environment (kilograms) can be expressed as the product of the emission rate into the environment (kilograms per year) and the residence time of the chemical in the environment (years). Persistence also retards removal from the environment once emissions are stopped. A legacy of “in place” contamination remains. This is the same equation that controls a human population. For example, the number of Canadians (about 30 million) is determined by the product or the rate at which Canadians are born (about 0.4 million per year) and the lifetime of Canadians (about 75 years). If Canadians were less persistent and lived for only 30 years, the population would drop to 12 million. Intuitively, the amount (and hence the concentration) of a chemical in the environment must control the exposure and effects of that chemical on ecosystems, because toxic and other adverse effects, such as ozone depletion, are generally a response to concentration. Unfortunately, it is difficult to estimate the environmental persistence of a chemical. This is because the rate at which chemicals degrade depends on which environmental media they reside in, on temperature (which varies diurnally and seasonally), on incidence of sunlight (which varies similarly), on the nature and number of degrading microorganisms that may be present, and on other factors such as acidity and the presence of reactants and catalysts. This variable persistence contrasts with radioisotopes, which have a half-life that is fixed and unaffected by the media in which they reside. In reality, a substance experiences a distribution of half-lives, not a single value, and this distribution varies spatially and temporally. Obviously, long-lived chemicals, such as PCBs, are of much greater concern than those, such as phenol, that may persist in the aquatic environment for only a few days as a result of susceptibility to biodegradation. Some estimate of persistence or residence time is thus necessary for priority setting purposes. Organo-halogen chemicals tend to be persistent and are therefore frequently found on priority lists. Later in this book, we develop methods of calculating persistence. 3.2.3

Bioaccumulation

The third factor is potential for bioaccumulation (i.e., uptake of the chemical by organisms). This is a phenomenon, not an effect; thus bioaccumulation per se is not necessarily of concern. It is of concern that bioaccumulation may cause toxicity to the affected organism or to a predator or consumer of that organism. Historically, it was the observation of pesticide bioaccumulation in birds that prompted Rachel Carson to write Silent Spring in 1962, thus greatly increasing public awareness of environmental contamination. As we discuss later, some chemicals, notably the hydrophobic or “water-hating” organic chemicals, partition appreciably into organic media and establish high concentrations in fatty tissue. PCBs may achieve concentrations (i.e., they bioconcentrate) in fish at factors of 100,000 times the concentrations that exist in the water in which the fish dwell. For some chemicals (notably PCBs, mercury, and DDT), there is also a food chain effect. Small fish are consumed by larger fish, at higher trophic levels, and by other animals such as gulls, otters, mink, and humans. These chemicals

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may be transmitted up the food chain, and this may result in a further increase in concentration such that they are biomagnified. Bioaccumulation tendency is normally estimated using an organic phase-water partition coefficient and, more specifically, the octanol-water partition coefficient. This, in turn, can be related to the solubility of the chemical in the water. Clearly, chemicals that bioaccumulate, bioconcentrate, and biomagnify have the potential to travel down unexpected pathways, and they can exert severe toxic effects, especially on organisms at higher trophic levels. The importance of bioaccumulation may be illustrated by noting that, in water containing 1 ng/L of PCB, the fish may contain 105 ng/kg. A human may consume 1000 L of water annually (containing 1000 ng of PCB) and 10 kg of fish (containing 106 ng of PCB), thus exposure from fish consumption is 1000 times greater than that from water. Particularly vulnerable are organisms such as certain birds and mammals that rely heavily on fish as a food source. 3.2.4

Toxicity

The fourth factor is the toxicity of the chemical. The simplest manifestation of toxicity is acute toxicity. This is most easily measured as a concentration that will kill 50% of a population of an aquatic organism, such as fish or an invertebrate (e.g., Daphnia magna), in a period of 24–96 hours, depending on test conditions. When the concentration that kills (or is lethal to) 50% (the LC50) is small, this corresponds to high toxicity. The toxic chemical may also be administered to laboratory animals such as mice or rats, orally or dermally. The results are then expressed as a lethal dose to kill 50% (LD50) in units of mg chemical/kg body weight of the animal. Again, a low LD50 corresponds to high toxicity. More difficult, expensive, and contentious are chronic, or sublethal, tests that assess the susceptibility of the organism to adverse effects from concentrations or doses of chemicals that do not cause immediate death but ultimately may lead to death. For example, the animal may cease to feed, grow more slowly, be unable to reproduce, become more susceptible to predation, or display some abnormal behavior that ultimately affects its life span or performance. The concentrations or doses at which these effects occur are often about 1/10th to 1/100th of those that cause acute effects. Ironically, in many cases, the toxic agent is also an essential nutrient, so too much or too little may cause adverse effects. Although most toxicology is applied to animals, there is also a body of knowledge on phytotoxicity, i.e., toxicity to plants. Plants are much easier to manage, and killing them is less controversial. Tests also exist for assessing toxicity to microorganisms. It is important to emphasise that toxicity alone is not a sufficient cause for concern about a chemical. Arsenic in a bottle is harmless. Disinfectants, biocides, and pesticides are inherently useful because they are toxic. The extent to which the organism is injured depends on the inherent properties of the chemical, the condition of the organism, and the dose or amount that the organism experiences. It is thus misleading to classify or prioritize chemicals solely on the basis of their inherent toxicity, or on the basis of the concentrations in the environment or exposures. Both must be considered. A major task of this book is to estimate exposure. A healthy tension often

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exists between toxicologists and chemists about the relative importance of toxicity and exposure, but fundamentally this argument is about as purposeful as squabbling over whether tea leaves or water are the more important constituents of tea. Most difficult is the issue of genotoxicity, including carcinogenicity, and teratogenicity. In recent years, a battery of tests has been developed in which organisms ranging from microorganisms to mammals are exposed to chemicals in an attempt to determine if they can influence genetic structure or cause cancer. A major difficulty is that these effects may have long latent periods, perhaps 20 to 30 years in humans. The adverse effect may be a result of a series of biochemical events in which the toxic chemical plays only one role. It is difficult to use the results of short-term laboratory experiments to deduce reliably the presence and magnitude of hazard to humans. There may be suspicions that a chemical is producing cancer in perhaps 0.1% of a large human population over a period of perhaps 30 years, an effect that is very difficult (or probably impossible) to detect in epidemiological studies. But this 0.1% translates into the premature death of 30,000 Canadians per year from such a cancer, and is cause for considerable concern. Another difficulty is that humans are voluntarily and involuntarily exposed to many toxic chemicals, including those derived from smoking, legal and illicit drugs, domestic and occupational exposure, as well as environmental exposure. Although research indicates that multiple toxicants act additively when they have similar modes of action, there are cases of synergism and antagonism. Despite these difficulties, a considerable number of chemicals have been assessed as being carcinogenic, mutagenic, or teratogenic, and it is even possible to assign some degree of potency to each chemical. Such chemicals usually rank high on priority lists. As was discussed earlier, endocrine modulating substances are of more recent concern. It seems likely that ingenious toxicologists will find other subtle toxic effects in the future. 3.2.5

Long-Range Transport

As lakes go, Lake Superior is fairly pristine, since there is relatively little industry on its shores. In the U.S. part of this lake is an island, Isle Royale, which is a protected park and is thus even more pristine. In this island is a lake, Siskiwit Lake, which cannot conceivably be contaminated. No responsible funding agency would waste money on the analysis of fish from that lake for substances such as PCBs. Remarkably, perceptive researchers detected substantial concentrations of PCBs. Similarly, surprisingly high concentrations have been detected in wildlife in the Arctic and Antarctic. Clearly, certain contaminants can travel long distances through the atmosphere and oceans and are deposited in remote regions. This potential for long-range transport (LRT) is of concern for several reasons. There is an ethical issue when the use of a chemical in one nation (which presumably enjoys social or economic benefit from it) results in exposure in other downwind nations that derive no benefit, only adverse effects. This transboundary pollution issue also applies to gases such as SO2, which can cause acidification of poorly buffered lakes at distant locations. A regulatory agency may then be in the position of having little or no control over exposures experienced by its public. The political implications are obvious.

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There is therefore a compelling incentive to identify those chemicals that can undertake long-range transport and implement international agreements to control them. A start on this process has been made recently by the United Nations Environment Program (UNEP), which has identified 12 substances or groups for international regulations or bans. These substances, listed in Table 3.3, are also identified as persistent, bioaccumulative, and toxic. Others are scheduled for restriction or reduction. They may represent merely the first group of chemicals that will be subject to international controls. Most contentious of the 12 is DDT, which is still widely and beneficially used for malaria control. Table 3.3 Substances Scheduled for Elimination, Restriction, or Reduction by UNEP

Scheduled for Elimination

Scheduled for Restriction

Scheduled for Reduction

Aldrin

DDT

PAHs

Chlordane

Hexachlorocyclohexanes

Dioxins/furans

DDT

Polychlorinated biphenyls

Hexachlorobenzene

Dieldrin Endrin Heptachlor Hexabromobiphenyl Hexachlorobenzene Mirex Polychlorinated biphenyls Toxaphene

3.2.6

Other Effects

Finally, there is a variety of other adverse effects that are of concern, including • the ability to influence atmospheric chemistry (e.g., freons) • alteration in pH (e.g., oxides of sulfur and nitrogen causing acid rain) • unusual chemical properties such as chelating capacity, which alters the availability of other chemicals in the environment • interference with visibility • odor (e.g., from organo-sulfur compounds) • color (e.g., from dyes) • the ability to cause foaming in rivers (e.g., detergents or surfactants) • formation of toxic metabolites or degradation products

3.2.7

Selection Procedures

A common selection procedure involves scoring these factors on some numeric hazard scale. The factors then may be combined to give an overall factor and

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determine priority. This is a subjective process, and it becomes difficult for two major reasons. First, chemicals that are subject to quite different patterns of use are difficult to compare. For example, chemical X may be produced in very large quantities, emitted into the environment, and found in substantial concentrations in the environment, but it may not be believed to be particularly toxic. Examples are solvents such as trichloroethylene or plasticizers such as the phthalate esters. On the other hand, chemical Y may be produced in minuscule amounts but be very toxic, an example being the “dioxins.” Which deserves the higher priority? Second, it appears that the adverse effects suffered by aquatic organisms and other animals, including humans, are the result of exposure to a large number of chemicals, not just to one or two chemicals. Thus, assessing chemicals on a caseby-case basis may obscure the cumulative effect of a large number of chemicals. For example, if an organism is exposed to 150 chemicals, each at a concentration that is only 1% of the level that will cause death, then death will very likely occur, but it cannot be attributed to any one of these chemicals. It is the cumulative effect that causes death. The obvious prudent approach is to reduce exposure to all chemicals to the maximum extent possible. The issue is further complicated by the possibility that some chemicals will act synergistically, i.e., they produce an effect that is greater than additive; or they may act antagonistically, i.e., the combined effect is less than additive. As a result, there will be cases in which we are unable to prove that a specific chemical causes a toxic effect but, in reality, it does contribute to an overall toxic effect. Indeed, some believe that this situation will be the rule rather than the exception. A compelling case can be made that the prudent course of action is for society to cast a fairly wide net of suspicion (i.e., assemble a fairly large list of chemicals) then work to elucidate sources, fate, and effects with the aim of reducing overall exposure of humans, and our companion organisms, to a level at which there is assurance that no significant toxic effects can exist from these chemicals. The risk from these chemicals then becomes small as compared to other risks such as accidents, disease, and exposure to natural toxic substances. This approach has been extended and articulated as the “Precautionary Principle,” the “Substitution Principle,” and the “Principle of Prudent Avoidance.” One preferred approach is to undertake a risk assessment for each chemical. Formal procedures for conducting such assessments have been published, notably by the U.S. Environmental Protection Agency (EPA). The process involves identifying the chemical, its sources, the environment in which it is present, and the organisms that may be affected. The toxicity of the substance is evaluated and routes of exposure quantified. Ultimately, the prevailing concentrations or doses are measured or estimated and compared with levels that are known to cause effects, and conclusions are drawn regarding the proximity to levels at which there is a risk of effect. This necessarily involves consideration of the chemical’s behavior in an actual environment. Risk is thus assessed only for that environment. Risk or toxic effects are thus not inherent properties of a chemical; they depend on the extent to which the chemical reaches the organism.

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3.3 3.3.1

KEY CHEMICAL PROPERTIES AND CLASSES

Key Properties

In Chapter 5, we discuss physicochemical properties in more detail and, in Chapter 6, we examine reactivities. It is useful at this stage to introduce some of these properties and identify how they apply to different classes of chemicals. It transpires that we can learn a great deal about how a chemical partitions in the environment from its behavior in an air-water-octanol (strictly 1-octanol) system as shown later in Figure 3.2. There are three partition coefficients, KAW , KOW , and KOA, only two of which are independent, since KOA must equal KOW/KAW. These can be measured directly or estimated from vapor pressure, solubility in water, and solubility in octanol, but not all chemicals have measurable solubilities because of miscibility. Octanol is an excellent surrogate for natural organic matter in soils and sediments, lipids, or fats, and even plant waxes. It has approximately the same C:H:O ratio as lipids. Correlations are thus developed between soil-water and octanol-water partition coefficients, as discussed in more detail later. An important attribute of organic chemicals is the degree to which they are hydrophobic. This implies that the chemical is sparingly soluble in, or “hates,” water and prefers to partition into lipid, organic, or fat phases. A convenient descriptor of this hydrophobic tendency is KOW. A high value of perhaps one million, as applies to DDT, implies that the chemical will achieve a concentration in an organic medium approximately a million times that of water with which it is in contact. In reality, most organic chemicals are approximately equally soluble in lipid or fat phases, but they vary greatly in their solubility in water. Thus, differences in hydrophobicity are largely due to differences of behavior in, or affinity for, the water phase, not differences in solubility in lipids. The word lipophilic is thus unfortunate and is best avoided. The chemical’s tendency to evaporate or partition into the atmosphere is primarily controlled by its vapor pressure, which is essentially the maximum pressure that a pure chemical can exert in the gas phase or atmosphere. It can be viewed as the solubility of the chemical in the gas phase. Indeed, if the vapor pressure in units of Pa is divided by the gas constant, temperature group RT, where R is the gas constant (8.314 Pa m3/mol K), and T is absolute temperature (K), then vapor pressure can be converted into a solubility with units of mol/m3. Organic chemicals vary enormously in their vapor pressure and correspondingly in their boiling point. Some (e.g., the lower alkanes) that are present in gasoline are very volatile, whereas others (e.g., DDT) have exceedingly low vapor pressures. Partitioning from a pure chemical phase to the atmosphere is controlled by vapor pressure. Partitioning from aqueous solution to the atmosphere is controlled by KAW, a joint function of vapor pressure and solubility in water. A substance may have a high KAW , because its solubility in water is low. Partitioning from soils and other organic media to the atmosphere is controlled by KAO (air/octanol), which is conventionally reported as its reciprocal, KOA. Partitioning from water to organic media, including fish, is controlled by KOW . Substances that display a significant tendency to partition into the air phase over other phases are termed volatile organic chemicals or VOCs. They have high vapor pressures.

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Another important classification of organic chemicals is according to their dissociating tendencies in water solution. Some organic acids, notably the phenols, will form ionic species (phenolates) at high pH. The tendency to ionize is characterized by the acid dissociation constant KA, often expressed as pKA, its negative base ten logarithm. In concert with partitioning characteristics, the other set of properties that determine environmental behavior is reactivity or persistence, usually expressed as a halflife. It is misleading to assign a single number to a half-life, because it depends on the intrinsic properties of the chemical and on the nature of the environment. Factors such as sunlight intensity, hydroxyl radical concentration, the nature of the microbial community, as well as temperature vary considerably from place to place and time to time. Here, we use a semiquantitative classification of half-lives into classes, assuming that average environmental conditions apply. Different classes are defined for air, water, soils, and sediments. The classification is that used in a series of “Illustrated Handbooks” by Mackay, Shiu, and Ma is shown below in Table 3.4. Table 3.4 Classes of Chemical Half-Life or Persistence, Adapted from the Handbooks of Mackay et al., 2000 Class 1 2 3 4 5 6 7 8 9

Mean half–life (hours) 5 17 (~ 1 day) 55 (~ 2 day) 170 (~1 week) 550 (~3 weeks) 1700 (~2 months) 5500 (~8 months) 17000 (~2 years) 55000 (~6 years)

Range (hours) <10 10–30 30–100 100–300 300–1000 1000–3000 3000–10,000 10,000–30,000 >30,000

The half-lives are on a logarithmic scale with a factor of approximately 3 between adjacent classes. It is probably misleading to divide the classes into finer groupings; indeed, a single chemical may experience half-lives ranging over three classes, depending on environmental conditions such as season. We examine, in the following sections, a number of classes of compounds that are of concern environmentally. In doing so, we note their partitioning and persistence properties. The structures of many of these chemicals are given in Figure 3.1. Table 3.5 gives suggested values of these properties for selected chemicals. Figure 3.2 is a plot of log KAW versus log KOW for the chemicals in Table 3.5 on which lines of constant KOA lie on the 45° diagonal. This graph shows the wide variation in properties. Volatile compounds tend to lie to the upper left, water-soluble compounds to the lower left, and hydrophobic compounds to the lower right. Assuming reasonable relative volumes of air (650,000), water (1300), and octanol (1), the percentages in each phase at equilibrium can be calculated. The lines of constant percentages are also shown. Lee and Mackay (1995) have used equilateral triangular diagrams to display the variation in partitioning properties in a format similar to that of Figure 3.2.

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Figure 3.1

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

Structures of selected chemicals of environmental interest (continues).

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ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES

Figure 3.1

(continued)

43

Chemical Name benzene 1,2,4-trimethylbenzene ethylbenzene n-propylbenzene styrene toluene nitrobenzene 2-nitrotoluene 4-nitrotoluene 2,4-dinitrotoluene chlorobenzene 1,4-dichlorobenzene 1,2,3-trichlorobenzene 1,2,3,4-tetrachlorobenzene pentachlorobenzene hexachlorobenzene fluorobenzene bromobenzene iodobenzene n-pentane n-hexane 1,3-butadiene 1,4-cyclohexadiene

Air Water 17 170 17 550 17 550 17 550 5 170 17 550 5 1700 17 55 17 55 17 55 170 1700 550 1700 550 1700 1700 5500 5500 17000 7350 55000 17 170 170 1700 170 1700 17 550 17 550 5 170 5 170

Soil Sediment 550 1700 1700 5500 1700 5500 1700 5500 550 1700 1700 5500 1700 5500 1700 5500 1700 5500 1700 5500 5500 17000 5500 17000 5500 17000 5500 17000 17000 17000 55000 55000 550 1700 5500 17000 5500 17000 1700 5500 1700 5500 550 1700 550 1700

Rat oral LD50 (mg/kg) 930 3550 5460 6040 2650 5000 349 891 1960 268 1110 500 756 1470 11000 3500 4399 2383 1749 90000 30000 5480 130

44

Melting Molar Vapor Aqueous mass (g/mol) pressure (Pa) solubility (g/m3) Log KOW point ((C) 78.11 12700 1780 2.13 5.53 120.2 270 57 3.6 –43.8 106.2 1270 152 3.13 –95 120.2 450 52 3.69 –101.6 104.14 880 300 3.05 –30.6 92.13 3800 515 2.69 –95 123.11 20 1900 1.85 5.6 137.14 17.9 651.42 2.3 –3.85 137.14 0.653 254.4 2.37 51.7 182.14 0.133 270 2.01 70 112.6 1580 484 2.8 –45.6 147.01 130 83 3.4 53.1 181.45 28 21 4.1 53 215.9 4 7.8 4.5 47.5 250.3 0.22 0.65 5 86 284.8 0.0023 0.005 5.5 230 96.104 10480 1430 2.27 –42.21 157.02 552 410 2.99 –30.8 204.01 130 340 3.28 –31.35 72.15 68400 38.5 3.45 –129.7 86.17 20200 9.5 4.11 –95 54.09 281000 735 1.99 –108.9 80.14 9010 700 2.3 –49.2

Degradation Half-lives (h)

Table 3.5 Physical Chemical properties of Selected Organic Chemicals at 25°C Including Estimated Half-Lives Classified as in Table 3.4 and Toxicity Expressed as Oral LD50 to the Rat. These data have been selected from a number of sources, including Mackay et al. (2000), RTECS (2000), and the Hazardous Substances Data Bank (2000).

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dichloromethane trichloromethane carbon tetrachloride tribromomethane bromochloromethane bromodichloromethane 1,2-dichloroethane 1,1,2,2-tetrachloroethane pentachloroethane hexachloroethane 1,2-dichloropropane 1,2,3-trichloropropane chloroethene (vinyl chloride) trichloroethylene tetrachloroethylene methoxybenzene bis(2-chloroethyl)ether bis(2-chloroisopropyl)ether 2-chloroethyl vinyl ether bis(2-chloroethoxy)methane 1-pentanol 1-hexanol benzyl alcohol cyclohexanol benzaldehyde 3-pentanone 2-heptanone cyclohexanone acetophenone

Table 3.5 (continued) 84.94 119.38 153.82 252.75 129.384 163.8 98.96 167.85 202.3 236.74 112.99 147.43 62.5 131.39 165.83 108.15 143.02 171.07 106.55 173.1 88.149 102.176 108.14 100.16 106.12 86.135 114.18 98.144 120.15

26222 26244 15250 727 19600 6670 10540 793 625 50 6620 492 354600 9900 2415 472 206 104 3566 21.6 300 110 12 85 174 4700 500 620 45

13200 8200 800 3100 14778 4500 8606 2962 500 50 2800 1896 2763 1100 150 2030 10200 1700 15000 8100 22000 6000 80 38000 3000 34000 4300 23000 5500

1.25 1.97 2.64 2.38 1.41 2.1 1.48 2.39 2.89 3.93 2 2.63 1.38 2.53 2.88 2.11 1.12 2.58 1.28 1.26 1.5 2.03 1.1 1.23 1.48 0.82 2.08 0.81 1.63

–95 –63.5 –22.9 –8.3 –87.95 –57.1 –35.36 –36 –29 186.1 –100.4 –14.7 –153.8 –73 –19 –37.5 –46.8 –97 –69.7 0 –78.2 –44.6 –15.3 25.15 –55.6 –38.97 –35 –32.1 19.62

1700 1700 17000 1700 550 550 1700 17000 17000 17000 550 550 55 170 550 17 17 17 17 17 55 55 55 55 5 55 55 55 550

1700 1700 1700 1700 550 550 1700 1700 1700 1700 5500 5500 550 5500 5500 550 550 550 550 550 55 55 55 55 55 170 170 170 170

5500 5500 5500 5500 1700 1700 5500 5500 5500 5500 5500 5500 1700 1700 1700 550 550 550 550 550 55 55 55 55 55 170 170 170 170

17000 17000 17000 17000 5500 5500 17000 17000 17000 17000 17000 17000 5500 5500 5500 1700 1700 1700 1700 1700 170 170 170 170 170 550 550 550 550

1600 1000 2350 933 5000 430 750 200 920 5000 1947 505 500 4920 2629 3700 75 240 210 65 3030 720 1230 1400 1300 2410 1670 1540 815

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ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES 45

86.09 102.13 100.12 169.23 93.12 129.16 84.14 122.13 116.1 136.15 138.12 178.2 252.3 228.3 128.19 178.2 106.2 202.3 134.19 142.2 154.2 223.1 223.1 257.5 292 326.4 360.9 498.7

naphthalene phenanthrene p-xylene pyrene benzo(b)thiophene 1-methylnaphthalene biphenyl PCB-7 PCB-15 PCB-29 PCB-52 PCB-101 PCB-153 PCB-209

14100 4500 5100 0.0612 65.19 1.21 10620 0.11 5 0.83 0.0208 0.001 7 x 10–7 5.7 x 10–7 10.4 0.02 1170 0.0006 26.66 8.84 1.3 0.254 0.0048 0.0132 0.0049 0.00109 0.000119 5.02x10–8 31 1.1 214.9488 0.132 130 28 7 1.25 0.06 0.14 0.03 0.01 0.001 10–6

20000 21000 15600 300 36070 6110 3015 3400 958 16600 2300 0.045 0.0038 0.002 3.37 4.57 3.18 5.18 3.12 3.87 3.9 5 5.3 5.6 6.1 6.4 6.9 8.26

0.73 1.24 1.38 3.45 0.9 2.06 1.81 1.89 1.92 1.41 2.2 4.54 6.04 5.61 80.5 101 13.2 156 30.85 –22 71 24.4 149 78 87 76.5 103 305.9

–92.8 –95 –42.8 52.8 –6.3 –14.85 –38 122.4 –3.44 77 159 216.2 175 255 17 55 17 170 170 17 55 170 170 550 1700 1700 5500 55000

55 55 17 5 5 55 55 55 55 55 55 55 170 170 170 550 550 1700 550 170 170 5500 5500 17000 55000 55000 55000 55000

55 55 55 55 170 170 55 55 55 55 55 550 1700 1700 1700 5500 1700 17000 1700 1700 550 17000 17000 55000 55000 55000 55000 55000

170 170 55 170 170 550 1700 170 170 170 170 5500 17000 17000 5500 17000 5500 55000 5500 5500 1700 17000 17000 55000 55000 55000 55000 55000

550 550 170 550 1700 1700 5500 550 550 550 550 17000 55000 55000

2400 n/a 4300 n/a 2200 1840 3280 n/a n/a n/a n/a n/a n/a n/a

2900 9370 7872 2000 250 331 1400 1700 6400 2250 891 8000 n/a n/a

46

vinyl acetate propyl acetate methyl methacrylate diphenylamine aniline quinoline thiophene benzoic acid hexanoic acid phenylacetic acid salicylic acid anthracene benzo[a]pyrene chyrsene

Table 3.5 (continued)

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total PCB dibenzo-p-dioxin 2,3,7,8-tetraCDD 1,2,3,4,7,8-hexaCDD 1,2,3,4,6,7,8-heptaCDD OCDD dibenzofuran 2,8-dichlorodibenzofuran 2,3,7,8-tetrachlorodibenzofuran octachlorodibenzofuran 4-chlorophenol 2,4-dichlorophenol 2,3,4-trichlorophenol 2,4,6-trichlorophenol 2,3,4,6-tetrachlorophenol pentachlorophenol 2,4-dimethylphenol p-cresol dimethylphthalate (DMP) diethylphthalate (DEP) dibutylphthalate (DBP) butyl benzyl phthalate di-(2-ethylhexyl)-phthalate (DEHP) aldicarb aldrin carbaryl carbofuran chloropyrifos

Table 3.5 (continued)

2x10–6 5x10–10 20 12 1 1.25 0.28 0.00415 13.02 13 0.22 0.22 0.00187 0.00115 1.33x10–5 0.004 0.005 0.0000267 0.00008 0.00227

190.25 364.93 201.22 221.3 350.6

0.0009 0.055 0.0000002 5.1x10–9 7.5x10–10 1.1x10–10 0.3 0.00039

326 184 322 391 425.2 460 168.2 237.1 306 443.8 128.56 163 197.45 197.45 231.89 266.34 122.17 108.13 194.2 222.26 278.34 312.39 390.54 1.1 6.50 2.36 2.32 4.92

6.6 4.3 6.8 7.8 8 8.2 4.31 5.44 6.1 8 2.4 3.2 3.8 3.69 4.45 5.05 2.35 1.96 2.12 2.47 4.72 4.68 5.11 99 104 142 151 41

0 123 305 273 265 322 86.5 184 227 258 43 44 79 69.5 70 190 26 34.8 5 –40.5 –35 –35 –50 5 55 55 5 17

550 5500 170 170 170

1700 17000 550 550 170

5500 55000 500000 55 55 1700 170 550 17000 550 1700 55000 550 1700 55000 550 5500 55000 55 170 1700 170 550 5500 170 550 17000 550 5500 55000 55 550 550 55 550 550 170 170 1700 170 170 1700 550 550 1700 550 550 1700 17 55 170 5 17 55 170 170 550 170 170 550 55 170 550 55 170 550 55 170 550 17000 55000 1700 1700 1700

500000 5500 55000 55000 55000 55000 5500 17000 55000 55000 1700 1700 5500 5500 5500 5500 550 170 1700 1700 1700 1700 1700

0.5 39 230 5 82

1900 1220 0.02 0.8 6.325 1 n/a n/a n/a n/a 500 2830 2800 2800 140 210 2300 207 2400 8600 8000 13500 25000

ENVIRONMENTAL CHEMICALS AND THEIR PROPERTIES

6000 0.02 120 351 0.73

0.024 0.865 1.93x10–5 4.42x10–6 2.4x10–6 7.4x10–8 4.75 0.0145 4.19x10–4 1.16x10–6 27000 4500 500 434 183 14 8795 20000 4000 1080 11.2 2.69 0.285

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0.0004 0.000866 0.00002 0.0005 0.008 0.00374 0.003 0.053 0.001 0.00013 0.0001 0.0006 0.002 0.00004 0.00008 0.0045 0.00031 0.0042 8.5x10–6 0.015 0.00133

409.8 319 354.5 380.93 304.36 290.85 290.85 373.4 330.36 345.7 545.59 291.27 263.5 215.68 221.04 221.04 214.6 283.8 201.7 335.5 240.4

4500 620 430 5 0.5 30

0.056 0.04 0.0055 0.17 60 7.3 1 0.056 145 0.045 0.000065 12.4 25 30 400 2.21 3.94 3.13 2.18 5.34 1.73

6 5.7 6.19 5.2 3.3 3.7 3.81 5.27 2.8 5.08 6.9 3.8 3 2.75 2.81 114 94 0 225 48.5 145

103 88 108.5 176 0 112 157 95 2.9 86 485 6 37 174 140.5 55 17 170 55 170 170

55 170 170 55 550 1040 1420 55 17 17 170 17 17 5 17 550 170 1700 550 1700 170

17000 55000 5500 17000 1700 17000 3364 550 55 170 170 550 550 550 55 550 170 1700 1700 1700 550

17000 55000 17000 17000 1700 17000 1687 1700 55 1700 55000 550 550 1700 550 1700 1700 5500 5500 5500 1700

55000 55000 55000 55000 5500 55000 55000 5500 550 5500 55000 1700 1700 1700 1700

1039 650 2200 971 1930 560

500 880 87 38.3 66 76 177 40 290 1855 235 2 6.01 672 375

48

cis-chlordane p,p’-DDE p,p’-DDT dieldrin diazinon γ–HCH (lindane) α–HCH heptachlor malathion methoxychlor mirex parathion methyl parathion atrazine 2-(2,4-dichlorophenoxy) acetic acid dicamba mecoprop metolachlor simazine trifluralin thiram

Table 3.5 (continued)

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Figure 3.2

49

Plot of log KAW vs. log KOW for the chemicals in Table 3.5 on which dotted lines of constant KOA line on the 45° diagonal. This graph shows the wide variation in properties. Volatile compounds tend to lie to the upper left, water-soluble compounds to the lower left, and hydrophobic compounds to the lower right. The thicker lines represent constant percentages present at equilibrium in air, water, and octanol phases, assuming a volume ratio of 656,000:1300:1, respectively. Modified from Gouin et al. (2000).

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3.3.2

3.3.2.1

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

Chemical Classes (see Fig. 3.1 for structures and Table 3.5 for properties) Hydrocarbons

Hydrocarbons are naturally occurring chemicals present in crude oil and natural gas. Some are formed by biogenic processes in vegetation, but most contamination comes from oil spills, effluents from petroleum and petrochemical refineries, and the use of fuels for transportation purposes. The alkanes can be separated into classes of normal, branched (or iso) species and cyclic alkanes, which range in molar mass from methane or natural gas to waxes. They are usually sparingly soluble in water. For example, hexane has a solubility of approximately 10 g/m3. This solubility falls by a factor of about 3 or 4 for every carbon added. The branched and cyclic alkanes tend to be more soluble in water, apparently because they have smaller molecular areas and volumes. Highly branched or cyclic alkanes such as terpenes are produced by vegetation. They are often sweet smelling and tend to be very resistant to biodegradation. The alkenes or olefins are not naturally occurring to any significant extent. They are mainly used as petrochemical intermediates. The alkynes, of which ethyne or acetylene is the first member, are also chemical intermediates that are rarely found in the environment. These unsaturated hydrocarbons tend to be fairly reactive and short-lived in the environment, whereas the alkanes are more stable and persistent. Of particular environmental interest are the aromatics, the simplest of which is benzene. The aromatics are relatively soluble in water, for example, benzene has a solubility of 1780 g/m3. They are regarded as fairly toxic and often troublesome compounds. A variety of substituted aromatics can be obtained by substituting various alkyl groups. For example, methyl benzene is toluene. When two benzene rings are fused, the result is naphthalene, which is also a chemical of considerable environmental interest. Subsequent fusing of benzene rings to naphthalene leads to a variety of chemicals referred to as the polycyclic aromatic hydrocarbons or polynuclear aromatic hydrocarbons (PAHs). These compounds tend to be formed when a fuel is burned with insufficient oxygen. They are thus present in exhaust from engines and are of interest because many are carcinogenic. Biphenyl is a hydrocarbon that is not of much importance as such, but it forms an interesting series of chlorinated compounds, the PCBs or polychlorinated biphenyls, which are discussed later. 3.3.2.2

Halogenated Hydrocarbons

If the hydrogen in a hydrocarbon is substituted by chlorine (or less frequently by bromine, fluorine, or iodine), the resulting compound tends to be less flammable, more stable, more hydrophobic, and more environmentally troublesome. Replacing a hydrogen with a chlorine usually causes an increase in molar volume and area and a corresponding decrease in solubility by a factor of about 3. The stability of many of these compounds makes them invaluable as solvents, examples being methylene chloride and tetrachloroethylene. The fluorinated and

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chlorofluoro compounds are very stable and are used as refrigerants. Because these molecules are quite small, they are fairly soluble in water and are therefore able to penetrate the tissues of organisms quite readily. They are thus used as anaesthetics and narcotic agents. The chlorinated aromatics are a particularly interesting group of chemicals. The chlorobenzenes are biologically active. 1,4 or paradichlorobenzene is widely used as a deodorant and disinfectant. The polychlorinated biphenyls, or PCBs, and their brominated cousins, the PBBs, are notorious environmental contaminants, as are chlorinated terpenes such as toxaphene, which is a very potent and long-lived insecticide. Many of the early pesticides, such as DDT, mirex, and chlordane, are chlorinated hydrocarbons. They possess the desirable properties of stability and a high tendency to partition out of air and water into the target organisms. Thus, application of a pesticide results in protection for a prolonged time. As Rachel Carson demonstrated in Silent Spring, the problem is that these chemicals persist long enough to affect non-target organisms and to drift throughout the environment, causing widespread contamination. Fluorinated chemicals also possess considerable stability and, because the fluorine atom is lighter than chlorine, they are generally more volatile. Polyfluorinated substances are very stable in the environment as a result of the strong C-F bond. Brominated chemicals are also stable, but with reduced volatility. A major use of brominated substances is in fire retardants, specifically polybrominated diphenyl ethers. 3.3.2.3

Oxygenated Compounds

The most common oxygenated organic compounds are the alcohols, ethanol being among the most widely used. Others are octanol, which is a convenient analytical surrogate for fat, and glycerol is of interest because it forms the backbone of fat molecules by esterification with fatty acids to form glycerides. The phenols consist of an aromatic molecule in which a hydrogen is replaced by an OH group. They are acidic and tend to be biologically disruptive. Phenol, or carbolic acid, was the first disinfectant. Substituting chlorines on phenol tends to increase the toxic potency of the substance and its tendency to ionize, i.e., its pKa is reduced. Pentachlorophenol (PCP) is a particularly toxic chemical and has been widely used for wood preservation. The ketones such as acetone, and aldehydes such as formaldehyde, are fairly reactive in the environment and can be of concern as atmospheric contaminants in regions close to sources of emission. Much of the smog problem is attributable to aldehydes formed in combustion processes. Organic acids such as acetic acid are also fairly reactive. They are not usually regarded as an environmental problem, but trifluoroacetic acid, which is formed by combustion of freons and from some pesticides, is very persistent. Some chlorinated organic acids, e.g., 2,4-D, are potent herbicides. Longer-chain acids, such as stearic acid, are mainly of interest because they esterify with glycerol to form fats. Humic and fulvic acids are of considerable environmental importance. These are substances

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of complex and variable structure that are naturally present in soils, water, and sediments. They are the remnants of living organic materials, such as wood, that has been subjected to prolonged microbial conversion. These acids are sparingly soluble in water, but the solubility can be increased at high pH. The esters or “salts” or organic acids and alcohols tend to be relatively innocuous and short-lived in most cases. A notable exception is the phthalate esters, which are very stable oily substances and are invaluable additives (plasticizers) for plastics, rendering them more flexible. Notable among the phthalate esters is diethylhexylphthalate (DEHP), the ester with two molecules of 2 ethylhexanol. The other esters of interest are the glycerides—for example, glyceryl trioleate, the ester of glycerine and oleic acid. This chemical has similar properties to fat and has been suggested as a convenient surrogate for measuring fat to water partitioning. The “dioxins” and “furans” are two series of organic compounds that have become environmentally notorious. The chlorinated dibenzo-p-dioxins were never produced intentionally but are formed under combustion conditions when chlorine is present. They form a series of very toxic chemicals, the most celebrated of which is 2,3,7,8 tetrachlorodibenzo-p-dioxin (TCDD). TCDD is possibly the most toxic chemical to mammals. A dose of 2 µg of TCDD per kg of body weight is sufficient to kill small rodents. A related series of chemicals is the dibenzofurans, which are similar in properties to the dioxins. It appears that molecules that are long and flat, with chlorine atoms strategically located at the ends, are particularly toxic. Examples are the chloronaphthalenes, DDT, the PCBs, and chlorinated dibenzo-p-dioxins and dibenzofurans. Other oxygenated compounds of interest include carbohydrates, cellulose, and lignins, which occur naturally. 3.3.2.4

Nitrogen Compounds

Nitrogen compounds of environmental interest include amines, amides, pyridines, quinolines, and amino acids, and various nitro compounds including nitro polycyclicaromatics and nitroso compounds. Many of these compounds occur naturally, are quite toxic, and are difficult to analyze. 3.3.2.5

Sulfur Compounds

Sulfur compounds, including thiols, thiophenes, and mercaptans, are well known because of their strong odor. One of the most prevalent classes of synthetic organic chemicals is the alkyl benzene sulfonates, which are widely used in detergents. 3.3.2.6

Phosphorus Compounds

Phosphorus compounds play a key role in energy transfer in organisms. Organophosphate compounds have been developed as pesticides (e.g., chloropyrifos), which have the very desirable properties of high biological activity but relatively short environmental persistence. They have therefore largely replaced organo-chlorine compounds in agriculture.

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53

Arsenic Compounds

Arsenic, which behaves somewhat similarly to phosphorus, is inadvertently liberated in mineral processing and has a long and celebrated history as a poison. It usually exists in anionic and organic forms. 3.3.2.8

Metals

Most metals are essential for human life in small quantities but can be toxic if administered in excessive dosages. The metals of primary toxicological interest here are those that form organo-metallic molecules. Notable is mercury, which can exist as the element in various ionic and organometallic forms. Other metals such as lead and tin behave similarly. A formidable literature exists on the behavior, fate, and effects of the “heavy” metals such as lead, copper, and chromium. These metals often have a complex environmental chemistry and toxicology that vary considerably, depending on their ionic state as influenced by acidity and redox status. 3.3.2.9

Pharmaceuticals and Personal Care Products

Considerable quantities of drugs are used by humans and for veterinary purposes on livestock. Antibiotics and steroids are examples. These substances are excreted and may pass through sewage treatment plants or enter soils or groundwater following agricultural use. There is a growing concern that these substances may have adverse effects or may cause an increase in antibiotic resistance in bacteria. Among personal care products of concern are detergents, fabric softeners, fragrances, and certain solvents. They may evaporate or be discharged with sewage, which may or may not be adequately treated. 3.3.2.10

Other Chemicals

Several other chemicals are of environmental concern including ozone, radon, chlorine, organic and inorganic sulfides and cyanides, as well as the indeterminate broad class of “conventional” pollutants or indicators of pollution such as biochemical oxygen demand (BOD) and chemical oxygen demand (COD). Finally, certain mineral substances such as asbestos are of concern, more because of their physical structure than their chemical composition. 3.3.2.11

The Future

It would be unwise to assume that current lists of priority chemicals are complete and will remain static. It may be that the chemicals on the lists reflect our present ability to detect and analyze them rather than their real environmental significance. The prevalence of organo-chlorine chemicals on lists is in part the result of the sensitive electron capture detector. As new analytical methods emerge, new chemicals will presumably be found, and priorities will change. Happy hunting grounds for environmental chemists include combustion gases, dyes, mine tailings, effluents

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from pulp and paper operations (especially those involving chlorine bleaching), landfill leachates, and a vast assortment of products of metabolic conversion in organisms ranging from bacteria to humans.

3.4

CONCLUDING EXAMPLE

Select five substances from Table 3.5 that range in their values of vapor pressure, aqueous solubility, and log KOW. Calculate KAW as vapor pressure (Pa) --------------------------------------------3 solubility (g/m )

( molar mass (g/mol ) ------------------------------------------------RT

where R is 8.314 Pa m3/mol K, and T is absolute temperature (298 K). Calculate how 100 kg of each of these chemicals would partition at equilibrium between three phases namely, 1 m3 octanol (representing perhaps 100 m3 of soil) 5000 m3 water 106 m3 air

Calculate all the concentrations and amounts (which should add to 100 kg!) and discuss briefly how each substance is behaving, i.e., its partitioning preference.

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CHAPTER

4

The Nature of Environmental Media 4.1

INTRODUCTION

The objective of this chapter is to present a qualitative description of environmental media, highlighting some of their more important properties. This is done because the fate of a chemical depends on two groups of properties: those of the chemical and those of the environment in which it resides. We find it useful to assemble “evaluative” environments, which are used in later calculations. We can consider, for example, an area of 1 × 1 km, consisting of some air, water, soil, and sediment. Volumes and properties can be assigned to these media, which are typical but purely illustrative and will, of course, require modification if chemical fate in a specific region is to be treated. The sequence is to treat the atmosphere, the hydrosphere (i.e., water), and then the lithosphere (bottom sediments and terrestrial soils), each with its resident biotic community. It transpires that it is convenient to define two evaluative environments. First is a simple four-compartment system that is easily understood and illustrates the application of the general principles of environmental partitioning. Second is a more complex, eight-compartment system that is more representative of real environments. It is correspondingly more demanding of data and leads to more lengthy calculations. The environments or “unit worlds” are depicted in Figure 4.1. Details are discussed by Neely and Mackay, 1982.

4.2 4.2.1

THE ATMOSPHERE

Air

The layer of the atmosphere that is in most intimate contact with the surface of the Earth is the troposphere, which extends to a height of about 10 km. The temperature, density, and pressure of the atmosphere fall steadily with increasing height, 55

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Figure 4.1

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

Evaluative environments.

which is a nuisance in subsequent calculations. If we assume uniform density at a pressure of one atmosphere, then the entire troposphere can be viewed as being compressed into a height of about 6 km. Exchange of matter from the troposphere

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through the tropopause to the stratosphere is a relatively slow process and is rarely important in environmental calculations, except in the case of chemicals such as the freons, which catalyze the destruction of stratospheric ozone, thus facilitating the penetration of UV light to the Earth’s surface. A reasonable atmospheric volume over our 1 km square world is thus 1000 × 1000 × 6000 or 6 × 109 m3. If our environmental model is concerned with a localized situation (e.g., a state, province, or metropolitan region), it is unlikely that most pollutants would manage to penetrate higher than about 500 to 2000 m during the time the air resides over the region. It therefore may be appropriate to reduce the height of the atmosphere to 500 to 2000 m in such cases. In extreme cases (e.g., over small ponds or fields), the accessible mixed height of the atmosphere may be as low as 10 m. The modeler must make a judgement as to the volume of air that is accessible to the chemical during the time that the air resides in the region of interest. 4.2.2

Aerosols

The atmosphere contains a considerable amount of particulate matter or aerosols that are important in determining the fate of certain chemicals. These particles may range in size and composition from water in the form of fog or cloud droplets to dust particles from soil and smoke from combustion. They vary greatly in size, but a diameter of a few µm is typical. Larger particles tend to deposit fairly rapidly. The concentration of these aerosols is normally reported in µg/m3. A rural area may have a concentration of about 5 µg/m3, and a fairly polluted urban area a concentration of 100 µg/m3. For illustrative purposes, we can assume that the particles have a density of 1.5 g/cm3 and are present at a concentration of 30 µg/m3. This corresponds to volume fraction of particles of 2 × 10–11. The density of these particles is usually unknown, thus the volume fractions are only estimates. It is, however, convenient for us to calculate this amount in the form of a volume fraction. In an evaluative air volume of 6 × 109 m3, there is thus 0.12 m3 or 120 L of solid material. These aerosols are derived from numerous sources. Some are mineral dust particles generated from soils by wind or human activity. Some are mainly organic in nature, being derived from combustion sources such as vehicle exhaust or wood fires, i.e., smoke. Some are generated from oxides of sulfur and nitrogen. Some “secondary” aerosols are formed by condensation as a result of oxidation of hydrocarbons in the atmosphere to less volatile species. These hydrocarbons can be generated by human activity such as fuel use, or they can be of natural origin. Forests often generate large quantities of isoprene that oxidize to give a blue haze, hence the terms “smokey” or “blue” mountains. These aerosols also contain quantities of water, the amount of which depends on the prevailing humidity. 4.2.3

Deposition Processes

Aerosol particles have a very high surface area and thus absorb (or adsorb or sorb) many pollutants, especially those of very low vapor pressure, such as the PCBs or polyaromatic hydrocarbons. In the case of benzo(a)pyrene, almost all the chemical present in the atmosphere is associated with particles, and very little exists in the gas phase. This is important, because chemicals associated with aerosol particles

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are subject to two important deposition processes. First is dry deposition, in which the aerosol particle falls under the influence of gravity to the Earth’s surface. This falling velocity, or deposition velocity, is quite slow and depends on the turbulent condition of the atmosphere, the size and properties of the aerosol particle, and the nature of the ground surface, but a typical velocity is about 0.3 cm/s or 10.8 m/h. The result is deposition of 10.8 m/h × 2 × 10–11 (volume fraction) × 106 m2 or 0.000216 m3/h or 1.89 m3/year. Second, the particles may be scavenged or swept out of the air by wet deposition with raindrops. As it falls, each raindrop sweeps through a volume of air about 200,000 times its volume prior to landing on the surface. Thus, it has the potential to remove a considerable quantity of aerosol from the atmosphere. Rain is therefore often highly contaminated with substances such as PCBs and PAHs. There is a common fallacy that rain water is pure. In reality, it is often much more contaminated than surface water. Typical rainfall rates lie in the range 0.3 to 1 m per year but, of course, vary greatly with climate. We adopt a figure of 0.8 m/year for illustrative purposes. This results in the scavenging of 200,000 × 0.8 m/year × 2 × 10–11 × 106 m2 or 3.2 m3/year, about twice the dry deposition. Snow is an even more efficient scavenger of aerosol particles. It appears that one volume of snow (as solid ice) may scavenge about one million volumes of atmosphere, five times more than rain, presumably because of its flaky nature with a high surface area and a slower, more tortuous downward journey. In the four-compartment evaluative environment, we ignore aerosols, but we include them in the eight-compartment version.

4.3 4.3.1

THE HYDROSPHERE OR WATER

Water

Seventy percent of the Earth’s surface is covered by water. In some evaluative models, the area of water is taken as 70% of the 1 million m2 or 700,000 m2. Similarly to the atmosphere, only near-surface water is accessible to pollutants in the short term. In the oceans, this depth is about 100 m but, since most situations of environmental interest involve fresh or estuarine water, it is more appropriate to use a shallower water depth of perhaps 10 m. This yields a water volume of about 7 × 106 m3. If the aim is to mimic the proportions of water and soil in a political jurisdiction, such as a state or province, the area of water will normally be considerably reduced to perhaps 10% of the total, or about 106 m3. We normally regard the water as being pure, i.e., containing no dissolved electrolytes, but we do treat its content of suspended particles. 4.3.2

Particulate Matter

Particulate matter in the water plays a key role in influencing the behavior of chemicals. Again, we do not normally know if the chemical is absorbed or adsorbed to the particles. We play it safe and use the vague term sorbed. A very clear natural water may have a concentration of particles as low as 1 g/m3 or the equivalent 1 mg/L.

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In most cases, however, the concentration is higher, in the range of 5 to 20 g/m3. Very turbid, muddy waters may have concentrations over 100 g/m3. Assuming a concentration of 7.5 g/m3 and a density of 1.5 g/cm3 gives a volume fraction of particles of about 5 × 10-6. Thus, in the 7 × 106 m3 of water, there is 35 m3 of particles. This particulate matter consists of a wide variety of materials. It contains mineral matter, which may be clay or silica in nature. It also contains dead or detrital organic matter, which is often referred to as humin, humic acids, and fulvic acids or, more vaguely, as organic matter. It is relatively easy to measure the total concentration of organic carbon (OC) in water or particles by converting the carbon to carbon dioxide and measuring the amount spectroscopically. Alternatively, the solids can be dried to remove water, then heated to ignition temperatures to burn off organic matter. The loss is referred to as loss on ignition (LOI) or as organic matter (OM). Thus, there are frequent reports of the amount of dissolved organic carbon (DOC) or total organic carbon (TOC) in water. These humic and fulvic acids have been the subject of intense study for many years. They are organic materials of variable composition that probably originate from the ligneous material present in vegetation. They contain a variety of chemical structures including substituted alkane, cycloalkane, and aromatic groups, and they have acidic properties imparted by phenolic or carboxylic acids. They are, therefore, fairly soluble in alkaline solution in which they are present in ionic form, but they may be precipitated under acidic conditions. The operational difference between humic and fulvic acids is the pH at which precipitation occurs. It is important to discriminate between organic matter (OM) and organic carbon (OC). Typically, OM contains 50 to 60% OC, thus an OM analysis of 10% may also be 5% OC. A mass basis, i.e., g/100 g, is commonly used. For convenience in our evaluative calculations, we will treat OM as 50% OC, and we will assume the density of both OM and OC as being equal to that of water. Concentrations of these suspended materials may be defined operationally by using filters of various pore size, for example, 0.45 µm. There is a tendency to describe material that is smaller than this, i.e., that passes through the filter, as being operationally “dissolved.” It is not clear how we can best discriminate between “dissolved” and “particulate” forms of such material, since there is presumably a continuous size spectrum ranging from molecules of a few nanometres to relatively large particles of 100 or 1000 nm. It transpires that the organic material in the suspended phases is of great importance, because it has a high sorptive capacity for organic chemicals. It is therefore common to assign an organic carbon content to these phases. In a fairly productive lake, the OM content may be as high as 50% but, for illustrative purposes, a figure of 33% for OM or 16.7% OC is convenient. In each cubic metre of water, there is thus 2.5 g or cm3 of OM and 5.0 g or 2.5 cm3 of mineral matter, totaling 7.5 g or 5.0 cm3, giving an average particle density of 1.5 g/cm3. 4.3.3

Fish and Aquatic Biota

Fish are of particular interest, because they are of commercial and recreational importance to users of water, and they tend to bioconcentrate or bioaccumulate

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metals and organic chemicals from water. They are thus convenient monitors of the contamination status of lakes. This raises the question, “What is the volume fraction of fish in a lake?” Most anglers and even aquatic biologists greatly overestimate this number. It is probably, in most cases, in the region of 10–8 to 10–9, but this is somewhat misleading, because most of the biotic material in a lake is not fish—it is material of lower trophic levels, on which fish feed. For illustrative purposes, we can assume that all the biotic material in the water is fish, and the total concentration is about 1 part per million, yielding a volume of “fish” of about 7 m3. It proves useful later to define a lipid or fat content of fish, a figure of 5% by volume being typical. In summary, the water thus consists of 7 × 106 m3 of water containing 35 m3 of particulate matter and 7 m3 of “fish” or biota. In shallow or near-shore water, there may be a considerable quantity of aquatic plants or macrophytes. These plants provide a substrate for a thriving microbial community, and they possess inherent sorptive capacity. Their importance is usually underestimated. Because of the present limited ability to quantify their sorptive properties, we ignore them here. 4.3.4

Deposition Processes

The particulate matter in water is important, because, like aerosols in the atmosphere, it serves as a vehicle for the transport of chemical from the bulk of the water to the bottom sediments. Hydrophobic substances tend to partition appreciably on to the particles and are thus subject to fairly rapid deposition. This deposition velocity is typically 0.5 to 2.0 m per day or 0.02 to 0.08 m/h. This velocity is sufficient to cause removal of most of the suspended matter from most lakes during the course of a year. Thus, under ice-covered lakes in the winter, the water may clarify. Some of the deposited particulate matter is resuspended from the bottom sediment through the action of currents, storms, and the disturbances caused by bottom-dwelling fish and invertebrates. During the summer, there is considerable photosynthetic fixation of carbon by algae, resulting in the formation of considerable quantities of organic carbon in the water column. Much of this is destined to fall to the bottom of the lake, but much is degraded by microorganisms within the water column. Assuming, as discussed earlier, 5 × 10–6 m3 of particles per m3 of water and a deposition velocity of 200 m per year, we arrive at a deposition rate of 0.001 m3/m2 of sediment area per year or, for an area of 7 × 105 m2, a flow of 700 m3/year. We examine this rate in more detail in the next section.

4.4 4.4.1

BOTTOM SEDIMENTS

Sediment Solids

Inspection of the state of the bottom of lakes reveals that there is a fairly fluffy or nepheloid active layer at the water–sediment interface. This layer typically consists of 95% water and 5% particles and is often highly organic in nature. It may consist of deposited particles and fecal material from the water column. It is stirred

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by currents and by the action of the various biota present in this benthic region. The sediment becomes more consolidated at greater depths, and the water content tends to drop toward 50%. The top few centimetres of sediment are occupied by burrowing organisms that feed on the organic matter (and on each other) and generally turn over (bioturbate) this entire “active layer” of sediment. Depending on the condition of the water column above, this layer may be oxygenated (aerobic or oxic) or depleted of oxygen (anaerobic or anoxic). This has profound implications for the fate of inorganic substances such as metals and arsenic, but it is relatively unimportant for organic chemicals except in that the oxygen status influences the nature of the microbial community, which in turn influences the availability of metabolic pathways for chemical degradation. The deeper sediments are less accessible, and ultimately the material becomes almost completely buried and inaccessible to the aquatic environment above. Most of the activity occurs in the top 5 cm of the sediment, but it is misleading to assume that sediments deeper than this are not accessible. There remains a possibility of bioturbation or diffusion reintroducing chemical to the water column. Bottom sediments are difficult to investigate, can be unpleasant, and have little or no commercial value. They are therefore often ignored. This is unfortunate, because they serve as the depositories for much of the toxic material discharged into water. They are thus very important, are valuable as a “sink” for contaminants, and merit more sympathy and attention. Fast-flowing rivers are normally sufficiently turbulent that the bottom is scoured, exposing rock or consolidated mineral matter. Thus, their sediments tend to be less important. Sluggish rivers have appreciable sediments. 4.4.2

Deposition, Resuspension, and Burial

It is possible to estimate the rate of deposition, i.e., the amount of material that falls annually to the bottom of the lake and is retained there. This can be done by sediment traps, which are essentially trays that collect falling particles, or by taking a sediment core and assigning dates to it at various depths using concentrations of various radioactive metals such as lead. Nuclear events provide convenient dating markers for sediment depths. The measurement of deposition is complicated by the presence of the reverse process of resuspension caused by currents and biotic activity. It is difficult to measure how much material is rising and falling, since much may be merely cycling up and down in the water column. Burial or net deposition rates vary enormously, but a figure of about 1 mm per year is typical. Much of this is water, which is trapped in the burial process. Chemicals present in sediments are primarily removed by degradation, burial, or resuspension back to the water column. For illustrative purposes we adopt a sediment depth of 3 cm and suggest that it consists of 67% water and 33% solids, and these solids consist of about 10% organic matter or 5% organic carbon. Living creatures are included in this figure. Some of this deposited material is resuspended to the water column, some of the organic matter is degraded (i.e., used as a source of energy by benthic or bottom-living organisms), and some is destined to be permanently buried. The low 5% organic

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carbon figure for deeper sediments compared to high 17% for the depositing material implies that about 75% of the organic carbon is degraded. It is now possible to assemble an approximate mass balance for the sediment mineral matter (MM) and organic matter (OM) and thus the organic carbon (OC). This is given in Table 4.1. Table 4.1 Illustrative Sediment–Water Mass Balance on a 1 m2 Area Basis

Mineral matter

Organic matter

Organic carbon

Total

cm3

g

cm3

g

cm3

g

g

Deposition

500

1200

500

500

1000

1700

250

Resuspension

200

480

200

200

400

680

100

OM conversion

–

–

233

233

233

233

117

300

720

67

67

367

787

33

Burial (solids)

Total burial is 1000 cm3/year or 1420 g/year, corresponding to a “velocity” of 1 mm/year. The sediment thus has a density of 1.42 g/cm3 or 1420 kg/m3. Assumed densities are: mineral matter 2.4 g/cm3, organic matter 1 g/cm3. Organic matter is 50% (mass) organic carbon.

On a 1 m2 basis, the deposition rate is 0.001 m3 per year or 1000 cm3 per year. With a particle density of 1.7 g/cm3, this corresponds to 1700 g/year of which 500 g is OM, and 1200 g is MM. We assume that 40% of this is resuspended, i.e., 200 g of OM and 480 g of MM. Of the remaining 300 g OM, we assume that 233 g is digested or degraded to CO2, and 67 g is buried along with the remaining 720 g of MM. Total burial is thus 1420 g, which consists of 720 g of MM, 67 g of OM, and 633 g of water. The total volumetric burial rate of solids is 367 cm3/year. Now, associated with these solids is 633 cm3 of pore water; thus, the total volumetric burial rate of solids plus water is approximately 1000 cm3/year, corresponding to a rise in the sediment-water interface of 1 mm/year. The mass percentage of OC in the depositing and resuspending material is 15%, while in the buried material it is 4.2%. The bulk sediment density, including pore water, is 1420 kg/m3. On a 7 × 105 m2 basis, the deposition rate is 700 m3/year, resuspension is 280 m3/year, burial is 257 m3/year, and degradation accounts for the remaining 163 m3/year. The organic and mineral matter balances are thus fairly complicated, but it is important to define them, because they control the fate of many hydrophobic chemicals. It is noteworthy that the burial rate of 1 mm/year coupled with the sediment depth of 3 cm indicates that, on average, it will take 30 years for sediment solids to become buried. During this time, they may continue to release sorbed chemical back to the water column. This is the crux of the “in-place contaminated sediments” problem, which is unfortunately very common, especially in the Great Lakes Basin. In the simple four-compartment environment, we treat only the solids but, in the eight-compartment version, we include the sediment pore water. In the interests of simplicity, we assign a density of 1500 kg/m3 to the sediment in the four-compartment model.

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4.5 4.5.1

63

SOILS

The Nature of Soil

Soil is a complex organic matrix consisting of air, water, mineral matter (notably clay and silica), and organic matter, which is similar in general nature to the organic matter discussed earlier for the water column. The surface soil is subject to diurnal and seasonal temperature changes and to marked variations in water content, and thus in air content. At times it may be completely flooded, and at other times almost completely dry. The organic matter in the soil plays a crucial role in controlling the retention of the water and thus in ensuring the viability of plants. The organic matter content is typically 1 to 5%, but peat soils and forest soils can have much higher organic matter contents. Depletion of organic matter through excessive agriculture tends to render the soil infertile, which is an issue of great concern in agricultural regions. Soils vary enormously in their composition and texture and consist of various layers, or horizons, of different properties. There is transport vertically and horizontally by diffusion in air and in water, flow, or advection in water and, of course, movement of water and nutrients into plant roots and thence into stems and foliage. Burrowing animals such as worms can also play an important role in mixing and transporting chemicals in soils. A typical soil may consist of 50% solid matter, 20% air, and 30% water, by volume. The dry soil thus has a porosity of 50%. The solid matter may consist of about 2% organic carbon or 4% organic matter. During and after rainfall, water flows vertically downward through the soil and may carry chemicals with it. During periods of dry weather, water tends to return to the surface by capillary action, again moving the chemicals. Later, we set up equations describing the diffusion or permeation of chemicals in soils. When doing so, we treat the soil as having a constant porosity. In reality, there are channels or “macroporous” areas formed by burrowing animals and decayed roots, and these enable water and air to flow rapidly through the soil, bypassing the more tightly packed soil matrix. This phenomenon is very difficult to address when compiling models of transport in soils and is the source of considerable frustration to soil scientists. Most soils are, of course, covered with vegetation, which stabilizes the soil and prevents it from being eroded by wind or water action. Under dry conditions, with poor vegetation cover, considerable quantities of soil can be eroded by wind action, carrying with it sorbed chemicals. Sand dunes are an extreme example. In populated regions, of more concern is the loss of soil in water runoff. This water often contains very high concentrations of soil, perhaps as much as a volume fraction of 1 part per thousand of solid material. This serves as a vehicle for the movement of chemicals, especially agricultural chemicals such as pesticides, from the soils into water bodies such as lakes. 4.5.2

Transport in Soils

In most areas, there is a net movement of water vertically from the surface soil to greater depths into a pervious layer of rock or aquifer through which groundwater

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flows. The quality of this groundwater has become of considerable concern recently, especially to those who rely on wells for their water supply. This water tends to move very slowly (i.e., at a velocity of metres per year) through the porous sub-surface strata. If contaminated, it can take decades or even centuries to recover. Of particular concern are regions in which chemical leachate from dumps or landfills has seeped into the groundwater and has migrated some distance into rivers, wells, or lakes. It is quite difficult and expensive to investigate, sample, and measure contaminant flow in groundwater. It may not even be clear in which direction the water is flowing or how fast it is flowing. Chemicals associated with groundwater generally move more slowly than the velocity of the groundwater. They are retarded by sorption to the soil to an extent expressed as a “retardation factor,” which is essentially the ratio of (a) the amount of chemical that is sorbed to the solid matrix to (b) the amount that is in solution. Sorption of organic chemicals is usually accomplished preferentially to organic matter; however, clays also have considerable sorptive capacity, especially when dry. Polar, and especially ionic, substances may interact strongly with mineral matter. The characterization of migration of chemicals in groundwater is difficult, and especially so when a chemical is present in an non-aqueous phase, for example, as a bulk oil or emulsified oil phase. Considerable effort has been devoted to understanding the fate of nonaqueous phase liquids (NAPLs) such as oils, and dense NAPLs (DNAPLs) such as chlorinated solvents that can sink in the aquifer and are very difficult to recover. For illustrative purposes, we treat the soil as covering an area 1000 m × 300 m × 15 cm deep, which is about the depth to which agricultural soils are plowed. This yields a volume of 45,000 m3. This consists of about 50% solids, of which 4% is organic matter content or 2% by mass organic carbon. The porosity of the soil, or void space, is 50% and consists of 20% air and 30% water. Assuming a density of the soil solids of 2400 kg/m3 and water of 1000 kg/m3 gives masses of 1200 kg solids and 300 kg water per m3 (and a negligible 0.2 kg air), totaling 1500 kg, corresponding to a bulk density of 1500 kg/m3. Rainwater falls on this soil at a rate of 0.8 m per year, i.e., 0.8 m3/m2 year. Of this, perhaps 0.3 m evaporates, 0.3 m runs off, and 0.2 m percolates to depths and contributes to groundwater flow. This results in water flows of 90,000 m3/year by evaporation, 90,000 m3/year by runoff, and 60,000 m3/year by percolation to depths totaling 240,000 m3/year. With the runoff is associated 90 m3/year of solids, i.e., an assumed high concentration of 0.1% by volume. Again, it must be emphasized that these numbers are entirely illustrative. This soil runoff rate of 90 m3/year does not correspond to the deposition rate of 700 m3/year, partly because of the contribution of organic matter generated in the water column, but mainly because of the low ratio of soil area to water area. 4.5.3

Terrestrial Vegetation

Until recently, most environmental models have ignored terrestrial vegetation. The reason for this is not that vegetation is unimportant, but rather that modelers currently have enormous difficulty calculating the partitioning of chemicals into plants. This topic is receiving more attention as a result of the realization that consumption of contaminated vegetation, either by humans, domestic animals, or wildlife, is a major route or vector for the transfer of toxic chemicals from one

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species to another, and ultimately to humans. Plants play a critical role in stabilizing soils and in inducing water movement from soil to the atmosphere, and they may serve as collectors and recipients of toxic chemicals deposited or absorbed from the atmosphere. They can also degrade certain chemicals and increase the level of microbial activity in the root zone, thus increasing the degradation rate in the soil. Amounts of vegetation, in terms of quantity of biomass per square metre, vary enormously from near zero in deserts to massive quantities that greatly exceed accessible soil volumes in tropical rain forests. They also vary seasonally. If it is desired to include vegetation, a typical “depth” of plant biomass might be 1 cm. This, of course, consists mainly of water, cellulose, starch, and ligneous material. There is little doubt that future, more sophisticated models will include chemical partitioning behavior into plants. But at the present state of the art, it is convenient (and rather unsatisfactory) to regard the plants as having a volume of 3000 m3, containing the equivalent of 1% lipid-like material and 50% water. We ignore vegetation in the simple four-compartment model, treating the soil as only a simple solid phase.

4.6

SUMMARY

These evaluative volumes, areas, compositions, and flow rates are summarized in Table 4.2. From them is derived a simple four-compartment version. Also suggested is an alternative environment that is more terrestrial and less aquatic, and it reflects more faithfully a typical political jurisdiction. It is emphasized again that the quantities are purely illustrative, and site-specific values may be quite different. All that is needed at this stage is a reasonable basis for calculation. Scientists who have devoted their lives to studying the intricacies of the structure, composition, and processes of the atmosphere, hydrosphere, or lithosphere will undoubtedly be offended at the simplistic approach taken in this chapter. The environment is very complex, and it is essential to probe the fine detail present in its many compartments. But, if we are to attempt broad calculations of multimedia chemical fate, we must suppress much of the media-specific detail. When the broad patterns of chemical behavior are established, it may be appropriate to revisit the media that are important for that chemical and focus on detailed behavior in a specific medium. At that time, a more detailed and site-specific description of the medium of interest will be justified and required. Our philosophy is that the model should be only as complex as is required to answer the immediate question, not every question that could be asked. As questions are answered, new questions will surface and new, more complex models can be developed to answer these questions.

4.7

CONCLUDING EXAMPLE

Select a region with which you are familiar; for example, a county, watershed, state, or province. Calculate the volumes of air to a height of 1000 m; soil to a depth

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Table 4.2 Evaluative Environments A. Four-compartment, 1 km2 environment Areas (m2)

× 105 3 × 105 7 × 105

Air–water

7

Air–soil Water–sediment

Depths (m)

Volumes (m3)

Densities (kg/m3)

6000

× 10 7 × 106 4.5 × 104 2.1 × 104

1.2

Air Water

9

6

10

Soil

0.15

Sediment

0.03

Compositions

1000 1500

2% OC

1500

5% OC

B. Eight-compartment, 1 km2 environment, areas as in A above Volumes (m3) Air Water Soil (50% solids, 20% air, 30% water)

× × 4.5 ×

Sediment (30% solids) Suspended Sediment Aerosols

6

10

106

1000

Water

104

1500

Soil (50% solids, 20% air, 30% water)

1500

Sediment (30% solids)

35

1500

16.7% OC

0.12

1500

2

2.1

×

104

1.2

Compositions

7

Aquatic Biota Vegetation

9

Densities (kg/m3) Air

× 10–11 volume fraction or 30 µg/m3

7

1000

5% lipid

3000

1000

1% lipid 0.8 m/year or 800,000 m3/year 560,000 m3 to water; 240,000 m3 to soil

Rain Rate Aerosol Deposition Rates (total) Dry deposition Wet deposition

× 10–6 m3 /h or 1.89 m3 /year 365 × 10–6 m3 /h or 3.2 m3 /year 216

Sediment Deposition Rates Deposition

700 m3 /year solids 17% OC

Resuspension

280 m3 /year solids 17% OC

Net deposition (burial)

257 m3 /year solids 5% OC

Fate of Water in Soil Evaporation

90,000 m3 /year

Runoff to water

90,000 m3 /year

Percolation to groundwater

60,000 m3 /year

Solids runoff

90 m3 /year

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Table 4.2 (continued) C. Regional, 100,000 km2 environment as used in the EQC model of Mackay et al. (1996b) Volume (m3) Air

1014

Aerosols

2000

Water

2

Soil

9

× 10 × 109

Suspended sediment Fish

100

×

× × 10 ×

Composition

109 (2 × 10–11 vol frn) 9

10

10

90

109

2% OC

109

4% OC

6

–

20% OC

105

–

5% lipid

10 2

× –

11

108

Sediment

Area (m2)

For details of other properties see Mackay et al. 1996b.

of 10 cm; water and bottom sediment to a depth of 3 cm, and vegetation. Obtain data on average temperature, rain rate, water flows, and wind velocity, and calculate air and water residence times. Attempt to obtain information on typical concentrations of aerosols, suspended solids in water, and the organic carbon contents of soils, bottom, and suspended sediments. Prepare a summary table of these data similar to Table 4.2. These basic environmental data can be used in subsequent assessments of the fate of chemicals in this region.

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CHAPTER

5

Phase Equilibrium 5.1 5.1.1

INTRODUCTION

The Nature of Partitioning Phenomena

There are two distinct tasks that must be addressed when predicting equilibrium partitioning of chemicals in the environment. First, we must fully understand how chemicals behave under ideal, laboratory conditions of controlled temperature and well defined, pure phases. This is the task of physical chemistry. Second is the translation of these partitioning data into the more complex and less defined conditions of the environment where phases vary in composition and properties. In both cases, we are concerned with the equilibrium distribution of a chemical between phases as illustrated in the simple two-compartment system of Figure 5.1. A small volume of nonaqueous phase (e.g., a particle of organic or mineral matter, a fish, or an air bubble) is introduced into water that contains a dissolved chemical such as benzene. There is a tendency for some of the benzene to migrate into this new phase and establish a concentration that is some multiple of that in the water. In the case of organic particles, the multiple may be 100 or, if the phase is air, the multiple may be only 0.2. Equilibrium becomes established in hours or days between the benzene dissolved in the water and the benzene in, or on, the nonaqueous phase. Analytical measurements may give the total or average concentration that includes the nonaqueous phase and may differ considerably from the actual dissolved water concentration. The phase may subsequently settle to the lake bottom or rise to the surface, conveying benzene with it. Clearly, it is essential to establish the capability of calculating these concentrations and thus the fractions of the total amount of benzene that remain in the water, and enter the second phase. In some cases, 95% of the benzene may migrate into the phase, and in others only 5%. These systems will behave quite differently. The aim is to answer the question, “Given a concentration in one phase, what will be the concentration in another phase that has been in contact with it long enough to achieve equilibrium?” This task is part of the science of thermodynamics 69

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Figure 5.1

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

Some principles and concept in phase equilibrium.

that is fully described in several excellent texts such as those of Denbigh (1966), Van Ness and Abbott (1982), Prausnitz et al. (1969), and for aquatic environmental systems by Stumm and Morgan (1981) and Pankow (1991). It is assumed here that the reader is familiar with the general principles of thermodynamics; therefore, no attempt is made to derive all the equations. The aim is rather to extract from the science of thermodynamics those parts that are pertinent to environmental chemical equilibria and explain their source, significance, and applications.

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Environmental thermodynamics or phase equilibrium physical chemistry applies to a relatively narrow range of conditions. Tropospheric or surface temperatures range only between –40° and +40°C and usually between the narrower limits of 0° and 25°C. Total pressures are almost invariably atmospheric but, of course, with an additional hydrostatic pressure at lake or ocean bottoms. Concentrations of chemical contaminants are (fortunately) usually low. Situations in which the concentration is high (as in spills of oil or chemicals) are best treated separately. These limited ranges are fortunate in that they simplify the equations and permit us to ignore large and complex areas of thermodynamics that deal with high and low pressures and temperatures, and with high concentrations. The presence of a chemical in the environment rarely affects the overall dominant structure, processes, and properties of the environment; therefore, we can take the environment “as is” and explore the behavior of chemicals in it with little fear of the environment being changed in the short term as a result. There are, however, certain notable exceptions, particularly when the biosphere (which can be significantly altered by chemicals) plays an important role in determining the landscape. An example is the stabilizing influence of vegetation on soils. Another is the role of depositing carbon of photosynthetic origin in lakes. A point worth emphasizing is that thermodynamics is based on a few fundamental “laws” or axioms from which an assembly of equations can be derived that relate certain useful properties to each other. Examples are the relationship between vapor pressure and enthalpy of vaporization, or concentration and partial pressure. In some cases, the role of thermodynamics is simply to suggest suitable relationships. Thermodynamics never defines the actual value of a property such as the boiling point of benzene; such data must be obtained experimentally. We thus process experimental data using thermodynamic relationships. Despite its name, thermodynamics is not concerned with process rates; indeed, none of the equations derived in this chapter need contain time as a dimension. It transpires that two approaches can be used to develop equations relating equilibrium concentrations to each other as shown in Figure 5.1. The simpler and most widely used is Nernst’s Distribution law, which postulates that the concentration ratio C1/C2 is relatively constant and is equal to a partition or distribution coefficient K12. Thus, C2 can be calculated as C1K12. K12 presumably can be expressed as a function of temperature and, if necessary, of concentration. Experimentally, mixtures are equilibrated, and concentrations measured and plotted as in Figure 5.1. Linear or nonlinear equations then can be fitted to the data. The second approach involves the introduction of an intermediate quantity, a criterion of equilibrium, which can be related separately to C1 and C2. Chemical potential, fugacity, and activity are suitable criteria, with fugacity being preferred for most organic substances because of the simplicity of the equations that relate fugacity to concentration. The advantage of the equilibrium criterion approach is that properties of each phase are treated separately using a phase-specific equation. Treating phases in pairs, as is done with partition coefficients, can obscure the nature of the underlying phenomena. We may detect a variability in K12 and not know from which phase the variability is derived. Further complications arise if we have 10 phases to consider. There are then 90 possible partition coefficients, of which only 9 are independent. Mistakes are less

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likely using an equilibrium criterion and the 10 equations relating it to concentration, one for each phase. It is useful to discriminate between partition coefficients and distribution coefficients. Although usage varies, a partition coefficient is strictly the ratio of the concentrations of the same chemical species in two phases. A distribution coefficient is a ratio of total concentrations of all species. Thus, if a chemical ionizes, the partition coefficient may apply to the unionized species, while the distribution coefficient applies to ionized and nonionized species in total. 5.1.2

Some Thermodynamic Fundamentals

There are four laws of thermodynamics. They are numbered 0, 1, 2, and 3, because the need for the zeroth was not realized until after the first was postulated. Although these laws cannot be proved mathematically, they are now universally accepted as true, or axiomatic, because they are supported by all available experimental evidence. On consideration, they are intuitively reasonable, and it now seems inconceivable that they are ever disobeyed. The zeroth law introduces the concept of temperature as a criterion of thermal equilibrium by stating that, when bodies are at thermal equilibrium, i.e., there is no net heat flow in either direction, their temperatures are equal. The first law was discovered largely as a result of careful experiments by Joule, and it establishes the concept of energy and its conservation. Energy takes several forms—potential, kinetic, heat, chemical, electrical, nuclear, and electromagnetic. There are fixed conversion rates among these forms. Furthermore, energy can neither be formed nor destroyed; it merely changes its form. Of particular importance are conversions between thermal energy (heat) and mechanical energy (work). The second law is intellectually more demanding and introduces the concept of entropy and a series of useful related properties, including chemical potential and fugacity. It is observed that, whereas there are fixed exchange rates between heat and work energy, it is not always possible to effect the change. The conversion of mechanical energy to heat (as in an automobile brake) is always easy, but the reverse process of converting heat to mechanical energy (as in a thermal power station) proves to be more difficult. If a quantity of heat is available at high temperature, then only a fraction of it, perhaps one third, can be converted into mechanical energy. The remainder is rejected as heat, but at a lower temperature. Most thermodynamics texts introduce hypothetical processes such as the Carnot cycle at this stage to illustrate these conversions. After some manipulation, it can be shown that there is a property of a system, called its entropy, that controls these conversions. Apparently, regardless of how it is arranged to convert heat to work, the overall entropy of the system cannot decrease. It must increase by what is termed an irreversible process, or in the limit, it could remain constant by what is called a reversible process. Although there may be a local entropy decrease, this must be offset by another and greater entropy increase elsewhere. Clausius summarized this law in the statement that the “entropy of the universe increases.” It can be shown that entropy is related to randomness or probability. An increase in entropy corresponds to a change to a more random or disordered or probable condition. The third law is not important for our immediate purposes.

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We are concerned with systems in which a chemical migrates from phase to phase. These phase changes involve input or output of energy, thus this energy exchange can compensate for entropy loss or gain. It can be shown that, whereas entropy maximization is the criterion of equilibrium for a system containing constant energy at constant volume, the criterion at constant temperature and pressure (the environmentally relevant condition) is minimization of the related function, the Gibbs free energy, which serves to combine energy and entropy in a common currency. Return to the example presented in Figure 5.1, of benzene diffusing from water into an air bubble and striving to achieve equilibrium. The basic concept is that, if we start with a benzene concentration in the water and none in air, the free energy of the system will decrease as benzene migrates from water to air, because the increase in free energy associated with the rise in benzene concentration in the air is less that of the decrease associated with benzene loss from the water. The process is thus spontaneous and irreversible. Benzene continues to diffuse from water into the air until it reaches a point at which the free energy increase in the air is exactly matched by the free energy decrease in the water. At this point, the system comes to rest or equilibrium. Likewise, if the system started with a higher benzene concentration in the air phase and approached equilibrium, it would reach exactly the same point of equilibrium with a particular ratio of concentrations in each phase. The system thus seeks a minimum in free energy at which its derivative with respect to moles of benzene is equal in both air and water phases. This derivative is of such importance that it is called the chemical potential. The underlying principle of phase equilibrium thermodynamics is that, when a solute such as benzene achieves equilibrium between phases such as air, water, and fish, it seeks to establish an equal chemical potential in all phases. The net diffusion flux will always be from high to low chemical potential. Thus, we can use chemical potential for deductions of mass diffusion in the same way that we use temperature in heat transfer calculations. 5.1.3

Fugacity

Unfortunately, chemical potential is logarithmically related to concentration, thus doubling the concentration does not double the chemical potential. A further complication is that a chemical potential cannot be measured absolutely, therefore it is necessary to establish some standard state at which it has a reference value. It was when addressing this problem that G.N. Lewis introduced a new equilibrium criterion in 1901, which he termed fugacity, and which has units of pressure and is assigned the symbol f. The term fugacity comes from the Latin root fugere, describing a “fleeing” or “escaping” tendency. It is identical to partial pressure in ideal gases and is logarithmically related to chemical potential. It is thus linearly or nearly linearly related to concentration. Absolute values can be established because, at low partial pressures under ideal conditions, fugacity and partial pressure become equal. Thus, we can replace the equilibrium criterion of chemical potential by that of fugacity. When benzene migrates between water and air, it is seeking to establish an equal fugacity in both phases; its escaping tendency, or pressures, are equal in both phases.

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Another useful quantity is the ratio of fugacity to some reference fugacity such as the vapor pressure of liquid benzene. This is a dimensionless quantity and is termed activity. Activity can also be used as an equilibrium criterion. This proves to be preferable for substances such as ions, metals, or polymers that do not appreciably evaporate and thus cannot establish vapor phase concentrations and partial pressures. Our task, then, is to start with a concentration of solute chemical in one phase, from this deduce the chemical potential, fugacity, or activity, argue that these equilibrium criteria will be equal in the other phase, and then calculate the corresponding concentration in the second phase. We therefore require recipes for deducing C from f and vice versa. This approach is depicted at the bottom of Figure 5.1. The partition coefficient approach contains the inherent assumption that, whatever the factors are that are used to convert C1 to f1 and C2 to f2, the ratio of these factors is constant over the range of concentration of interest. Thus, it is not actually necessary to calculate the fugacities; their use is sidestepped. In the fugacity approach, no such assumption is made, and the individual calculations are undertaken. We can illustrate these approaches with an example. Worked Example 5.1 Benzene is present in water at a specified temperature and a concentration C1 of 1 mol/m3 (78 g/m3). What is the equilibrium concentration in air C2? 1. Partition coefficient approach

K21 is 0.2, i.e., C2/C1 Therefore, C2 = K21C1 = 0.2 × 1 = 0.2 mol/m3 = 15.6 g/m3 1. Fugacity approach

Using techniques devised later, we find that, for water under these conditions, f1

=

C1/Z1

=

C1/0.002 =

500 Pa

=

f2

C2

=

Z2f2

=

0.0004f2

0.2 mol/m3

=

15.6 g/m3

=

Clearly, the problem is to determine the conversion factors Z2 and Z1, or K21, which is their ratio. Care must be taken to avoid confusing K21 with its reciprocal K12 or C1/C2, which in this case has a value of 5. We therefore face the task of developing methods of estimating Z values that relate concentration and fugacity, and partition coefficients that are ratios of Z values. The theoretical foundations are set out in Section 5.3 and result in a set of working equations applicable to the air-water-octanol system. The three solubilities (or

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pseudo-solubilities) in these media and the three partition coefficients are then discussed in more detail in Section 5.4. Armed with this knowledge we then address how this “laboratory” information can be applied to environmental media such as soils and aerosols.

5.2

PROPERTIES OF PURE SUBSTANCES

For reasons discussed later, it is important to ascertain if the substance of interest is solid, liquid, or vapor at the environmental temperature. This is obviously done by comparing this temperature with the melting and boiling points. Figure 5.2 is the familiar P-T diagram that enables the state of a substance to be determined. Of particular interest for solids is the supercooled liquid vapor pressure line, shown as a dashed line. This is the vapor pressure that a solid (such as naphthalene, which melts at 80°C) would have if it were liquid at 25°C. The reason it is not liquid at 25°C is that naphthalene is able to achieve a lower free energy state by forming a crystal. Above 80°C, this lower energy state is not available, and the substance remains liquid. Above the boiling point, the liquid state is abandoned in favor of a vapor state. It is not possible to measure the supercooled liquid vapor pressure by direct experiment. It can be calculated as discussed shortly, and it can be measured

Figure 5.2

P-T diagram for a pure substance.

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experimentally, but not directly, using gas chromatographic retention times. It is possible to measure the vapor pressure above the boiling point by operating at high pressures. Beyond the critical point, the vapor pressure cannot be measured, but it can be estimated. The triple-point temperature at which solid, liquid, and vapor phases coexist is usually very close to the melting point at atmospheric pressure, because the solidliquid equilibrium line is nearly vertical; i.e., pressure has a negligible effect on melting point. Melting point is easily measured for stable substances, and estimation methods are available as reviewed by Tesconi and Yalkowsky (2000). High melting points result from strong intermolecular bonds in the solid state and symmetry of the molecule. Ice (H20) has a high melting point compared to H2S because of strong hydrogen bonding. The symmetrical three-ring compound anthracene has a higher melting point (216°C) than the similar but unsymmetrical phenanthrene (101°C). The critical point temperature is of environmental interest only for gases, since it is usually well above environmental temperatures. For example, it is 305 K for ethane and 562 K for benzene. Its principal interest lies in its being the upper limit for measurement of vapor pressure. The location of the liquid-vapor equilibrium or vapor pressure line is very important, since it establishes the volatility of the substances, as does the boiling point, which is the temperature at which the vapor pressure equals 1 atmosphere. Methods of estimating boiling point have been reviewed by Lyman (2000), and methods of using boiling point to estimate vapor pressures at other temperatures have been reviewed by Sage and Sage (2000). For many substances, correlations exist for vapor pressure as a function of temperature. The simplest correlation is the two-parameter Clapeyron equation, ln P = A – B/T A and B are constants, and T is absolute temperature (K). B is ∆H/R, where ∆H is the enthalpy of vaporization (J/mol), and R is the gas constant. A better fit is obtained with the three-parameter Antoine equation, lnP = A – B/(T + C) Care must be taken to check the units of P, whether base e or base 10 logs are used, and whether T is K or °C in the Antoine equation. Several other equations are used as reviewed by Reid et al. (1987). Correlations also exist for the vapor pressure of solids and supercooled liquids. Of particular environmental interest is the relationship between these vapor pressures, which can be used to calculate the unmeasurable supercooled liquid vapor pressure from that of the solid. The reason for this is that, when a solid such as naphthalene is present in a dilute, subsaturated, dissolved, or sorbed state at 25°C, the molecules do not encounter each other with sufficient frequency to form a crystal. Thus, the low-energy crystal state is not accessible. The molecule thus behaves as if it were a liquid at 25°C. It “thinks” it is a liquid, because it has no

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access to information about the stability of the crystalline state, i.e., does not “know” its melting point. As a result, it behaves in a manner corresponding to the liquid vapor pressure. A similar phenomenon occurs above the critical point where a gas such as oxygen, when in solution in water, behaves as if it were a liquid at 25°C, not a gas. No liquid vapor pressure can be measured for either naphthalene or oxygen at 25°C; it can only be calculated. Later, we term this liquid vapor pressure the reference fugacity. We may need to know this fictitious vapor pressure for several reasons. The ratio of the solid vapor pressure to the supercooled liquid vapor pressure is termed the fugacity ratio, F. To estimate F, we need to know how much energy is involved in the solid-liquid transition, i.e., the enthalpy of melting or fusion. The rigorous equation for estimating F at temperature T(K) is (Prausnitz et al., 1986) ln F = –∆S(TM – T)/RT + ∆CP(TM – T)/RT – ∆CP ln(TM/T)/R where ∆S (J/mol K) is the entropy of fusion at the melting point TM (K), ∆CP (J/mol K) is the difference in heat capacities between the solid and liquid substances, and R is the gas constant. The heat capacity terms are usually small, and they tend to cancel, so the equation can be simplified to ln F = –∆S(TM – T)/RT = –(∆H/TM)(TM – T)/RT = –(∆H/R)(1/T – 1/TM) where ∆H (J/mol) is the enthalpy of fusion and equals TM∆S. Note that, since TM is greater than T, the right-hand side is negative, and F is less than one, except at the melting point, when it is 1.0. F can never exceed 1.0. A convenient method of estimating ∆H is to exploit Walden’s rule that the entropy of fusion at the melting point ∆S, which is ∆H/TM, is often about 56.5 J/mol K. It follows that ln F = –(∆S/R)(TM/T – 1) The group ∆S/R is often assigned a value of 56/8.314 or 6.79. Thus, F is approximated as F = exp[–6.79(TM/T – 1)] If base 10 logs are used and T is 298 K, this equation becomes log F = –6.79(TM/298 – 1)/2.303 = –0.01(TM – 298) This is useful as a quick and easily remembered method of estimating F. If more accurate data are available for ∆H or ∆S, they should be used, and if the substance is a high melting point solid, it may be advisable to include the heat capacity terms.

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5.3 5.3.1

PROPERTIES OF SOLUTES IN SOLUTION

Solution in the Gas Phase

Equations are needed to deduce the fugacity of a solute in solution from its concentration. We first treat nonionizing substances that retain their structure when in solution. It transpires that, at low concentrations, a substance’s fugacity and concentration are linearly related, i.e., fugacity is proportional to concentration. This suggests using a relationship of the following form: C = Zf where C is concentration (mol/m3), f is fugacity (Pa), and Z, the proportionality constant (termed the fugacity capacity) has units of mol/m3Pa. The aim is then to deduce Z for the substance in air, water, and other phases. Later, we examine the significance of Z in more detail, because it becomes a key quantity when assessing environmental partitioning. The easiest case is a solution in a gas phase (air) in which there are usually no interactions between molecules other than collisions. The basic fugacity equation as presented in thermodynamics texts (Prausnitz et al., 1986) is f = y φ PT where y is mole fraction, φ is a fugacity coefficient, PT is total (atmospheric) pressure, and P is yPT, the partial pressure. If the gas law applies, PTV = nRT or PV = ynRT Here, n is the total number of moles present, R is the gas constant, V is volume (m3), and T is absolute temperature (K). Now the concentration of the solute in the gas phase CA will be yn/V or P/RT mol/m3. CA = yPT/RT = (1/ φRT) f = ZAf Fortunately, the fugacity coefficient φ rarely deviates appreciably from unity under environmental conditions. The exceptions occur at low temperatures, high pressures, or when the solute molecules interact chemically with each other in the gas phase. Only this last class is important environmentally. Carboxylic acids such as formic and acetic acid tend to dimerize, as do certain gases such as NO2. The constant ZA is thus usually (1/RT) or about 4 × 10–4 mol/m3Pa and is the same for all noninteracting substances. The fugacity is thus numerically equal to the partial pressure of the solute P or yPT. This raises a question as to why we use the term fugacity in preference to partial pressure. The answers are that (1) under conditions when φ is not unity, fugacity

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and partial pressure are not equal, and (2) there is some conceptual difficulty about referring to a “partial pressure of DDT in a fish” when there is no vapor present for a pressure to be present in—even partially. 5.3.2

Solution in Liquid Phases

The fugacity equation (Prausnitz et al., 1986) for solute i in solution is given in terms of mole fraction xi activity coefficient γi and reference fugacity fR on a Raoult’s law basis. fi = xiγifR Now, xi, the mole fraction of solute, can be converted to concentration C mol/m3 using molar volumes v (m3/mol), amounts n (mol), and volumes V (m3) of solute (subscript i) and solution (subscript w for water as an example). Assuming that the solute concentration is small, i.e., Vi<
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1.0. This, then, is the fugacity or vapor pressure of pure liquid solute at the temperature (and strictly the pressure) of the system. The activity coefficient γ is defined here on a “Raoult’s law” basis such that γ is 1.0 when x is 1.0. In most cases, γ values exceed 1.0 and, for hydrophobic chemicals, values may be in the millions. An alternative convention, which we do not use here, is to define γ on a Henry’s law basis such that γ is 1.0 when x is zero. The activity coefficient is thus a very important quantity. It can be viewed as the ratio of the activity or fugacity of the solute to the activity or fugacity that the solute would have if it were in a solution consisting entirely of its own kind. It depends on the concentration of the solute with a dependence of the type log γ = log γO (1 – x)2 where γO is the activity coefficient at infinite dilution, i.e., when x the mole fraction approaches zero. Another useful way of viewing activity coefficients is that they can be regarded as an inverse expression of solubility, i.e., an insolubility. A solute that is sparingly soluble in a solvent will have a high activity coefficient, an example being hexane in water. For a liquid solute such as hexane, at the solubility limit, when excess pure hexane is present, the fugacity equals the reference fugacity fR and fi = fR = xiγifR Therefore, xi = 1/γi or γi = 1/xi The activity coefficient is thus the reciprocal of the solubility when expressed as a mole fraction. For solids at saturation, fi is the fugacity of the pure solid fS. Thus, fS = xiγifR and xi = (fS/fR)/γi = F/γi where F is the fugacity ratio discussed earlier. Solid solutes of high melting point thus tend to have low solubilities, because F is small. It is more common to express solubilities in units such as g/m3. Under dilute conditions, the solubility Si mol/m3 is xi/vS, where vS is the molar volume of the solution (m3/mol) and approaches the molar volume of the solvent. Si is thus 1/γivS for liquids and F /γivS for solids. In the gas phase, the solubility is essentially the vapor pressure in disguise, i.e.,

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Si = n/V = PS/RT Invaluable information about how a substance will behave in the environment can be obtained by considering its three solubilities, namely those in air, water, and octanol. These solubilities express the substance’s relative preferences for air, water, and organic phases. Returning to the definition of ZW as (1/vWγifR), it is apparent that ZW also can be expressed in terms of aqueous solubility, SW , and vapor pressure PS. For liquid solutes, SW is 1/γivW . For solid solutes it is F/γivw . The reference fugacity fR is the vapor pressure of the liquid, i.e., it is PS for a liquid and PS/F for a solid. Substituting gives ZW as SW/PS in both cases, the F cancelling for solids. The ratio PS/S is the Henry’s law constant H in units of Pa m3/mol, thus ZW is 1/H. Polar solutes such as ethanol do not have measurable solubilities in water, because they are miscible. This generally occurs when γ is less than about 20. We can still use the concept of solubility and call it a “hypothetical or pseudo-solubility” if it is defined as 1/γivS. For a liquid substance that behaves nearly ideally, i.e., γi is 1.0, the solubility approaches 1/vS, which is the density of the solvent in units of mol/m3. For water, this is about 55,500 mol/m3, i.e., 106 g/m3 divided by 18 g/mol. For a solid solute under ideal conditions, the solubility approaches F/vS mol/m3. These equations are general and apply to a nonionizing chemical in solution in any liquid solvent, including water and octanol. The solution molar volumes and the activity coefficients vary from solvent to solvent. The Z value for a chemical in octanol is, by anology, 1/vOγifR, where vO is the molar volume of octanol. 5.3.3

Solutions of Ionizing Substances

Certain substances, when present in solution, adopt an equilibrium distribution between two or more chemical forms. Examples are acetic acid, ammonia, and pentachlorophenol, which ionize by virtue of association with water releasing H+ (strictly H3O+) or OH– ions. Some substances dimerize or form hydrates. For ionizing substances, the distribution is pH dependent, thus the solubility and activity are also pH dependent. This could be accommodated by defining Z as being applicable to the total concentration, but it then becomes pH dependent. A more rigorous approach is to define Z for each chemical species, noting that, for ionic species, Z in air must be zero under normal conditions, because ions as such do not evaporate. In any event, it is useful to know the relative proportions of each species, because they will partition differently. This issue is critical for metals in which only a small fraction may be in free ionic form. For acids, an acid dissociation constant Ka is defined as Ka = H+ A–/HA where H+ is hydrogen ion concentration, A– is the dissociated anionic form, and HA is the parent undissociated acid. The ratio of ionic to nonionic forms I is thus I = A–/HA = Ka/H+ = 10(pH – pKa)

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where pH is –log H+ and pKa is –log Ka This is the Henderson-Hasselbalch relationship. For acids, when pKa exceeds pH by 2 units or more, ionization can be ignored. When considering substances which have the potential to ionize it is essential to obtain pKa and determine the relative proportions of each form. The handbook by Lyman et al. (1982) and the text by Perrin et al. (1981) can be consulted for more details of estimation methods for pKa, and for applications. Dissociation can be regarded as causing an increase in the Z value of a substance in aqueous solution. The total Z value is the sum of the nonionic and ionic contributions, which will have respective fractions 1/(I + 1) and I/(I + 1). The Z value of the nonionic form ZW can be calculated by measuring solubility, activity coefficient, or another property under conditions when I is very small, i.e., pH << pKa. The same ZW value applies to the nonionic form at all pH levels. The additional contribution of the ionic form is then calculated at the pH of interest as IZW , and the total effective Z value is ZW(I +1), which can be used to calculate the total concentration. An inherent assumption here is that the presence of the ionic form does not affect ZW. For example, phenol has a pKa of 9.90, and pentachlorophenol (PCP) has a pKa of 4.74 (Mackay et al., 1995). At a pH of 6.0, the corresponding values of I are phenol 0.00013 and PCP 18.2. For phenol, ionization can be ignored for pH values up to about 8. For PCP, the dominant species in solution is the ionic form, and the total Z value in aqueous solution is 19.2 times that of the nonionic form. As a result, partitioning of PCP from solution in water to other media, such as air or octanol, is very pH dependent, and the issue of whether the ionic form also partitions must be addressed. KOW is thus pH dependent if (as is usual) total concentrations are used to calculate it. 5.3.4

Solutions in Solids

It is not usual to regard substances as having solubilities in solids. If the structure of the solid and the size of the solute molecule are such that the molecule can diffuse into and out of the solid matrix in a reasonable time period, then a solubility can be defined and measured. Organic molecules can diffuse into polymers such as polyethylene. Indeed, plasticizers such as phthalate esters are essentially in solution in polymers such as PVC to render the plastic flexible. Over time, they evaporate or leach from the plastic, rendering it more brittle. Semipermeable membrane devices (SPMDs) are increasingly used in water analysis. They consist of organic solvent (often triolein) contained in polyethylene bags that are submerged in the water. Hydrophobic molecules partition into the polymer, migrate through it, and accumulate in the solvent, providing a convenient integrated sample for analysis. To some extent, they simulate fish. In this case, the solutes have a finite solubility in the solid polymer. The organic matter discussed in Chapter 4 is solid. The sorption phenomenon can be regarded as simply partitioning into solid solution in this organic matter matrix. Solubilities and Z values can thus be calculated for solutions in solids.

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5.4 5.4.1

PARTITION COEFFICIENTS

Fugacity and Solubility Relationships

If we have two immiscible phases or media (e.g., air and water or octanol and water), we can conduct experiments by shaking volumes of both phases with a small amount of solute such as benzene to achieve equilibrium, then measure the concentrations and plot the results as was shown in Figure 5.1. It is preferable to use identical concentration units in each phase of amount per unit volume but, when one phase is solid, it may be more convenient to express concentration in units such as amounts per unit mass (e.g., µg/g) to avoid estimating phase densities. The plot of the concentration data is often linear at low concentrations; therefore, we can write C2/C1 = K21 and the slope of the line is K21. Some nonlinear systems are considered later. Now, since C2 is Z2f2 and C1 is Z1f1, and at equilibrium f1 equals f2, it follows that K21 is Z2/Z1. A Z value can be regarded as “half” a partition coefficient. If we know Z for one phase (e.g., Z1 as well as K21), we can deduce the value of Z2 as K21Z1. This proves to be a convenient method of estimating Z values. The line may extend until some solubility limit or “saturation” is reached. In water, this is the aqueous solubility, but, for some substances such as lower alcohols, there is no “solubility,” because the solute is miscible with water. In air, the “solubility” is related to the vapor pressure of the pure solute, which is the maximum partial pressure that the solute can achieve in the air phase. Partition coefficients are widely available and used for systems of air-water, aerosol-air, octanol-water, lipid-water, fat-water, hexane-water, “organic carbon”water, and various minerals with water. Applying the theory that was developed earlier and noting that, at equilibrium, the solute fugacities will be equal in both phases, we can define partition coefficients for air-liquid and liquid-liquid systems. For air-water as an example at a total atmospheric pressure PT , f = xiγifR = yiPT = Pi Thus, yi/xi = γifR/PT But if we use concentrations Ci (mol/m3) instead of mole fraction, yi is CiAvA or CiART/PT where vA is the molar volume of air. Similarly, xi is CiWvW where vW is the molar volume of the solution and is approximately that of water. Since fR is also PS, the partition coefficient KAW is then given by KAW = CiA/CiW = γi vW fR /RT = γi vW PS/RT = SiA/SiW

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In terms of solubilities, SiA and SiW, KAW is simply SiA/SiW, the ratio of the two solubilities. The pioneering work on air-water partitioning was done by Henry, who measured Pi as a function of xi and discovered that the solubility in water was proportional to the partial pressure Pi. The proportionality constant H´ is γifR and has units of pressure (Pa). Interestingly, for super-critical gases such as oxygen, fR cannot be measured, but γifR can be measured. If concentration is expressed as mol/m3, i.e., Ci instead of mole fraction xi, another and more convenient Henry’s law constant H can be defined as Pi/Ci and is γivWfR. KAW is then obviously H/RT, and it is also ZA/ZW . Note that H is also PSL/SiW and is 1/ZW , as was shown earlier. KAW is sometimes (wrongly) referred to as a Henry’s law constant. Atmospheric scientists, who are concerned with partitioning from air to water (e.g., into rain) use KWA, the reciprocal of KAW , and often refer to it as a Henry’s law constant. Extreme care thus must be taken when using reported values of Henry’s law constants because of these different definitions. For a liquid solute in a liquid-liquid system such as octanol-water, f = xiW γiW fR = xiO γiO fR where subscripts W and O refer to water and octanol phases. It follows that xiO/xiW = γiW/γiO and CiO/CiW = KOW = γiW vW/γiO vO = SiO/SiW If the solute is solid the same final equation applies because F, like fR, cancels. Because vW and vO are relatively constant, the variation in KOW between solutes is a reflection of variation in the ratio of activity coefficients γiW/γiO. Hydrophobic substances such as DDT have very large values of γiW and low solubilities in water. The solubility in octanol is usually fairly constant for organic solutes, thus KOW is approximately inversely proportional to SiW . Numerous correlations have been proposed between log KOW and log SiW , which are based on this fundamental relationship. Finally, for completeness, the octanol-air partition coefficient can be shown to be KOA = SiO/SiA = RT /γi vO PSL where γi applies to the octanol phase. It can be shown that ZO is 1/γi vO PSL and that KOA is ZO/ZA. Measurements of solubilities and partition coefficients are subject to error, as is evident by examining the range of values reported in handbooks. An attractive approach is to measure the three partition coefficients, KAW , KOW , and KOA, and

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perform a consistency check that, for example, KOA is KOW/KAW. Further checks are possible if solubilities can be measured to confirm that KAW is SA/SW or PS/SWRT. These checks are also useful for assessing the “reasonableness” of data. For example, if an aqueous solubility SW is reported as 1 part per million or 1 g/m3 or (say) 10–2 mol/m3, and KOW is reported to be 107, then the solubility in octanol must be SWKOW or 100,000 mol/m3. Octanol has a solubility in itself, i.e., a density of about 820 kg/m3 or 6300 mol/m3. It is inconceivable that the solubility of the solute in octanol exceeds the solubility of octanol in octanol by a factor of 100,000/6300 or 16; therefore, either SW or KOW or both are likely erroneous. The relationships between the three solubilities and the partition coefficients are shown in Figure 5.3. Two points worthy of note. There are numerous correlations for quantities such as KAW, KOW, SW, SA as a function of molecular structure and properties. They are generally derived independently, so it is possible to estimate SW, SA, and KAW and obtain inconsistent results, i.e., KAW will not equal SA/SW. It is preferable, in principle, to correlate SA, SW , and SO independently and use the values to estimate KAW, KOW , and KOA. There is then no possible inconsistency. It must be easier to correlate S (which depends on interactions in only one phase) than K (which depends on interactions in two phases). Finally, all activity coefficients, solubilities, and partition coefficients are temperature dependent.

Figure 5.3

Illustration of the relationships between the three solubilities, CA, CW, and CO, and the three partition coefficients, KAW, KOW, and KOA, with values for four substances. Note the wide substance variation in concentrations corresponding to unit concentration in the air phase.

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The temperature coefficient is the enthalpy of phase transfer, e.g., pure solute to solution for solubility or from solution to solution for partition coefficients. The enthalpies must be consistent around the cycle air-water-octanol such that their sum is zero. This provides another consistency check. It should be noted that the enthalpy change refers to the solubility or partition coefficient variation when expressed in mole fractions, not mol/m3 concentrations. This is particularly important for partitioning to air, where a temperature increase causes a density decrease, thus C or S will fall while x remains constant. For details of the merits of applying the “three solubility” approach, the reader is referred to Cole and Mackay (2000). We discuss these partition coefficients individually in more detail in the following sections. 5.4.2

Air-Water Partitioning

The nature of air-water partition coefficients or Henry’s law constants has been reviewed by Mackay and Shiu (1981), and estimation methods have been described by Mackay et al. (2000) and Baum (1997), and only a brief summary is given here. Several group contribution and bond contribution methods have been developed, and estimation methods are available from websites such as the EPIWIN programs of the Syracuse Research Corporation site at www.syrres.com. As was discussed above, the simplest method of estimating Henry’s law constants of organic solutes is as a ratio of vapor pressure to water solubility. It must be recognized that this contains the inherent assumption that water is not very soluble in the organic material, because the vapor pressure that is used is that of the pure substance (normally the pure liquid) whereas, in the case of solubility of a liquid such as benzene in water, the solubility is not actually that of pure benzene but is inevitably of benzene saturated with water. When the solubility of water in a liquid exceeds a few percent, this assumption may break down, and it is unwise to use this relationship. If a solute is miscible with water (e.g., ethanol), it is preferable to determine the Henry’s law constant directly; that is, by measuring air and water concentrations at equilibrium. This can be done by various techniques, e.g., the EPICS method described by Gossett (1987) or a continuous stripping technique described by Mackay et al. (1979). A desirable strategy is to measure vapor pressure PS, solubility CS, and H or KAW and perform an internal consistency check that H is indeed PS/CS or close to it. KAW is, of course, ZA/ZW. Care must be taken when calculating Henry’s law constants to ensure that the vapor pressures and solubilities apply to the same temperature and to the same phase. In some cases, reported vapor pressures are estimated by extrapolation from higher temperatures. They may be of a liquid or subcooled liquid, whereas the solubility is that of a solid. As was discussed earlier, subcooled conditions are not experimentally accessible but prove to be useful for theoretical purposes. Henry’s law constants vary over many orders of magnitude, tending to be high for substances such as the alkanes (which have high vapor pressures, low boiling points, and low solubilities) and very low for substances such as alcohols (which have a high solubility in water and a low vapor pressure). There is a common misconception that substances that are “involatile,” such as DDT, will have a low Henry’s law constant. This is not necessarily the case, because these substances also

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have very low solubilities in the water, i.e., they are very hydrophobic; thus, their low vapor pressure is offset by their very low water solubility, and they have relatively large Henry’s law constants. They may thus partition appreciably from water into the atmosphere through evaporation from rivers and lakes. The solubility and activity of a solute in water are affected by the presence of electrolytes and other co-solvents; thus, the Henry’s law constant is also affected. The magnitude of the effect is discussed later in Section 5.4.5. Worked Example 5.2 Deduce H and KAW for benzene, DDT, and phenol given the following data at 25°C: Molar Mass (g/mol) benzene DDT phenol

Solubility (g/m3)

78

1780

354.5

12700

0.0055

94.1

Vapor Pressure (Pa)

88360

0.00002 47

In each case, the solubility CS in mol/m3 is the solubility in g/m3 divided by the molar mass, e.g., 1780/78 or 22.8 mol/m3 for benzene. H is then PS/CS or 556 Pa m3/mol for benzene. KAW is H/RT or 556/(8.314 × 298) or 0.22. H benzene

556

KAW 0.22

DDT

1.29

5.2 × 10–4

phenol

0.050

2 × 10–5

Note that these substances have very different H and KAW values because of their solubility and vapor pressure differences. The vapor pressure of DDT is about 600 million times less than that of benzene, but H is only 400 times less, because of DDT’s very low water solubility. Phenol has a much higher vapor pressure than DDT, but it has a much lower H and KAW. Benzene tends to evaporate appreciably from water into air, and DDT less so but still to a significant extent, while phenol does not evaporate significantly. Inherent in this calculation for phenol is the assumption that it does not ionize appreciably. 5.4.3

Octanol-Water Partitioning

The dimensionless octanol-water partition coefficient (KOW) is one of the most important and frequently used descriptors of chemical behavior in the environment. In the pharmaceutical and biological literature, KOW is given the symbol P (for partition coefficient), which we reserve for pressure. The use of 1-octanol has been popularized by Hansch and Leo, who have tested its correlations with many biochemical phenomena and have compiled extensive databases. Various methods are available for calculating KOW from molecular structure, as reviewed by Lyman et al.

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(1982), Baum (1997) and Leo (2000). Extensive databases are also available as reviewed by Baum (1997). Octanol was selected because it has a similar carbon to oxygen ratio as lipids, is readily available in pure form, and is only sparingly soluble in water (4.5 mol/m3). The solubility of water in octanol of 2300 mol/m3, however, is quite large (Baum, 1997). The molar volumes of these phases are 18 × 10–6 m3/mol and 120 × 10–6 m3/mol, a ratio of 0.15. It follows that KOW is 0.15 γW/γO. KOW is a measure of hydrophobicity, i.e., the tendency of a chemical to “hate” or partition out of water. As was discussed earlier, it can be viewed as a ratio of solubilities in octanol and water but, in most cases of liquid chemicals, there is no real solubility, because octanol and the liquid are miscible. The “solubility” of organic chemicals in octanol tends to be fairly constant in the range 200 to 2000 mol/m3, thus variation in KOW between chemicals is primarily due to variation in water solubility. It is therefore misleading to assert that KOW describes lipophilicity or “love for lipids,” because most organic chemicals “love” lipids equally, but they “hate” water quite differently. Viewed in terms of Z values, KOW is ZO/ZW . ZO is (relatively) constant for organic chemicals; however, ZW varies greatly and is very small (relatively) for hydrophobic substances. Because KOW varies over such a large range, from approximately 0.1 to 107, it is common to express it as log KOW. It is a disastrous mistake to use log KOW in a calculation when KOW should be used! KOW is usually measured by equilibrating layers of water and octanol containing the solute of interest at subsaturation conditions and analyzing both phases. If KOW is high, the concentration in water is necessarily low, and even a small quantity of emulsified octanol in the aqueous phase can significantly increase the apparent concentration. A “slow stirring” method is usually adopted to avoid emulsion formation. An alternative is to use a generator column in which water is flowed over a packing containing octanol and the dissolved chemical. 5.4.4

Octanol-Air Partition Coefficients

This partition coefficient is invaluable for predicting the extent to which a substance partitions from the atmosphere to organic media including soils, vegetation, and aerosol particles. It can be estimated as KOW/KAW or measured directly, usually by flowing air through a column containing a packing saturated with octanol with the solute in solution. Values of KOA can be very large, i.e., up to 1012 for substances of very low volatility such as DDT, and values are especially high at low temperatures. Harner et al. (2000) have reported data for this coefficient and cite other data and measurement methods. 5.4.5

Solubility in Water

This property is of importance as a measure of the activity coefficient in aqueous solution, which in turn affects air-water and octanol-water partitioning. It can be regarded as a partition coefficient between the pure phase and water, but the ratio of concentrations is not calculated. A comprehensive discussion is given in the text by Yalkowsky and Banerjee (1992), and estimation methods are described by Mackay

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(2000) and Baum (1997). Extensive databases are available, for example, the handbooks by Mackay et al. (2000), which also give details of methods of experimental determination. It is important to appreciate that solubility in water is affected by temperature and the presence of electrolytes and other solutes in solution. It is often convenient to increase the solubility of a sparingly soluble organic substance by addition of a cosolvent to the water. Methanol and acetone are common cosolvents. To a first approximation, a “log-linear” relationship applies in that, if the solubility in water is SW and that in pure cosolvent is SC , then the solubility in a mixture SM is given by log SM = (1 – vC) log SW + vC log SC where vC is the volume fraction cosolvent in the solution. Electrolytes generally decrease the solubility of organics in water, the principal environmentally relevant issue being the solubility in seawater. The Setschenow equation is usually applied for predictive purposes, namely log (SW/SE) = kCS where SW is solubility in water, SE is solubility in electrolyte solution, k is the Setschenow constant specific to the ionic species, and CS is the electrolyte concentration (mol/L). Values of k generally lie in the range 0.2 to 0.3 L/mol; thus, in seawater, which is approximately 33 g NaCl/L or CS is about 0.5 mol/L, the solubility is about 70 to 80% of that in water. Xie et al. (1997) have reviewed this literature, especially with regard to seawater. 5.4.6

Solubility in Octanol

There are relatively few data on this solubility, and for many substances, especially liquids, the low activity coefficients render the solute-octanol system miscible; thus, no solubility is measurable. Pinsuwan and Yalkowsky (1995) have reviewed available solubility data and the relationships between KOW and solubilities in octanol and water. 5.4.7

Solubility of a Substance in Itself

The fugacity of a pure solute is its vapor pressure PS, and its “concentration” is the reciprocal of its molar volume vS (m3/mol) (typically, 10–4 m3/mol). Thus, C = (1/vS) = Zpf = ZPPS and ZP = 1/PS vS

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Although it may appear environmentally irrelevant to introduce ZP , there are situations in which it is used. If there is a spill of PCB or an oil into water of sufficient quantity that the solubility is exceeded, at least locally, the environmental partitioning calculations may involve the use of volumes and Z values for water, air, sediment, biota, and a separate pure solute phase. Indeed, early in the spill history, most of the solute will be present in this phase. The difference in behavior of this and other phases is that the pure phase fugacity (and, of course, concentration) remains constant, and as the chemical migrates out of the pure phase, the phase volume decreases until it becomes zero at total dissolution or evaporation. In the case of other phases, the concentration changes at approximately constant volume as a result of migration. It can be useful to compare a set of calculated Z values with ZP to gain an impression of the degree of nonideality in each phase. Rarely does a Z value of a chemical in a medium exceed ZP , but they may be equal when ideality applies and activity coefficients are close to 1.0. 5.4.8

Partitioning to Interfaces

Chemicals tend to adsorb from air or water to the surface of solids. An extensive literature exists on this subject as reviewed in texts in chemistry and environmental processes. A good review with environmental applications is given by Valsaraj (1995) in which the fundamental Gibbs equation is developed into the commonly used adsorption isotherms. These isotherms relate concentration at the surface to concentration in the bulk phase. Examples are the Langmuir, BET, and Freudlich isotherms. Generally, a linear isotherm applies at low concentrations as are usually encountered in the environment in relatively uncontaminated situations. Nonlinear behavior occurs at high concentrations in badly contaminated systems and in process equipment such as carbon adsorption units. It is often not realized that partitioning also occurs at the air-water interface, where an excess concentration may exist. This is exploited in the solvent sublation process for removing solutes from water using fine bubbles. If an area of the surface is known, a surface concentration in units of mol/m2 can be calculated, but more commonly the concentration is given in mol/mass of sorbent, which is essentially the product of the surface concentration and a specific area expressed in m2 surface per unit mass of sorbent. Solids such as activated carbon have very high specific areas and are thus effective sorbents. Partitioning to the airwater interface can become very important when the area of that interface is large compared to the associated volume of air or water. This occurs in fog droplets and snow where the ratio of area to water volume is very large, or in fine bubbles where the ratio of area to air volume is large. These ratios are (6/diameter) m2/m3. A Z value can be defined on an area basis (mol/m2 Pa) or for the bulk phase by including the specific area. Schwarzenbach et al. (1993) have reviewed mechanisms of sorption and have summarized reported data. This partitioning is important for ionizing substances but less important for nonpolar compounds, which sorb more strongly to organic matter.

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91

Quantitative Structure Property Relationships

An invaluable feature of many series of organic chemicals is that their properties vary systematically, and therefore predictably, with changes in molecular structure. This relationship is illustrated for the chlorobenzenes in Figure 5.4. Figure 5.4A is a plot of log subcooled liquid solubility versus chlorine number from 0 (benzene) to 6 (hexachlorobenzene), which shows the steady drop in solubility as a result of substituting a chlorine for a hydrogen. The magnitude is a decrease in log solubility of about 0.65 units (factor of 4.5) per chlorine. Vapor pressure (Figure 5.4B) behaves similarly, with a drop of 0.72 units (factor of 5.2). KOW (Figure 5.4C) shows an increase of 0.53 units (factor of 3.4). The Henry’s law constant (Figure 5.4D) decreases by 0.16 units (factor of 1.4). These plots are invaluable as a method of interpolating to obtain values for unmeasured compounds. They provide a consistency check for newly reported data. They form the basis of estimation methods in which these properties are calculated for a variety of atomic and group fragments. An extension of the QSPR concept is to use the same principles to correlate and estimate toxicity. This is referred to as a quantitative structure activity relationship or QSAR. The best environmental example is the correlation of fish toxicity data expressed as a LC50 (µmol/L) versus KOW as obtained by Konemann (1981). log (1/LC50) = 0.87 log KOW – 4.87 This and other correlations have been reviewed and discussed by Veith et al. (1983), Kaiser (1984, 1987), Karcher and Devillers (1990), and Abernethy et al. (1986, 1980). The fundamental relationship expressed by this correlation is best explained by an example. Consider two chemicals of log KOW 3 and 5. The LC50 values will be 182 and 3.3 µmol/L, a factor of 55 different. If the target site is similar to octanol in solvent properties, and equilibrium is reached, then the concentrations at the target site will be the product LC50 × KOW or 182,000 and 330,000 µmol/L, only a difference of a factor of 1.8. The chemical of lower KOW appears to be less toxic (it has a higher LC50) when viewed from the point of view of water concentration. When viewed from the target site concentration, it is slightly more toxic. The chemicals in the correlation display similar toxicities when evaluated from the target site concentrations. The correlation therefore expresses two processes, partitioning and toxicity, with most of the chemical-to-chemical variation being caused by partitioning difference. Such chemicals are referred to as narcotics in which the effect seems to be induced by high lipid concentrations. 5.4.10

Summary

The key properties of a pure substance for our purposes are its vapor pressure (i.e., its solubility in air), its solubility in water, its solubility in octanol, and the three partition coefficients KAW , KOW , and KOA. The magnitudes of these quantities are controlled by vapor pressure and activity coefficients.

Illustration of quantitative structure property relationships for benzene and the chlorobenzenes showing the systematic changes in properties with chlorine substitution.

92

Figure 5.4

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We can also relate equilibrium concentrations in these phases using the three Z values and fugacity. The partition coefficients are simply the ratio of the respective Z values; e.g., KAW is ZA/ZW. The use of Z values at this stage brings little benefit, but they become very useful when we calculate partitioning to environmental media. We use ZA, ZW , and ZO to calculate Z in phases such as soils, sediments, fish, and aerosol particles, and it proves useful to have ZP as a reference point when examining the magnitude of Z values. It is enlightening to calculate all the physical chemical properties of a solid and a liquid substance as shown in the following example. Worked Example 5.3 Deduce all relevant thermodynamic air-water partitioning properties for benzene (liquid) and naphthalene (a solid of melting point 80°C) at 25°C. Benzene Vapor pressure = 12700 Pa (PSL), molar mass = 78 g/mol, solubility in water = 1780 g/m3. ZA = 1/RT = 1/(8.314 × 298) = 4.04 × 10–4 Solubility CSL = 1780/78 = 22.8 mol/m3 Activity coefficient γ = 1/vwCSL = 1/(18 × 10–6 × 23.1) = 2440 H = PSL/CSL = 556, also = vWγPSL ZW = 1/H or 1/vwγPSL = 0.0018 KAW = H/RT or ZA/ZW = 0.22 Naphthalene Solubility = 33 g/m3, molar mass = 128 g/mol, vapor pressure = 10.9 Pa ZA = 4.04 × 10–4 as before F = exp(–6.79(353/298–1)) = 0.286 (mp = 353 K) PSL = PSS/F = 38.1 CSS = 33/128 = 0.26 mol/m3, CSL = 0.90 mol/m3 γ = 1/vwCSL = 61700 H = PSS /CSS or PSL/CSL = 42 ZW = 1/H or 0.024 = 1/vwγPSL KAW = H/RT or ZA/ZW = 0.017 Note that naphthalene has a higher activity coefficient corresponding to its lower solubility and greater hydrophobicity.

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5.5 5.5.1

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

ENVIRONMENTAL PARTITION COEFFICIENTS AND Z VALUES Introduction

Our aim is now to use the physical chemical data to predict how a chemical will partition in the environment. Information on air-water-octanol partitioning is invaluable and can be used directly in the case of air and pure water, but the challenge remains of treating other media such as soils, sediments, vegetation, animals, and fish. The general strategy is to relate partition coefficients involving these media to partitioning involving octanol. We thus, for example, seek relationships between KOW and soil-water or fish-water partitioning. 5.5.2

Organic Carbon-Water Partition Coefficients

Studies by agricultural chemists have revealed that hydrophobic organic chemicals tended to sorb primarily to the organic matter present in soils. Similar observations have been made for bottom sediments. In a definitive study, Karickhoff (1981) showed that organic carbon was almost entirely responsible for the sorbing capacity of sediments and that the partition coefficient between sediment and water expressed in terms of an organic carbon partition coefficient (KOC) was closely related to the octanol-water partition coefficient. Indeed, the simple relationship was established to be KOC = 0.41 KOW This relationship is based on experiments in which a soil-water partition coefficient was measured for a variety of soils of varying organic carbon content (y) and chemicals of varying KOW . The soil concentration was measured in units of µg/g or mg/kg (usually of dry soil) and the water in units of µg/cm3 or mg/L. The ratio of soil and water concentration (designated KP) thus has units of L/kg or reciprocal density. KP = CS/CW (mg/kg)/(mg/L) = L/kg If a truly dimensionless partition coefficient is desired, it is necessary to multiply KP by the soil density in kg/L (typically 2.5), or equivalently multiply CS by density to give a concentration in units of mg/L. A plot of KP versus organic carbon content, y (g/g), proves to be nearly linear and passes close to the origin, suggesting the relationship KP = y KOC where KOC is an organic carbon-water partition coefficient. In practice, there is usually a slight intercept, thus the relationship must be used with caution when y is less than 0.01, and especially when less than 0.001. Since y is dimensionless, KOC, like KP also has units of L/kg. Measurements of KOC for a

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variety of chemicals show that KOC is related to KOW as discussed above. KOW is dimensionless, thus the constant 0.41 has dimensions of L/kg. Care must be taken to use consistent units in these calculations. For example, if the water concentration has units of mol/m3, and KP is applied, the soil concentration will be in mol/Mg, i.e., moles per 106 grams. The usual units used are mg/L in water and mg/kg in soil. Units of either mass (g) or amount (mol) of solute can be used, but they must be consistent in both water and soil. The relationship between KOW and KOC has been the subject of considerable investigation, and it appears to be variable. For example, DiToro (1985) has suggested that, for suspended matter in water, KOC approximately equals KOW. Other workers, notably Gauthier et al. (1987), have shown that the sorbing quality of the organic carbon varies and appears to be related to its aromatic content as revealed by NMR analysis. Gawlik et al. (1997) have recently reviewed some 170 correlations between KOC and KOW, solubility in water, liquid chromatographic retention time, and various molecular descriptors. They could not recommend a single correlation as being applicable to all substances. Seth et al. (1999) analyzed these data and suggested that KOC is best approximated as 0.35 KOW (a coefficient slightly lower than Karickhoff’s 0.41) but that the variability is up to a factor of 2.5 in either direction. It is thus expected that, depending on the nature of the organic carbon, KOC can be as high as 0.9 KOW and as low as 0.14 KOW. Values outside this range may occur because of unusual combinations of chemical and organic matter. Doucette (2000) has given a very comprehensive review of this issue, and Baum (1997) has reviewed estimation methods. In summary, Z values can be calculated for soils and sediments containing organic carbon of 0.35 ZOy(ρ/1000) or 0.41 ZOy(ρ/1000), where ZO is for octanol, y is the organic carbon content, and ρ is the solid density, typically 2500 kg/m3. If an organic matter content is given, the organic carbon content can be estimated as 56% of the organic matter content. These relationships provide a very convenient method of calculating the extent of sorption of chemicals between soils or sediments and water, provided that the organic carbon content of the soil and the chemical’s octanol-water partition coefficient are known. This is illustrated in Example 5.4 below. Worked Example 5.4 Estimate the partition coefficient between a soil containing 0.02 g/g of organic carbon for benzene (KOW of 135) and DDT (log KOW of 6.19), and the concentrations in soil in equilibrium with water containing 0.001 g/m3, using the Karickhoff (0.41) correlation. benzene DDT

KOC = 0.41 KOW = 55, KOC = 0.41 KOW = 635000,

KP = 0.02 KOC = 1.1 L/kg KP = 0.02 KOC = 12700 L/kg

KP and KOC have units of L/kg or m3/Mg, i.e., reciprocal density thus when applying the equation below CS the soil concentration will have units of g/Mg or µg/g.

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CS = KPCW

benzene DDT

= 1.1 × 0.001 = 0.0011 µg/g = 10320 × 0.001 = 12.7 µg/g

Note the much higher DDT concentration because of its hydrophobic character. The concentrations in the organic carbon are CWKOC or 0.055 µg/g for benzene and 635 µg/g for DDT. If octanol was exposed to this water, similar concentrations of CWKOW or 0.135 µg/cm3 for benzene and 1549 µg/cm3 for DDT would be established in the octanol. 5.5.3

Lipid-Water and Fish-Water Partition Coefficients

Studies of fish-water partitioning by workers such as Spacie and Hamelink (1982), Neely et al. (1974), Veith et al. (1979), and Mackay (1982) have shown that the primary sorbing or dissolving medium in fish for hydrophobic organic chemicals is lipid or fat. A similar approach can be taken as for soils, but there is a more reliable relationship between KOW and KLW , the lipid-water partition coefficient. For most purposes, they can be assumed to be equal although, for the very hydrophobic substances, Gobas et al. (1987) suggest that this breaks down, possibly because of the structured nature of the lipid phases. It is thus possible to calculate an approximate fish-to-water bioconcentration factor or partition coefficient if the lipid content of the fish is known. Mackay (1982) reanalyzed a considerable set of bioconcentration data and suggested the simple linear relationship, KFW = 0.048 KOW This can be viewed as containing the assumption that fish is about 4.8% lipid. Lipid contents vary considerably, and it is certain that there is some sorption to nonlipid material, but it appears that, on average, the fish behaves as if it is about 4.8% octanol by volume. In summary, Z for lipids can be equated to ZO for octanol. For a phase such as a fish of lipid volume fraction L, ZF is LZO. 5.5.4

Mineral Matter-Water Partition Coefficients

Partition coefficients of hydrophobic organics between mineral matter and water are generally fairly low and do not appear to be simply related to KOW. Typical values of the order of unity to 10 are observed as reviewed by Schwarzenbach et al. (1993). A notable exception occurs when the mineral surface is dry. Dry clays display very high sorptive capacities for organics, probably because of the activity of the inorganic sorbing sites. This raises a problem that some soils may display highly variable sorptive capacities as they change water content as a result of heating, cooling, and rainfall during the course of diurnal or seasonal variations. Some pesticides are supplied commercially in the form of the active ingredient sorbed to an inorganic clay such as bentonite. In the environment, most clay surfaces appear to be wet and thus of low sorptive capacity. Most mineral surfaces that are accessible to the biosphere also appear to

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be coated with organic matter probably of bacterial origin. They thus may be shielded from the solute by a layer of highly sorbing organic material. It is thus a fair (and very convenient) assumption that the sorptive capacity of clays and other mineral surfaces can be ignored. Notable exceptions to this are subsurface environments in which there may be extremely low organic carbon contents and when the solute ionizes. In such cases, the inherent sorptive capacity of the mineral matter may be controlling. 5.5.5

Aerosol-Air Partition Coefficient

One of the most difficult, and to some extent puzzling, sorption partition coefficients is that between air and aerosol particles. These particles have very high specific areas, i.e., area per unit volume. They also appear to be very effective sorbents. The partition coefficient is normally measured experimentally by passing a volume of air through a filter then measuring the concentrations before and after filtration, and also the concentrations of the trapped particles. Relationships can then be established between the ratio of gaseous to aerosol material and the concentration of total suspended particulates (TSP). There has been a profound change over the years in our appreciation of this partitioning phenomenon. The pioneering work was done by Junge and later by Pankow resulting in the Junge-Pankow equation, which takes the form φ = C θ/(PSL + C θ) where φ is the fraction on the aerosol, θ is the area of the aerosol per unit volume of air, C is a constant, and PSL is the liquid vapor pressure. This is a Langmuir type of equation, which implies that sorption is to a surface, and the maximum extent of sorption is controlled by the available area. Experimental data were better correlated by calculating KP. It can be shown that φ = KP TSP/(1 + KP TSP) from which KP = φ/[TSP(1 – φ)] The units of TSP are usually µg/m3, thus it is convenient to express KP in units of m3/µg. KP is usually correlated against PSL for a series of structurally similar chemicals using the relationship log KP = m log PSL + b where m and b are fitted constants, and m is usually close to –1 in value. Bidleman and Harner (2000) list 21 such correlations and present a more detailed account of this theory.

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Mackay (1986), in an attempt to simplify this correlation, forced m to be –1 and obtained a one-parameter equation, using the liquid vapor pressure, KQA = 6 × 106/PSL where KQA is a dimensionless partition coefficient, i.e., a ratio of (mol/m3)/(mol/m3), and PSL has units of Pascals. Here, we use subscript Q to designate the aerosol phase. This enables ZQ, the Z value of the chemical in the aerosol particle, to be estimated as KQAZA. It can be shown that KQA is 1012 KP(ρ/1000) where ρ is the density of the aerosol (kg/m3), i.e., typically 1500 kg/m3. The 1012 derives from the conversion of µg to Mg. The fraction on the aerosol ϕ can then be calculated as ϕ = KQA vQ/(1 + KQA vQ) where vQ is the volume fraction of aerosol and is 10–12 TSP/(ρ/1000) when TSP has units of µg/m3. TSP is typically about 30 µg/m3, plus or minus a factor of 5; thus, vQ is about 20 × 10–12, plus or minus the same factor. Equipartitioning between air and aerosol phases occurs when ϕ is 0.5 or KP·TSP and KQAvQ equals 1.0. This implies a chemical with a KP of 0.03 m3/µg or KQAof 0.05 × 1012 and a vapor pressure of about 10–4 Pa. It is noteworthy that ZQ has a value of KQAZA or about (6 × 106/PSL)(4 × 10–4) or 2400/PSL. This is comparable to ZP , the pure substance Z value of 1/v PSL, where v is the chemical’s molar volume and is typically 100 cm3/mol or 10–4 m3/mol, giving a ZP of about 10,000/PSL. This implies that the solute is behaving near-ideally in the aerosol, i.e., the solubility in the aerosol is about 24% of the solubility of a substance in itself. This casts doubt on the surface sorption model, since it seems a remarkable coincidence that the area is such that it gives this near-ideal behavior. It further suggests that ZQ may correlate well with ZO for octanol. This was explored by Finizio et al. (1997), Bidleman and Harner (2000), and Pankow (1998), leading ultimately to a suggestion that KP = 10–12 KOA y (1000/820) = 10–11.91 KOA y where 820 kg/m3 is the density of octanol, and y is the fraction organic matter in the aerosol, which is typically 0.2. This reduces to KP = 10–12.61 KOA

or

0.25 × 10–12 KOA m3/µg

the use of KOA is advantageous, because it eliminates the need to deduce the fugacity ratio, F, when calculating the subcooled liquid vapor pressure. It is also possible that, for a series of chemicals, the activity coefficients in octanol and aerosol organic matter are similar, or at least have a fairly constant ratio. This approach is appealing, because it mimics the Karickhoff method of calculating soil-water partitioning, except that partitioning is now to air instead of water; thus, KOA replaces KOW.

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KQA thus can be calculated by the analogous equation, KQA = yKOA(ρ/1000) where y is organic matter mass fraction. This is equivalent to ZQ = yZO (ρ/1000), where ZO is the chemical’s Z value in octanol and ρ is the aerosol density. In summary, ZQ can be deduced using the above equation, using the simple oneparameter expression for KQA or one of the two-parameter equations for KP . Bidleman and Harner (2000) discuss the merits of these approaches in more detail.

5.5.6

Other Partition Coefficients

In principle, partition coefficients can be defined and correlated for any phase of environmental interest, usually with respect to the fluid media air or water. For example, vegetation or foliage-air partition coefficients KFA can be measured and correlated against KOA. Since KFA is ZF/ZA and KOA is ZO/ZA, the correlation is essentially of ZF versus ZO. Hiatt (1999) has suggested that, for foliage, KFA is approximately 0.01 KOA, implying that ZF is about 0.01 ZO, or foliage has a content of octanol-equivalent material of 1%. It is thus possible to estimate Z values for chemicals in any phase of environmental interest, provided that the appropriate partition coefficient has been measured or can be estimated. Figure 5.5 summarizes the relationships between fugacity, concentrations, partition coefficients, and Z values.

5.6 5.6.1

MULTIMEDIA PARTITIONING CALCULATIONS

The Partition Coefficient Method

The calculation of one phase concentration from another by the use of a simple partition coefficient is the most direct and convenient method. Care must be taken that the concentration units and the partition coefficient dimensions are consistent, especially when dealing with solid phases. There may also be inadvertent inversion of a partition coefficient, i.e., the use of K12 instead of K21. It is also possible to deduce certain partition coefficients from others; e.g., if an air/water and a soil/water partition coefficient are available, then the air/soil or soil/air partition coefficient can be deduced as follows: KAS = KAW/KSW If we are treating 10 phases, then it is possible to define 9 independent interphase partition coefficients, the 10th being dependent on the other 9. In principle, with 10 phases, it is possible to define 90 partition coefficients, half of which are reciprocals of the others. When dealing with very complex multicompartment envi-

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Definition of Fugacity Capacities Definition of Z (mol/m3 Pa)

Compartment Air

1/RT

R = 8.314 Pa m3/mol K

Water

1/H or CS/PS

CS = aqueous solubility (mol/m3) PS = vapor pressure (Pa) H = Henry’s law constant (Pa m3/mol)

Solid sorbent (e.g., soil, sediment, particles)

KPρS/H

KP = partition coeff. (L/kg) ρS = density (kg/L)

Biota

KBρS/H

KB = bioconcentration factor (L/kg) ρB = density (kg/L)

Pure solute

1/PSv

v = solute molar volume (m3/mol)

Figure 5.5

T = temp. (K)

Relationships between Z values and partition coefficients and summary of Z value definitions.

ronmental media, extreme care must be taken to avoid over- or underspecifying partition coefficients and to ensure that the ratios are not inverted. We have developed the capability of performing our first multimedia partitioning calculations. If we have a series of phases of volume V1, V2, V3, and V4, and we know the partition coefficients K12, K13, K14, and we introduce a known amount of

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chemical M mol into this hypothetical environment, then we can argue that M must be the sum of the concentration-volume products as follows: M = C1V1 + C2V2 + C3V3 + C4V4 = C1V1 + (K21C1)V2 + (K31C1)V3 + (K41C1)V4 = C1[V1 + K21V2 + K31V3 + K41V4] Therefore, C1 = M/[V1 + K21V2 + K31V3 + K41V4] and C2 = K21C1, C3 = K31C1 and C4 = K41C1 It is thus possible to calculate the concentrations in each phase, the amounts in each phase, and the percentages, and obtain a tentative picture of the behavior of this chemical at equilibrium in an evaluative environment. This is best illustrated by an example. Worked Example 5.5 Benzene partitions in a hypothetical environment between air, water, sediment, and fish (subscripted A, W, S, and F). The volumes of each phase are given below. The dimensionless partition coefficients are also given below. Calculate the concentrations, amounts, and percentages in each phase, assuming that a total of 10 moles of benzene is introduced into this system. VA = 1000 KAW = 0.2

VW = 20 KSW = 15

VS = 10 KFW = 20

VF = 0.05 m3

Using the equation developed above, CW = 10/(20 + 0.2 × 1000 + 15 × 10 + 20 × 0.05) = 10/371 = 0.027 mol/m3 Therefore, CA = 0.0054, CS = 0.405, CF = 0.54 The amounts in each phase are the products CV, namely, air = 5.39,

water = 0.539,

sediment = 4.04,

fish = 0.03

from which the percentages are, respectively, 53.9, 5.4, 40.4, and 0.3. It is clear that, in this system, benzene partitions primarily into air, mainly because of the large volume of air. The concentration in the air is lower than in any other phase; thus, we must discriminate between phases where the amount is large, which depends on the product CV, and where the concentration is large. There is a

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high concentration in the fish, but only a negligible fraction of the benzene is associated with fish. Such calculations are invaluable, because they establish the dominant medium into which the chemical is likely to partition, and they even give approximate concentrations. 5.6.2

The Fugacity Method

We now repeat these calculations using the fugacity concept and replacing C by Zf. We know that Z will depend on 1. 2. 3. 4. 5.

the nature of the solute (i.e., the chemical) the nature of the medium or compartment temperature pressure (but the effect is usually negligible) concentration (but the effect is negligible at low concentrations)

We have developed procedures by which Z values can be estimated for any given environmental situation. Equilibrium concentrations can then be deduced using f as a common criterion of equilibrium. We can repeat the previous partitioning example using the fugacity method and demonstrate the equivalence of the two approaches as before, but now applying the same fugacity to each phase. M = C1V1 + C2V2 + C3V3 + C4V4 = Z1fV1 + Z2fV2 + Z3fV3 + Z4fV4 = f (V1Z1 + V2Z2 + V3Z3 + V4Z4) Therefore, f = M/(V1Z1 + V2Z2 + V3Z3 + V4Z4) from which each C can be calculated as Zf, and the amount in each phase m is CV or VZf. In general, f = M/ΣViZi

Ci = Zif

mi = ViZif

Worked Example 5.6 Using the data in Example 5.5, recalculate the distribution using fugacity and assuming that ZA is 4 × 10–4 mol/m3 Pa. ZW = ZA/KAW = 0.002 ZS = KSWZW = 0.03

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ZF = KFWZW = 0.04 f = 10/(1000 × 4 × 10–4 + 20 × 0.002 + 10 × 0.03 + 0.05 × 0.04) = 10/0.742 = 13.48 CA = ZAf = 0.0054, CW = 0.027, CS = 0.405, CF = 0.54 And the amounts and percentages are as before. The procedure is simply to tabulate the volumes, the Z values, calculate and sum the VZ products, and divide this into the total amount to obtain the prevailing fugacity. This is readily done using a computer spreadsheet or program, and there is no increase in mathematical complexity with increasing numbers of phases. 5.6.3

A Digression: The Heat Capacity Analogy to Z

The fugacity capacity Z is at first a difficult concept to grasp, since it has unfamiliar units of mol/(volume × pressure). Heat capacity calculations provide a precedent for introducing Z and may help to illustrate the fundamental nature of this quantity. The traditional heat capacity equation is written in the form heat content (J) = mass of phase (kg) × heat capacity (J/kg K) × temperature (K) For example, water has a heat capacity of 4180 J/kg K, which is more familiar as 1 cal/g°C. We can rearrange this equation in terms of volumes instead of masses to give heat concentration (J/m3) = heat capacity (J/m3 K) × temperature (K) This new volumetric heat capacity for water is 4,180,000 J/m3 K. The use of mass rather than volume in heat capacities is an “accident” resulting from the greater ease and accuracy of mass measurements compared to volume measurements, and the complication that volumes change on heating, while mass remains constant. The equilibrium criterion used above is temperature (K), whereas we are concerned with fugacity (Pa). The quantity that partitions above is heat (J), whereas we are concerned with amount of matter (moles). Replacing K by Pa and J by mol leads to the analogous fugacity equation, C (mol/m3) = Z (mol/m3Pa) × f (Pa) Z is thus analogous to a heat capacity. Experience with heat calculations leads to a mental concept of heat capacity as a property describing the “capacity of a phase to absorb heat for a certain temperature rise.” For example, if 1 g of water (heat capacity 4.2 J/g°C) absorbs 4.2 J, its temperature will rise 1°C. Copper with a lower

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heat capacity of 0.38 J/g°C requires absorption of only 0.38 J to cause the same rise in temperature. Hydrogen gas has a large heat capacity of 14.3 J/g°C and thus requires a great deal of heat to raise its temperature. These substances differ markedly in their temperature response when heat is added. If 1000 J are added to equal masses of 1 g of these substances, the copper becomes much hotter by 263°C (or 100/0.38), while the water only heats up by 24°C (or 100/4.2) and the hydrogen by only 7°C (or 100/14.3). Hydrogen and water can thus absorb or “soak up” larger quantities of heat without becoming much hotter. The fugacity capacity is similar. Phases of high Z (possibly sediments or fish) are able to absorb a greater quantity of solute yet retain a low fugacity. It follows that solutes will tend to partition into these high Z phases and build up a substantial concentration yet retain a relatively low fugacity. Conversely, phases with low Z values will tend to experience a large increase in f following absorption of a small quantity of solute. A substance such as DDT is readily absorbed by fish and achieves a high concentration at low fugacity. The Z value of DDT in fish is large. On the other hand, DDT is not readily absorbed by water; indeed, it is hydrophobic or “water hating.” Its Z value in water is very low. This analogy between heat and fugacity capacity is perhaps best illustrated by the following pair of numerically identical examples, the fugacity quantities being given in parentheses. Worked Example 5.7 A system consists of three phases 10 g of water (10 m3 of water) of heat capacity 4.2 J/g°C (fugacity capacity 4.2 mol/m3 Pa), 5g of copper (5 m3 of air) of heat capacity 0.38 J/g°C (fugacity capacity 0.38 mol/m3 Pa), and 1 g of hydrogen (1 m3 of sediment) of heat capacity 14.3 J/g°C (i.e., fugacity capacity mol/m3 Pa). To this system is added 582 J of heat (582 mol of solute). What is the heat (solute) distribution at equilibrium, and what is the rise in temperature (fugacity) and heat concentrations in J/g (concentrations in mol/m3). We assume for simplicity that the initial temperature is 0°C, and the initial concentrations are also zero. (Note that Z for a solute in air never has the above value.) When approaching equilibrium, the temperatures (fugacities) will rise equally to a new common value at T °C, (f) such that the amount of heat (solute) in each phase will be mass (g) × heat capacity × T or [volume (m3) × Zf] Thus, the total will be the summation over the three phases, i.e., 582 = 10 × 4.2 × T + 5 × 0.38 × T + 1 × 14.3 × T Thus, T = 582/(10 × 4.2 + 5 × 0.38 + 1 × 14.3) = 10°C (Pa)

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1. Heat (moles) in water 2. Heat (moles) in copper (air) 3. Heat (moles) in hydrogen (sediment)

= 10 × 4.2 × 10 = 5 × 0.38 × 10 = 1 × 14.3 × 10

= 420 J (moles) 72% = 19 J (moles) 33% = 143 J (moles) 25%

Total

= 582 J and 100%

The concentrations are Water

4.2

×

10

=

42 J/g (mol/m3)

Copper (air)

0.38

×

10

=

3.8 J/g (mol/m3)

Hydrogen (sediment)

14.3

×

10

=

143 J/g (mol/m3)

The distribution of heat (moles) is influenced by the relative phase masses (volumes) and the heat capacities (Z values). Despite the fact that the third phase is small, its much larger heat capacity (Z) results in accumulation of a substantial fraction of the total (25%), and its concentration is a factor of 3.4 and 38 greater than the other two phases—which is, of course, the ratio of the heat capacities (the ratio of Z values, this ratio being the partition coefficient). This example could have been solved using heat capacity partition coefficients but, of course, no such quantities are tabulated in handbooks. Indeed, any suggestion that heat partition coefficients are useful would be treated with derision. In environmental calculations, on the other hand, the use of Z is less conventional, and the use of partition coefficients is routine. In essence, the use of fugacity capacities is an attempt to bring to environmental calculations some of the procedural benefits that are routinely enjoyed by the use of heat capacities. Worked Example 5.8 A three-phase system has Z values Z1 = 5 × 10–4, Z2 = 1.0, and Z3 = 20 (all mol/m3 Pa), and volumes V1 = 1000, V2 = 10, and V3 = 0.1 (all m3). Calculate the distributions, concentrations, and fugacity when 1 mol of solute is distributed at equilibrium between these phases. It is suggested that the calculations be done in tabular form. Phase 1 2 3 Total

Z 5

×

V

10–4 1.0

20

VZ

1000 10 0.1

0.5

C = Zf 4

×

10–5

VC

%

0.04

4

10

0.08

0.80

80

2

1.6

0.16

16

1.0

100

12.5

M = V1Z1f + V2Z2f + V3Z3f = f ΣViZi

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Therefore, f = M/ΣViZi = 1.0/12.5 = 0.08 Again, a large value of Z or C does not necessarily imply a large quantity. Quantity is controlled by VZ. Concentration is controlled by Z. 5.6.4

Sorption by Dispersed Phases

A frequently encountered environmental calculation is the estimation of the fraction of a chemical that is present in a fluid that is sorbed to some dispersed sorbing phases within that fluid. This is a special case of multimedia partitioning involving only two phases. Examples are the estimation of the fraction of material attached to aerosols in air or associated with suspended solids or with biotic matter in water. The reason for this calculation is that the measured concentration is often of the total (i.e., dissolved and sorbed) chemical, and it is useful to know what fractions are in each phase. This is particularly useful when subsequently calculating uptake of chemical by fish from water in which the partitioning may be only from the dissolved solute. It is useful to establish the general equations describing sorption in such cases as follows. We designate the continuous phase by subscript A and the dispersed phase by subscript B. The dispersed phase volume is typically a factor of 10–5 or less, as compared to that of the continuous phase. • • • •

The volumes (m3) are denoted VA and VB, and usually VA is much greater than VB. The equilibrium concentrations are denoted CA and CB mol/m3. The dimensionless partition coefficient KBA is CB/CA. The total amount of solute M moles is distributed between the two phases.

M = VACA + VBCB = VTCT where CT is the total concentration. It can be assumed that VT , the total volume is approximately VA. Now, CB = KBACA Therefore, M = CA(VA + VBKBA) = CTVA Therefore, CA = CT/(1 + KBAVB/VA) = CT/(1 + KBAvB) where vB is the volume fraction of phase B and is approximately VB/VA. The fraction dissolved (i.e., in the continuous phase) is

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ϕA = CA /CT = 1/(1 + KBAvB) and that sorbed is ϕB = (1–CA/CT) = KBAvB/(1 + KBAvB) The key quantity is thus KBAvB or the product of the dimensionless partition coefficient and the volume fraction of the dispersed sorbing phase. When this product is 1.0, half the solute is in each state. When it is smaller than 1.0, most is dissolved, and when it exceeds 1.0, more is sorbed. When the phase B is solid, it is usual to express the concentration CB in units of moles or grams per unit mass of B in which case KBA has units of volume/mass or reciprocal density. For example, it is common to use mg/L for CA, mg/kg for CB, and L/kg for KP; then, with M in mg and VA in L, then it can be shown that M = CA(VA + mBKP) = CTVA where mB is the mass of sorbing phase (kg) from which CA = CT/(1 + KPmB/VA) = CT/(1 + KPXB) where XB is the concentration of sorbent in kg/L. The units of the partition coefficient KBA or KP and concentration of sorbent vB or XB do not matter as long as their product is dimensionless and consistent, i.e., the amounts of sorbing phase, continuous phase, and chemical are the same in the definition of both the partition coefficient and the sorbent concentration. Care must be taken when interpreting sorbed concentrations to ascertain if they represent the amount of chemical per unit volume or mass of sorbent, or the per unit volume of the environmental phase such as water. The analogous fugacity equations are simply ϕA = VAZA/(VAZA + VBZB) ϕB = VBZB/(VAZA + VBZB) In some cases, it is preferable to calculate a Z value for a bulk phase consisting of other phases in equilibrium. Examples are air plus aerosols; water plus suspended solids; and soils consisting of solids, air, and water. If the total volume is VT, the effective bulk Z value is ZT, and equilibrium applies, then the total amount of chemical must be VTZTf = ΣviZif Thus,

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ZT = Σ(Vi/VT)Zi =ΣviZi where vi is the volume fraction of each phase. The key point is that the component Z values add in proportion to their volume fractions. The use of bulk Z values helps to simplify calculations by reducing the number of compartments, but it does assume that equilibrium exists within the bulk compartment. Worked Example 5.9 An aquarium contains 10 m3 of water and 200 fish, each of volume 1 cm3. How will 0.01 g (i.e., 10 mg) of benzene and the same mass of DDT partition between water and fish, given that the fish are 5% lipids, and log KOW is 2.13 for benzene and 6.19 for DDT? KFW will be 0.05 KOW or 6.7 for benzene and 77440 for DDT. CT is 0.001 g/m3 or 1 mg/m3 in both cases. The fraction dissolved ϕ2 is 1/(1 + KFWvF), where vF is the volume fraction of fish, i.e., 200 × 10–6/10 = 2 × 10–5. For benzene, KFWvF is 0.00013. For DDT, KFWvF is 1.55.

The fractions dissolved are 0.99987, essentially 1.0, for benzene and 0.39 for DDT. The dissolved concentrations are thus 0.00099987 g/m3 or 0.99987 mg/m3 (benzene) and 0.00039 g/m3 or 0.39 mg/m3 (DDT), and the sorbed concentrations (per m3 of water) are 0.00013 mg/m3 and 0.61 mg/m3, respectively. The sorbed concentrations per m3 of fish are 0.0067 g/m3 and 30 g/m3, respectively. Example 5.10 A lake of volume 106 m3 contains 15 mg/L of sorbing material. The total concentration of a chemical of KP equal to 105 L/kg is 1 mg/L. What are the dissolved and sorbed concentrations? Answer 0.4 mg/L dissolved and 0.6 mg/L sorbed. 5.6.5

Maximum Fugacity

When fugacities are calculated, it is advisable to check that the value deduced is lower than the fugacity of the pure phase, i.e., the solid or liquid fugacity or, in the case of gases, of atmospheric pressure. If these fugacities are exceeded, supersaturation has occurred, a “maximum permissible fugacity” has been exceeded, and the system will automatically “precipitate” a pure solute phase until the fugacity drops to the saturation value. It is possible to calculate inadvertently and use (i.e.,

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misuse) these “over-maximum” fugacities. For example, a chemical may be spilled into a lake. The fugacity can be calculated as the amount spilled divided by VZ for water. If the resulting fugacity exceeds the vapor pressure, the water has insufficient capacity to dissolve all the chemical, and a separate pure chemical phase must be present. A similar situation can apply when a pesticide is applied to soils. It is likely that the maximum Z value that a solute can ever achieve is that of the pure phase Zp. It may be useful to calculate ZP to ensure that no mistakes have been made by grossly overestimating other Z values. 5.6.6

Solutes of Negligible Volatility

A problem arises when calculating values of the fugacity and fugacity capacity of solutes that have a negligible or zero vapor pressure. Thermodynamically, the problem is that of determining the reference fugacity. The practical problem may be that no values of vapor pressure or air-water partition coefficients are published or even exist. Examples are ionic substances, inorganic materials such as calcium carbonate or silica and polymeric, or high-molecular-weight substances including carbohydrates and proteins. Intuitively, no vapor pressure determination is needed (or may be possible), because the substance does not partition into the atmosphere, i.e., its “solubility” in air is effectively zero. Ironically, its air fugacity capacity can still be calculated as (1/RT), but all the other (and the only useful) Z values cannot be calculated, since H cannot be determined and indeed may be zero. Apparently, the other Z values are infinite or at least are indeterminably large. This difficulty is more apparent than real and is a consequence of the selection of fugacity rather than activity as an equilibrium criterion. There are two remedies. The first method, which is convenient but somewhat dishonest, is to assume a fictitious and reasonable, but small, value for vapor pressure (such as 10–6 Pa) and proceed through the calculations using this value. The result will be that Z for air will be very small compared to the other phases, and negligible concentrations will result in the air. It is obviously essential to recognize that these air concentrations are fictitious and erroneous. The relative values of the other concentrations and Z values will be correct, but the absolute fugacity will be meaningless. The second method, which is less convenient but more honest, is to select a new equilibrium criterion. We can illustrate this for air, water, and another phase(s) by equating fugacities as follows: f = CA/ZA = CW/ZW = CS/ZS f = CART = CwPS/CS = CSPS/(CSKSW) We can divide through by PS to give f/PS = a = CART/PS = CW/CS = CS/CSKSW The equilibrium criterion is now a, an activity that is dimensionless and is the ratio of fugacity to vapor pressure. The new Z values with units of mol/m3 can be defined as

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air Z = PS/RT, water Z = CS, sorbed Z = CSKSW A saturated solution thus has an activity of 1.0. A zero or near-zero vapor pressure can be used to calculate Z for air as zero or near zero. In some cases, we may have to go farther, because we are uncertain about the solubility CS. The simple expedient is then to multiply through by CS to give a new equilibrium criterion A as fCS/PS = A = CARTCS/PS = CW = CS/KSW or, for air, Z = PS/RTCS, water Z = 1.0, sorbed phase Z = KSW. We are now using the water concentration or the equivalent equilibrium water concentration as the criterion of equilibrium. This has been termed the aquivalent concentration (Mackay and Diamond, 1989) and can be used for metals in ionic form when the solubility is meaningless. The essential procedure is that, for most organic substances, Z is defined in air as 1/RT, then all other Z values are deduced from it. In the “aquivalence” approach, Z is arbitrarily set to 1.0 in water, and all other Z values are deduced from this basis using partition coefficients. This approach is used in the EQC model for involatile substances (Mackay et al., 1996). 5.6.7

Some Environmental Implications

Viewing the behavior of a solute in the environment in terms of Z introduces new and valuable insights. A solute tends to migrate into (or stay in) the phase of largest Z. Thus, SO2 and phenol tend to migrate into water, freons into air, and DDT into sediment or biota. The phenomenon of bioconcentration is merely a manifestation of Z in biota, which is much higher (by the bioconcentration factor) than Z in the water. Occasionally, a solute such as inorganic mercury changes its chemical form becoming organometallic (e.g., methylmercury). Its Z values change, and the mercury now sets out on a new environmental journey with a destination of the new phase in which Z is now large. In the case of mercury, the ionic form will sorb to sediments or dissolve in water but will not appreciably bioconcentrate. The organic form experiences a large Z in biota and will bioconcentrate. The metallic form tends to evaporate. Some solutes, such as DDT or PCBs, have very low Z values in water because of their highly hydrophobic nature; i.e., they exert a high fugacity even at low concentration, reflecting a large “escaping tendency.” They will therefore migrate readily into any neighboring phase such as sediment, biota, or the atmosphere. Atmospheric transport should thus be no surprise, and the contamination of biota in areas remote from sites of use is expected. With this hindsight, it is not surprising that these substances are found in the tissues of Arctic bears and Antarctic penguins! From the environmental monitoring and analysis viewpoint, it is preferable to sample and analyze phases in which Z is large, because it is in these phases that

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concentrations are likely to be large and thus easier to determine accurately. When monitoring for PCBs in lakes, it is thus common to sample sediment or fish rather than water, since the expected concentrations in water are very low. Likewise, those concerned with PCB behavior in the atmosphere may measure the PCBs on aerosols or in rainfall containing aerosols, since concentrations are higher than in the air. In general, when assessing the likely environmental behavior of a new chemical, it is useful to calculate the various Z values and from them identify the larger ones, since it is likely that the high Z compartments are the most important. It is no coincidence that solutes such as halogenated hydrocarbons, about which there is great public concern, have high Z values in humans! It should be borne in mind that, when calculating the environmental behavior of a solute, Z values are needed only for the phases of concern. For example, if no atmospheric partitioning is considered, it is not necessary to know the air-water partition coefficient or H. An arbitrary value of H can be used to define Z for water and other phases, because H cancels. Intuitively, it is obvious that H, or vapor pressure, play no role in influencing water-fish-sediment equilibria. In summary, in this chapter we have introduced the concept of equilibrium existing between phases and have shown that this concept is essentially dictated by the laws of thermodynamics. Fortunately, we do not need to use or even understand the thermodynamic equations on which equilibrium relationships are based. However, it is useful to use these relationships for purposes such as correlation of partition coefficients. It transpires that there are two approaches that can be used to conduct equilibrium calculations. First is to develop and use empirical correlations for partition coefficients. Using these coefficients, it is possible to calculate the partitioning of the chemical in a multimedia environment. The second approach, which we prefer, is to use an equilibrium criterion such as fugacity or, in the case of involatile chemicals, an aquivalent concentration. The criterion can be related to concentration for each chemical and for each medium using a proportionality constant or Z value. The Z value can be calculated from fundamental equations or from partition coefficients. We have established recipes for the various Z values in these media using information on the nature of the media and the physical chemical properties of the substance of interest. This enables us to undertake simple multimedia partitioning calculations.

5.7

LEVEL I CALCULATIONS

Calculation of the equilibrium Level I distribution of a chemical is simple, but it can be tedious. It is ideal for implementation on a computer. The obvious steps are 1. 2. 3. 4. 5.

Definition of the environment, i.e., volumes and compositions Input of relevant physical chemical properties Calculation of Z values for each medium (see Table 5.1) Input of chemical amount Calculation of fugacity, and hence concentrations, amounts, and percent distribution

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Table 5.1 Table 5.1 Summary of Definitions of Z Values and Equations Used in Level I Calculations Definitions of Z values ZA = 1/RT ZW = 1/H = CS/PS = ZA/KAW ZO = ZW KOW (octanol) ZP = 1/vPPS (pure phase) ZS = yOCKOCZW (ρS/1000) (soils, sediments) KOC = 0.41 KOW ZQ = ZAKQA (aerosols) KQA = 6 × 106/PSL or yOMKOA (ρQ/1000) ZB = LZO (fish, biota)

where

R is the gas constant (8.314 Pa m3/mol K) T is absolute temperature (K) H is Henry’s law constant (Pa m3/mol) CS is solubility in water (mol/m3) PS is vapor pressure (Pa) KAW is air–water partition coefficient KOW is octanol–water partition coefficient KOC is organic carbon–water partition coefficient vP is molar volume of pure chemical (m3/mol) yOC is mass fraction organic carbon yOM is mass fraction organic matter ρS is density of soil, etc., (kg/m3) ρQ is density of aerosols (kg/m3) KQA is aerosol–air partition coefficient PSL is vapor pressure of liquid or subcooled liquid L is lipid content (volume fraction)

Note that the Z value of a bulk phase consisting of continuous and dispersed material, e.g., water plus suspended solids, is given by the volume fraction weighted Z values. ZT = Σ viZi where vi is the volume fraction of phase i.

Fugacity equation f = M/ΣViZi where

f is fugacity (Pa) M is total amount of chemical (mol) V is volume (m3) Ci = Zif

mi = CiVi = ViZif

mi is amount in phase i (mol)

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Programs to accomplish this calculation are available on the website www.trentu.ca/envmodel. The “Level I” calculation (Figure 5.6) is the simplest multimedia environmental calculation possible. The EQC model contains a Level I calculation for a regional environment as well as other, more advanced, calculations. To assist the reader to understand the nature of this calculation, two “fugacity forms” (Figures 5.7 and 5.8) are included at the end of this chapter. They contain a worked example for a hypothetical chemical. Blank forms are provided in the Appendix that may be reproduced for use in other examples. The results of the computer calculations should be consistent with these hand calculations.

5.8

CONCLUDING EXAMPLES

The concepts presented in this chapter are best grasped by working through examples. A chemical can be selected from those listed in Chapter 3 and the

Figure 5.6

Specimen output of a Level I calculation.

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Figure 5.7

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

Fugacity form 1 for deducing Z values.

properties used with assumed media volumes to deduce the distribution of a defined quantity of the substance between these media. Worked Example 5.11 A chemical has Z values in air of 4 × 10–4 mol/m3Pa, in water of 10–3 mol/m3Pa, and in sediment of 5 mol/m3Pa. What are the air and sediment concentrations in equilibrium with a water concentration of 2 mol/m3? Using subscripts W for water, A for air, and S for sediment, CW = 2.0 CA = f ZA CS = f ZS

ZW = 10–3 = 2000 × 4 × 10–4 = 2000 × 5

f = CW/ZW = 2000 Pa = 0.8 mol/m3 = 10000 mol/m3

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Figure 5.8

115

Fugacity form 2 for deducing Z values.

This example illustrates the fundamental simplicity of the equilibrium calculation and shows that the Z values are intimately related to the partition coefficients. The dimensionless air-water partition coefficient KAW , which is defined as CA/CW , is clearly 0.8/2.0 or 0.4. Likewise, CS/CW is 10000/2.0 or 5000. These ratios are also ratios of Z values.

5.9

CONCLUDING EXAMPLE

Select two nonionizing substances from Table 3.5, one a liquid and the other a solid, preferably with a melting point exceeding 100°C, and with log KOW in the range 3 to 6.

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Calculate the following physical-chemical properties for these substances at 25°C: fugacity ratio (1.0 for the liquid); Henry’s law constant; solubilities (mol/m3) in air, water, and octanol; the three partition coefficients between these phases and the activity coefficients in water and octanol. Assume a molar volume of 18 cm3/mol for water and 120 cm3/mol for octanol. In the case of the solid, calculate both the solid and supercooled liquid solubilities. Calculate the Z values in air, water, octanol, and in the pure chemical phase (assuming a density of 1 g/cm3 if the chemical’s density is not readily available from a handbook). Using Fugacity Form 1 as a template, calculate Z values in the following media: • • • • •

soil solids containing 1% organic carbon suspended sediment solids containing 15% organic carbon bottom sediment solids containing 5% organic carbon aerosol particles fish containing 5% lipid

Assume KOC to be 0.41 KOW and all solid densities to be 2000 kg/m3. Using Fugacity Form 2, calculate the fugacity, concentrations, and distribution of 100 kg of each chemical in an environment consisting of these volumes: air 109 m3 water 106 m3 soil solids 104 m3 suspended sediment solids 50 m3 bottom sediment solids 103 m3 aerosols 1 m3 fish 5 m3

Alternatively, use the environment that was deduced in the concluding example from Chapter 4. Write a short account of the partitioning behavior of these substances. Where would you analyze for monitoring purposes? Why? At what fraction of the saturation conditions is the chemical present, i.e., the ratio of fugacity to vapor pressure? In the water and atmosphere, what fractions of the total concentration are present in each of the dispersed phases of aerosols, suspended sediment, and fish?

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CHAPTER

6

Advection and Reactions 6.1

INTRODUCTION

In Level I calculations, it is assumed that the chemical is conserved; i.e., it is neither destroyed by reactions nor conveyed out of the evaluative environment by flows in phases such as air and water. These assumptions can be quite misleading when determining of the impact of a given discharge or emission of chemical. First, if a chemical, such as glucose, is reactive and survives for only 10 hours as a result of its susceptibility to rapid biodegradation, it must pose less of a threat than PCBs, which may survive for over 10 years. But the Level I calculation treats them identically. Second, some chemical may leave the area of discharge rapidly as a result of evaporation into air, to be removed by advection in winds. The contamination problem is solved locally, but only by shifting it to another location. It is important to know if this will occur. Indeed, recently, considerable attention is being paid to substances that are susceptible to long-range transport. Third, it is possible that, in a given region, local contamination is largely a result of inflow of chemical from upwind or upstream regions. Local efforts to reduce contamination by controlling local sources may therefore be frustrated, because most of the chemical is inadvertently imported. This problem is at the heart of the Canada–U.S., and Scandinavia–Germany–U.K. squabbles over acid precipitation. It is also a concern in relatively pristine areas such as the Arctic and Antarctic, where residents have little or no control over the contamination of their environments. In this chapter, we address these issues and devise methods of calculating the effect of advective inflow and outflow and degrading reactions on local chemical fate and subsequent exposures. It must be emphasized that, once a chemical is degraded, this does not necessarily solve the problem. Toxicologists rarely miss an opportunity to point out reactions, such as mercury methylation or benzo(a)pyrene oxidation, in which the product of the reaction is more harmful than the parent compound. For our immediate purposes, we will be content to treat only the parent compound. Assessment of degradation products is best done separately by having the degradation product of one chemical treated as formation of another. 117

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A key concept in this discussion that was introduced earlier, and is variously termed persistence, lifetime, residence time, or detention time of the chemical. In a steady-state system, as shown in Figure 6.1a, if chemical is introduced at a rate of E mol/h, then the rate of removal must also be E mol/h. Otherwise, net accumulation or depletion will occur. If the amount in the system is M mol, then, on average, the amount of time each molecule spends in the steady-state system will be M/E hours. This time, τ, is a residence time and is also called a detention time or persistence. Clearly, if a chemical persists longer, there will be more of it in the system. The key equation is τ = M/E or M = τ E This concept is routinely applied to retention time in lakes. If a lake has a volume of 100,000 m3, and if it receives an inflow of 1000 m3 per day, then the retention time is 100,000/1000 or 100 days. A mean retention time of 100 days does not imply

Figure 6.1

Diagram of a steady-state evaluative environment subject to (a) advective flow, (b) degrading reactions, (c) both, and (d) the time course to steady state.

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that all water will spend 100 days in the lake. Some will bypass in only 10 days, and some will persist in backwaters for 1000 days but, on average, the residence time will be 100 days. The reason that this concept is so important is that chemicals exhibit variable lifetimes, ranging from hours to decades. As a result, the amount of chemical present in the environment, i.e., the inventory of chemical, varies greatly between chemicals. We tend to be most concerned about persistent and toxic chemicals, because relatively small emission rates (E) can result in large amounts (M) present in the environment. This translates into high concentrations and possibly severe adverse effects. A further consideration is that chemicals that survive for prolonged periods in the environment have the opportunity to make long and often tortuous journeys. If applied to soil, they may evaporate, migrate onto atmospheric particles, deposit on vegetation, be eaten by cows, be transferred to milk, and then consumed by humans. Chemicals may migrate up the food chain from water to plankton to fish to eagles, seals, and bears. Short-lived chemicals rarely survive long enough to undertake such adventures (or misadventures). This lengthy justification leads to the conclusion that, if we are going to discharge a chemical into the environment, it is prudent to know 1. how long the chemical will survive, i.e., τ, and 2. what causes its removal or “death”

This latter knowledge is useful, because it is likely that situations will occur in which a common removal mechanism does not apply. For example, a chemical may be potentially subject to rapid photolysis, but this is not of much relevance in long, dark arctic winters or in deep, murky sediments. In the process of quantifying this effect, we will introduce rate constants, advective flow rates and, ultimately, using the fugacity concept, quantities called D values, which prove to be immensely convenient. Indeed, armed with Z values and D values, the environmental scientist has a powerful set of tools for calculation and interpretation. It transpires that there are two primary mechanisms by which a chemical is removed from our environment: advection and reaction, which we discuss individually and then in combination.

6.2

ADVECTION

Strangely, “advection” is a word rarely found in dictionaries, so a definition is appropriate. It means the directed movement of chemical by virtue of its presence in a medium that happens to be flowing. A lazy canoeist is advected down a river. PCBs are advected from Chicago to Buffalo in a westerly wind. The rate of advection N (mol/h) is simply the product of the flowrate of the advecting medium, G (m3/h), and the concentration of chemical in that medium, C (mol/m3), namely, N = GC mol/h

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Thus, if there is river flow of 1000 m3/h (G) from A to B of water containing 0.3 mol/m3 (C) of chemical, then the corresponding flow of chemical is 300 mol/h (N). Turning to the evaluative environment, it is apparent that the primary candidate advective phases are air and water. If, for example, there was air flow into the 1 square kilometre evaluative environment at 109 m3/h, and the volume of the air in the evaluative environment is 6 × 109 m3, then the residence time will be 6 hours, or 0.25 days. Likewise, the flow of 100 m3/h of water into 70,000 m3 of water results in a residence time of 700 hours, or 29 days. It is easier to remember residence times than flow rates; therefore, we usually set a residence time and from it deduce the corresponding flow rate. Burial of bottom sediments can also be regarded as an advective loss, as can leaching of water from soils to groundwater. Advection of freons from the troposphere to the stratosphere is also of concern in that it contributes to ozone depletion. 6.2.1

Level II Advection Algebra Using Partition Coefficients

If we decree that our evaluative environment is at steady state, then air and water inflows must equal outflows; therefore, these inflow rates, designated G m3/h, must also be outflow rates. If the concentrations of chemical in the phase of the evaluative environment is C mol/m3, then the outflow rate will be G C mol/h. This concept is often termed the continuously stirred tank reactor, or CSTR, assumption. The basic concept is that, if a volume of phase, for example air, is well stirred, then, if some of that phase is removed, that air must have a concentration equal to that of the phase as a whole. If chemical is introduced to the phase at a different concentration, it experiences an immediate change in concentration to that of the well mixed, or CSTR, value. The concentration experienced by the chemical then remains constant until the chemical is removed. The key point is that the outflow concentration equals the prevailing concentration. This concept greatly simplifies the algebra of steadystate systems. Essentially, we treat air, water, and other phases as being well mixed CSTRs in which the outflow concentration equals the prevailing concentration. We can now consider an evaluative environment in which there is inflow and outflow of chemical in air and water. It is convenient at this stage to ignore the particles in the water, fish, and aerosols, and assume that the material flowing into the evaluative environment is pure air and pure water. Since the steady-state condition applies, as shown in Figure 6.1a, the inflow and outflow rates are equal, and a mass balance can be assembled. The total influx of chemical is at a rate GACBA in air, and GWCBW in water, these concentrations being the “background” values. There may also be emissions into the evaluative environment at a rate E. The total influx I is thus I = E + GACBA + GWCBW mol/h Now, the concentrations within the environment adjust instantly to values CA and CW in air and water. Thus, the outflow rates must be GACA and GWCW. These outflow concentrations could be constrained by equilibrium considerations; for example, they may be related through partition coefficients or through Z values to a common fugacity.

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This enables us to conceive of, and define, our first Level II calculation in which we assume equilibrium and steady state to apply, inputs by emission and advection are balanced exactly by advective emissions, and equilibrium exists throughout the evaluative environment. All the phases are behaving like individual CSTRs. Of course, starting with a clean environment and introducing these inflows, it would take the system some time to reach steady-state conditions, as shown in Figure 6.1d. At this stage, we are not concerned with how long it takes to reach a steady state, but only the conditions that ultimately apply at steady state. We can therefore develop the following equations, using partition coefficients and later fugacities. I = E + GACBA + GWCBW = GACA + GWCW But CA = KAWCW Therefore, I = CW[GAKAW + GW] and CW = I/[GAKAW + GW] Other concentrations, amounts (m), and the total amount (M) can be deduced from CW. The extension to multiple compartment systems is obvious. For example, if soil is included, the concentration in soil will be in equilibrium with both CA and CW. 6.2.2

Level II Advection Algebra Using Fugacity

We assume a constant fugacity f to apply within the environment and to the outflowing media, thus, I = GAZAf + GWZWf = f(GAZA + GWZW) f = I/(GAZA + GWZW) or, in general, f = I/ΣGiZi from which the fugacity and all concentrations and amounts can be deduced. Worked Example 6.1 An evaluative environment consists of 104 m3 air, 100 m3 water, and 1.0 m3 soil. There is air inflow of 1000 m3/h and water inflow of 1 m3/h at respective chemical concentrations of 0.01 mol/m3 and 1 mol/m3. The Z values are air 4 × 10–4, water

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0.1, and soil 1.0. There is also an emission of 4 mol/h. Calculate the fugacity concentrations, persistence amounts and outflow rates. I = E + GACBA + GWCBW = 4 + 1000 × 0.01 + 1 × 1 = 15 mol/h ΣGZ = 1000 × 4 × 10–4 +1 × 10–1 = 0.5 CA = 0.012

CW = 3

CS = 30 mol/m3

mA = 120

mW = 300

mS = 30

GACA = 12

GWCW = 3

f = I/ΣGZ = 30 Pa

M (total) = 450 mol

GSCS = 0

Total = 15 = I mol/h

τ = 450/15 =30 h In this example, the total amount of material in the system, M, is 450 mol. The inflow rate is 15 mol/h, thus the residence time or the persistence of the chemical is 30 hours. This proves to be a very useful time. Note that the air residence time is 10 hours, and the water residence time is 100 hours; thus, the overall residence time of the chemical is a weighted average, influenced by the extent to which the chemical partitions into the various phases. The soil has no effect on the fugacity or the outflow rates, but it acts as a “reservoir” to influence the total amount present M and therefore the residence time or persistence. 6.2.3

D values

The group G Z, and other groups like it, appear so frequently in later calculations that it is convenient to designate them as D values, i.e., G Z = D mol/Pa h The rate, N mol/h, then equals D f. These D values are transport parameters, with units of mol/Pa h. When multiplied by a fugacity, they give rates of transport. They are thus similar in principle to rate constants, which, when multiplied by a mass of chemical, give a rate of reaction. Fast processes have large D values. We can write the fugacity equation for the evaluative environment in more compact form, as shown below: f = I/(DAA + DAW) = I/ΣDAi where DAA = GAZA, DAW = GWZW, and the first subscript A refers to advection. Recalculating Example 6.1, DAA = 0.4 and DAW = 0.1 and ΣDAi = 0.5 Therefore,

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f = 15/0.5 = 30 and the rates of output, Df, are 12 and 3 mol/h, totaling 15 mol/h as before. It is apparent that the air D value is larger and most significant. D values can be added when they are multiplied by a common fugacity. Therefore, it becomes obvious which D value, and hence which process, is most important. We can arrive at the same conclusion using partition coefficients, but the algebra is less elegant. Note that how the chemical enters the environment is unimportant, all sources being combined or lumped in I, the overall input. This is because, once in the environment, the chemical immediately achieves an equilibrium distribution, and it “forgets” its origin. 6.2.4

Advective Processes

In an evaluative environment, there are several advective flows that convey chemical to and from the environment, namely, 1. 2. 3. 4. 5.

inflow and outflow of air inflow and outflow of water inflow and outflow of aerosol particles present in air inflow and outflow of particles and biota present in water transport of air from the troposphere to the stratosphere, i.e., vertical movement of air out of the environment 6. sediment burial, i.e., sediment being conveyed out of the well mixed layer to depths sufficient that it is essentially inaccessible 7. flow of water from surface soils to groundwater (recharge)

It also transpires that there are several advective processes which can apply to chemical movement within the evaluative environment. Notable are rainfall, water runoff from soil, sedimentation, and food consumption, but we delay their treatment until later. In situations 1 through 4, there is no difficulty in deducing the rate as GC or Df, where G is the flowrate of the phase in question, C is the concentration of chemical in that phase, and the Z value applies to the chemical in the phase in which it is dissolved or sorbed. For example, aerosol may be transported to an evaluative world in association with the inflow of 1012 m3/h of air. If the aerosol concentration is 10–11 volume fraction, then the flowrate of aerosol GQ is 10 m3/h. The relevant concentration of chemical is that in the aerosol, not in the air, and is normally quite high, for example, 100 mol/m3. Therefore, the rate of chemical input in the aerosol is 1000 mol/h. This can be calculated using the D and f route as follows, giving the same result. If ZQ = 108, then f = CQ/ZQ = 100/108 = 10–6 Pa DAQ = GQZQ = 10 × 108 = 109

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Therefore, N = Df = 109 × 10–6 = 1000 mol/h Treatment of transport to the stratosphere is somewhat more difficult. We can conceive of parcels of air that migrate from the troposphere to the stratosphere at an average, continuous rate, G m3/h, being replaced by clean stratospheric air that migrates downward at the same rate. We can thus calculate the D value. As discussed by Neely and Mackay (1982), this rate should correspond to a residence time of the troposphere of about 60 years, i.e., G is V/τ. Thus, if V is 6 × 109 and τ is 5.25 × 105 h, G is 11400 m3/h. This rate is very slow and is usually insignificant, but there are situations in which it is important. We may be interested in calculating the amount of chemical that actually reaches the stratosphere, for example, freons that catalyze the decomposition of ozone. This slow rate is thus important from the viewpoint of the receiving stratospheric phase, but is not an important loss from the delivering, or tropospheric, phase. Second, if a chemical is very stable and is only slowly removed from the atmosphere by reaction or deposition processes, then transfer to the troposphere may be a significant mechanism of removal. Certain volatile halogenated hydrocarbons tend to be in this class. If we emit a chemical into the evaluative world at a steady rate by emissions and allow for no removal mechanisms whatsoever, its concentrations will continue to build up indefinitely. Such situations are likely to arise if we view the evaluative world as merely a scaled-down version of the entire global environment. There is certainly advective flow of chemical from, for example, the United States to Canada, but there is no advective flow of chemical out of the entire global atmospheric environment, except for the small amounts that transfer to the stratosphere. Whether advection is included depends upon the system being simulated. In general, the smaller the system, the shorter the advection residence time, and the more important advection becomes. Sediment burial is the process by which chemical is conveyed from the active mixed layer of accessible sediment into inaccessible buried layers. As was discussed earlier, this is a rather naive picture of a complex process, but at least it is a starting point for calculations. The reality is that the mixed surface sediment layer is rising, eventually filling the lake. Typical burial rates are 1 mm/year, the material being buried being typically 25% solids, 75% water. But as it “moves” to greater depths, water becomes squeezed out. Mathematically, the D value consists of two terms, the burial rate of solids and that of water. For example, if a lake has an area of 107 m2 and has a burial rate of 1 mm/year, the total rate of burial is 10,000 m3/year or 1.14 m3/h, consisting of perhaps 25% solids, i.e., 0.29 m3/h of solids (GS) and 0.85 m3/h of water (GW). The rate of loss of chemical is then GSCS + GWCW = GSZSf + GWZWf = f(DAS + DAW) Usually, there is a large solid to pore water partition coefficient; therefore, CS greatly exceeds CW or, alternatively, ZS is very much greater than ZW, and the term

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DAS dominates. A residence time of solids in the mixed layer can be calculated as the volume of solids in the mixed layer divided by GS. For example, if the depth of the mixed layer is 3 cm, and the solids concentration is 25%, then the volume of solids is 75,000 m3 and the residence time is 260,000 hours, or 30 years. The residence time of water is probably longer, because the water content is likely to be higher in the active sediment than in the buried sediment. In reality, the water would exchange diffusively with the overlaying water during that time period. As discussed in Chapter 5, there are occasions in which it is convenient to calculate a “bulk” Z value for a medium containing a dispersed phase such as an aerosol. This can be used to calculate a “bulk” Z value, thus expressing two loss processes as one. D is then GZ where G is the total flow and Z is the bulk value.

6.3

DEGRADING REACTIONS

The word reaction requires definition. We regard reactions as processes that alter the chemical nature of the solute, i.e., change its chemical abstract system (CAS) number. For example, hydrolysis of ethyl acetate to ethanol and acetic acid is definitely a reaction, as is conversion of 1,2-dichlorobenzene to 1,3-dichlorobenzene, or even conversion of cis butene 2 to trans butene 2. In contrast, processes that merely convey the chemical from one phase to another, or store it in inaccessible form, are not reactions. Uptake by biota, sorption to suspended material, or even uptake by enzymes are not reactions. A reaction may subsequently occur in these locations, but it is not until the chemical structure is actually changed that we consider reaction to have occurred. In the literature, the word reaction is occasionally, and wrongly, applied to these processes, especially to sorption. We have two tasks. The first is to assemble the necessary mathematical framework for treating reaction rates using rate constants, and the second is to devise methods of obtaining information on values of these rate constants. 6.3.1

Reaction Rate Expressions

We prefer, when possible, to use a simple first-order kinetic expression for all reactions. The basic rate equation is rate N = VCk = Mk mol/h where V is the volume of the phase (m3), C is the concentration of the chemical (mol/m3), M is the amount of chemical, and k is the first-order rate constant with units of reciprocal time. The group VCk thus has units of mol/h. The classical application of this equation is to radioactive decay, which is usually expressed in the forms dM/dt = –kM

or

dC/dt = –Ck

The use of C instead of M implies that V does not change with time.

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Integrating from an initial condition of CO at zero time gives the following equations: ln(C/CO) = –kt

or

C = CO exp(–kt)

Rate constants have units of frequency or reciprocal time and are therefore not easily grasped or remembered. A favorite trick question of examiners is to ask a student to convert a rate constant of 24 h–1 into reciprocal days. The correct answer is 576 days–1, so beware of this conversion! It is more convenient to store and remember half-lives, i.e., the time, τ1/2, which is the time required for C to decrease to half of CO. This can be related to the rate constant as follows. When C = 0.5 CO, then t = τ1/2 ln (0.5) = –kτ1/2, therefore, τ1/2 = 0.693/k For example, an isotope with a half-life of 10 hours has a rate constant, k, of 0.0693 h–1. 6.3.2

Non-First-Order Kinetics

Unfortunately, there are many situations in which the real reaction rate is not a first-order reaction. Second-order rate reactions occur when the reaction rate is dependent on the concentration of two chemicals or reactants. For example, if A+B→D+E then the rate of the reaction is dependent on the concentration of both A and B. Therefore, the reaction rate is as follows: N = Vk CA CB Reactant “B” is often another chemical, but it could be another environmental reactant such as a microbial population or solar radiation intensity. Third-order reaction rates, when the rate of reaction is dependent on the concentration of three reactants (N = Vk CA CB CC), are very rare and are unlikely to occur under environmental conditions. We can often circumvent these complex reaction rate equations by expressing them in terms of a pseudo first-order rate reaction. The primary assumption is that the concentration of reactant “B” is effectively constant and will not change appreciably as the reaction proceeds. Thus, the constant k and concentration of reactant “B” can be lumped into a new rate constant, kP , and the second-order reaction becomes a pseudo first-order reaction. Therefore, N = Vk CACB

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and kP = k CB Therefore, N = VkP CA which has the form of a simple first-order reaction. Examples of pseudo first-order reactions include photolysis reactions where reactant “B” is the solar radiation intensity (I, in photons/s) or microbial degradations processes where “B” is the populations of microorganisms. Reactions between two chemicals can also be considered a pseudo first-order reaction when CA << CB, so the concentration of B does not change as the reaction proceeds. Second-order rate expressions also arise when a chemical reacts with itself, giving rise to a messy quadratic equation. Thus, if A + A → D, the rate equation is N = Vk CACA = Vk CA2 Fortunately, most pollutants are present at low concentrations and tend not to react with themselves, so these types of reactions are rare. Zero-order expressions occasionally occur in which the rate is independent of the concentration of the chemical and is thus proportional to concentration to the power zero. Including zero-order expressions in mass balance models is potentially dangerous, because the equations can now predict a positive rate of reaction, even when there is no chemical present. It is embarrassing when computer models calculate negative concentrations of chemicals. Our strategy is to use every reasonable excuse to force first-order kinetics on systems by lumping parameters in k. The dividends that arise are worth the effort, because subsequent calculations are much easier. Perhaps most worrisome are situations in which we treat the kinetics of microbial degradation of chemicals. It is possible that, at very low concentrations, there is a slower or even no reaction, because the required enzyme systems are not “turned on.” At very high concentrations, the enzyme may be saturated; therefore, the rate of degradation ceases to be controlled by the availability of the chemical and becomes controlled by the availability of enzyme. In other cases, the rate of conversion may be influenced by the toxicity of the chemical to the organism or by the presence of co-metabolites, chemicals that the enzyme recognizes as being similar to that of the chemical of interest. Microbiologists have no difficulty conceiving of a multitude of situations in which chemical kinetics become very complicated and very difficult to predict and express. They seem to obtain a certain perverse delight in finding these situations. Saturation kinetics is usually treated by the Michaelis–Menten equation, which can be derived from first principles or, more simply, by writing down the basic firstorder equation and multiplying the rate expression by the group shown below.

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Basic expression

N = VCk

Group

CM/(C + CM)

Combined expression

N = VCCMk/(C + CM)

When C is small compared to CM, the rate reduces to VCk. When C is large compared to CM, it reduces to VCMk, which is independent of C, is constant, and corresponds to the maximum, or zero-order rate. The concentration, CM, therefore corresponds to the concentration that gives the maximum rate using the basic expression. When C equals CM, the rate is half the maximum value. This can be (and usually is) expressed in terms of other rate constants for describing the kinetics of the association of the chemical with the enzyme. The rate expression is usually written in biochemistry texts in the form N/V = C vM/(C + kM) where vM is a maximum rate or velocity equivalent to kCM, and kM is equivalent to CM and is viewed as a ratio of rate constants. A somewhat similar expression, the Monod equation, is used to describe cell growth. If kinetics are not of the first order, it may be necessary to write the appropriate equations and accept the increased difficulty of solution. A somewhat cunning but unethical alternative is to guess the concentration, calculate the rate N using the non-first-order expression, then calculate the pseudo first-order rate constant in the expression. For example, if a reaction is second order and C is expected to be about 2 mol/m3, V is 100 m3, and the second-order rate constant, k2, is 0.01 m3/mol·h, then N equals 4 mol/h. We can set this equal to VCk; then, k is 0.02 h–1. Essentially, we have lumped Ck2 as a first-order rate constant. This approach must be used, of course, with extreme caution, because k depends on C. 6.3.3

Additivity of Rate Constants

A major advantage of forcing first-order kinetics on all reactions is that, if a chemical is susceptible to several reactions in the same phase, with rate constants kA, kB, kC, etc., then the total rate constant for reaction is (kA + kB + kC), i.e., the rate constants are simply added. Another favorite trick of perverse examiners is to inform a student that a chemical reacts by one mechanism with a half-life of 10 hours, and by another mechanism with a half-life of 20 hours, and asks for the total half-life. The correct answer is 6.7 hours, not 30 hours. Half-lives are summed as reciprocals, not directly. 6.3.4

Level II Reaction Algebra Using Partition Coefficients

We can now perform certain calculations describing the behavior of chemicals in evaluative environments. The simplest is a Level II equilibrium steady-state

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reaction situation in which there is no advection, and there is a constant inflow of chemical in the form of an emission, as depicted in Figure 6.1b. When a steady state is reached, there must be an equivalent loss in the form of reactions. Starting from a clean environment, the concentrations would build up until they reach a level such that the rates of degradation or loss equal the total rate of input. We further assume that the phases are in equilibrium, i.e., transfer between them is very rapid. As a result, the concentrations are related through partition coefficients, or a common fugacity applies. The equations are as follows: E = V1C1k1 + V2C2k2 etc. = ΣViCiki Using partition coefficients, E = ΣViCwKiwki = CwΣViKiwki from which Cw can be deduced, followed by other concentrations, amounts, rates of reaction, and the persistence. In the general expression, KWW , the water-water partition coefficient is unity. Worked Example 6.2 The evaluative environment in Example 6.1 is subject to emission of 10 mol/h of chemical, but no advection. The reaction half-lives are air, 69.3 hours; water, 6.93 hours; and soil, 693 hours. Calculate the concentrations. Recall that KAW = 0.004 and KSW = 10. The rate constants are 0.693/half-lives or air, 0.01; water, 0.1; soil, 0.001; h–1. E = VACAkA + VWCWkW + VSCSkS = CW(VAKAWkA + VWkW + VSKSWkS) = CW(0.4 + 10 + 0.01) = CW(10.41) = 10 Therefore, CW = 0.9606 mol/m3, CA = 0.0038, CS = 9.606 The rates of reaction then are air = 0.38 water = 9.61 soil = 0.01 which add to the emission of 10.

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It is important to note that the reaction rate is controlled by the product V, C, and k. A large value of any one of these quantities may convey the wrong impression that the reaction is important. 6.3.5

Level II Using Fugacity and D Values for Reaction

We can now follow the same process as used when treating advection and define D values for reactions. If the rate is V C k or V Z f k, it is also DRf, where DR is V Z k. Note that DR has units of mol/m3 Pa identical to those of DA or G Z, discussed earlier. If there are several reactions occurring to the same chemical in the same phase, then each reaction can be assigned a D value, and these D values can be added to give a total D value. This is equivalent to adding the rate constants. The Level II mass balance becomes E = ΣViCiki = ΣViZifki = fΣViZik = fΣDR Thus, f can be deduced, followed by concentrations, amounts, the total amount M, and the rates of individual reactions as V C k or D f. We can repeat Example 6.2 in fugacity format. ×

VA = 104

ZA = 4

Water

VW = 100

ZW = 0.1

kW = 0.1

DRW = 1.0

Sediment

VS = 1.0

ZS = 1.0

kS = 0.001

DRS = 0.001

Air

10–4

kA = 0.01

DRA = 0.04

Total = 1.041 f = E/ΣDRi = 10/1.041 = 9.606 CA = 0.0038

rate = D f = 0.384

CW = 0.9606

= 9.606

CS = 9.6060

= 0.010

Worked Example 6.3 An evaluative environment consists of 10000 m3 air, 100 m3 water, and 10 m3 soil. There is input of 25 mol/h of chemical, which reacts with half-lives of 100 hours in air, 75 hours in water, and 50 hours in soil. Calculate the concentrations and amounts given the Z values below:

Phase Air Water Soil Total

Volume V (m3)

Z

k

VZk or D

C (mol/m3)

m (mol)

Rate (mol/h)

4 × 10–4

0.00693

0.0277

0.0386

386

2.68

100

0.1

0.00924

0.0924

9.66

966

8.93

10

1.0

0.0139

0.1386

96.6

966

13.39

2318

25.0

10000

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The rate constants in each case are 0.693/half-life. The sum of the V Z k terms or D values is 0.2587, thus, f = E/ΣD = 96.6 Pa Thus, each C is Z f and each amount m is VC, totaling 2318 mol. Each rate is V C k or D f, totaling 25 mol/h. It is clear that the D value V Z k controls the overall importance of each process. Despite its low volume and relatively slow reaction rate, the soil provides a fairly fast-reacting medium because of its large Z value. It is not until the calculation is completed that it becomes obvious where most reaction occurs. The overall residence time is 2318/25 or 93 hours. Note that the persistence or M/E is a weighted mean of the persistence or reciprocal rate constants in each phase. It is also ΣVZ/ΣD.

6.4

COMBINED ADVECTION AND REACTION

Advective and reaction processes can be included in the same calculation as shown in the example below, which is similar to those presented earlier for reaction. We now have inflow and outflow of air and water at rates given below and with background concentrations as shown in Figure 6.1c. The mass balance equation now becomes I = E + GACBA + GWCBW = GACA + GWCW + ΣViCiki This can be solved either by substituting KiWCW for all concentrations and solving for CW, or calculating the advective D values as GZ and adding them to the reaction D values. The equivalence of these routes can be demonstrated by performing both calculations. Worked Example 6.4 The environment in Example 6.3 has advective flows of 1000 m3/h in air and 1 m3/h in water as in Example 6.1 and reaction D values as in Example 6.3, with a total input by advection and emission of 40 mol/h. Calculate the fugacity concentrations, amounts, and chemical residence time.

Z

10000

4 × 10–4

0.4

0.0277

0.021

210

22.55

Water

100

0.1

0.1

0.0924

5.27

527

10.14

Soil

10

1.0

0.0

0.1386

527

7.31

0.5

0.2587

Phase Air

Total

DA (advection)

DR (reaction)

C (mol/m3)

52.7

m (mol)

Rate (mol/h) f(DA + DR)

Volume (m3)

1264

40

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The total of all D values is 0.7587. E = 40 Therefore, f = 40/ΣD = 52.7 The total amount is 1264 mols, giving a mean residence time of 31.6 hours. The most important loss process is advection in air, which accounts for 21.08 mol/h. Next is soil reaction at 7.31 mol/h, the water advection at 5.27 mol/h, etc. Each individual rate is D f mol/h. 6.4.1

Advection as a Pseudo Reaction

Examination of these equations shows that the group G/V plays the same role as a rate constant having identical units of h–1. It may, indeed, be convenient to regard advective loss as a pseudo reaction with this rate constant and applicable to the phase volume of V. Note that the group V/G is the residence time of the phase in the system. Frequently, this is the most accessible and readily remembered quantity. For example, it may be known that the retention time of water in a lake is 10 days, or 240 hours. The advective rate constant, k, is thus 1/240 h–1, and the D value is V Z k, which is, of course, also G Z. It is noteworthy that this residence time is not equivalent to a reaction half-time, which is related to the rate constant through the constant 0.693 or ln 2. Residence time is equivalent to 1/k. 6.4.2

Residence Times and Persistence

Confusion may arise when calculating the residence time or persistence of a chemical in a system in which advection and reaction occur simultaneously. The overall residence time in Example 6.4 is 31.6 hours and is a combination of the advective residence time and the reaction time. The presence of advection does not influence the rate constant of the reaction; therefore, it cannot affect the persistence of the chemical. But, by removing the chemical, it does affect the amount of chemical that is available for reaction, and thus it affects the rate of reaction. It would be useful if we could establish a method of breaking down the overall persistence or residence time into the time attributable to reaction and the time attributable to advection. This is best done by modifying the fugacity equations as shown below for total input I. I = ΣDAif + ΣDRif But I = M/τO, where M is the amount of chemical and τO is the overall residence time. Furthermore, M = ΣVZf or fΣVZ. Thus, dividing both sides by M and cancelling f gives

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1/τO = ΣDAi/ΣVZ + ΣDRi/ΣVZ = 1/τA + 1/τR The key point is that the advective and reactive residence times τA and τR add as reciprocals to give the reciprocal overall time. These are the residence times that would apply to the chemical if only that process applied. Clearly, the shorter residence time dominates, corresponding, of course, to the faster rate constant. It can be shown that the ratio of the amounts removed by reaction and by advection are in the ratio of the overall rate constants or the reciprocal residence times. Example 6.5 Calculate the individual and overall residence times in Example 6.4. Each residence time is VZ/D and the rate constant is D/VZ. VZ Air

ΣVZ/D (advection)

VZ/D (reaction)

4

60

866

10

240

260

Soil

10

∞

173

Total

24

Water

Adding the reciprocals, i.e., the rate constants, gives 1/60 + 1/240 + 1/866 + 1/260 + 1/∞ + 1/173 = 0.0167 + 0.0042 + 0.0012 + 0.0038 + 0 + 0.0058 = 0.0209 + 0.0108 = 0.0317 = 1/31.5 The advection residence time is 1/0.0209 or 47.8 h, and for reaction it is 1/0.0108 or 92.6 h. Each residence time (e.g., 60, 866, etc.) contributes to give the overall residence time of 31.5 hours, reciprocally. In mass balance models of this type, it is desirable to calculate the advection, reaction, and overall residence times. An important observation is that these residence times are independent of the quantity of chemical introduced; in other words, they are intensive properties of the system. Concentrations, amounts, and fluxes are dependent on emissions and are extensive properties. These concepts are useful, because they convey an impression of the relative importance of advective flow (which merely moves the problem from one region to another) versus reaction (which may help solve the problem). These are of particular interest to those who live downwind or downstream of a polluted area. 6.5

UNSTEADY-STATE CALCULATIONS

A related calculation can be done in unsteady-state mode in which we introduce an amount of chemical, M, into the evaluative environment at zero time, then allow

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it to decay in concentration with time, but maintain equilibrium between all phases at the same time. This is analogous to a batch chemical reaction system. Although it is possible to include emissions or advective inflow, we prefer to treat first the case in which only reaction occurs to an initial mass M. We assume that all volumes and Z values are constant with time. dM/dt = –ΣViCiki = –fΣViZiki = –fΣDRi But, M = ΣViZif = fΣViZi df/dt = –fΣViZiki/ΣViZi = –fΣDRi/ΣViZi Solving gives f = fO exp(–kOt) where kO = ΣViZiki/ΣViZi = ΣDRi/ΣViZi, and fO is the initial fugacity. Note that kO, the overall rate constant, is the reciprocal of the overall residence time. Worked Example 6.6 Calculate the time necessary for the environment in Example 6.3 to recover to 50%, 36.7%, 10%, and 1% of the steady-state level of contamination after all emissions cease. Here, ΣVZ is 24 and ΣD is 0.2587. Thus, f = fO exp (–0.2587t/24) = fO exp (–0.01078t) Since M is proportional to f, and fO is 96.6 Pa, we wish to calculate t at which f is 48.3, 35.4, 9.66, and 0.966 Pa. Substituting and rearranging gives t = –1/0.01078 ln (48.3/96.6), etc., or t is, respectively, 64 h, 93 h, 214 h, and 427 h. The 93-hour time is significant as both the steady-state residence time and the time of decay to 36.7% or exp(–1) of the initial concentration. It is possible to include advection and emissions with only slight complications to the integration. The input terms may no longer be zero. This example raises an important point, which we will address later in more detail. The steady-state situations in the Level II calculations are somewhat artificial and contrived. Rarely is the environment at a steady state; things are usually getting worse or better. A valid criticism of Level II calculations is that steady-state analysis does not convey information about the rate at which systems will respond to changes. For example, a steady-state analysis of salt emission into Lake Superior may demonstrate what the ultimate concentration of salt will be, but it will take 200 years for this steady state to be achieved. In a much smaller lake, this steady state may

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be achieved in 10 days. Detractors of steady-state models point with glee to situations in which the modeler will be dead long before steady state is achieved. Proponents of steady-state models respond that, although they have not specifically treated the unsteady-state situation, their equations do contain much of the key “response time” information, which can be extracted with the use of some intelligence. The response time in the unsteady-state Example 6.5 was 93 hours, which was ΣVZ/ΣD. This is identical to the overall residence time, t, in Example 6.2. The response time of an unsteady-state Level II system is equivalent to the residence time in a steady-state Level II system. By inspection of the magnitude of groups, VZ/D, or the reciprocal rate constants that occur in steady-state analysis, it is possible to determine the likely unsteady-state behavior. This is bad news to those who enjoy setting up and solving differential equations, because “back-of-theenvelope” calculations often show that it is not necessary to undertake a complicated unsteady-state analysis. Indeed, when calculating D values for loss from a medium, it is good practice to calculate the ratio VZ/D, where VZ refers to the source medium. This is the characteristic time of loss, or specifically the time required for that process to reduce the concentration to e–1 of its initial value if it were the only loss process. In some cases, we have an intuitive feeling for what that time should be. We can then check that the D value is reasonable.

6.6

THE NATURE OF ENVIRONMENTAL REACTIONS

The most important environmental reaction processes are biodegradation, hydrolysis, oxidation, and photolysis. We treat each process briefly below with the view to establishing methods by which the rate of the reaction can be characterized, and giving references to authoritative reviews. 6.6.1

Biodegradation

Microbiologists are usually quick to point out that the process of microbial conversion of chemicals in the environment is exceedingly complex. The rate of conversion depends on the nature of the chemical compound; on the amount and condition of enzymes that may be present in various organisms in various states of activation and availability to perform the chemical conversion; on the availability of nutrients such as nitrogen, phosphorus, and oxygen; as well as pH, temperature, and the presence of other substances that may help or hinder the conversion process. Virtually all organic chemicals are susceptible to microbial conversion or biodegradation. Notable among the slowly degrading or recalcitrant compounds are highmolecular-weight compounds such as the humic acids, certain terpenes that appear to have structures that are too difficult for enzymes to attack, and many organohalogen substances. Generally, water-soluble organic chemicals are fairly readily biodegraded. Over evolutionary time, enzymes have adapted and evolved the capability of handling most naturally occurring organic compounds. When presented with certain synthetic organic compounds that do not occur in nature (notably the

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halogentated hydrocarbons), they experience considerable difficulty, and they may or may not be able to perform useful chemical conversions. In such cases, if environmental degradation does take place, it is often the result of abiotic processes such as photolysis or reaction with free radicals. Our aim is to be able to define a half-life or rate constant for microbial conversion of the chemical, usually in water but often also in soil and in sediments. These rate constants may be measured by introducing the chemical into the medium of interest and following its decay in concentration. If first-order behavior is observed, a rate constant and half-life may be established. Care must be taken to ensure that the decay is truly attributable to biodegradation and not to other processes such as volatilization. In many cases, non-first-order behavior occurs. For example, it is suspected that, in some situations, the concentration of chemical is so low that the enzymes necessary for conversion do not become adequately activated, and the chemical is essentially ignored. At high concentrations, the presence of the chemical may result in toxicity to the microorganisms, and therefore the conversion process ceases. The number of active enzymatic sites may also be limited, thus the rate of conversion of the chemical species becomes controlled not by the concentration of the species but by the number of active sites and the rate at which chemicals can be transferred into and out of these sites. Under these conditions of saturation, a Michaelis–Menten type equation can be applied as described earlier. Much to the chagrin of microbiologists, we will adopt a simple expedient assuming that a first-order rate constant (or half-life) applies and that the rate constant can be estimated by experiment or from experience. This is necessarily an approximation to the truth and often involves merely a judgement that, in a particular type of water or soil, this compound is subject to biodegradation with a half-life of approximately x hours. The rate constant is therefore 0.693/x hours. Valiant efforts have been made to devise experimental protocols in which chemicals are subjected to microbial degradation conditions in the field or in the laboratory using, for example, innoculated sewage sludge. Such estimates are of particular importance in the prediction of chemical fate in sewage treatment plants. Even more valiant attempts are being made to predict the rate of biodegradation of chemicals purely from a knowledge of their molecular structure. Others have been content to categorise organic chemicals into various groups that have similar biodegradation rates or characteristics. Several standard and near-standard tests exist for determining biodegradation rates under aerobic and anaerobic conditions in water and in soils. Simplest is the biochemical oxygen demand (BOD) test as described in various standard methods compilations by agencies such as ASTM and APHA. More complex systems involve the use of chemostats and continuous flow systems, which are analogous to benchtop sewage treatment plants. An important characterization of biodegradation relates to whether the organism requires an oxygenated environment to thrive. All organisms require energy, which is obtained by performing chemical reactions. The most common reaction is oxidation, which is performed by aerobic organisms when oxygen is present. Oxidation of ethanol to acetic acid is an example. When oxygen is absent and anaerobic conditions prevail, the organism can obtain energy by processes such as reducing

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sulfate to sulfide or by dechlorinating a molecule. The latter is very important as a method of degrading organo-chlorine compounds, which are recalcitrant to direct oxidation. Howard (2000) has reviewed the principles surrounding biodegradation processes, the laboratory and field test methods that are employed, and a variety of methods by which biodegradation half-lives or classes can be estimated. One of the most popular and accessible biodegradation estimation methods is the BIODEG program, which is available from the Syracuse Research Corp. website (www.syrres.com). It is well established that certain groupings of atoms impart reactivity or recalcitrance to a molecule, thus a molecular structure can be examined to identify how fast it is likely to degrade. Computer programs such as BIODEG can do this automatically and assign a structure to a class such as “biodegrades fast” with a half-life of days to weeks. The half-life may be reported for primary degradation, i.e., loss of the parent compound, but also if interest is the time for complete mineralization to CO2 and water. The science of biodegradation is still a long way from being able to estimate half-lives within an accuracy of a factor of three; indeed, it may not be possible to estimate half-lives with greater accuracy. In addition to the excellent review by Howard (2000), the reader will find valuable material in the texts by Alexander (1994), Pitter and Choduba (1990), and Schwarzenbach et al. (1993). Howard (2000) also lists databases, notably the BIOLOG database of some 6000 chemicals. 6.6.2

Hydrolysis

In this process, the chemical species is subject to addition of water as a result of reaction with water, hydrogen ion, or hydroxyl ion. All three mechanisms may occur simultaneously at different rates; therefore, the overall rate can be very sensitive to pH. Rates of environmental hydrolysis have been thoroughly reviewed by Mabey and Mill (1978) and Wolfe and Jeffers (2000). For many organic compounds, hydrolysis is not applicable. A systematic method of testing for susceptibility to hydrolysis is to subject the chemical to pH levels of 3, 7, and 11; observe the decay; and deduce rate constants for acid, base, and neutral hydrolysis. These rate constants can be combined to give an expression for the rate at any desired pH, namely, dC/dt = –kH[H+]C – kOH[OH–]C – kW[H2O]C Structure activity approaches can be used to correlate and predict these rate constants. Often, the best approach is to seek data on a structurally similar substance. Other useful references on hydrolysis include the Wolfe (1980), Pankow and Morgan (1981), Zepp et al. (1975), Wolfe et al. (1977), and Jeffers et al. (1989). 6.6.3

Photolysis

The energy present in sunlight (photons) is often sufficient to cause chemical reactions or the rupture of chemical bonds in molecules that are able to absorb this

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light. Sunburn and photosynthesis are examples of such reactions. This process is primarily of interest when considering the fate of chemicals in solution in the atmosphere and in water. The radiation that is most likely to effect chemical change is high-energy, short-wavelength photons at the blue and near UV end of the spectrum, i.e., shorter than 400 nm. The relationships between energy, wavelength, and frequency are readily deduced using the fundamental constants of the speed of light c (3.0 × 108 m/s), Planck’s constant h (6.6 × 10–34 Js), and Avogadro’s Number N (6.0 × 1023). The energy of a photon of wavelength λ nm (frequency c/λ Hz) is hc/λ J/molecule or hcN/λ J/mol or Einsteins. A photon of wavelength 307 nm has a frequency of 9.8 × 1014Hz and energy of 387,000 J/mol or Einsteins. This is approximately the dissociation energy of the tertiary C-H bond in isobutane (2 methyl propane); thus, in principle, if the energy in such a photon could be applied to that bond, dissociation would occur. Short-wavelength photons are more energetic and are more likely to induce chemical reactions. There are two general concerns. Will the photon be absorbed such that reaction will occur? Will the quantity of photons be such that the reaction rate will be significant? To be absorbed directly, the molecule must have a chromophore that imparts suitable absorption characteristics. These properties can be measured using a spectrophotometer. As discussed later, there may be indirect absorption of the energy from another species that absorbs the photon then passes on the energy to the substance of interest. The issue of quantity can be assessed by calculating the amount of energy absorbed, recognizing that there are competitive absorbing substances such as natural organic matter present in the environment. The extent of absorption can be calculated from the Beer–Lambert Law such that log I = log IO – εCL = log IO – A where IO is the incident radiation, I is the surviving radiation at distance L, concentration C, extinction coefficient ε, and absorbance A. The quantity of light absorbed is (IO – I), and the fraction that is absorbed by the chemical can be deduced by comparing A for the chemical with A for the natural organic matter. In neartransparent or clear water when A is small, the quantity of light absorbed approaches 2.3IOεCL Einsteins/m2·h. Note that (1 – 10–x) approaches 2.3x when x is small. If each photon absorbed causes ϕ molecules (the quantum yield) to react, then the reaction rate will be 2.3ϕIOεCL mol/m2·h and, in principle, the first-order rate constant is 2.3ϕIOε, IO having units of mol/m2 h and ε units of m2/mol. In practice, IO and ε are functions of wavelength. Not only is there direct absorption of sunlight from the sun, but diffuse radiation from the sky also contributes. IO also depends on latitude, time of day and year, and cloud cover. If ε is known as a function of wavelength, computer programs can be used to integrate over the solar spectrum to give the total photolysis rate constant. The quantum yield may be quite small, e.g., 0.1 or, in the case of chain reactions, it can be larger than 1.0. Computer programs such as SOLAR are available to undertake these calculations. The reader is referred

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to Zepp and Cline (1977) for the original work in this area; to Leifer (1988) for an update; and to Calvert and Pitts (1966), Mill (2000), and Schwarzenbach et al. (1993) for more details and examples of photochemical reactions and computer programs. For our purposes, it is sufficient to appreciate that, knowing the absorbance properties of the molecule, the quantum yield and the local insolation conditions, it is possible to calculate a rate constant and a half-life for direct photolysis. Relatively simple experiments can be conducted in which the chemical is dissolved in distilled or natural water in a suitable container and exposed to natural sunlight or to artificial light for a period of time, and the concentration decay is monitored. Test methods have been described by Svenson and Bjarndahl (1988), Lemaire et al. (1982), and Dulin and Mill (1982). The issue is complicated by the presence of photosensitizing molecules or substances. These substances absorb light then pass on the energy to the chemical of interest, resulting in subsequent chemical reaction. It is therefore not necessary for the chemical to absorb the photon directly. It can receive it second hand from a photosensitizer. This is a troublesome complication, because it raises the possibility that chemicals may be subject to photolysis due to the unexpected presence of a photosensitizer. Of particular interest are the naturally occuring organic matter photosensitizers that are present in water and give it its characteristic brown color, especially in areas in which there is peat and decaying vegetation. 6.6.4

Atmospheric Oxidation Reactions

A chemical present in the atmosphere may react with oxygen, an activated form of oxygen such as singlet oxygen, ozone, hydrogen peroxide, or with various radicals, notably OH radicals. Fortunately, we live in a world with an abundance of oxygen, and it is not surprising that a suite of oxygen compounds exists that are eager to oxidize organic chemicals. The rates of these reactions can be estimated by conducting conventional chemical kinetic experiments in which the substance is contacted with known concentrations of the oxidant, the decay of chemical is followed, and a kinetic law and rate constant established. The most important oxidative process is the reaction of hydroxyl radicals with chemical species in the atmosphere. The concentration of sunlight-induced hydroxyl radicals is exceedingly small, averaging only about 1 million molecules per cubic centimetre. Peak concentrations approach 8 million per cm3 in urban areas. Concentrations in rural or remote areas are much lower. They are extremely reactive and are responsible for the reaction of many organic chemicals in the environment that would otherwise be persistent. Ozone is produced by UV radiation in the stratosphere and by certain hightemperature and photolytic processes in the troposphere. The average mixing ratio, i.e., the ratio of ozone to non-ozone molecules, is in the range of 10 to 40 × 10–9. Oxides of nitrogen produced at high temperature include NO, NO2, and the reactive NO3 radical. The latter has an average concentration of about 500 million molecules per cm3 and peaks in concentration at night.

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A formidable literature exists on the kinetics of gas phase organic substances, notably hydrocarbons, with OH radicals. Quantitative structure activity relationships have been developed in which each part of the molecule is assigned a rate constant for abstraction of H by OH radicals, or for addition of OH radicals to unsaturated bonds. Atkinson (2000) has reviewed these estimation methods and provides references to compilations of rate constant data. Computer programs exist to estimate these rate constants from molecular structure, for example from the Syracuse Research Corporation website (www.syrres.com). It is important to appreciate that the atmosphere is a very reactive medium in which large quantities of chemical species are converted into oxidized products. This is fortunate, because otherwise there would be more severe air pollution and problems associated with the transport of these chemicals to remote regions. 6.6.5

Aqueous Oxidation and Reduction

Natural oxidizing agents include oxygen, hydrogen peroxide, ozone, and “engineered” oxidants include chlorine, hypochlorite, chlorine dioxide, permanganate, chromate, and ferrate. Natural reducing agents include sulphide, ferrous and manganous ion, and organic matter, while “engineered” reductants include dithionite and zero-valent (metal) iron. Oxidation usually involves the addition of oxygen but, in more general terms, it is the removal of or abstraction of an electron. Reduction involves electron addition. The potential or feasibility of such a reaction occurring can be readily evaluated from the standard potential of the half reactions. The kinetics are usually expressed using a second-order expression including the concentration of the substance and the oxidant or reductant. In some cases, the reactant is a solid (e.g., zero-valent iron), and an area-normalized value can be used. Tratnyek and Macalady (2000) provide an excellent review of this literature and give several examples of oxidation and reduction processes. Again, for our purposes, a first-order rate constant can be estimated that includes the concentration of the oxidising or reducing agent. This can be used to calculate the corresponding halflife and D value. 6.6.6

Summary

It has been possible to provide only a brief account of the vast literature relating to chemical reactivity in the environment. The air pollution literature is particularly large and detailed. References have been provided to give the reader an entry to the literature. The susceptibility of a chemical in a specific medium to degrading reaction depends both on the inherent properties of the molecule and on the nature of the medium, especially temperature and the presence of candidate reacting molecules or enzymes. In this respect, environmental chemicals are fundamentally different from radioisotopes, which are totally unconcerned about external factors. Translation and extrapolation of reaction rates from environment to environment and laboratory to environment is therefore a challenging and fascinating task that will undoubtedly keep environmental chemists busy for many more decades.

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6.7

141

LEVEL II COMPUTER CALCULATIONS

As with Level I calculations, it is desirable to reduce the tedium of calculations by using the computer. Figure 6.2 gives an illustrative fugacity form calculation, a blank form being provided in the appendix. Computer programs that conduct Level II calculations are available from the Internet. The input data include the properties of the environment, the chemical properties, input rates by emission and advection, and information on reaction and advection rates. The fugacity is calculated, followed by a complete mass balance. Since equilibrium is assumed to apply within the environment, it is immaterial into which phase the chemical is introduced. The user is encouraged to test the environmental behavior of some of the chemicals introduced earlier, assuming or obtaining literature data on reaction rates. Worked Example 6.7 Calculate the partitioning of the hypothetical chemical in Figure 5.6 assuming that the rate constants for reaction are 0.001 h–1 in water, 0.01 h–1 in soil, and 0.0001 h–1 in sediment, and with no reaction in air. Assume advective inputs in air at 10–6 mol/m3 (flow 107 m3/h) and in water at 0.01 mol/m3 (flow 1000 m3/h). The emission rate is 100 mol/h. The hand calculation is fairly tedious and is reproduced in Figure 6.2. It involves calculation of the total inputs of 100 mol/h (emission), 10 mol/h (advection in air), and 10 mol/h (advection in water) totaling 120 mol/h (I). The reaction and advection D values are then deduced and added to give a total (ΣD) of approximately 10,390 mol/Pa h. The fugacity is then I/ΣD or 0.0115 Pa. Concentrations, amounts, and process rates can then be deduced and added to check the mass balance. The computed output is given in Figure 6.3. Note that it is not possible to input an infinite half life in air to give a zero rate of reaction. A fictitious, large value of 1011 h is used instead.

6.8

SUMMARY

In this chapter, we have learned to include advection and reaction rates in evaluative Level II calculations. These calculations can be done using concentrations and partition coefficients or fugacities and D values. The concepts of residence time and persistence have been reintroduced. These are invaluable descriptors of environmental fate. We have briefly reviewed the essential environmental chemistry of biodegradation, photolysis, hydrolysis, and other reactions, and provided references to studies, reviews, and estimation methods. Critics will be eager to point out a major weakness in these calculations. Environmental media are rarely in equilibrium; therefore, a use of a common fugacity or the use of equilibrium partition coefficients to relate concentrations between phases or media is often not valid. Treating nonequilibrium situations is the task of Chapter 7.

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Fugacity Form 3

Level II

Chemical:

Hypothene

Direct emission rate E

100 mol/h

Advective input rates Compartment

Water

Air

Volume m (V)

6 × 10

7 × 106

Residence time h (t)

600

7000

Flow rate m3/h = V/t = G

107

1000

3

9

3

–6

Inflow concentration mol/m CB

10

10–2

Chemical inflow rate mol/h = GCB

10

10

Total input rate E + ΣGCB = I =

100 + 10 + 10 = 120

Compartment

Air

Water

Soil

Sediment

Volume m3 (V)

6 × 109

7 × 106

45000

21000

Z

4 × 10–4

0.1

12.3

24.6

VZ

2.4 × 106

7 × 105

5.5 × 105

5.17 × 105

Reaction half life (h)t

∞

693

69.3

6930

Rate constant k = 0.693/t (h–1)

0

0.001

0.01

0.0001

Advective flow G m3/h

107

1000

0

0

D reaction = VZk = DR

0

700

5535

51.7

D advection = GZ = DA

4000

100

0

0

DR + DA = DT

4000

800

5535

51.7

Total D value = ΣDT =

10387

Fugacity f = I/ΣD =

120/10387 = 1.15 × 10–2

C = Z f mil/m3

4.6 × 10–6

1.15 × 10–3

0.14

0.28

m = C V mol

27731

8087

6394

5968

Percent

57.5

16.8

13.3

12.4

CG g/m , i.e., CW

9.2 × 10

0.23

28.4

56.8

Density ρ kg/m3

1.18

1000

1500

1500

CU µg/g, i.e., CG × 1000/ρ

0.79

0.23

19

38

Reaction rate DRf

0

8.1

63.9

0.6

Advection rate DAf

46.2

1.2

0

0

Total DTf

46.2

9.3

63.9

0.6

Total amount M = Σm =

48180

Total reaction rate = ΣDRf =

72.6

Reaction residence time (h) = M/ΣDRf

663

Total advection rate = ΣDAf =

47.4

Advection residence time (h) = M/ΣDAf

1017

Total output rate (mol/h) = I =

120

Overall residence time (h) = M/I =

401

3

Figure 6.2

–4

Fugacity form for completing a Level II calculation.

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Figure 6.3

Fugacity Level II calculation corresponding to Figure 6.3.

143

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6.9

CONCLUDING EXAMPLE

For the two substances selected from Table 3.5, which were the subject of the concluding example in Chapter 5, perform a Level II calculation for the air, water, soil, and bottom sediment phases either as defined in that example or using the environment deduced in the concluding example from Chapter 4. Use Fugacity Form 3 and ignore other phases. Assume reasonable residence times in air, water, and bottom sediment for the purpose of calculating advection rates. There is no advection from soil. Use the degradation half-lives from Table 3.5. Assume first a total input by emission of 100 kg/h and calculate the fugacity, concentrations, amounts, and the three chemical residence times (overall, reaction, and advection). Second, recalculate this Level II example assuming that the inflow air and water both contain the chemical at a concentration that is 20% of the air and water concentrations calculated above. Discuss the results and present them in a diagrammatic form. Discuss which reaction and advection processes are most important. Are the residence times in these two examples equal or not? Explain why they are equal or different.

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CHAPTER

7

Intermedia Transport 7.1

INTRODUCTION

The Level II calculations described in Chapter 6 contain the major weakness that they assume environmental media to be in equilibrium. This is rarely the case in the real environment; therefore, the use of a common fugacity (or concentrations related by equilibrium partition coefficients) is usually, but not always, invalid. Reasons for this are best illustrated by an example. Suppose we have air and water media as illustrated in Figure 7.1, with emissions of 100 mol/h of benzene into the water. There is only slow reaction in the water (say, 20 mol/h), but there is rapid reaction (say, 80 mol/h) in the air. This implies that benzene is evaporating from water to air at a rate of 80 mol/h. The question arises: is benzene capable of evaporating at 80 mol/h, or will there be a resistance to transfer that prevents evaporation at this rate? If only 40 mol/h could evaporate, the evaporated benzene may react in the air phase at 40 mol/h, but it will tend to build up in the water phase to a higher concentration and fugacity until the rate of reaction in the water increases to 60 mol/h. The benzene fugacity in the air will thus be lower than the fugacity in water, and a nonequilibrium situation will have developed. The ability to calculate how fast chemicals can migrate from one phase to another is the challenging task of this chapter. The topic is one in which there still remain considerable uncertainty and scope for scientific investigation and innovation. We begin it by listing and categorizing all the transport processes that are likely to occur.

7.2 7.2.1

DIFFUSIVE AND NONDIFFUSIVE PROCESSES

Nondiffusive Processes

The first group of processes consists of nondiffusive, or piggyback, or advective processes. A chemical may move from one phase to another by piggybacking on 145

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Figure 7.1

Illustration of nonequilibrium behavior in an air-water system. In the lower diagram, the rate of reaction in air is constrained by the rate of evaporation.

material that has decided, for reasons unrelated to the presence of the chemical, to make this journey. Examples include advective flows in air, water, or particulate phases, as discussed in Chapter 6; deposition of chemical in rainfall or sorbed to aerosols from the atmosphere to soil or water; and sedimentation of chemical in association with particles that fall from the water column to the bottom sediments. These are usually one-way processes. The rate of chemical transfer is simply the product of the concentration C mol/m3 of chemical in the moving medium, and the flowrate of that medium, G, m3/h. We can thus treat all these processes as advection and calculate the D value and rate as follows: N = GC = GZf = Df mol/h The usual problem is to measure or estimate G and the corresponding Z value or partition coefficient. We examine these rates in more detail later, when we focus on individual intermedia transfer processes. 7.2.2

Diffusive Processes

The second group of processes are diffusive in nature. If we have water containing 1 mol/m3 of benzene and add some octanol to it as a second phase, the benzene will

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diffuse from the water to the octanol until it reaches a concentration in octanol that is KOW, or 135, times that in the water. We could rephrase this by stating that, initially, the fugacity of benzene in the water was (say) 500 Pa, and the fugacity in the octanol was zero. The benzene then migrates from water to octanol until both fugacities reach a common value of (say) 200 Pa. At this common fugacity, the ratio CO/CW is, of course, ZO/ZW or KOW. We argue that diffusion will always occur from high fugacity (for example, fW in water) to low fugacity (fO in octanol). Therefore, it is tempting to write the transfer rate equation from water to octanol as N = D(fW – fO) mol/h This equation has the correct property that, when fW and fO are equal, there is no net diffusion. It also correctly describes the direction of diffusion. In reality, when the fugacities are equal, there is still active diffusion between octanol and water. Benzene molecules in the water phase do not know the fugacity in the octanol phase. At equilibrium, they diffuse at a rate, DfW, from water to benzene, and this is balanced by an equal rate, DfO, from octanol to water. The escaping tendencies have become equal, and N is zero. The term (fW – fO) is termed a departure from equilibrium group, just as a temperature difference represents a departure from thermal equilibrium. It quantifies the diffusive driving force. Other areas of science provide good precedents for using this approach. Ohm’s law states that current flows at a rate proportional to voltage difference times electrical conductivity. Electricians prefer to use resistance, which is simply the reciprocal of conductivity. The rate of heat transfer is expressed by Fourier’s law as a thermal conductivity times a difference in temperature. Again, it is occasionally convenient to think in terms of a thermal resistance (the reciprocal of thermal conductivity), especially when buying insulation. These equations have the general form rate = (conductivity) × (departure from equilibrium) or rate = (departure from equilibrium)/(resistance) Our task is to devise recipes for calculating D as an expression of conductivity or reciprocal resistance for a number of processes involving diffusive interphase transfer. These include the following: 1. Evaporation of chemical from water to air and the reverse process of absorption. Note that we consider the chemical to be in solution in water and not present as a film or oil slick, or in sorbed form. 2. Sorption from water to suspended matter in the water column, and the reverse desorption. 3. Sorption from the atmosphere to aerosol particles, and the reverse desorption. 4. Sorption of chemical from water to bottom sediment, and the reverse desorption.

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5. Diffusion within soils, and from soil to air. 6. Absorption of chemical by fish and other organisms by diffusion through the gills, following the same route traveled by oxygen. 7. Transfer of chemical across other membranes in organisms, for example, from air through lung surfaces to blood, or from gut contents to blood through the walls of the gastrointestinal tract, or from blood to organs in the body.

Armed with these D values, we can set up mass balance equations that are similar to the Level II calculations but allow for unequal fugacities between media. To address these tasks, we return to first principles, quantify diffusion processes in a single phase, then extend this capability to more complex situations involving two phases. Chemical engineers have discovered that it is possible to make a great deal of money by inducing chemicals to diffuse from one phase to another. Examples are the separation of alcohol from fermented liquors to make spirits, the separation of gasoline from crude oil, the removal of salt from sea water, and the removal of metals from solutions of dissolved ores. They have thus devoted considerable effort to quantifying diffusion rates, and especially to accomplishing diffusion processes inexpensively in chemical plants. We therefore exploit this body of profit-oriented information for the nobler purpose of environmental betterment.

7.3 7.3.1

MOLECULAR DIFFUSION WITHIN A PHASE

Diffusion As a Mixing Process

In liquids and gases, molecules are in a continuous state of relative motion. If a group of molecules in a particular location is labeled at a point in time, as shown in the upper part of Figure 7.2, then at some time later it will be observed that they have distributed themselves randomly throughout the available volume of fluid. Mixing has occurred. Since the number of molecules is large, it is exceedingly unlikely that they will ever return to their initial condition. This process is merely a manifestation of mixing in which one specific distribution of molecules gives way to one of many other statistically more likely mixed distributions. This phenomenon is easily demonstrated by combining salt and pepper in a jar, then shaking it to obtain a homogeneous mixture. It is the rate of this mixing process that is at issue. We approach this issue from two points of view. First is a purely mathematical approach in which we postulate an equation that describes this mixing, or diffusion, process. Second is a more fundamental approach in which we seek to understand the basic determinants of diffusion in terms of molecular velocities. Most texts follow the mathematical approach and introduce a quantity termed diffusivity or diffusion coefficient, which has dimensions of m2/h, to characterize this process. It appears as the proportionality constant, B, in the equation expressing Fick’s first law of diffusion, namely N = –B A dC/dy

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Figure 7.2

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The fundamental nature of molecular diffusion.

Here, N is the flux of chemical (mol/h), B is the diffusivity (m2/h), A is area (m2), C is concentration of the diffusing chemical (for example, benzene in water) (mol/m3), and y is distance (m) in the direction of diffusion. The group dC/dy is thus the concentration gradient and is characteristic of the degree to which the solution is unmixed or heterogeneous. The negative sign arises because the direction of diffusion is from high to low concentration, i.e., it is positive when dC/dy is negative. Here, we use the symbol B for diffusivity to avoid confusion with D values. Most texts sensibly use the symbol D. The equation is really a statement that the rate of diffusion is proportional to the concentration gradient and the proportionality constant is diffusivity. When the equation is apparently not obeyed, we attribute this misbehavior to deviations or changes in the diffusivity, not to failure of the equation. As was discussed earlier, there are differences of opinion about the word flux. We use it here to denote a transfer rate in units such as mol/h. Others insist that it should be area specific and have units of mol/m2h. We ignore their advice. Occa-

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sionally, the term flux rate is used in the literature. This is definitely wrong, because flux contains the concept of rate just as does speed. Flux rate is as sensible as speed rate. It is worthwhile digressing to examine how the mixing process leads to diffusion and eventually to Fick’s first law. This elucidates the fundamental nature of diffusivity and the reason for its rather strange units of m2/h. Much of the pioneering work in this area was done by Einstein in the early part of this century and arose from an interest in Brownian movement—the erratic, slow, but observable motion of microscopic solid particles in liquids, which is believed to be due to multiple collisions with liquid molecules. 7.3.2

Fick’s Law and Diffusion at Steady State

We consider a square tunnel of cross-sectional area A m2 containing a nonuniform solution, as shown in the middle of Figure 7.2, having volumes V1, V2, etc., separated by planes 1–2, 2–3, 3–4, etc., each y metres apart. We assume that the solution consists of identical dissolved particles that move erratically, but on the average travel a horizontal distance of y metres in t hours. In time t, half the particles in volume V3 will cross the plane 2–3, and half the plane 3–4. They will be replaced by (different) particles that enter volume V3 by crossing these planes in the opposite direction from volumes V2 and V4. Let the concentration of particles in V3 and V4 be C3 and C4 mol/m3 such that C3 exceeds C4. The net transfer across plane 3–4 will be the sum of the two processes: C3 yA/2 moles from left to right, and C4 yA/2 moles from right to left. The net amount transferred in time t is then C3yA/2 – C4 yA/2 = (C3 – C4) yA/2 mol Note that CyA is the product of concentration and volume and is thus an amount (moles). The concentration gradient that is causing this net diffusion from left to right is (C3 – C4)/y or, in differential form, dC/dy. The negative sign below is necessary, because C decreases in the direction in which y increases. It follows that (C3 – C4) = –ydC/dy The flux or diffusion rate is then N or N = (C3 – C4) yA/2t = –(y2A/2t) dC/dy = –BAdC/dy mol/h which is referred to as Fick’s first law. The diffusivity B is thus (y2/2t), where y is the molecular displacement that occurs in time t. In a typical gas at atmospheric pressure, the molecules are moving at a velocity of some 500 m/s, but they collide after traveling only some 10–7 m, i.e., after 10–7/500 or 2 × 10–10 s. It can be argued that y is 10–7 m, and t is 2 × 10–10; therefore, we

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expect a diffusivity of approximately 0.25 × 10–4 m2/s or 0.25 cm2/s or 0.1 m2/h, which is borne out experimentally. The kinetic theory of gases can be used to calculate B theoretically but, more usefully, the theory gives a suggested structure for equations that can be used to correlate diffusivity as a function of molecular properties, temperature, and pressure. In liquids, molecular motion is more restricted, collisions occur almost every molecular diameter, and the friction experienced by a molecule as it attempts to “slide” between adjacent molecules becomes important. This frictional resistance is related to the liquid viscosity µ (Pa s). It can be shown that, for a liquid, the group (Bµ/T) should be relatively constant and (by the Stokes-Einstein equation) approximately equal to R/(6πNr), where N is Avogadro’s number, R is the gas constant, and r is the molecular radius (typically 10–10 m). B is therefore T R/(µ6πNr), where the viscosity of water µ is typically 10–3 Pa s. Substituting values of R, T, µ, and r suggests that B will be approximately 2 × 10–9 m2/s or 2 × 10–5 cm2/s or 7 × 10–6 m2/h, which is also borne out experimentally. Again, this equation forms the foundation of correlation equations. The important conclusion is that, during its diffusion journey, a molecule does not move with a constant velocity related to the molecular velocity. On average, it spends as much time moving backward as forward, thus its net progress in one direction in a given time interval is not simply velocity/time. In t seconds, the distance traveled (y) will be 2tB m. Taking typical gas and liquid diffusivities of 0.25 × 10–4 m2/s and 2 × 10–9 m2/s respectively, a molecule will travel distances of 7 mm in a gas and 0.06 mm in a liquid in one second. To double these distances will require four seconds, not two seconds. It thus may take a considerable time for a molecule to diffuse a “long” distance, since the time taken is proportional to the square of the distance. The most significant environmental implication is that, for a molecule to diffuse through, for example, a 1 m depth of still water requires (in principle) a time on the order of 3000 days. A layer of still water 1 m deep can thus effectively act as an impermeable barrier to chemical movement. In practice, of course, it is unlikely that the water would remain still for such a period of time. The reader who is interested in a fuller account of molecular diffusion is referred to the texts by Reid et al. (1987), Sherwood et al. (1975), Thibodeaux (1996), and Bird et al. (1960). Diffusion processes occur in a large number of geometric configurations from CO2 diffusion through the stomata of leaves to large-scale diffusion in ocean currents. There is thus a considerable literature on the mathematics of diffusion in these situations. The classic text on the subject is by Crank (1975), and Choy and Reible (2000) have summarized some of the more environmentally useful equations. 7.3.3

Mass Transfer Coefficients

Diffusivity is a quantity with some characteristics of a velocity but, dimensionally, it is the product of velocity and the distance to which that velocity applies. In many environmental situations, B is not known accurately, nor is y or ∆y; therefore, the flux equation in finite difference form contains two unknowns, B and ∆y. Ignoring the negative sign,

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N = AB∆C/∆y mol/h Combining B and ∆y in one term kM, equal to B/∆y, with dimensions of velocity thus appears to decrease our ignorance, since we now do not know one quantity instead of two. Hence we write N = AkM∆C mol/h Term kM is termed a mass transfer coefficient, has units of velocity (m/h), and is widely used in environmental transport equations. It can be viewed as the net diffusion velocity. The flux N in one direction is then the product of the velocity, area, and concentration. For example, if, as in the lower section of Figure 7.2, diffusion is occurring in an area of 1 m2 from point 1 to 2, C1 is 10 mol/m3, C2 is 8 mol/m3, and kM is 2.0 m/h, we may have diffusion from 1 to 2 at a velocity of 2.0 m/h, giving a flux of kMAC1 of 20 mol/h. There is an opposing flux from 2 to 1 of kMAC2 or 16 mol/h. The net flux is thus the difference or 4 mol/h from 1 to 2, which of course equals kMA(C1 – C2). The group kMA is an effective volumetric flowrate and is equivalent to the term G m3/h, introduced for advective flow in Chapter 6. 7.3.4

Fugacity Format, D Values for Diffusion

The concentration approach is to calculate diffusion fluxes N as ABdC/dy or AB∆C/∆y or kMA∆C. In fugacity format, we substitute Zf for C and define D values as BAZ/∆y or kMAZ, and the flux is then D∆f, since ∆C is Z∆f. Note that the units of D are mol/Pa h, identical to those used for advection and reaction D values. D = BAZ/∆y or D = kMAZ N = Df1 – Df2 = D(f1 – f2) Worked Example 7.1 A chemical is diffusing through a layer of still water 1 mm thick, with an area of 200 m2 and with concentrations on either side of 15 and 5 mol/m3. If the diffusivity is 10–5 cm2/s, what is the flux and the mass transfer coefficient? y = 10–3 m, B = 10–5 cm2/s × 10–4 m2/cm2 = 10–9 m2/s Thus, kM is B/∆y = 10–6 m/s The flux N is thus kMA(C1–C2) = 10–6 (200(15 – 5)) = 0.002 mol/s

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This flux of 0.002 mol/s can be regarded as a net flux consisting of kMAC1 or 0.003 mol/s in one direction and kMAC2 or 0.001 mol/s in the opposing direction. Worked Example 7.2 Water is evaporating from a pan of area 1 m2 containing 1 cm depth of water. The rate of evaporation is controlled by diffusion through a thin air film 2 mm thick immediately above the water surface. The concentration of water in the air immediately at the surface is 25 g/m3 (this having been deduced from the water vapor pressure), and in the room the bulk air contains 10 g/m3. If the diffusivity is 0.25 cm2/s, how long will the water take to evaporate completely? B is 0.25 cm2/s or 0.09 m2/h ∆y is 0.002 m ∆C is 15 g/m3 N = AB∆C/∆y = 675 g/h To evaporate 10000 g will take 14.8 hours

Note that the “amount” unit in N and C need not be moles. It can be another quantity such as grams, but it must be consistent in both. In this example, the 2 mm thick film is controlled by the air speed over the pan. Increasing the air speed could reduce this to 1 mm, thus doubling the evaporation rate. This ∆y is rather suspect, so it is more honest to use a mass transfer coefficient, which, in the example above is 0.09/0.002 or 45 m/h. This is the actual net velocity with which water molecules migrate from the water surface into the air phase. 7.3.5

Sources of Molecular Diffusivities

Many handbooks contain compilations of molecular diffusivities. The text by Reid et al. (1987) contains data and correlations, as does the text on mass transfer by Sherwood, Pigford, and Wilke (1975). The handbook by Lyman et al. (1982) and the text by Schwarzenbach et al. (1994) give correlations from an environmental perspective. The correlations for gas diffusivity are based on kinetic theory, while those for liquids are based on the Stokes–Einstein equation. In most cases, only approximate values are needed. In some equations, the diffusivity is expressed in dimensionless form as the Schmidt number (Sc) where Sc = µ/ρB where µ is viscosity and ρ is density.

7.4

TURBULENT OR EDDY DIFFUSION WITHIN A PHASE

So far, we have assumed that diffusion is entirely due to random molecular motion and that the medium in which diffusion occurs is immobile or stagnant, with

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no currents or eddies. In practice, of course, the environment is rarely stagnant, there being currents and eddies induced by the motion of wind, water, and biota such as fish and worms. This turbulent motion, illustrated in Figure 7.3, also promotes mixing by conveying an element or eddy of fluid from one region to another. The eddies may vary in size from millimetres to kilometres, and a large eddy may contain a fine structure of small eddies. Intuitively, it is unreasonable for an eddy to penetrate an interface, thus in regions close to interfaces, eddies tend to be damped, and only slippage parallel to the interface is possible. There may, therefore, be a thin layer of relatively quiescent fluid close to the interface that can be referred to as a laminar sublayer. In this layer, movement of solute to and from the interface may occur only by molecular diffusion. Under certain conditions, eddies in fluids may be severely damped, or their generation may be prevented. This occurs in a layer of air or water when the fluid density decreases with increasing height. This may be due to the upper layers being warmer or, in the case of sea water, less saline. An eddy that is attempting to move upward immediately finds itself entering a less dense fluid and experiences a hydrostatic “sinking” force. Conversely, a companion eddy moving downward experiences a “floating” force, which also tends to restore it to its original position. This inherent resistance to eddy movement damps out most fluid movement, and stable, stagnant conditions prevail. Thermoclines in water and inversions in the atmosphere are examples of this phenomenon. These stagnant or near-stagnant layers may act as diffusion barriers in which only molecular diffusion or slight eddy diffusion can occur. Conversely, situations in which density increases with height tend to be unstable, and eddy movement is enhanced and accelerated by the density field. An attractive approach is to postulate the existence of an eddy diffusivity, or a turbulent diffusivity, BT , which is defined identically to the molecular diffusivity, BM. The flux equation within a phase then becomes N = –A(BM + BT)dC/dy The task is then to devise methods of estimating BT for various environmental conditions. We expect that, in many situations, such as in winds or fast rivers, BT

Figure 7.3

The nature of turbulent or eddy diffusion in which chemical is conveyed in eddies within a fluid to a surface.

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is much greater than BM, and the molecular processes can be ignored. In stagnant regions, such as thermoclines or in deep sediments, BT may be small or zero, and BM dominates. As we move closer to a phase boundary, BT tends to become smaller; thus, it is possible that much of the resistance to diffusion lies in the layer close to the interface. The roughness of the interface plays a role in determining the thickness of this layer. For example, grass may damp out wind eddies and retard the rate of diffusion from soil to air. Animal fur retards both diffusion of heat and water vapor. A complicating factor is that we have no guarantee that BT is isotropic, i.e., that the same value applies vertically and horizontally. In Figure 7.3, we postulate that some eddies may be constrained to form elongated “roll cells.” The horizontal BT will therefore exceed the vertical value. In practice, this nonisotropic situation is common and even leads to conditions in rivers where three BT values must be considered: vertical, upstream-downstream, and cross-stream. To give an order of magnitude appreciation of turbulent diffusivities, it is observed that a vertical eddy diffusivity in air is typically 3600 m2/h plus or minus a factor of 3, thus the time for moving a distance of 1 m is of the order of 1 s. Molecular diffusion is clearly negligible in comparison. In lakes, a vertical eddy diffusivity may be 36 m2/h near the surface, corresponding to a velocity over a distance of 1 m of 1 cm/s. At greater depths, diffusion is much slower, possibly by a factor of 100. To estimate eddy diffusivities, one can watch a buoyant particle and time its transport over a given distance. The diffusivity is then that distance squared, divided by the time. Turbulent processes in the environment are thus quite complex and difficult to describe mathematically. The interested reader can consult Thibodeaux (1996) or Csanady (1973) for a review of the mathematical approaches adopted. We sidestep this complex issue here, but certain generalizations that emerge from the study of turbulent diffusion are worth noting. In the bulk of most fluid masses (air and water) that are in motion, turbulent diffusion dominates. We can measure and correlate these diffusivities. Generally, vertical diffusion is slower than horizontal diffusion. Often, diffusion is so fast that near-homogeneous conditions exist, which is fortunate, because it eliminates the need to calculate diffusion rates. In the atmosphere and oceans, there is a spectrum of eddies of varying size and velocity. The larger eddies move faster. Consequently, when a plume in the atmosphere or a dye patch in an ocean expands in size, it becomes subject to dispersion by larger, faster eddies, and the diffusivity increases. If the velocity of expansion of the plume or patch is constant, this implies that diffusivity increases as the square of distance. At phase interfaces (e.g., air-water, water-bottom sediment), turbulent diffusion is severely damped or is eliminated, thus only molecular diffusion remains. One can even postulate the presence of a “stagnant layer” in which only molecular diffusion occurs and calculate its diffusion resistance. This model is usually inherently wrong in that no such layer exists. It is more honest (and less trouble) to avoid the use of diffusivities and stagnant layer thicknesses close to the phase interfaces and invoke mass transfer coefficients that combine the varying eddy diffusivities, the molecular

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diffusivity, and some unknown layer thickness, into one parameter, kM. We then measure and correlate kM as a function of fluid conditions (e.g., wind speed) and seek advice from the turbulent transport theorists as to the best form of the correlation equations. In some diffusion situations, such as bottom sediments, the eddy diffusion may be induced by burrowing worms or creatures that “pump” water. This is termed bioturbation and is difficult to quantify. Its high variability and unpredictability is a source of delight to biologists and irritation to physical scientists. The study of turbulent diffusion in the atmosphere includes aspects such as the micrometeorology of diffusion near the ground as it influences evaporation of pesticides, the uptake of contaminants by foliage, and the dispersion of plumes from stacks, in which case the plume is treated by the Gaussian dispersion equations. In lakes, rivers, and oceans it is important to calculate concentrations near sewage and industrial outfalls and in intensively used regions such as harbors. In each case, a body of specialized knowledge and calculation methods has evolved.

7.5

UNSTEADY-STATE DIFFUSION

Those who dislike calculus, and especially partial differential equations, can skip this section, but the two concluding paragraphs should be noted. In certain circumstances, we are interested in the transient or unsteady-state situation, which exists when diffusion starts between two volumes that are brought into contact. This is shown conceptually in Figure 7.4, in which a “shutter” is removed, exposing a concentration discontinuity. The two regions proceed to mix and chemical diffuses, eventually achieving homogeneity. Environmentally, this situation is encountered when a volume of fluid (e.g., water) moves to an interface and there contacts another phase (e.g., air) containing a solute with a different fugacity. Volatilization may then occur over a period of time. There are now three variables: concentration (C), position (y), and time (t). If we consider a volume of A∆y, as shown in Figure 7.4, then the flux in is –BA dC/dy, and the flux out is –BA(dC/dy + ∆yd2C/dy2), while the accumulation is A∆y∆C in the time increment ∆t. It follows that –BA dC/dy + BA(dC/dy + ∆yd2C/dy2) = A∆y∆C/∆t or as ∆y and ∆t tend to zero, Bd2C/dy2 = dC/dt This is Fick’s second law. Solution of this partial differential equation requires two boundary conditions, usually initial concentrations at specified positions. A particularly useful solution is the “penetration” equation, which describes diffusion into a slab of fluid that is brought into contact with another slab of constant concentration CS. The boundary conditions are

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Figure 7.4

157

Unsteady-state or penetration diffusion.

C = CS at y = 0 at all times C = 0 for y > 0 at t = 0 Solution is easiest if some hindsight is invoked to suggest that the dimensionless group X or (y/ 4Bt ) will occur in the solution. Interestingly, this is of the same form as the initial definition of B as y2/2t. It can be shown that C = CS (1 – (2/ π ) 0∫X exp(–X2) dX) = Cs [1–erf(X)]

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where X = y/ 4Bt Unfortunately, this integral, which is known as the Gauss Error integral or probability function or error function, cannot be solved analytically, thus tabulated values must be used. The error function has the property that it is zero when X is zero, and it approaches unity when X is 3 or larger. Its value can be found in tables of mathematical functions, or it can be evaluated using built-in approximations in spreadsheet software. A convenient approximation is erf(X) = 1 – exp(–0.746X – 1.101 X2) which is quite accurate when X exceeds 0.75. When X is less than 0.5, erf(X) is approximately 1.1X. The penetration solution shown in Figure 7.4 illustrates the very rapid initial transfer close to the interface, followed by slower penetration that occurs later as the concentration gradient becomes smaller. Now the transfer rate at the boundary (y = 0) can be shown to be B(dC/dy)y=0 = CsA

B/π t

Over a time t, the total flux (mol) becomes CSA 4Bt/π The average flux is then obtained by dividing by t CsA 4B/π t mol/h But, since the average flux is CSAkM, the average mass transfer coefficient kM, which applies over this time, must be 4B/π t . The mass transfer coefficient, kM, under these transient conditions, thus depends on the time of exposure (short exposures giving a large kM) and on the square root of diffusivity. This contrasts with the steady-state solution, in which kM is independent of time and proportional to diffusivity. The reason for this behavior is that kM is apparently very large initially, because the concentration gradient is large. It falls in inverse proportion to t , thus the average also falls in this proportion. The lower dependence on diffusivity (to the power of 0.5 instead of 1.0) arises, because not all the transferring mass has to diffuse the total distance; much of it goes into “storage” during the transient concentration buildup. A problem now arises in environmental calculations: which definition of kM applies, B/∆y or 4B/π t ? Contact time is the key determinant. If the contact time between phases is long, and the amount transferred exceeds the capacity of the phases, it is likely that a steady-state condition applies, and we should use B/∆y.

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Conversely, if the contact time is short, we can expect to use 4B/π t . If we measure the transfer rates at several temperatures, and thus different diffusivities, or measure the transfer rate of different chemicals of different B, then plot kM versus B on loglog paper, the slope of the line will be 1.0 if steady-state applies, and 0.5 if unsteadystate applies. In practice, an intermediate power of about 2/3 often applies, suggesting that we have mostly penetration diffusion followed by a period of near-steady-state diffusion.

7.6

DIFFUSION IN POROUS MEDIA

When a solute is diffusing in air or water, its movement is restricted only by collisions with other molecules. If solid particles or phases are also present, the solid surfaces will also block diffusion and slow the net velocity. Environmentally, this is important in sediments in which a solute may have been deposited at some time in the past, and from which it is now diffusing back to the overlying water. It is also important in soils from which pesticides may be volatilizing. It is therefore essential to address the question, “By how much does the presence of the solid phase retard diffusion?” We assume that the solid particles are in contact, but there remains a tortuous path for diffusion (otherwise, there is no access route, and the diffusivity would be zero). The process of diffusion is shown schematically in Figure 7.5, in which it is apparent that the solute experiences two difficulties. First, it must take a more tortuous path, which can be defined by a tortuosity factor, FY, the ratio of tortuous distance to direct distance. Second, it does not have available the full area for diffusion, i.e., it is forced to move through a smaller area, which can be defined using an area factor, FA. This area factor FA, is equal to the void fraction, i.e., the

Figure 7.5

Diffusion in a porous medium in which only part of the area is accessible, and the diffusing molecule must take a longer, tortuous path.

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fraction of the total volume that is fluid, and thus is accessible to diffusion. It can be argued that the tortuousity factor, FY, is related to void fraction, v, raised possibly to the power –0.5; therefore, in total, we can postulate that the effective diffusivity in the porous medium, BE, is related to the molecular diffusivity, B, by BE = BFA/FY = Bv1.5 Such a relationship is found for packings of various types of solids, as discussed by Satterfield (1970). This equation may be seriously in error since (1) the effective diffusivity is sensitive to the shape and size distribution of the particles, (2) there may also be “surface diffusion” along the solid surfaces, and (3) the solute may become trapped in “cul de sacs” or become sorbed on active sites. At least the equation has the correct property that it reduces to intuitively correct limits that BE equals B when v is unity, and BE is zero when v is zero. There is no substitute for actual experimental measurements using the soil or sediment and solute in question. For soils, it is usual to employ the Millington–Quirk (MQ) expression for diffusivity as a function of air and water contents. An example is in the soil diffusion model of Jury et al. (1983). The MQ expression uses air and water volume fractions vA and vW and calculates effective air and water diffusivities as follows: BAE= BAvA10/3/(vA + vW)2 BWE= BWvW10/3/(vA + vW)2 where BA and BW are the molecular diffusivities, and BAE and BWE are the effective diffusivities. Inspection of these equations shows that they reduce to a similar form to that presented earlier. If vW is zero, BAE is proportional to vA to the power 1.33 instead of 1.5. Occasionally, there is confusion when selecting the concentration driving force that is to be multiplied by BE. This should be the concentration in the diffusing medium, not the total concentration including sorbed form. In sediments, the pore water concentration may be 0.01 mol/m3, but the total sorbed plus pure water, i.e., bulk concentration, is 10 mol/m3. BE should then be multiplied by 0.01 not 10. In some situations (regrettably), the total concentration (10) is used, in which case BE must be redefined to be a much smaller “effective diffusivity,” i.e., by a factor of 1000. The problem is that diffusivity is then apparently controlled by the extent of sorption. In sediments, it is suspected that much of the chemical present in the pore or interstitial water, and therefore available for diffusion, is associated with colloidal organic material. These colloids can also diffuse; consequently, the diffusing chemical has the option of diffusing in solution or piggy backing on the colloid. From the Stokes–Einstein equation, the diffusivity B is approximately inversely proportional to the molecular radius. A typical chemical may have a molecular mass of 200 and a colloid an equivalent molar mass of 6000 g/mol, i.e., it is a factor of 30

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larger in mass and volume, but only a factor of 300.33 or about 3 in radius. The colloid diffusivity will thus be about one-third that of the dissolved molecule. But if 90% of the chemical is sorbed, the colloidal diffusion rate will exceed that of the dissolved form. As a result, is necessary to calculate and interpret the component diffusion processes, since it may not be obvious which route is faster.

7.7 DIFFUSION BETWEEN PHASES: THE TWO-RESISTANCE CONCEPT 7.7.1

Derivation Using Concentrations

So far in this discussion, we have treated diffusion in only one phase, but in reality, we are most interested in situations where the chemical is migrating from one phase to another. It thus encounters two diffusion regimes, one on each side of the interface. Environmentally, this is discussed most frequently for air-water exchange, but the same principles apply to diffusion from sediment to water, soil to water and to air, and even to biota-water exchange. An immediate problem arises at the interface, where the chemical must undergo a concentration “jump” from one equilibrium value to another. The chemical may even migrate across the interface from low to high concentration. Clearly, whereas concentration difference was a satisfactory “driving force” for diffusion within one phase, it is not satisfactory for describing diffusion between two phases. When diffusion is complete, the chemical’s fugacities on both sides of the interface will be equal. Thus, we can use fugacity as a “driving force” or as a measure of “departure from equilibrium.” Indeed, fugacity is the fundamental driving force in both cases, but it was not necessary to introduce it for one-phase systems, because only one Z applies, and the fugacity difference is proportional to the concentration difference. Traditionally, interphase transfer processes have been characterized using the Whitman Two-Resistance mass transfer coefficient (MTC) approach (Whitman, 1923), in which departure from equilibrium is characterized using a partition coefficient, or in the case of air-water exchange, a Henry’s law constant. We derive the flux equations for air-water exchange using the Whitman approach and following Liss and Slater (1974), who first applied it to transfer of gases between the atmosphere and the ocean, and Mackay and Leinonen (1975), who applied the same principles to other organic solutes. We will later derive the same equations in fugacity format. Unfortunately, the algebra is lengthy, but the conclusions are very important, so the pain is justified. Figure 7.6 illustrates an air-water system in which a solute (chemical) is diffusing at steady-state from solution in water at concentration CW (mol/m3) to the air at concentration CA mol/m3, or at a partial pressure P (Pa), equivalent to CART. We assume that the solute is transferred relatively rapidly in the bulk of the water by eddies, thus the concentration gradient is slight. As it approaches the interface, however, the eddies are damped, diffusion slows, and a larger concentration gradient is required to sustain a steady diffusive flux. A mass transfer coefficient, kW, applies over this region. The solute reaches the interface at a concentration CWI, then abruptly changes to CAI, the air phase value. The question arises as to whether there is a

Mass transfer at the interface between two phases as described by the two-resistance concept. Note the concentration discontinuity on the right, whereas, in the equivalent fugacity profile on the left, there is no discontinuity.

162

Figure 7.6

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significant resistance to transfer at the interface. It appears that if it does exist, it is small and unmeasurable. In any event, we do not know how to estimate it, so it is convenient to ignore it and assume that equilibrium applies. We thus argue that there is no interfacial resistance, and CWI and CAI are in equilibrium. CAI/CWI = KAW = ZA/ZW = H/RT and CAI/KAW = CWI The solute then diffuses in the air from CAI to CA in the bulk air with a mass transfer coefficient kA. We can write the flux equations for each phase, noting that the fluxes N must be equal, otherwise there would be net accumulation or loss at the interface. N = kWA(CW – CWI) = kAA(CAI – CA) mol/h or more conveniently, CW – CWI = N/kWA CAI – CA = N/kAA or CAI/KAW – CA/KAW = N/(kAAKAW) which is CWI – CA/KAW = N/(kAAKAW) Adding the first and last equations to eliminate CWI gives CW – CA/KAW = N(1/kWA + 1/kAAKAW) = N/kOWA or N = kOWA(CW – CA/KAW) = kOWA(CW – P/H) where 1/kOW = 1/kW + 1/kAKAW = 1/kW + RT/HkA The term kOW is an “overall” mass transfer coefficient that contains the individual kW and kA terms and KAW. It should not be confused with KOW, the octanol-water

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partition coefficient. The significance of the addition of reciprocal k terms is perhaps best understood by viewing the process in terms of resistances rather than conductivities, where the resistance, R, is 1/k in the same sense that the electrical resistance (ohms) is the reciprocal of conductivity (siemens or mhos). The overall resistance, RO, is then the sum of the water phase resistance RW and the air phase resistance RA. RW = 1/(kW A) RA = RT/(H kA A) = 1/(KAW A kA) Thus, RO = RW + RA = 1/(kOW A) which is equivalent to the equation for 1/kOW above. Because the resistances are in series, they add, and the total reciprocal conductivity is the sum of the individual reciprocal conductivities. The reason that KAW enters the summation of resistances is that it controls the relative values of the concentrations in air and water. If KAW is large, CWI is small compared to CAI, thus the concentration difference (CW – CWI) will be constrained to be small compared to (CAI – CA), and the flux N will be constrained by the small value of kW(CW – CWI). In general, diffusive resistances tend to be largest in phases where the concentrations are lowest, and thus the concentration gradients are lowest. Typical values of kA and kW are, respectively, 10 and 0.1 m/h; thus, the resistances become equal when KAW is 0.01 or H is approximately 25 Pa m3/mol. If H exceeds 250 Pa m3/mol, the concentration in the air is relatively large, and the air resistance, RA, is probably less than one-tenth of RW and may be ignored. Conversely, if H is less than 2.5 Pa m3/mol, the water resistance RW is less than one-tenth of RA, and it can be ignored. Interestingly, when H is large, kW tends to equal kOW, and if CA or P/H is small, the flux N becomes simply kWACW. This group does not contain H, thus the evaporation rate becomes independent of H or of vapor pressure. At first sight, this is puzzling. The reason is that, if H or vapor pressure is high enough, its value ceases to matter, because the overall rate is limited only by the diffusion resistance in the water phase. An overall mass transfer coefficient kOA can also be defined as 1/kOA = 1/kA + H/RTkW = 1/kA + KAW/kW and N = kOA(CwKAW – CA) = kOA(CwH – P)/RT It follows that kOW = kOAKAW = kOAH/RT

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If H is low, kOA approaches kA, and when P/H is small, the flux approaches kACWKAW or kACWH/RT. In such cases, volatilization becomes proportional to H and may be negligible if H is very small. In the limit, when H is zero (as with sodium chloride), volatilization does not occur at all. Figure 7.7 is a plot of log vapor pressure, PS, versus log solubility in water on which the location of certain solutes is indicated. Recalling that H or KAWRT is the ratio of these solubilities, compounds of equal H or KAW will lie on the same 45° diagonal. Compounds of H > 250 Pa m3/mol lie to the upper left, are volatile, and are water phase diffusion controlled. Those of H < 2.5 or KAW < 0.001 lie to the lower right, are relatively involatile, and are air phase diffusion controlled. There is an intermediate band in which both resistances are important. It is interesting to note that a homologous series of chemicals, such as the chlorobenzenes or PCBs, tends to lie along a 45° diagonal of constant KAW . Substituting methyl groups or chlorines for hydrogen tends to reduce both vapor pressure and solubility by a factor of 4 to 6, thus KAW tends to remain relatively constant,

Figure 7.7

Plot of log vapor pressure versus log solubility in water for selected chemicals. The diagonals are lines of constant Henry’s law constant. The dashed line corresponds to a Henry’s law constant of 25 Pa m3/mol at which there are approximately equal resistances in the water and air phases.

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and the series retains a similar ratio of air and water resistances. Paradoxically, reducing vapor pressure as one ascends such a series does not reduce evaporation rate from solution, since it is KAW that controls the rate of evaporation, not vapor pressure. It is noteworthy that oxygen and most low-molecular-weight hydrocarbons lie in the water phase resistant region, whereas most oxygenated organics lie in the air phase resistant region. The H for water can be deduced from its vapor pressure of 2000 Pa at 20°C and its concentration in the water phase of 55,000 mol/m3 to be 0.04 Pa m3/mol. If a solute has a lower H than this, it may concentrate in water as a result of faster water evaporation but, of course, humidity in the air alters the water evaporation rate. Water evaporation is entirely air phase resistant, because the water need not, of course, diffuse through the water phase to reach the interface. It is already there. Certain inferences can be made concerning the volatilization rate of one solute from another, provided that (1) their H values are comparable, i.e., the same resistance or distribution of resistances applies, and (2) corrections are applied for differences in molecular diffusivity. For example, rates of oxygen transfer can be estimated using noble gases or propane as tracers, because all are gas phase controlled. Particularly elegant is the use of stable isotopes and enantiomers as tracers, since the partition coefficients and diffusivities are nearly identical. 7.7.2

Derivation Using Fugacity

We can use D values instead of mass transfer coefficients and diffusivities. These two-resistance equations can be reformulated in fugacity terms to yield an algebraic result equivalent to the concentration version. The derivation is less painful when fugacity is used. If the water and air fugacities are fW and fA and the interfacial fugacity is fI, then replacing C by Zf in the steady-state Fick’s law equation yields N = kWA(CW – CWI) = kWAZW(fW – fI) = DW(fW – fI) mol/h and N = kAA(CAI – CA) = kAAZA(fI – fA) = DA(fI – fA) mol/h where DW = kWAZW

and

DA = kAAZA

This is illustrated in Figure 7.6. Now, the interfacial fugacity fI is not known or measureable, thus it is convenient to eliminate it by adding the equations in rearranged form, namely, fW – fI = N/DW

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and fI – fA = N/DA Adding gives (fW – fA) = N(1/DW + 1/DA) = N/DV and N = DV(fW – fA) where 1/DV = 1/DW + 1/DA = 1/kWAZW + 1/kAAZA The groups 1/DA and 1/DW are effectively resistances that add to give the total resistance 1/DV. It can be shown that DV = kOWAZW = kOAAZA Thus, kOW/kOA = ZA/Zw = KAW as before. The net volatilization rate, DV(fW – fA), can be viewed as the algebraic sum of an upward volatilization rate, DVfW, and a downward absorption rate, DVfA. Expressions for intermedia diffusion become very simple and transparent when written in fugacity form. The selection of one of two possible overall MTCs is avoided. Each conductivity is expressed in identical units containing its own Z value. The conductivities add reciprocally, as do electrical conductivities in series.

7.8

MEASURING TRANSPORT D VALUES

Measuring nondiffusive D values is, in principle, simply a matter of measuring Z and G, the latter usually being the problem. Flows of air, water, particulate matter, rain, and food can be estimated directly. The more difficult situations involve estimations of the rate of deposition of aerosols and sedimenting particles in the water column. The obvious approach is to place a bucket, tray, or a sticky surface at the depositing surface and measure the amount collected. This method can be criticized, because the presence of the bucket alters the hydrodynamic regime and thus the settling rate. This problem is acute when estimating aerosol deposition rates on foliage in a field or forest where the boundary layer is highly disturbed. Measure-

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ments of resuspension rates are particularly difficult, because the resuspension event may be triggered periodically by a storm or flood or by an especially energetic fish chasing prey at the bottom of the lake. Regrettably, the sediment-water interface is not easily accessible, thus measurements are few, difficult, and expensive. Measurement of diffusion D values usually involves setting up a system in which there is a known fugacity driving force (f1 – f2) and the capacity to measure N, leaving the overall transport D value as the only unknown in the flux equation. A difficulty arises because, for a two-resistance in series system, it is impossible to measure the concentrations or fugacity at the interface; therefore, it is not possible to deduce the individual D values that combine to give the overall D value. The subterfuge adopted is to select systems in which one of the resistances dominates and that resistance can be equated to the total resistance. Guidance on chemical selection can be obtained from the location of the substance on Figure 7.7. Air phase mass transfer coefficients (MTCs) can be determined directly by measuring the evaporation rate of a pool of pure liquid, or even the sublimation rate of a volatile solid. The interfacial partial pressure, fugacity, or concentration of the solute can be found from vapor pressure tables. The concentration some distance from the surface can be zero if an adequate air circulation is arranged, thus ∆C or ∆f is known. The pool can be weighed periodically to determine N, and area A can be measured directly, thus the MTC or evaporation D value is the only unknown. Worked Example 7.3 A tray (50 × 30 cm in area) contains benzene at 25°C (vapor pressure 12,700 Pa). The benzene is observed to evaporate into a brisk air stream at a rate of 585 g/h. What are D and kM, the mass transfer coefficient? Since the molecular mass is 78 g/mol, N is 585/78 or 7.5 mol/h. ∆f = (12700 – 0) Pa D = 7.5/12700 = 5.9 × 10–4 mol/Pa h A is 0.5 × 0.3 or 0.15 m2. ZA is 1/RT or 4.04 × 10–4. Since D is kMAZ, kM is 9.7 m/h. In conventional units, ∆C is 12,700/RT or 5.13 mol/m3. N = kMA∆C Thus, kM = 9.7 as before. Obviously, the two approaches are algebraically equivalent. Using an experimental system of this type, the dependence of kM on wind speed can be measured. Measurement of overall intermedia D values or MTCs is similar in principle, ∆f applying between two bulk phases. A convenient method of measuring water-to-air transfer is to dissolve the solute in a tank of water, blow air across the surface to simulate wind, and measure the evaporation rate indirectly by following the decrease in concentration in the water with time. If the water volume is V m3, area is A m2, and depth is Y m, then

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N = VdCW/dt = –kOWA(CW – CA/KAW) where CA and CW are concentrations in air and water and kOW the overall MTC. Assuming CA to be zero, integrating gives CW = CWO exp(–kOWAt/V) = CWO exp (–kOWt/Y) or fW = fWO exp(–DVt/VZW) Plotting CW on semilog paper vs. linear time gives a measurable slope of –kOW/Y, hence kOW can be estimated. A system of this type has been described by Mackay and Yuen (1983) and is illustrated in Figure 7.8. A very useful quantity is the evaporation half-life, which is 0.693Y/kOW and 0.693 VZW/DV. Often, an order of magnitude estimate of this time is sufficient to show that volatilization is unimportant or that it dominates other processes, such as reaction. As noted earlier, measurement of the individual contributing air and water D values or MTCs is impossible, because the interfacial concentrations cannot be measured. If, however, the evaporation rates of a series of chemicals of different KAW are measured, it is possible to deduce kW and kA or DW and DA. The relationship 1/kOW = 1/kW + 1/kAKAW suggests plotting, as in Figure 7.8, 1/kOW versus 1/KAW for a series of chemicals. The intercept will be 1/kW and the slope 1/kA. A correction may be necessary for molecular diffusivity differences. kW or DW is measured by selecting chemicals of high KAW for which the term 1/kAKAW or 1/DA is negligible. Alkanes, oxygen, or inert gases are convenient. kA or DA is measured by choosing chemicals of low KAW such that 1/kAKAW or 1/DA is large compared to 1/kW or 1/DW. Alcohols are convenient for this purpose. Worked Example 7.4 A tank contains 2 m3 of water at 25°C, 50 cm deep, with dissolved benzene (KAW = 0.22) and naphthalene (KAW = 0.017), each at a concentration of 0.1 mol/m3. After 2 hours, these concentrations have dropped to 47.1 and 63.9% of their initial value, respectively. What are the overall and individual MTCs and D values? In each case, CW = CWO exp(–kOWt/Y), Y being 0.5 m. Thus, kOW = –(Y/t) ln (CW/CWO). Substituting gives Benzene kOW = 0.188 m/h Naphthalene kOW = 0.112 m/h

Assuming each kOW to be made up of identical kW and kA values, i.e., 1/kOW = 1/kW + 1/(KAWkA)

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Movement of air phase (kA), water phase (kW), and overall (kOV) mass transfer coefficients by following the volatilization of substances with different air-water partition coefficients, KAW.

This equation can be written twice, once for benzene and once for naphthalene, using the specific values of kOW and KAW . The two equations can be solved for kW and kA, giving kW = 0.20

kA = 15 m/h

In fugacity format, ZA is 4.04 × 10–4 for both substances, and ZW is 1.836 × 10–3 for benzene and 23.7 × 10–3 for naphthalene, thus the fugacities are initially 54 and 4.22 Pa, falling to 25 and 2.72 Pa. The DV values are obtained from

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fW = fWO exp(–DVt/VZW) DV for benzene DV for naphthalene

= =

1.38 × 10–3 10.6 × 10–3

(note that this is kOWAZW)

Now, 1/DV equals (1/DA + 1/DW), the DA value being common to both chemicals. DW contains the variable ZW; therefore, DW for naphthalene is DW for benzene times 23.7 × 10–3/1.836 × 10–3 or 12.9. Benzene Naphthalene

DA = 0.0242 DA = 0.0242

DW = 0.00146 DW = 0.0189

In practice, it is unwise to rely on only two chemicals, it being better to use at least five, covering a wide range of KAW values. The air phase resistance, when viewed as 1/DA, is 41.3 units in both cases, but the water phase resistance for benzene is 685, while for naphthalene it is 52.9. Benzene experiences 5.7% of the transfer resistance in the air, while naphthalene experiences 44% resistance in the air, because it has a much lower KAW. Example 7.5 Ten kilograms each of benzene, 1,4 dichlorobenzene, and p cresol are spilled into a pond 5 m deep with an area of 1 km2. If kW is 0.1 m/h, and kA is 10 m/h, what will be the times necessary for half of each chemical to be evaporated? Use the property data from Chapter 3, and ignore other loss processes. Answer Benzene, 36 h, dichlorobenzene 38 h, p cresol 12400 h. Some of the earliest environmental modeling was of oxygen transfer to oxygendepleted rivers in which a “reaeration constant,” k2, was introduced (with units of reciprocal time) using the equation dCW/dt = k2(CE – CW) mol/m3h where CE is the equilibrium solubility of oxygen in water. Another term is usually included for oxygen consumption, but we ignore it here. Now, if the volume in question is 1 m2 in horizontal area and Y m deep, it will have a volume of Y m3, and the flux N will be YdCw/dt mol/h. But N = kM(CE – CW) = YdCW/dt = Yk2(CE – CW) kM is thus equivalent to Yk2. A typical k2 of 1 day–1 in a river of depth 2.4 m corresponds to a mass transfer coefficient of 2.4 m/day or 0.1 m/h. Oxygen reaeration

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rates can thus be used to estimate mass transfer coefficients for other solutes having similar (large) H such as alkanes. Indeed, an ingenious experimental approach for determining k2 for oxygen is to use a volatile hydrocarbon, such as propane, as a tracer, thus avoiding the complications of biotic oxygen consumption or generation, which confound environmental measurements of oxygen concentration change. It is erroneous to use k2 to estimate the rate of volatilization of a chemical with low H, since k2 contains negligible air phase resistance information. A correction should also be applied for the effect of molecular diffusivity, preferably using the dimensionless form of diffusivity, the Schmidt number raised to a power such as –0.5 or –0.67. This technique of probing interfacial MTCs by measuring N for various chemicals can be applied in other areas. When chemical is taken up by fish, it appears that it passes through one or more water layers and one or more organic membranes in series. By analogy with air-water transfer, we can write an organic membranewater transfer equation simply by replacing subscript A by subscript M, giving N = kOWA(CW – CM/KOW) where 1/kOW = 1/kW + 1/kMKMW or more conveniently changing to an overall organic phase MTC, kOM, N = kOMA(CWKMW – CM) where 1/kOM = 1/kM + KMW/kW A plot of 1/kOM versus KMW, the organic-water or octanol-water partition coefficient, gives 1/kM as intercept and 1/kW as slope. This is essentially the fish bioconcentration equation (discussed in more detail in Chapter 8) in disguise, which is conventionally written dCF/dt = k1CW – k2CF where VF is fish volume and CO is CF/L, where L is the volume fraction lipid (equivalent to octanol) in the fish. It follows that dCF/dt = (kOMA/VF)(CWKMW – CF/L) k1 is obviously KOWkOMA/VF, and k2 is [kOMA/(VFL)]. k1/k2 is then LKOW, the bioconcentration factor. The area of the respiring gill surface is uncertain as is kOM, so it is convenient to lump these uncertainties in one unknown k2. This suggests plotting

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1/k2 versus KOW to obtain quantities containing kM and kW. Such a plot was compiled by Mackay and Hughes (1984), yielding estimates of the two-resistances expressed as characteristic uptake times. Another example is the penetration of chemicals through the waxy cuticles of leaves in which there are air and wax resistances in series. Kerler and Schonherr (1988) have measured such penetration rates for a variety of chemicals, and Schramm et al. (1987) have attempted to model chemical uptake by trees using this tworesistance approach. A plant’s principal problem in life is to manage its water budget and avoid excessive loss of water through leaves. It accomplishes this by forming a waxy layer through which water has only a very slow diffusion rate. Diffusivities are very low, leading to very low MTCs and D values for water. The plant thus exploits this two-resistance approach to conserve water. If only governments could manage their budgets with the same efficiency!

7.9

COMBINING SERIES AND PARALLEL D VALUES

Having introduced these transport D values and shown how they combine when describing resistances in series, it is useful to set out the general flux equation for any combination of transport processes in series or parallel. Each transport process is quantified by a D value (deduced as GZ, kAZ, or BAZ/Y) that applies between two points in space such as a bulk phase and an interface, or between two bulk phases. It is helpful to prepare an arrow diagram of the processes showing the connections, as illustrated in Figure 7.9. Diffusive processes are reversible, so they actually consist of two arrows in opposing directions with the same D value but driven by different source fugacities. When processes apply in parallel between common points, the D values add. An example is wet and dry deposition from bulk air to bulk water. DTOTAL = D1 + D2 + D3, etc. When processes apply in series, the resistances add or, correspondingly, the reciprocal D values add to give a reciprocal total. 1/DTOTAL = 1/D1 + 1/D2 + 1/D3, etc. An example is the addition of air and water boundary layer resistances, which in total control the rate of volatilization from water. It is possible to assemble numerous combinations of series and parallel processes linking bulk phases and interfaces. These situations can be viewed as electrical analogs, with voltage being equivalent to fugacity, resistance equivalent to 1/D, and current equivalent to flux (mol/h). Figure 7.9 gives some examples. In air-water exchange, there can be deposition by the parallel processes of (1) dry particle deposition, (2) wet particle deposition, (3) rain dissolution, and (4) diffusive absorption-volatilization.

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Combination of D values and resistances in series, parallel, and combined configurations.

The soil-air exchange example involves parallel diffusive transport from bulk soil to the interface in water and air, followed by a series air boundary layer diffusion step. The sediment-water example is similar, having parallel diffusive paths for chemical transport in water and in association with organic colloids. The difficulty is to estimate the diffusivity of the colloids. Even more complex combinations can be compiled for transport processes into and within organisms, this being essentially the science of pharmacokinetics.

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7.10 7.10.1

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LEVEL III CALCULATIONS

Level III D Values

In this chapter, we have examined the nature of molecular and eddy diffusivity, introduced the concept of mass transfer coefficients (k), and treated the problem of resistances occurring in series and parallel as material diffuses from one phase to another. Two new D values have been introduced, a kAZ product and a BAZ/∆Y product. We can treat situations in which various D values apply in series and in parallel. In some situations, diffusion D values may be assisted or countered by advective transfer D values. For example, PCB may be evaporating from a water surface into the atmosphere only to return by association with aerosol particles that fall by wet or dry deposition. We can add D values when the fugacities with which they are multiplied are identical, i.e., the source is the same phase. This is convenient, because it makes the equations algebraically simple and enables us to compare the rates at which materials move by various mechanisms between phases. We thus have at our disposal an impressive set of tools for calculating transport rates between phases. We need Z values, mass transfer coefficients, diffusivities, path lengths, and advective flow rates. Quite complicated models can be assembled describing transfer of a chemical between several media by a number of routes. In general, the total D value for movement from phase A to phase B will not be the same as that from B to A. The reason is that there may be an advective process moving in only one direction. Diffusive processes always have identical D values applying in each direction. D values for loss by reaction can also be included in the mass balance expression. We are now able to use these concepts to perform a Level III calculation. These calculations were suggested and illustrated in a series of papers on fugacity models (Mackay, 1979; Mackay and Paterson 1981, 1982; and Mackay et al. 1985). It is important to emphasize that these models will give the same results as other concentration-based models, provided that the intermedia transport expressions are ultimately equivalent. A major advantage of the fugacity approach is that an enormous amount of detail can be contained in one D value, which can be readily compared with other D values for different processes. It is quite difficult, on the other hand, to compare a reaction rate constant, a mass transfer coefficient, and a sedimentation rate and identify their relative importance. Figure 7.10 depicts the simple four-compartment evaluative environment with the intermedia transport processes indicated by arrows. In addition to the reaction and advection D values, which were introduced in Level II, there are seven intermedia D values. The emission rates of chemicals must now be specified on a medium-bymedium basis whereas, in Level II, only the total emission rate was needed. Table 7.1 lists the intermedia D values and gives the equations in terms of transport rate parameters. Subscripts are used to designate air, 1; water, 2; soil, 3; and sediment, 4. Table 7.2 gives order-of-magnitude values for parameters used to calculate intermedia transport D values. These values depend on the environmental conditions and

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Figure 7.10 Four-compartment Level III diagram.

to some extent on chemical transport properties such as diffusivities. The variation in diffusivity is usually small compared to the variation in Z values, thus the use of chemical-specific diffusivities is justified only for the most accurate simulations. Most of these transport parameters can be expressed as velocities. The values given vary considerably from place to place and time to time. If the aim is to simulate conditions in a specific region, appropriate transport rate parameters for that region can be sought. The air-side and water-side mass transfer coefficients kVA and kVW have been measured in wind-wave tanks and in lakes as a function of wind speed. Schwarzenbach et al. (1993) have reviewed these correlations. The following correlations are suggested by Mackay and Yuen (1983). Note that units are m/s. kVA = 10–3 + 0.0462 U*(ScA)–0.67 m/s kVW = 10–6 + 0.0034 U*(ScW)–0.5 m/s U* = 0.01(6.1 + 0.63 U10)0.5U10 m/s where U* is the friction velocity, which characterizes the drag of the wind on the water surface. ScA is the Schmidt number in air and ranges from 0.6 for water to about 2.5, and ScW applies to the water phase and is generally about 1000. U10 is the wind velocity at 10 m height. Changing the velocity units to m/h, substituting typical values for the Schmidt number, and taking into account other studies, the following correlations are suggested.

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Table 7.1 Intermedia Transfer D Value Equation Compartments

Process

air(1) – water(2)

D Values

diffusion

DV = 1/(1/kVAA12ZA + 1/kVWA12ZW)

rain dissolution

DRW2 = A12 UQ ZW

wet deposition

DQW2 = A12 UR Q vQ ZQ

dry deposition

DQD2 = A12 UQ vQ ZQ D12 = DV + DRW2 + DQD2 + DQW2 D21 = DV

air(1) – soil(3)

diffusion

DE =1/(1/kEAA13Z + Y3/(A13(BMAZA + BMWZW)))

rain dissolution

DRW3 = A13 UR vQZW

wet deposition

DQW3 = A13 UR Q vQ ZQ

dry deposition

DQD3 = A13 UQ vQ ZQ D13 = DE + DRW3 + DQW3 + DQD3 D31 = DE

soil(3) – water(2)

soil runoff

DSW = A13 UEW ZE

water runoff

DWW = A13 UWW ZW D32 = DSW + DWW D23 = 0

sediment(4) – water(2)

diffusion

DY = 1/(1/kSWA24ZW + Y4/BW4A24ZW)

deposition

DDS = UDP A23 ZP

resuspension

DRS = URS A23 ZS D24 = DY + DDS D42 = DY + DRS

reaction either bulk phase i or sum of all phases

DRi = kRi Vi Zi DRi = Σ(kRij Vij Zij)

advection bulk phase

DAi = Gi Zi or Ui AI Zi

Aij is the horizontal area between media i and j. Subscripts on Z are A, water W, aerosol Q, soil E, sediments S and particles in water P.

kVA = 3.6 + 5 U101.2 m/h

U10 in m/s

kVW = 0.0036 + 0.01 U101.2 m/h

U10 in m/s

These correlations will underestimate the mass transfer coefficients under turbulent conditions of breaking waves or in rivers where there is “white water.” Sedimentation rates can be estimated by assuming a deposition velocity of about 1 m/day. Therefore, a lake containing 15 g/m3 of suspended solids is probably depositing 15 g/m2 day of solids, which corresponds to about 10 cm3/m2 day if the density is 1.5 g/cm3. This corresponds to 0.4 cm3/m2 h or 40 × 10–8 m3/m2 h or 40 × 10–8 m/h. Of this, a fraction is buried, and the remainder is resuspended. In waters of lower solids concentration, the deposition rate is correspondingly slower. The air-to-water D value (D12) consists of diffusive absorption (DV) and nondiffusive wet and dry aerosol deposition. Each D value can be estimated and summed to give D12.

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Table 7.2 Order of Magnitude Values of Transport Parameters Parameter

Symbol

Suggested typical value

Air side MTC over water

kVA

3 m/h

Water side MTC

kVW

0.03 m/h

Transfer rate to higher altitude

US

0.01 m/h (90m/y)

Rain rate (m3rain/m2area.h)

UR

9.7 × 10–5 m/h (0.85m/y)

Scavenging ratio

Q

200000

Vol. fraction aerosols

vQ

30

Dry deposition velocity

UQ

10.8 m/h (0.003 m/s)

Air side MTC over soil

kEA

1 m/h

Diffusion path length in soil

Y3

0.05 m

Molecular diffusivity in air

BMA

0.04 m2/h

Molecular diffusivity in water

BMW

4.0

Water runoff rate from soil

UWW

Solids runoff rate from soil

UEW

× 3.9 × 2.3 ×

Water side MTC over sediment

kSW

0.01 m/h

Diffusion path length in sediment

Y4

0.005 m

Sediment deposition rate

UDP

4.6

Sediment resuspension rate

URS

Sediment burial rate

UBS

Leaching rate of water from soil to ground water UL

×

× 1.1 × 3.4 × 3.9 ×

10–12

10–6 m2/h 10–5 m/h (0.34 m/y) 10–8 m3/m2h (0.0002 m/y)

10–8 m3/m2h (0.0004 m/y) 10–8 m3/m2h (0.0001 m/y) 10–8 m3/m2h (0.0003 m/y) 10–5 m3/m2h (0.34 m/y)

The water-to-air D value (D21) is DV for diffusive volatilization and is, of course, the same DV as for absorption. The air-to-soil D value (D13) is similar to D12, but the areas differ, and the absorptionvolatilization D value is also different. The soil-to-air D value (D31) is for volatilization. The water-to-sediment D value (D24) represents diffusive transfer plus nondiffusive sediment deposition. The sediment-to-water D value (D42) represents diffusive transfer plus nondiffusive resuspension. Finally, the soil-to-water D value (D32) consists of nondiffusive water and particle runoff.

There is no water-to-soil transfer, nor is there sediment-air exchange. The half-life for loss from a phase of volume V and Z value Z by process D is clearly 0.693 VZ/D. If a half-life τ1/2 is suggested, D is 0.693 VZ/τ1/2. Short halflives represent large D values and fast, important processes. It is always useful to calculate a half-life or a characteristic time VZ/D to ensure that it is reasonable. 7.10.2

Level III Equations

We now write the mass balance equations for each medium as follows.

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Air (subscript 1) E1 + GA1CB1 + f2D21 + f3D31 = f1(D12 + D13 + DR1 + DA1) = f1DT1 Water (subscript 2) E2 + GA2CB2 + f1D12 + f3D32 + f4D42 = f2(D21 + D24 + DR2 + DA2) = f2DT2 Soil (subscript 3) E3 + f1D13 = f3(D31 + D32 + DR3) = f3DT3 Sediment (subscript 4) E4 + f2D24 = f4(D42 + DR4 + DA4) = f4DT4 In each case, Ei is the emission rate (mol/h), GA is the advective inflow rate (m3/h), CBi is the advective inflow concentration (mol/m3), DRi is the reaction rate D value, and DAi is the advection rate D value. DTi is the sum of all loss D values from medium i. Sediment burial and air-to-stratospheric transfer can be included as an advection process or as a pseudo reaction. These four equations contain four unknowns (the fugacities), thus solution is possible. After some algebra, it can be shown that f2 = (I2 + J1J4/J3 + I3D32/DT3 + I4D42/DT4)/(DT2 – J2J4/J3 – D24·D42/DT4) f1 = (J1 + f2J2)/J3 f3 = (I3 + f1D13)/DT3 f4 = (I4 + f2D24)/DT4 where J1 = I1/DT1 + I3D31/(DT3DT1) J2 = D21/DT1 J3 = 1 – D31D13/(DT1DT3) J4 = D12 + D32D13/DT3 and Ii = Ei + GAiCBi i.e., the total of emission and advection inputs into each medium.

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Unlike the Level II calculation, it is now necessary to specify the emissions into each compartment separately. Different mass distributions, concentrations, and residence times result if 100 mol/h is emitted to air, water, or soil; thus, “mode of entry” is an important determinant of environmental fate and persistence. Having obtained the fugacities, all process rates can be deduced as Df, and a steady-state mass balance should emerge in which the total inputs to each medium equal the outputs. The amounts and concentrations can be calculated. An overall residence time can be calculated as the sum of the amounts present divided by the total input (or output) rate. A reaction residence time can be calculated as the amount divided by the total reaction rate, and a corresponding advection residence time can also be deduced. Doubling emissions simply doubles fugacities, masses, and concentrations, but the residence times are unchanged. An important property of this model is its linear additivity. This is also called the principle of superposition. Because all the equations are linear, the fugacity in, for example, water, deduced as a result of emissions to air, water, and soil, is simply the sum of the fugacities in water deduced from each emission separately. It is thus possible to attribute the fugacity to sources, e.g., 50% is from emission to water, 30% is from emission to soil, and 20% from emission to air. The masses and fluxes are also linearly additive. Figure 7.11 is a schematic representation of the results corresponding to the computed output in Figure 7.12. This is a comprehensive multimedia picture of chemical emission, advection, reaction, intermedia transport, and residence time or persistence. The important processes are now clear, and it is possible to focus on them when seeking more accurate rate data. Figure 7.11 contains information about 21 processes, some of which, such as air-water transfer, consist of several contributing processes. The human mind is incapable of making sense of the vast quantity of physical chemical and environmental data without the aid of a conceptual tool such as a Level III program. It is possible to add more compartments and to subdivide the existing compartments. It may be advantageous to add vegetation as a separate compartment. The atmosphere or water column could be segmented vertically. The soil can be treated as several layers. If information is available to justify these changes, they can be implemented, albeit at the expense of greater algebraic complexity. If the number of compartments becomes large and highly connected, it is preferable to solve the equations by matrix algebra. Computer programs are provided on the Internet, as discussed in Chapter 8, that undertake the Level III calculation of the multimedia fate of a specified chemical. The user must provide physical chemical (partitioning) properties, reaction halflives, and sufficient information to deduce intermedia transport D values. Assembly of an entire Level III model for a chemical is a fairly demanding task, since there are numerous areas, flows, mass transfer coefficients, and diffusivities to be estimated. To assist in this task, Table 7.2 gives suggested order-of-magnitude values for the various parameters. Such values are included as defaults in some programs, but they can be modified as desired. The user is encouraged to conduct Level III calculations for chemicals of interest, or those specified in Chapter 3. It is instructive to prepare a mass balance diagram,

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Figure 7.11 Schematic representation of the results corresponding to computed output.

check that the balance is correct (i.e., input equals output for each compartment) and identify the primary processes which control environmental fate. It may then be appropriate to examine these processes in more detail, seeking more accurate parameter values. Usually, the chemical’s fate is controlled by a few key processes, but these are not always obvious until a Level III calculation is performed. 7.11

LEVEL IV CALCULATIONS

It is relatively straightforward to extend the Level III model to unsteady-state conditions. Instead of writing the steady-state mass balance equations for each medium, we write a differential equation. In general, for compartment i, this takes the form ViZidfi/dt = Ii + Σ(Djifj) – DTifi

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Figure 7.12 Sample Level III output.

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where Vi is volume, Zi is bulk Z value, Ii is the input rate (which may be a function of time), each term Djifj represents intermedia input transfers, and DTifi is the total output. If an initial fugacity is defined for each medium, these four equations can be integrated numerically to give the fugacities as a function of time, thus quantifying the time response characteristics of the system. It is noteworthy that the characteristic response time of a compartment is ViZi/DTi, which can be deduced from the Level III steady-state version. These characteristic times provide advance insight into how a Level IV system should respond to changing emissions. This calculation is most useful for estimating recovery times of a contaminated system that is now experiencing zero or reduced emissions.

7.12

CONCLUDING EXAMPLES

Hand Calculation Using only air, water, and sediment from the four-compartment environments and the chemicals treated in the Level II example at the conclusion of Chapter 6, draw a Level III diagram similar to Figure 7.11, showing the compartments, the VZ values for each, and the advection and reaction D values. Write in somewhat arbitrary values for the six intermedia transport D values, but assigning values that lie in the range of 0.1 to 1% of VZ of the source phase. This gives rate constants for transport of 10–3 to 10–2 h–1. Feel free to round off all VZ and D values to facilitate calculation. Assume total inputs into air of 100 mol/h, and into water of 20 mol/h. Write down the three mass balance equations and solve by hand for the three fugacities. Calculate all the fluxes and check the mass balance for each compartment and the system as a whole. Calculate the three chemical residence times. Confirm the validity of the linear additivity assertion by calculating the fugacity in water for emissions only to air, and only to water, and show that their sum is the fugacity calculated when both emissions apply simultaneously. Do the residence times depend on the chemicals’ mode of entry to the environment? Computer Calculation Using the Level III program described in Chapter 8, compile a Level III mass balance diagram for a chemical using data from Table 3.5 and postulated emission rates in the range of 0.1 to 10 g per hour per square kilometre into air, water, and soil. Discuss the results, including the primary media of accumulation, the important processes, the relative media fugacities, and the residence times. EQC Calculation Using the EQC model described in Chapter 8, compile Level I, II, and III mass balances for a chemical and discuss the results.

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CHAPTER

8

Applications of Fugacity Models 8.1

INTRODUCTION, SCOPE, AND STRATEGIES

The ability to define Z values for a variety of media, and D values for processes such as advection, reaction, and intermedia transport, enables us to set up mass balance equations and then deduce fugacities, concentrations, fluxes, and amounts. We thus have the capability of addressing a series of environmental modeling problems in addition to the Level I, II, and III calculations described earlier. The aim of this chapter is to provide the reader with a description of the calculation of chemical fate in a variety of environmental situations in the expectation that the parameter values describing the environment and the chemical can be modified to simulate specific situations. It may be desirable to add or delete processes or change the model structure to suit individual requirements. Many of the models apply to steady-state conditions and can be reformulated to describe time-varying conditions by writing differential rather than algebraic equations. These differential equations can be solved algebraically or integrated numerically, depending on their complexity. Some of the most satisfying moments in environmental science come when a model is successfully fitted to experimental or observed data and it becomes apparent that the important chemical transport and transformation processes are being represented with fidelity. Even more satisfying is the subsequent use of the model to predict chemical fate in as yet uninvestigated situations leading to gratifying and successful “validation.” Failure of the model may be disappointing, but it is a positive demonstration that our fundamental understanding of environmental processes is flawed and further investigation is needed. For a review of the history of environmental mass balance models, the reader is referred to Wania and Mackay (1999). 8.1.1

Scope

In this chapter, several models are described. We start with a recapitulation of the Level I, II, and III models, including descriptions of various software and 185

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applications. Citations are given to enable the reader to download these models from the internet site of the Canadian Environmental Modelling Centre at Trent University, namely http://www.trentu.ca/envmodel. Included are DOS and Windows Level I, Level II, and Level III models, the EQC (EQuilibrium Criterion) model, the Generic model, and ChemCAN, a Level III model that has data for regions of Canada but can be (and has been) adapted to other regions. The next group of models is used to explore how a chemical is migrating or exchanging across the interface between two media, given the concentrations or fugacity in both. No mass balance is necessarily sought—merely a knowledge of how fast, and by what mechanism, the chemical is migrating. Compartment volumes are not necessary, but they may be included for the purpose of calculating half-lives. An example is air-water exchange in which both concentrations are defined and the aim is to deduce in what direction and at what rate the chemical is moving. Often, it is not clear if a substance in a lake is experiencing net input or output as a result of exchange with the atmosphere. An important conclusion is that zero net flux does not necessarily correspond to equilibrium or equifugacity. We refer to these as intermedia exchange models. The simplest mass balance model is a one-compartment “box” that receives various defined inputs either as an emission term or as the product of a D value and a fugacity from an adjoining compartment. The various D values for output or loss processes are then calculated. The steady-state fugacity at which inputs and outputs are equal is then deduced. An unsteady-state version of the model can also be devised. Examples are a “box” of soil to which sludge or pesticide is applied, a one-compartment fish with input of chemical from respired water and food, and a mass balance for the water in a lake. The complexity can be increased by adding more connected compartments. The QWASI (Quantitative Water, Air, Sediment Interaction) model includes mass balances in two compartments (water and sediment), the concentration in air being defined. A river, harbour, or estuary can be treated as a series of connected Eulerian QWASI boxes or using Lagrangian (follow a parcel of water as it flows) coordinates. A sewage treatment plant (STP) model is described in which the compartments are the three principal vessels in the activated sludge process. This illustrates that the modeling concepts can also be applied to engineered systems. Indeed, such systems are often easier to model, because they are well defined in terms of volumes, flows, and other operating conditions such as temperature. This multicompartment approach can be applied to chemical fate in organisms ranging from plants to humans and whales. These are physiologically based pharmacokinetic (PBPK) models. Fairly complex models containing multiple compartments can be assembled, an example being the POPCYCLING–BALTIC model of chemical fate in the Baltic region. The ultimate model is one of chemical fate in the entire global environment, GloboPOP. These models are available from a website at the University of Toronto, to which a link is provided from the Trent University address given earlier. Where possible, references are given to published studies in which the models have been applied. These reports give more detail than is possible here.

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8.1.2

187

Model-Building Strategies

The general model-building strategy is first to evaluate the system being simulated, then to decide how many compartments and thus mass balances are required. There is a compelling incentive to start with a simple model then build up complexity only when justified. The volumes and bulk Z values are deduced for each compartment. All inputs and outputs are identified, preferably as arrows on a mass balance diagram. Equations are written for each flux, either as an emission or a Df product. For a steady-state model, the inputs are equated to outputs for each of the n compartments, leading to n equations with n unknown fugacities. These equations are solved, either algebraically or using a matrix method. For a dynamic system, the differential equations for each medium are written in the form d(VZf)/dt = inputs – outputs mol/h These equations are then solved, either analytically or numerically, for a defined initial condition and defined inputs. The integration time step can be selected as 5% of the shortest half-time for transport or transformation and the stability of the result checked by decreasing the time step systematically. Integration is best done using a Runge–Kutta method, but the simple Euler’s method may be adequate. Results should be checked for a mass balance. For steady-state models, this is simply a comparison of inputs and outputs for each compartment. For dynamic models, the initial mass plus the cumulative inputs should equal the final mass and the cumulative outputs. To gain a pictorial appreciation of the results, a mass balance diagram should be drawn listing all inputs and outputs beside the appropriate arrows. The dominant processes then become apparent. It is often useful to play “sensitivity games” with the model to gain an appreciation of how variation in an input quantity such as an emission rate, Z value, or D value propagates through the calculation and affects the results. An input quantity can be increased by 1% and the effect on the desired output quantity determined. The best way to quantify this sensitivity S is to deduce S = (∆output/output)/(∆input/input) For example, if the input quantity of 100 is increased to 101 and the output quantity changes from 1000 to 1005, then S is (5/1000)/(1/100) or 0.5. This is actually an estimate of the partial derivative of log output with respect to log input, and it is dimensionless. For linear systems of the types treated here, all values of S should be less than 1.0. It may be useful to list the input parameters, deduce S for each, then rank them in order of decreasing S. The most sensitive parameters, for which the most accurate data are necessary, are then identified. Often, the sensitivity of a parameter is surprisingly small, and only a rough estimate is needed. It may be desirable to revisit and improve the accuracy of the estimates of most sensitive parameters.

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Another approach is to employ Monte Carlo analysis and run the model repeatedly, allowing the input data to vary between prescribed limits and deducing the variation in the output quantities. This gives an impression of the likely variability in the output, but it does not necessarily reveal the individual quantitative sources of this variability. The model results can then be compared with measured values to achieve a measure of validation. Complete validation is impossible, because chemicals or conditions can always be found for which the model fails. For a philosophical discussion of the feasibility of validation, the reader is referred to a review by Oreskes et al. (1994). A model can be useful, even if not validated, because it can give reliable results for a restricted set of conditions. A final note on transparency. It is unethical for an environmental scientist to assert that a chemical experiences certain fate characteristics as a result of model calculations unless the full details of the calculations inherent in the model are made available. The scientific basis on which the conclusions are reached must be fully transparent. For various reasons, the modeler may elect to prevent the user from modifying the code, but the calculations themselves must be readable. For this reason, all model calculations described here are fully transparent. Table 8.1 Summary of Z Value Equations All partition coefficients are dimensionless unless otherwise noted. All densities are kg/m3. Air Water Octanol Lipid Aerosols

ZA = 1/RT

R is gas constant, 8.314 Pa m3/mol K T is temperature

ZW = 1/H = ZA/KAW

H is Henry’s Law Constant Pa m3/mol KAW is air-water partition coefficient

ZO = ZWKOW Z L = ZO

KOW is octanol-water partition coefficient Lipid is equivalent to octanol

ZQ = KQAZA

KQA is aerosol-air partition coefficient

Organic carbon

ZOC = KOCZW (ρOC/1000)

Organic matter

ZOM = KOMZW (ρOM/1000) KOM is organic matter partition coefficient (L/kg) ρOM is density or organic matter (1000)

Mineral matter

ZMM = KMMZW (ρMM/1000) KMM is mineral-water partition coefficient (L/kg) ρMM is density of mineral matter

Biota

ZB = LZL

KOC is organic carbon partition coefficient (L/kg) ρOC is density or organic carbon (~1000)

L is lipid volume fraction

To aid in formulating models, Table 8.1 summarizes the expressions for estimating Z values, and Table 8.2 summarizes equations for estimating D values. Bulk Z values are ΣviZi, where vi is the volume fraction of phase i, and Zi is its Z value. For example, for bulk air, ZTA = vAZA + vQZQ For a solid phase (subscript s) containing organic carbon of mass fraction yOC, in which sorption to mineral matter is negligible and the partitioning coefficient with respect to water is KD L/kg,

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Table 8.2 Summary of D Value Equations Advection or flow

D = GZ = UAZ

G is medium flow rate (m3/h) and may be given as a product of velocity U (m/h) and area A m2.

Reaction

D = VZk

V is volume (m3) and k is rate constant (h–1).

Diffusion

D = BAZ/Y

B is molecular or effective diffusivity (m2/h). A is area (m2) and Y is path length (m).

Mass transfer

D = kAZ

k is mass transfer coefficient or velocity (m/h).

Growth dilution

D = ZdV/dt = VZk

dV/dt is growth rate (m3/h), and k is the growth rate constant (h–1) or (dV/dt)/V.

In all cases, Z refers to the medium in which the process occurs. The rate is Df (mol/h). For series processes 1/D = Σ1/Di. For parallel processes D = ΣDi. Characteristic times are VZ/D (h) where V and Z refer to the source phase. Half–times are 0.693 VZ/D (h). Rate constants are D/VZ (h–1).

KD = yOC KOC L/kg

KSW = KD(ρS/1000)

ZS = ZWKSW

KOC can be estimated as 0.41 KOW (Karickhoff, 1981) or as 0.35 KOW plus or minus a factor of 2.5 (Seth et al., 1999). Note that KOM is typically 0.56 KOC, i.e., OM is typically 56% OC. KQA can be estimated as 6 × 106/PS, where PS is the liquid vapor pressure, or from other correlations using vapor pressure or KOA as described in Chapter 5.

8.2

LEVEL I, II, AND III MODELS

These models have been described earlier in Chapters 5, 6, and 7. These programs are available from http://www.trentu.ca/envmodel in two formats. First are BASIC models, which can be run directly on DOS systems or using GWBASIC or QBASIC. Second are more user-friendly Windows® models in which input parameters are more easily changed, and output is available on the screen, and it can be printed or saved to a file. The code of the BASIC models can be changed, and they can serve as a template for building other models. The calculations in the Windows models cannot be modified by the user. In all cases, the code can be inspected; it is fully transparent. The following BASIC models are available. LEVEL1A A Level I program treating four compartments. It prompts for chemical properties and amount. The phase volumes and properties can be changed by editing the program.

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LEVEL1B A six-compartment Level I program, similar to LEVEL1A. LEVEL2A A four-compartment Level II program that prompts for the same information as Level I programs, but also for reaction and advection rate data. LEVEL2B A six-compartment Level II program, similar to LEVEL2A. LEVEL3A A four-compartment Level III program that requires all the Level II data and prompts for D values in the form of transfer half-lives. LEVEL3B A six-compartment Level III program, similar to LEVEL3A, but prompts for D values directly. The following Windows models are available: Level I Level II Level III The DOS-based Generic model contains all the Level I, II, and III calculations. It is described in a paper by Mackay et al. (1992). It is also described in Chapter 1 of the five-volume Illustrated Handbook of Physical Chemical Properties and Environmental Fate of Organic Chemicals, by Mackay, Shiu, and Ma, which is also available in CD ROM format. The EQC (EQuilibrium Criterion) model is described in a series of papers by Mackay et al. (1996). It applies to a 100,000 km2 land area (about the size of Ohio). It is a Windows model and has been used in several assessments of the likely behavior of a chemical in an evaluative context. An example is the application to the chlorobenzenes in Ontario by MacLeod and Mackay (1999). Booty and Wong (1996) have also applied the model to the same region for a variety of chemicals. The Level III ChemCAN model is in Windows format and contains databases of 24 regions of Canada and a number of chemicals. It can be applied to other regions, and it has been modified to apply to regions of France (Devillers and Bintein, 1995), Germany (Berding et al., 2000), and the U.K. The CalTOX model of McKone, which is available from the California Environmental Protection Agency (http://www.cwo.com/~herd1/), is a Level III model that

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includes an excellent treatment of human exposure. It was originally designed to assess exposure to chemicals present in hazardous waste sites. The environmental fate equations are formulated using fugacities. Maddalena et al. (1995) compared the output of CalTOX and ChemCAN and found similar results, indicating that both models were using similar predictive equations. The reader should consult the Trent website for current versions of the model software described in this text.

8.3 8.3.1

AN AIR-WATER EXCHANGE MODEL

Introduction

Air-water exchange process calculations are useful when estimating chemical loss from treatment lagoons, ponds, and lakes; for estimating deposition rates of atmospheric contaminants, and for interpreting observed air and water concentrations to establish the direction and rate of transfer. The complexity of the several processes and the widely varying physical chemical properties of chemicals of interest leads to situations in which chemical behavior is not necessarily intuitively obvious. The simple model derived here provides a rational method of estimating exchange characteristics and exploring the sensitivity of the results to assumed values of the various chemical and environmental parameters. Bidleman (1988) gives an excellent review of atmospheric processes treated by the model. An elegant application of the fugacity concept to elucidating chemical exchange in the air-water system is that of Jantunen and Bidleman (1997). Samples of air and surface water from the Bering and Chukchi Seas (between Alaska and Russia) were analysed for α-hexachlorocyclohexane (α-HCH) over a multiyear period, and the ratios of air to water fugacities were deduced using the Henry’s law constant for seawater at the appropriate temperature. Initially, in the mid 1980s, this ratio was greater than 1, indicating that α-HCH was being absorbed by the ocean. This is consistent with the source of α-HCH being evaporation of technical lindane following application in South East Asia, India, and China, with subsequent atmospheric transport. Later, in the mid 1990s, after the use of technical lindane was greatly reduced, the fugacity ratio became less than 1 (because of the drop in air fugacity), and net volatilization of α-HCH started. Essentially, the ocean acted first as a “sponge,” absorbing α-HCH, then it desorbed the α-HCH in response to changes in the concentration in air. Interpretation of data using of the fugacity ratio illustrated this clearly. It is an example to be followed in cases where there is doubt about the direction of net transport of chemicals between air and water. 8.3.2

Process Description

The situation treated here, and the resulting model, are largely based on the study of air-water exchange by Mackay et al. (1986) as is depicted in Figure 8.1. The water phase area and depth (and hence volume) are defined, it being assumed that the water is well mixed. The water contains suspended particulate matter, to

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Figure 8.1

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

Air-water exchange processes.

which the chemical can be sorbed, and which may contain mineral and organic material. The concentration (mg/L or g/m3) of suspended matter is defined, as is its organic carbon (OC) content (g OC per g dry particulates). By assuming a 56% OC content of organic matter, the masses of mineral and organic matter can be deduced. Densities of 1000, 1000, and 2500 kg/m3 are assumed for water, organic matter, and mineral matter respectively, thus enabling the volumes and volume fractions to be deduced. The air phase is treated similarly, having the same area as the water and a defined (possibly arbitrary) height and containing a specified concentration (ng/m3) of aerosols or atmospheric particulates to which the chemical may sorb. By assuming an aerosol density of 1500 kg/m3, the volume fraction of aerosols can be deduced. No information on aerosol composition, size distribution, or area is sought or used. If the concentration of aerosols or total suspended particulates is TSP ng/m3, this corresponds to 10–12 TSP kg/m3 and to a volume fraction vQ of 10–12 TSP/ρQ, where ρQ is the aerosol density (1500 kg/m3). Thus, a typical TSP of 30,000 ng/m3 or 30 µg/m3 is equivalent to a volume fraction of 20 × 10–12. The volumes of particles can be calculated as the product of total volume V, and respective volume fractions. Physical chemical properties of the chemical are requested. The total or bulk concentrations in the water and air phases are requested in mass/volume units. These are converted to mol/m3 and divided by the bulk Z values to give the water and air fugacities. The quantities and concentrations in dissolved, or gaseous, and sorbed form are then calculated, i.e., the input concentrations are apportioned to sorbed and nonsorbed forms. Z and D values are calculated using the expressions in Tables 8.1 and 8.2.

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A check should be made of the magnitude of f/PS, where PS is the solid or liquid vapor pressure. When this ratio equals 1, saturation is achieved. When the ratio exceeds 1, the chemical will precipitate as a pure phase, i.e., its solubility in air or water is exceeded, and the fugacity will drop to the saturation value indicated by the vapor pressure. Normally, the ratio is much less than unity. Four processes are considered as shown in Figure 8.1: (1) diffusive exchange by volatilization and the reverse absorption, (2) dry deposition of aerosols, (3) wet dissolution of chemical, and (4) wet deposition of aerosols. In each case, a D value (mol/Pa h) is used to characterize the rate, which is Df mol/h. For diffusion, the two-resistance approach is used, and D values are deduced for the air and water boundary layers, DA = kA A ZA

DW = kW A ZW

where kA and kW are mass transfer coefficients with units of m/h, and A is area (m2). Illustrative values of 5 m/h for kA and 5 cm/h for kW can be used, but it should be appreciated that environmental values can vary widely, especially with wind speed, and a separate calculation may be needed for the situation being simulated. The overall resistance (1/DV) is obtained by adding the series resistances (1/D) as 1/DV = 1/DA + 1/DW The rate of vaporization is then fWDV, the rate of absorption is fADV, and the net rate of vaporization is DV(fW – fA). An overall mass transfer coefficient is also calculated. For dry deposition, a dry deposition velocity UD of particles is used, a typical value being 0.3 cm/s or 10 m/h. The total dry deposition rate is thus UDvQA m3/h, the corresponding D value DD is UDvQAZQ, and the rate is DDfA mol/h. For wet dissolution, a rain rate is defined, usually in units of m/year, a typical value being 0.5 m/year or 6 × 10–5 m/h, designated UR. The total rain rate is then URA (m3/h), the D value, DR, is URAZW, and the rate is DRfA mol/h. For wet aerosol deposition, a scavenging ratio Q is used, representing the volume of air efficiently scavenged by rain of its aerosol content, per unit volume of rain. A typical value of Q is 200,000. The volume of air scavenged per hour is thus URAQ (m3/h), which will contain URAQvQ (m3/h) of aerosol (vQ is the volume fraction of aerosol). The D value DQ is thus URAQvQZQ, and the rate is DQfA (mol/h). A washout ratio is often employed in such calculations. This is the dimensionless ratio of concentration in rain to total concentration in air, usually on a volumetric (g/m3 rain per g/m3 air) basis, but occasionally on a gravimetric (mg/kg per mg/kg) basis. The total rate of chemical deposition in rain is (DR + DQ)fA; thus, the concentration in the rain is (DR + DQ)fA/URA or fA(ZW + QvQZQ) mol/m3. The total air concentration is fA(ZA + vQZQ), and therefore the volumetric washout ratio is (ZW + QvQZQ)/(ZA + vQZQ). The gravimetric ratio is smaller by the ratio of air to water densities, i.e., approximately 1.2/1000. If the chemical is almost entirely aerosol associated, as is the case with metals such as lead, the volumetric washout ratio approaches Q. These washout ratios are calculated and can be compared to reported values.

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The total rates of transfer are thus fWDV mol/h

water to air

fA(DV + DD + DR + DQ) = fADT mol/h

air to water

The total amounts of chemical in each phase may be calculated as VAZTAfA and VWZTWfW, where the subscript T refers to the total or bulk phase. The rate constants (h–1) and half-times (h) for transfer from each phase are respectively from air

DT/(VAZTA) h–1

from water

DV/(VWZTW) h–1

and and

0.693VAZTA/DT h 0.693VWZTW/DV h

These quantities are useful as indicators of the rapidity with which chemical can be cleared from one phase to the other, thus enabling the significance of these exchange processes to be assessed relative to other processes such as reaction. Inspection of the individual D values shows which processes are most important. It is noteworthy that a steady-state (i.e., no net transfer) condition may apply in which the air and water fugacities are unequal, i.e., a nonequilibrium, steady-state condition applies. The steady-state condition will apply when fWDV = fADT The steady-state water fugacity and concentration with respect to the air, and the steady-state air fugacity and concentration with respect to the water, can thus be calculated to give an impression of the extent to which the actual concentration departs from the steady-state values, as distinct from the equilibrium (equifugacity) value. It is noteworthy that, because DT exceeds DV, the fugacity in water will tend to exceed the fugacity in air; however, this will be affected by removal processes in water. 8.3.3

Model

The AirWater model is available from the website as Windows software and in the older DOS-based BASIC format. In both cases, the calculations can be viewed by the user, and sufficient comments are included to enable the logic to be followed. A sample chemical and set of air and water properties are included as an example for the user. Given the concentrations in the air and water, the steady-state fluxes are calculated.

8.4 8.4.1

A SURFACE SOIL MODEL

Introduction

Chemicals are frequently encountered in surface soils as a result of deliberate application of agrochemicals and sewage sludge, and by inadvertent spillage and

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leakage. It is often useful to assess the likely fate of the chemical, i.e., how fast the rates of degradation, volatilization, and leaching in water are likely to be, and how long it will take for the soil to “recover” to a specified or acceptable level of contamination. Persistence is an important characteristic for pesticide selection. Remedial measures such as excavation may be needed when recovery times are unacceptably long. Most modeling efforts in this context have been for agrochemical purposes, the most comprehensive recent effort being described in a series of publications by Jury, Spencer, and Farmer (1983, 1984a, 1984b, 1984c). Other notable models are reviewed in these papers. The Soil model is essentially a very simplified version of the Jury model (1983) and is a modification of a published herbicide fate model (Mackay and Stiver, 1990). The reader is referred to the texts by Sposito (1989) and Sawhney and Brown (1989) and the chapter by Green (1988) for fuller accounts of chemical fate in soils. Cousins et al. (1999) have reviewed and modeled these processes. In the Soil model, only soil-to-air processes are treated; no air-to-soil transport is considered. A second, more complex fugacity model SoilFug was developed by DiGuardo et al. (1994a), which allows the user to calculate the fate of the pesticide in a defined agricultural area over time with changing rainfall. The model gave satisfactory predictions of pesticide runoff in agricultural regions in Italy and the U.K. (Di Guardo et al., 1994a, 1994b). Both models are available from the website. Only the Soil model is described here. 8.4.2

Process Description

In the Soil model, the soil matrix illustrated in Figure 8.2 is considered to consist of four phases: air, water, organic matter, and mineral matter. The organic matter is considered to be 56% organic carbon. The volume fractions of air and water are defined, either by the user or by default values, as is the mass fraction organic carbon (OC) content on soil basis. Assuming densities of 1.19, 1000, 1000, and 2500 kg/m3 for air, water, organic matter, and mineral matter, respectively, enables the mass and volume fractions of each phase, and the overall soil density, to be calculated. The soil area and depth are specified, thus enabling the total volumes and mass of soil and its component phases to be deduced. The amount of chemical present in the soil is specified as a concentration or as an amount in units of kg/ha, which is convenient for agrochemicals. The chemical is assumed to be homogenously distributed throughout the entire soil volume. The individual phase Z values are calculated, then the bulk Z value of the soil ZTS is deduced. From the concentration, the fugacity is deduced, and the individual phase quantities and concentration are calculated. It is prudent to examine the fugacity to check that it is less than the vapor pressure. If it exceeds the vapor pressure, phase separation of pure chemical will occur; i.e., the capacity of all phases to “dissolve” chemical is exceeded. This can occur in heavily contaminated soils that have been subject to spills, or when there is heavy application of a pesticide. Essentially, the “solubility” of the chemical in the soil is exceeded. This calculation of partitioning behavior provides an insight

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Figure 8.2

MULTIMEDIA ENVIRONMENTAL MODELS AND FUGACITY

Chemical transport and transformation processes in a surface soil.

into the amounts present in the air and water phases. It also shows the extent to which organic matter dominates the sorptive capacity of the soil. Three loss processes are considered: degrading reactions, volatilization, and leaching, each rate being characterized by a D value. An overall reaction half-life τ(h) is specified from which an overall rate constant kR (h–1) is deduced as 0.693/τ. The reaction D value DR is then calculated from the total soil volume and the bulk Z value as kRVTZTS. In principle, if a rate constant ki is known for a specific phase in the soil, the phase-specific D value can be deduced as kiViZi, but normal practice is to report an overall rate constant applicable to the total amount of chemical in the entire soil matrix. If no reaction occurs, an arbitrarily large value for the half-life, such as 1010 hours, should be input. A water leaching rate is specified in units of mm/day. This may represent rainfall (which is typically 1 to 2 mm/day) or irrigation. This rate is converted into a total water flow rate GL (m3/h), which is combined with the water Z value to give the advection leaching D value DL as GLZW. This assumes that the concentration of chemical in the water leaving the soil is equal to that in the water in the soil; i.e., local equilibrium has become established, and no bypassing or “short circuiting” occurs. The “solubilizing” effect of dissolved or colloidal organic matter in the soil water is ignored, but it could be included by increasing the Z value of the water to account for this extra capacity.

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Volatilization is treated using the approach suggested by Jury et al. (1983). Three contributing D values are deduced. An air boundary layer D value, DE, is deduced as the product of area A, a mass transfer coefficient kV, and the Z value of air, i.e., A kV ZA. Jury has suggested that kV be calculated as the ratio of the chemical’s molecular diffusivity in air (0.43 m2/day or 0.018 m2/h being a typical value), and an air boundary layer thickness of 4.75 mm (0.00475 m); thus, kV is typically 3.77 m/h. Another kV value may be selected to reflect different micrometerological conditions. An air-in-soil diffusion D value characterizes the rate of transfer of chemical vapor through the soil in the interstitial air phase. The Millington–Quirk equation is used to deduce an effective diffusivity BEA from the air phase molecular diffusivity BA as outlined in Chapter 7, namely, BEA = BA vA10/3/(vA + vW)2 where vA is the volume fraction of air, and vW is the volume fraction of water. If vW is small, this reduces to a dependence on vA to the power 1.33. A diffusion path length Y must be specified, which is the vertical distance from the position of the chemical of interest to the soil surface; i.e., it is not the “tortuous” distance. The air diffusion D value DA is then BEA A ZA/Y A similar approach is used to calculate the D value for chemical diffusion in the water phase in the soil, except that the molecular diffusivity in water BW is used (a value of 4.3 × 10–5 m2/day being assumed), and the water volume fraction and Z value being used, namely, DW = BEWAZW/Y where BEW = BWvW10/3/(vA + vW)2 Since the diffusion D values DA and DW apply in parallel, the total D value for chemical transfer from bulk soil to the soil surface is (DA + DW). The boundary layer D value then applies in series so that the overall volatilization D value, DV, is given as illustrated in Figure 8.2 as 1/DV = 1/DE + 1/(DA + DW) Selection of the diffusion path length Y involves an element of judgement. If, for example, chemical is equally distributed in the top 20 cm of soil, an average value of 10 cm for Y may be appropriate as a first estimate. This will greatly underestimate the volatilization rate of chemical at the surface. Since the rate is inversely proportional to Y, a more appropriate single value of Y as the average between two depths Y1 and Y2 is the log mean of Y1 and Y2, i.e., (Y1 – Y2)/ln(Y1/Y2). Unfortunately, a zero (surface) value of Y cannot be used when calculating the log

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mean. For chemical between depths of 1 and 10 cm, a log mean depth of 3.9 cm is more appropriate than the arithmetic mean of 5.5 cm. It may be useful to consider layers of soil separately, e.g., 2 to 4 cm, 4 to 6 cm, etc., and calculate separate volatilization rates for each. Chemical present at greater depths will thus volatilize more slowly, leaving the remaining chemical more susceptible to other removal processes. It is acceptable to specify a mean Y of, say, 10 cm to examine the fate of chemical in the 2 cm depth region from 9 to 11 cm. This depth issue is irrelevant to reaction or leaching, but it must be appreciated that, if the soil is treated as separate layers, the leaching rate is applicable to the total soil, not to each layer independently. The total rate of chemical removal is then f DT , where the total D value is: DT = DR + DL + DV the individual rates being f DR, f DL, and f DV. The overall rate constant kO is thus DT/VTZT, where VTZT is the sum of the ViZi products, and the overall half-life τO is 0.693/kO hours. The half-life τi attributable to each process individually is 0.693VTZT/Di, thus, 1/τO = 1/τR + 1/τL + 1/τV It is illuminating to calculate the rates of each process, the percentages, and the individual half-lives. Obviously, the shorter half-lives dominate. The situation being simulated is essentially the first-order decay of chemical in the soil by three simultaneous processes, thus the amount remaining from an initial amount M (mol) at any time t (h) will be M exp(–DTt/VTZT) = M exp(–kOt) mols This relatively simple calculation can be used to assess the potential for volatilization or for groundwater contamination. Implicit in this calculation is the assumption that the chemical concentration in the air, and in the entering leaching water, is zero. If this is not the case, an appropriate correction must be included. In principle, it is possible to estimate atmospheric deposition rates as was done in the air-water example and couple these processes to the soil fate processes in a more comprehensive air-soil exchange model. It may prove desirable to segment the soil into multiple layers, especially if evaporation or input from the atmosphere is important. Models of this type have been reported by Cousins et al. (1999) for PCBs in soils. 8.4.3

Model

The Soil model is available from the website in both Windows software and in the older DOS-based BASIC format, similarly to the AirWater model. The SoilFug model is also available and can be used to explore the effects of varying precipitation on soil runoff.

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Users are encouraged to modify the various parameter values and are cautioned that the values given are not necessary widely applicable. It should be noted again that varying the input temperature will not vary physical chemical properties such as vapor pressure. Temperature dependence must be entered “by hand.”

8.5 8.5.1

A SEDIMENT-WATER EXCHANGE MODEL

Introduction

Exchange of chemical at the sediment-water interface can be important for the estimation of (1) the rates of accumulation or release from sediments, (2) the concentration of chemicals in organisms living in, or feeding from, the benthic region, (3) which transfer processes are most important in a given situation, and (4) the likely recovery times in the case of “in-place” sediment contamination. The complexity of the system and the varying properties of chemicals of possible concern lead to a situation in which a specific chemical’s behavior is not necessarily obvious. This situation treated here, and the resulting model are largely based on a discussion of sediment-water exchange by Reuber et al. (1987), Eisenreich (1987), Diamond et al. (1990), and in part on a report by Formica et al. (1988). It is depicted in Figure 8.3.

Figure 8.3

Sediment-water exchange processes.

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8.5.2

Process Description

The water phase area and depth (and hence volume) are defined, it being assumed that the water is well mixed. The water contains suspended particulate matter, which may contain mineral and organic material. The concentration (mg/L or g/m3) of suspended matter is defined, as is its organic carbon (OC) content (g OC per g dry particulates). The volume fractions are calculated similarly to those for air-water exchange. The sediment phase is treated similarly, having the same area, a defined well mixed depth, and a specified concentration of solids and interstitial or pore water. Rates of sediment deposition, resuspension, and burial are specified, as are firstorder reaction rates in the sediment phase. Allowance is made for infiltration of ground water through the sediment in either vertical direction. Lipid contents of organisms present in the water and sediment are specified for later illustrative bioconcentration calculations. The equilibrium partitioning distribution is calculated using Z values for the water and sediment phases using specified total chemical concentrations (g/m3 or mg/L) in the water, and µg/g of dry sediment solids in the sediment. Since no air phase appears in the calculation, the vapor pressure is not strictly necessary. Identical concentration, but not fugacity, results are obtained when an arbitrary vapor pressure is used. Illustrative biotic Z values can be deduced for both water and sediment as KBZW, where the bioconcentration factor KB is estimated from the product of lipid content LB (e.g., 0.05) and KOW, i.e., LBKOW. Biota are included only for illustrative purposes and are not included in the mass balance. The total and contributing concentrations in all phases and the fugacities can thus be deduced. From the biotic Z values, the corresponding concentrations can also be deduced for biota resident in water and sediment. Several transport and transformation processes are considered: (1) sediment deposition, (2) sediment resuspension, (3) sediment burial, (4) diffusive exchange of water between the water column and the pore water, and (5) sediment reaction. Irrigation, i.e., net flow of groundwater into or out of the sediment, could be added as a sixth process. 1. The deposition D value DD is calculated as the product of volumetric deposition rate GD m3/h, and the particle Z value ZP, i.e., GD ZP. 2. The resuspension D value DR is calculated similarly as GR ZS. 3. The burial D value DB is calculated as GB ZS. 4. For diffusive exchange of water, the D value DT is calculated from an overall water phase mass transfer coefficient (MTC) kW, the area A, and the Z value for water as kW A ZW. This mass transfer coefficient can be calculated from an effective diffusivity for the sediment solids content as discussed by Formica et al. (1988) and a path length. 5. For reaction, the overall rate constant is kR, and DS is VS ZTS kR. 6. It is also possible to include a water flow or irrigation D value DI, which is calculated from an irrigation velocity UI (m/h), and which is converted into a water flow rate GI as UI A (m3/h) and to a D value as GI ZW.

The individual and total rates of transfer can be calculated as the D f products.

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8.5.3

201

Model

The Sediment model is available from the website in Windows- and DOS-based versions. Input data are requested on the properties of the chemical, the dimensions and properties of the media, and the prevailing concentrations. The Z and D values are calculated, followed by fugacities and fluxes. It is also of interest to calculate the overall steady-state mass balance, which is given by fW (DD + DT) = fS (DR + DT + DB + DS) The steady-state water and sediment fugacities corresponding to the defined sediment and water fugacities are deduced. Response times can be calculated for each medium if the volumes are known. It is noteworthy that, for a persistent, hydrophobic substance, it is likely that the steady-state sediment fugacity will exceed that of the water. The principal loss process of a persistent chemical from the sediment is likely to be DR, which must be less than DD, because some sediment is buried, and the organic carbon content of the resuspended material will be less than the deposited material because of mineralization. As a result, a benthic organism that respires sediment pore water may reach a higher fugacity and concentration than a corresponding organism in the water column above. A compelling case can be made for monitoring benthic organisms, because they are less mobile than fish and they are likely to build up higher tissue concentrations of contaminants. These sediment-water calculations can be invaluable for estimating the rate at which “in-place” sediment concentrations, resulting from past discharges of persistent substances, are falling. Often, the memory of past stupidities lingers longer in sediments than in the water column.

8.6 8.6.1

QWASI MODEL OF CHEMICAL FATE IN A LAKE

Introduction

Having established air-water and sediment-water exchange models, it is relatively straightforward to combine them in a lake model by adding reaction and advective inflow and outflow terms. The result is the QWASI (Quantitative Water Air Sediment Interaction) model of Mackay et al. (1983), which was applied to Lake Ontario (Mackay, 1989). Other reports include an application to a variety of chemicals by Mackay and Diamond (1989), to organochlorine chemicals produced by the pulp and paper industry by Mackay and Southwood (1992), the use of spreadsheets to aid fitting parameter values to the model (Southwood et al., 1989), to situations in which surface microlayers are important (Southwood et al., 1999), and to metals by Woodfine et al. (2000). Mackay et al. (1994) took the QWASI fugacity model and replaced all fugacities by C/Z and all Z values by partition coefficients, then converted all D values to rate constants. This “new” model, called the “rate constant” model, gives identical results and is suitable for use by people who are too lazy to

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learn the benefits of using fugacity. This RateConstant model is available from the Trent University website, but only in DOS format. In principle, the QWASI model can be applied to any well mixed body of water for which the hydraulic and particulate flows are defined. Figure 8.4 shows the transport and transformation processes treated, and Table 8.3 lists the D values and the corresponding fugacity in the rate expressions. Figure 8.5 gives the mass balance equations in steady-state and unsteady-state or differential form. The steady-state solution describes conditions that will be reached after prolonged exposure of the lake to constant input conditions, i.e., emissions, air fugacity, and inflow water fugacity. Also given in Figure 8.5 is the solution to the differential equations from a defined initial condition, assuming that the input terms remain constant with time. If it is desired to vary these inputs, or any other terms, as a function of time, the differential equations must be solved numerically. The subscript refers to the steady-state solution, which applies at infinite time. Since D values add, simple inspection reveals which are important and control the overall chemical fate. For example, if DV greatly exceeds DQ, DC, and DM, it is apparent that most transfer from air is by absorption. The relative magnitudes of the processes of removal from water are particularly interesting. These occur in the denominator of the fW equation as volatilization (DV), reaction (DW), water outflow (DJ), particle outflow (DY), and a term describing net loss to the sediment. The gross loss to the sediment is (DD + DT), but only a fraction of this (DS + DB)/(DR + DT + DS + DB) is

Figure 8.4

Transport and transformation processes treated in the QWASI model, consisting of a defined atmosphere with water and sediment compartments.

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Table 8.3 D Values in the QWASI Model and Their Multiplying Fugacity

Process

D Value

Definition of D Value

Multiplying Fugacity

Sediment burial

DB

GBZS

fS

Sediment transformation

DS

V SZ Sk S

fS

Sediment resuspension

DR

G RZ S

fS

Sediment to water diffusion

DT

kTASZW

fS

Water to sediment diffusion

DT

kTASZW

fW

Sediment deposition

DD

G DZ P

fW

Water transformation

DW

VWZWkW

fW

Volatilization

DV

kVAWZW

fW

Absorption

DV

kVAWZW

fA

Water outflow

DJ

G JZ W

fW

Water particle outflow

DY

GYZP

fW

Rain dissolution

DM

GMZW

fA

Wet particle deposition

DC

G CZ Q

fA

Dry particle deposition

DQ

GQZQ

fA

Water inflow

DI

GIZW

fI

Water particle inflow

DX

GXZP

fI

Direct emissions

—

EW

Nomenclature and Explanation The rate (mol/h) is the product of the D Value and the multiplying fugacity, e.g., DBfS. G values are flows (m3/h) of a phase, e.g., GB is m3/h of sediment that is buried. fW, fS, fA, and fI are the fugacities of water, sediment, air and water inflow. Z values are fugacity capacities (mol/m3 Pa), the subscript being S sediment, W water, A air, Q aerosol, P water particles. The advective flows are subscripted I water inflow, X water particle inflow, J water outflow, and Y water particle outflow. kS and kW are sediment and water transformation rate constants (h–1). kT is a sediment–water mass transfer coefficient and kV an overall (water–side) air–water mass transfer coefficient (m/h). AW and AS are air–water and water–sediment areas (m2). VW and VS are water and sediment volumes (m3).

retained in the sediment, with the remaining fraction (DR + DT)/(DR + DT + DS + DB) being returned to the water. The three terms in the numerator of the fW equation give the inputs from emissions, inflow, and transfer from air. When the equations are solved, the concentrations, amounts, and fluxes can be calculated. An illustration of such an output is given in Figure 8.6 for PCBs in Lake Ontario (Mackay, 1989). Such mass balance diagrams clearly show which processes are most important for the chemical of interest. Windows- and DOS-based BASIC programs are provided that process the various Z values, volumes, areas, flows, D values, and the chemical input parameters to give the steady-state solution. The conditions simulated in the BASIC program

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QWASI Equations Steady-state solutions (i.e., derivatives are equal to zero) Since Sum of all input rates = sum of all output rates The sediment mass balance is fW(DD + DT) = fS(DR + DT + DS + DB) and can be rewritten as fS = fW (DD + DT)/(DR + DT + DS + DB) The water mass balance is EW + fI(DI + DX) + fA(DV + DQ + DC + DM) + fS(DR + DT) = fW(DV + DW + DJ + DY + DD + DT) and can be rewritten as E W + f I ( Di + D X ) + f A ( DV + DQ + DC + D M ) + f S ( DR + DT ) f W = ------------------------------------------------------------------------------------------------------------------------------------------------------------DV + DW + DJ + DY + DD + DT To solve water mass balance, eliminate fS. E W + f I ( Di + D X ) + f A ( DV + DQ + DC + D M ) f W = --------------------------------------------------------------------------------------------------------------------( DD + DT ) ( DS + DB ) D V + D W + D J + D Y + ----------------------------------------------------DR + DT + DS + DB The sediment fugacity can then be calculated.

Unsteady-state analytical solutions Since VZdf/dt = (total input rate – total output rate) the sediment differential equation is V S Z BS d f S --------------------------- = f W ( D D + D T ) – f s ( D R + D T + D S + D B ) dt The water differential equation is V W Z BW d f W --------------------------------- = E W + f I ( D I + D X ) + f A ( D V + D Q + D C + D M ) + f s ( D R + D T ) – f W ( D V + D W + D J + D Y + D D + D T ) dt Here, subscript B refers to the bulk or total phase including dissolved and sorbed material. This pair of equations can be written more compactly as d fW ----------- = I 1 + I 2 f S – I 3 f W dt d fS --------- = I 4 f W – I 5 f S dt where E W + f I ( DI + D X ) + f A ( DV + DQ + DC + D M ) I 1 = ---------------------------------------------------------------------------------------------------------------------V W Z BW DR + DT I 2 = --------------------V W Z BW DV + DW + DJ + DY + DD + DT I 3 = -------------------------------------------------------------------------------V W Z BW DD + DT I 4 = ---------------------V S Z BS DR + DT + DS + DB I 5 = ------------------------------------------------V S Z BS The solution with the initial conditions fSO and fWO and final conditions fS∞ and fW∞ is f W = f W∞ + I 8 exp [ – ( I 6 – I 7 )t ] + I 9 exp [ – ( I 6 + I 7 ) t ] ( I 3 – I 6 + I 7 )I 8 exp [ – ( I 6 – I 7 )t ] + ( I 3 – I 6 – I 7 )I 9 exp [ – ( I 6 + I 7 )t ] f S = f S∞ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------I 2

where fW∞ fS∞ I6 I7 I8 I9

Figure 8.5

= = = = = =

I1I5/(I3I5 – I2I4), as in the steady-state solution above I1I4/(I3I5 – I2I4), as in the steady-state solution above (I3 + I5)/2 [(I3 – I5)2 + 4I2 I4]0.5/2 [–I2(fS∞ – fSO) + (I3 – I6 – I7)(fW∞ – fWO)]/2I7 [+I2(fS∞ – fSO) – (I3 – I6 + I7)(fW∞ – fWO)]/2I7

QWASI model: steady-state and unsteady-state solutions.

Figure 8.6

Illustrative results from a QWASI model calculation of steady-state behavior of PCBs in Lake Ontario (Mackay, 1989).

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are similar to those described by Mackay (1989) for the fate of PCBs in Lake Ontario. To obtain unsteady-state solutions requires programming the equations in Figure 8.5 or solving the differential equations by a numerical method. Worked Example 8.1 For chemical X, determine for both water and sediment, the D values, total inputs and outputs, fugacities at steady state, concentrations, total amounts, and residence times. Also estimate the concentration of chemical X in fish and benthos. Z Values mol/m3 Pa

Properties of Chemical X Molar mass = 250 g/mol

ZA = 4 × 10–4

Vapor pressure = 0.5 Pa

ZW = 0.8

Solubility = 100 g/m3

ZP = 1600

H = 1.25 Pa m3/mol

ZS = 600

log KOW = 4.47

ZF = 1200 (fish and benthos) ZQ = 5000 (aerosol) (as controlled by organic carbon and lipid contents)

Volumes water 106 m3

particles in water 25 m3

sediment 104 m3

D values Transport: For convenience, we express them as GZ, where G is an equivalent flow m3/h. Refer to Table 8.3 for details. GB = 0.1

GR = 0.2

GT = 50

GD = 0.3

GI = GJ = 400

GY = 0.01

GM = 0.1

GV = 500 (DV is GV ZW)

GC = 0.001

GQ = 0.0005

GI = 400

GX = 0.02 (there is net sedimentation)

Transformation:

D = VZk

water k = 0.0001 h–1 (dissolved only) Inputs Emissions

EW = 1 mol/h

Fugacities (derived from observed concentrations) fA (air) = 10–4 Pa fI (input water) = 10–3 Pa

sediment = 0.00001 h–1

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207

Calculation of D values Note that rates are calculated later as D f. Rates (mol/h)

Multiplying Fugacity

DB = GBZS = 0.1 × 600 = 60

0.149

fS

DR = GRZS = 0.2 × 600 = 120

0.300

fS

DT = GTZW = 50 × 0.8 = 40

0.100

fS (sediment to water)

0.054

fW (water to sediment)

DD = GDZP = 0.3 × 1600 = 480

0.643

fW

DW = VWZWkW = 106 × 0.8 × 10–4 = 80

0.107

fW (water phase only, not on particles)

DV = GVZW = 500 × 0.8 = 400

0.536

fW (evaporation)

0.040

fA (absorption)

0.429

fW

DY = GYZP = 0.01 × 1600 = 16

0.021

fW

DM = GMZW = 0.1 × 0.8 = 0.08

8 × 10–6

fA

DC = GCZQ = 0.001 × 5000 = 5

5 × 10

fA

DJ = GJZW = 400 × 0.8 = 320

DQ = GQZQ = 0.0005 × 5000 = 2.5

–4

2.5 × 10–4

fA

DI = GIZW = 400 × 0.8 = 320

0.320

fI

DX = GXZP = 0.02 × 1600 = 32

0.032

fI

DS = VSZSkS = 104 × 600 × 10–5 = 60

0.149

fS

EW = 1.0

fI = 10–3

fA = 10–4

Calculation of fugacities fW = [EW

+

/[DV

+

DW

DJ

= [1.0

+

10–3(320

/[400

fI(DI + DX)

+

+

+

DY

+

(DD

+

32)

+

10–4(400

+

80

+

320

+

fA(DV

16

+

+

DC

+

DM)]

DT)(DS

+

DB)/(DR

DQ +

(480

+

+

+

+

+

2.5

+

5

40)(60

+

60)/(120

DT

+

DS

+

DB)]

0.08)] +

40

+

60

+

60)]

= (1.0 + 0.352 + 0.041)/(400 + 80 + 320 + 16 + 223) = 1.393/1039 = 0.00134 Pa fS = fW(DD +DT)/(DR + DT + DS + DB) = 0.00134(480 + 40)/(120 + 40+ 60 + 60) = 0.00134 × 520/280 = 0.00249 Pa Concentrations CW = ZWfW = 0.00107 mol/m3 = 0.27 g/m3 (dissolved only) CS = ZSfS = 1.49 mol/m3 = 373 g/m3

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CF ≈ ZFfW ≈ 1.608 mol/m3 ≈ 402 g/m3 (fish) CB ≈ ZFfS ≈ 2.988 mol/m3 ≈ 747 g/m3 (benthos) Total Amounts Water

VWZWfW

= 1072 mol (dissolved only)

Particles

VPZPfW

= 54 mol

Sediment

VSZSfS

= 14940 mol

Total

= 16070 mol

Residence Times Total input to water is 1.393 mol/h from emissions, advective inflow and the atmosphere and 0.4 mol/h from sediment. Total input from sediment is by deposition and diffusion from water. Water

(1072 + 54)/(1.393 + 0.4)

= 628 h or 26 days

Sediment

14940/(0.643 + 0.054)

= 21400 h or 893 days

Total

16070/1.393

= 11540 h or 481 days

(does not include fish or benthos which are probably negligible) Total inputs to and outputs from both water and sediment and the entire system balance within round-off error. This example, while tedious to do by hand, is readily implemented on a spreadsheet, or the software available on the internet can be used. It gives a clear quantification of all the fluxes and demonstrates which processes are most important. A mass balance diagram such as Figure 8.6 illustrates the process rates clearly.

8.7

QWASI MODEL OF CHEMICAL FATE IN RIVERS

The QWASI lake equations can be modified to describe chemical fate in rivers by one of two methods. The river can be treated as a series of connected lakes or reaches, each of which is assumed to be well mixed, with unique water and sediment concentrations. There can be varying discharges into each reach, and tributaries can be introduced as desired. The larger the number of reaches, the more closely simulated is the true “plug flow” condition of the river. Figure 8.7 illustrates the approach. The second approach is to set up and solve the Lagrangian differential equation for water concentration as a function of river length, as was discussed when comparing Eulerian and Lagrangian approaches in Chapter 2. This has been discussed

Chemical fate in a river, as treated by QWASI models in series with no downstream-to-upstream flows.

APPLICATIONS OF FUGACITY MODELS

Figure 8.7

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209

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by Mackay et al. (1983), and an application to surfactant decay in a river has been described by Holysh et al. (1985). A differential equation is set up for the water column as a function of flow distance or time, and steady-state exchange with the sediment is included. The equation can then be solved from an initial condition with zero or constant inputs of chemical. The practical difficulty is that changes in flow volume, velocity, or river width or depth cannot be easily included, therefore the equation necessarily applies to idealized conditions. This equation may be useful for calculating a half-time or half-distance of a substance in a river as the concentration decays as a result of volatilization or degradation. A version is the oxygen sag equation. This contains an additional term for oxygen consumption by organic matter added to the river. This model was first developed by Streeter and Phelps in 1925 and is described in texts such as that of Thibodeaux (1996). This is historically significant as being among the first successful applications of mathematical models to the fate of a chemical (oxygen) in the aquatic environment.

8.8

QWASI MULTI-SEGMENT MODELS

A lake, river, or estuary rarely can be treated as a single well mixed “box” of water, and a more accurate simulation is obtained if the system is divided into a series of connected boxes. In the case of a river, it may be acceptable to use the output from one box as input to the next downstream box and treat any downstreamupstream flow as being negligible compared to upstream-downstream flow. In slowly moving water, this will be invalid if there are significant flows in both directions. In principle, if there are n water and n sediment boxes, there are 2n mass balance equations containing 2n fugacities, and the equations can be solved algebraically for steady-state conditions or numerically for dynamic conditions. In the case of steady-state conditions, a major simplification is possible if it is assumed that there is no direct sediment-sediment transfer between reaches; i.e., all transfer is via the water column. Each sediment mass balance equation then can be written to express the sediment fugacity as a function of the fugacity in the overlying water. The fugacity in sediment then can be eliminated entirely, leaving only n water fugacities to be solved. This is equivalent to calculating the water fugacity as in the QWASI model by including loss to the sediment in the denominator. A total loss D value can thus be calculated for the water and sediment. Algebraic solution of the equations is straightforward, provided the number of boxes is small and there is minimal branching. For a set of boxes connected in series with both “upstream and downstream” flows, the solution becomes simple, elegant, general, and intuitively satisfying because of its transparency. This is illustrated in Figure 8.8. For a given box, the numerator consists of a series of terms, each reflecting inputs (designated I) to this box and Q, including input from other boxes. For each box, the input I (by discharge and from the atmosphere) is included directly. For adjacent boxes, the input to that box is multiplied by the fraction that migrates to the box in question. This fraction,

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QWASI Multi-segment Equations I(i) is total input (mol/h) to each reach from emissions, the atmosphere, and any tributaries. It does not include advective inputs from other reaches. DT(i) is the sum of D values for losses by reaction, burial, and volatilization plus all advective losses from water, including net loss to sediment. The (i) refers to reach 1, 2, 3, or 4. D(i,j) is water and particle flow between reaches i and j.

J(i) is a D value for the net output from each reach. J(1) = DT(1) J(2) = DT(2) – D(2,1) D(1,2)/J(1) J(3) = DT(3) – D(3,2) D(2,3)/J(2) J(4) = DT(4) – D(4,3) D(3,4)/J(3) X(i) is the ratio of D values and is the fraction of the chemical in water and particle flow that enters downstream reach (reach 1 is upstream of reaches 2, 3, and 4). X(1) = D(1,2)/J(1) X(2) = D(2,3)/J(2) X(3) = D(3,4)/J(3) Fraction of Q(1) = I(1) Q(2) = I(2) Q(3) = I(3) Q(4) = I(4)

total input received by each reach, including upstream reaches. + I(1) X(1) = I(2) + Q(1) X(1) + I(2) X(2) + I(1) X(1) X(2) = I(3) + Q(2) X(2) + I(3) X(3) + I(2) X(2) X(3) + I(1) X(1) X(2) X(3) = I(4) + Q(3) X(3)

The solution for water fugacities fW(i) for each reach is as follows: fW(4) = {Q(4) + [fW(5) D(5,4)]}/J(4) fW(3) = {Q(3) + [fW(4) D(4,3)]}/J(3) fW(2) = {Q(4) + [fW(3) D(3,2)]}/J(2) fW(1) = {Q(4) + [fW(2) D(2,1)]}/J(1) The sediment fugacities fS(i) can then be calculated from the steady-state equation in Fig. 8.5. fs(i) = fw(i)(DD(i) + DT(i))/(DB(i) + DS(i) + DR(i) + DT(i)) with D values specific to each segment.

Figure 8.8

Steady-state mass balance equations for a series of four QWASI models with flows in both directions.

X, is a ratio of D values, namely the ratio of the box-to-box advective transfer D value and the total loss D value from the source. The denominators, J, contain terms for losses from each box, again modified by fractions undergoing box-to-box transfer. Inspection of the equations reveals the significance of each term. When the connections are more complex with branching, the equations must be modified

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accordingly, but the final solution can still be inspected to reveal the significance of each term. If the number of boxes is very large (above about 8) and there is appreciable branching, it may be easier to set up the equations in differential form and solve them numerically in time, with constant inputs to reach a steady-state solution. Details of how this can be accomplished are given in Figure 8.9. The dynamic version allows the user to observe changes in concentrations with time. The steady-state version gives the concentrations when the system has reached a nonchanging condition with respect to time. If a long enough integration period is used for the dynamic version, the concentrations approach those in the steadystate version. It is best to use the steady-state version if the user is concerned only with the end result, and the dynamic version if it is desired to track changes in the system over time or how long it will take the system to approach a steady state. It is good practice to check the consistency between steady-state and dynamic solutions by comparing the steady-state output with the dynamic output obtained after integrating for a prolonged period at constant input rates, such that a steady state has been achieved. The following papers are examples of multi-QWASI model applications. Lun et al. (1998) describe the fate of PAHs in the Saguenay River in Quebec. Ling et al. (1993) treat the fate of chemicals in a harbor including vertical segmentation. Hickie

Numerical Integration of QWASI Differential Equations Define time interval between sampling, e.g., 10 h.

DTIM = 10

Note: It is recommended that DTIM be selected as about 5% of the smallest response time VZ/D. Define number of iterations, e.g., 5000. N = 1 to 5000 Note: This should cover a sufficiently long period that the system will reach steady state. Changes in fugacity in water (dfW) and sediment (dfS) as a function of time for each reach, as pseudocode. For I = 1 to 4 dfW(I) = DTIM ((I(I) + fA(I) (DM(I) + DQ(i) + DC(I) + DV(I)) + fS(I)(DT(I) + DR(I)) + fW(I + 1) D(I + 1, I) + fW(I – 1) D(I – 1, I)) – fW(I)(DW(I) + DV(I) + DD(I) + DT(I) + D(I, I – 1) + D(I, I + 1))}/(VW(I) ZWT(I)) dfS(I) = DTIM ((fW(I) (DD(I) + DT(I))) – fS(I) (DB(I) + DS(I) + DR(I) + DT(I)))/(VS(I) ZST(I)) Next I For I = 1 to 4 fW(I) = fW(I) + dfW(I) fS(I) = fS(I) + dfS(I) Next I Note: VW is volume of water, VS is volume of sediment, ZWT is Z for bulk water, and ZST is Z for bulk sediment.

Figure 8.9

Dynamic mass balance equations for a four-reach multi-QWASI system.

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and Mackay (2000) describe the fate of PAHs from atmospheric sources to Lac Saint Louis in the St. Lawrence River. Diamond et al. (1994) treat the fate of a variety of organic chemicals and metals in a highly segmented model of the Bay of Quinte which is connected to Lake Ontario.

8.9 8.9.1

A FISH BIOACCUMULATION MODEL

Introduction

The fish bioaccumulation phenomenon is very important as a means by which chemicals present at low concentration in water become concentrated by many orders of magnitude, thus causing a potential hazard to the fish and other creatures, especially to the birds and humans who consume these fish. For example, DDT may be found in fish at concentrations a million times that of water. The primary cause of this effect is simply the difference in Z values between water and fish lipids as characterized by KOW, but there are other, more subtle effects at work. The kinetics of uptake are also important, because a fish may never reach thermodynamic equilibrium. There is also a fascinating biomagnification phenomenon that is not yet fully understood in which concentrations increase progressively through food chains. It is useful to define some terminology, although opinions differ on the correct usage. Bioconcentration refers here to uptake from water by respiration from water, usually under laboratory conditions when the fish are not fed. Bioaccumulation is the total (water plus food) uptake process and can occur in the laboratory or field. Biomagnification is a special case of bioaccumulation in which there is an increase in concentration or fugacity from food to fish. This situation may occur for nonmetabolizing chemicals of log KOW exceeding 5. A comprehensive review of methods of estimating bioaccumulation is that of Gobas and Morrison (2000). Other reviews are the texts by Connell (1990) and Hamelink et al. (1994) and the paper by Mackay and Fraser (2000). The models described here are based on those of Clark et al. (1990), Gobas (1993), and Campfens and Mackay (1997). Figure 8.10 shows the processes of uptake and clearance. The approach taken here is to set out the mass balance equations in conventional rate constant form, then show that they are equivalent to the fugacity forms using D values. The final model gives both rate constants and D values. 8.9.2

Equations in Rate Constant Format

Here, we treat the fish as one “box.” The conventional concentration expression for uptake of chemical by fish from water, through the gills only under laboratory conditions, was first written by Neely et al. (1974) as dCF/dt = k1CW – k2CF where CF and CW are concentrations in fish and water, k1 is an uptake rate constant and k2 is the clearance rate constant. The fish is regarded as a single compartment.

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Figure 8.10 Fish bioaccumulation processes expressed as concentration/rate constants and fugacity/D values.

Apparently, the chemical passively diffuses into the fish along much the same route as oxygen. In the laboratory, it is usual to expose a fish to a constant water concentration for a period of time during which the concentration in the fish should rise from zero to CF according to the integrated version of the differential equation with CF initially zero and CW constant. CF = (k1/k2)CW[1 – exp(–k2t)] After prolonged exposure, when k2t is large, i.e., >4, CF approaches (k1/k2)CW or KFWCW where KFW is the bioconcentration factor. The fish is then placed in clean water, and loss or clearance or depuration is followed, the corresponding equation being CF = CFO exp(–k2t) where CFO is the concentration at the start of clearance. It is apparent that there are three parameters, k1, k2, and KFW or k1/k2, thus only two can be defined independently. The most fundamental are k1 (which is a kinetic rate constant term quantifying the volume of water that the fish respires and from which it removes chemical, divided by the volume of the fish) and KFW, which is a thermodynamic term reflecting equilibrium partitioning. The loss rate constant k2 is best regarded as k1/KFW. The uptake and clearance half-times are both 0.693 KFW/k1 or 0.693/k2. This equation can be expanded to include uptake from food with a rate constant kA and food concentration CA, loss by metabolism with a rate constant kM, and loss by egestion in feces with rate constant kE, namely, dCF/dt = k1CW + kACA – CF(k2 + kM + kE)

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If the fish is growing, there will be growth dilution, which can be included as an additional loss rate constant kG, which is the fractional increase in fish volume per hour. At steady-state conditions, the left side is zero, and CF = (k1CW + kACA)/(k2 + kM + kE) Gobas (1993) has suggested correlations for these rate constants as a function of fish size. The mass balance around the fish can be deduced and the important processes identified. The rate constant k1 is much larger than kA, typically by a factor of 5000. Thus, uptake from water and food become equal when CA is about 5000 times CW, which corresponds to a KOW of about 105 and 5% lipid. For lower KOW chemicals, uptake from water dominates whereas, for higher KOW chemicals, uptake from food dominates. 8.9.3

Equations in Fugacity Format

We can rewrite the bioconcentration equation in the equivalent fugacity form as VFZFdfW/dt = DV(fW – fF) where DV is a gill ventilation D value analogous to k1 and applies to both uptake and loss. This form implies that the fish is merely seeking to establish equilibrium with its surrounding water. The corresponding uptake and clearance equations are fF = fW(1 – exp[–DVt/VFZF)] fF = fFO exp(–DVt/VFZF) The following expressions relate the rate constants and D values, showing that the two approaches are ultimately identical algebraically. k1 = DV/VFZW k2 = DV/VFZF KFW = ZF/ZW As was discussed earlier, ZF can be approximated as LZO, where L is the volume fraction lipid content of the fish, and ZO is the Z value for octanol or lipid. KFW is then LKOW, where L is typically 0.05 or 5%. From an examination of uptake data, Mackay and Hughes (1984) suggested that DV is controlled by two resistances in series, a water resistance term DW, and an organic resistance term DO. Since the resistances are in series, 1/DV = 1/DW + 1/DO

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The nature of the processes controlling DW and DO is not precisely known, but it is suspected that they are a combination of flow (GZ) and mass transfer (kAZ) resistances. If we substitute GZ for each D, recognizing that G may be fictitious, we obtain 1/k2 = VFZF/DV = VFLZO(1/GWZW + 1/GOZO) = (VFL/GW)KOW + (VFL/GO) = τWKOW + τO By plotting 1/k2 against KOW for a series of chemicals taken up by goldfish, Mackay and Hughes (1984) estimated that τW was about 0.001 hours and τO 300 hours. This is another example of probing the nature of series or “two-film” resistances using chemicals of different partition coefficient as discussed in Chapter 7. The times τW and τO are characteristic of the fish species and vary with fish size and their metabolic or respiration rate, as discussed by Gobas and Mackay (1989). The uptake and clearance equilibria and kinetics, i.e., bioconcentration phenomena, of a conservative chemical in a fish are thus entirely described by KOW, L, τW, and τO. The bioconcentration equation can be expanded as before to include uptake from food with a D value DA, loss by egestion (DE) and loss by metabolism (DM), giving VFZFdfF/dt = DVfW + DAfA – fF (DV + DM + DE) A growth dilution D value DG defined as ZFdVF/dt can be included as an additional loss term. This term can become very important for hydrophobic chemicals for which the DV and DM terms are small. The primary determinant of concentration is then how fast the fish can grow and thus dilute the chemical. It should be noted that this treatment of growth is simplistic in that growth is assumed to be first order and does not change other D values. A mass balance envelope problem arises when treating the food uptake or digestive process. The entire fish, including gut contents, can be treated as a single compartment. In this case, the food uptake D value is simply the GZ product of the food consumption rate and its Z value, i.e., GAZA, subscript A applying to food. ZA can be estimated as LAZO, where LA is the lipid content of the food. Often, the fugacity of a chemical in the food fA will approximate the fugacity in water. The rate of chemical uptake into the body of the fish is then EADAfA or DAEfA, where EA is the uptake efficiency of the chemical. In reality, the gut is “outside” the epithelial tissue of the fish, and it may be better to treat the fish as only the volume inside the epithelium. In this case, the uptake D value is GAZAEA. To avoid confusion, we define two uptake D values, DAE, which includes the efficiency, and DA, which does not. The same problem applies to egestion where we define DEE as including a transport efficiency and DE, which does not. The digestive process that controls EA and thus DAE is more complicated and less understood than gill uptake. The first problem is quantifying the uptake efficiency, i.e., the ratio of quantity of chemical absorbed by the fish to chemical consumed. It is generally about 50% to 90%. Gobas et al. (1989) have suggested

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that the uptake efficiency EA from food in the gastrointestinal tract of a “clean” fish can also be described by a two-film approach yielding 1/EA = AWKOW + AO where AW and AO are water and organic resistance terms similar in principle to τW and τO, but are dimensionless. AO appears to have a magnitude of about 2, and AW a magnitude of about 10–7, thus, for all but the most hydrophobic chemicals, EA is about 50%. When KOW exceeds 107, the efficiency drops off, because of a high water phase resistance in the gut. A major difficulty is encountered when describing the loss of chemical in feces and urine. In principle, D values can be defined, but it is quite difficult and messy to measure G and Z; therefore, neither are known. It is probable that the digestion process, which removes both mass and lipid content to provide matter and energy to the fish, reduces both GA and ZA so that DE for egestion is smaller than DA. The simplest expedient is to postulate that it is reduced by a factor Q, thus we estimate DE for loss by egestion as DA/Q or DAE/Q, i.e., DE or DEE. The resistances causing EA for uptake are assumed to apply to loss by egestion. This assumption is probably erroneous, but it is acceptable for most purposes, especially because there is presently an insufficiency of data to justify different values for uptake and loss. The steady-state solution to the differential equation for the entire fish becomes fF = (DVfW + DAfA)/(DV + DM + DA/Q) For the fish inside the epithelium, DA is replaced by DAE. Assuming that fA equals fW, it is clear that fF will approach fW only when the DV term dominates in both the numerator and denominator. If KOW is large, e.g., 106, the term DA will exceed DV (because ZA will greatly exceed ZW), and the uptake of chemical in food becomes most important. The fish fugacity then tends toward QfA or QfW, i.e., the fish achieves a biomagnification factor of Q. Q is thus a maximum biomagnification factor as well as being a ratio of D values. This biomagnification behavior was first clearly documented in terms of fugacity by Connolly and Pedersen (1988), Q typically having a value of 3 to 5. Biomagnification is not immediately obvious until the fugacities are examined instead of the concentrations. At each step in the food chain, or at each trophic level, there is a possibility of a fugacity multiple applying. It is thus apparent that fish fugacities and concentrations are a reflection of a complex combination of kinetic and equilibrium terms that can in principle be described by D values. The detailed physiology of the factors controlling Q has been investigated in a series of elegant experiments by Gobas and colleagues (1993, 1999). The fugacity change in the gut contents as they journey through the gastrointestinal tract was followed by head space analysis. These experiments showed convincingly that the hydrolysis and absorption of lipids reduces the Z value, causing the fugacity to increase as a result of loss of the lipid “solvent.” Additionally, the mass of food is reduced, thus G also decreases. The net effect is a decrease in GZ by about a factor Q of 4. It is noteworthy that Q for mammals and birds is much larger, e.g., 30, thus

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biomagnification is more significant for these animals, rendering them more vulnerable to toxic effects of persistent chemicals. Worked Example 8.2 Calculate the fugacity of a fish at steady-state when exposed to uptake of a hydrophobic chemical from water and food. Deduce the fluxes and determine if biomagnification occurs. What would be the implication it the metabolic rate constant increases by a factor of 8? Input data fW = 10–6 fA

= 2 × 10–6 (slight biomagnification in food)

DV = 10–4 DA = 10–3 DM = 10–4 Q

=4

fF

= (DV fW + DA fA)/(DV + DM + DA /Q) = (10–10 + 2 × 10–9)/(4.5 × 10–4) = 4.67 × 10–6 Pa

There is food to fish biomagnification by a factor of 2.33. If DM is increased to 8 × 10–4, the fugacity in the fish drops to 1.83 × 10–6 Pa, which eliminates biomagnification entirely. The fluxes in the two cases are (in units of 10–9 mol/h or n mol/h) as follows. Input from water

0.1

Input from food

2.0

(total input is 2.1)

Loss to water

0.467

(slow metabolism),

0.183

(fast metabolism)

Loss by egestion

1.168

(slow metabolism),

0.457

(fast metabolism)

Loss by metabolism

0.467

(slow metabolism),

1.460

(fast metabolism)

Total loss

2.1

(slow metabolism),

2.1

(fast metabolism)

In both cases, because fugacity in the fish exceeds that of the water, there is net loss by respiration. The presence of biomagnification depends on whether the fish can clear the chemical fast enough to maintain a fugacity less than that of the food. In this case, if the metabolic rate was zero, the fish would reach a fugacity of 6 × 10–6 Pa, 3 times that of the food, losing 0.6 to water and 1.5 by egestion.

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8.9.4

219

Model

Available on the website are Windows- and DOS-based Fish models that calculate the fish fugacity, concentration and the various flux terms from input data on the chemical’s properties, concentration in water, and various physiological constants. They include a “bioavailability” calculation in the water by estimating sorption to organic matter in particles. The models are particularly useful for exploring how variation in KOW and metabolic half-life affect bioaccumulation, and they show the relative importance of food and water as sources of chemical. An overall residence time is calculated that indicates the time required for contamination or decontamination to take place. 8.9.5

Food Webs

The fish bioaccumulation model can be applied to an aquatic food web starting with water and then moving successively to phytoplankton, zooplankton, invertebrates, small fish, and to various levels of larger fish. Each level becomes food for the next higher level. If KOW is relatively small, i.e., <105, the DV terms dominate, and equifugacity is probable throughout the web; i.e., fA, fW and fF will be equal. For larger KOW chemicals, biomagnification is likely. For “superhydrophobic” chemicals of KOW>107, the EA term becomes small, uptake is slowed, and the growth and metabolism terms become critical. Association with suspended organic matter in the water column becomes important, i.e., “bioavailability’ is reduced. A falloff in observed BCFs is (fortunately) observed for such chemicals; thus, there appears to be a “window” in KOW of about 106 to 107 in which bioaccumulation is most significant and most troublesome. Chemicals such as DDT and PCBs lie in this “window.” This issue has been discussed in detail and modeled by Thomann (1989). Several features of food web biomagnification are worthy of note. Humans usually eat creatures close to the top of food webs, and strive to remain at the top of food webs, avoiding being eaten by other predators. Fish consumption is often the primary route of human exposure to hydrophobic chemicals. Creatures high in food webs are invaluable as bioindicators or biomonitors of contamination of lakes by hydrophobic chemicals. But to use them as such requires knowledge of the D values, especially the D value for metabolism. A convincing argument can be made that, if we live in an ecosystem in which wildlife at all trophic levels is thriving, we can be fairly optimistic that we humans are not being severely affected by environmental chemicals. This is a (selfish) social incentive for developing, testing, and validating better environmental fate models, especially those employing fugacity. A food web model treating multiple species can be written by applying the general bioaccumulation equation to each species (with appropriate parameters). The final set of equations for n organisms has n unknown fugacities that can be solved sequentially, starting at the base of the food web and proceeding to other species, with smaller animals becoming food for larger animals.

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An alternative and more elegant method is to set up the equations in matrix form as described by Campfens and Mackay (1997) and solve the equations by a routine such as Gaussian elimination. This permits complex food webs to be treated with no increase in mathematical difficulty. A DOS-based BASIC model Foodweb is available that performs these calculations as described in detail by Campfens and Mackay (1997). It is an expansion of the Fish model, and it treats any number of aquatic species that consume each other according to a dietary preference matrix. A steady-state condition is calculated using matrix algebra. All organism-to-organism fluxes (i.e., food consumption rates) are given. The model is also useful as a means of testing how concentrations in top predators respond to changes in food web structure. It is essentially a multibox model with one-way transfers from box to box. An obvious extension is to include nonaquatic species such as birds and mammals. This has been discussed by Clark et al. (1989), who showed that fish-eating birds could be included in an aquatic food web model. In the long term, it may be possible to build models containing all relevant biota, including fish, birds, mammals, insects, and vegetation. A framework for accomplishing this has been described by Sharpe and Mackay (2000). The primary difficulties are in the development of species-specific mass balance equations, determining appropriate parameters for the organisms and obtaining validation data. There is little doubt that comprehensive food web models will be developed and validated in the future, even models including humans.

8.10

SEWAGE TREATMENT PLANTS

Many chemicals are discharged to sewers and are subsequently treated biologically in sewage treatment plants (STPs), also called publicly owned treatment works (POTWs). Treatment configurations vary from simple lagoons to more complex systems in which the concentration and activity of the biomass are optimized by recycling the biomass or sludge. Such activated sludge STPs essentially consist of a series of connected vessels, the contents of which are well mixed, having contact with air either at the surface (as in a lake) or by forced aeration. A typical STP flow diagram is shown in Figure 8.11 with illustrative chemical fluxes. The influent sewage is settled by primary sedimentation, followed by secondary treatment under aeration conditions with subsequent settling and recycling. The flows of water, solids, and air are defined by the plant operating conditions. The task is then to deduce the corresponding fluxes of the chemical present in the influent. Steady-state mass balance equations can be set up for each vessel and solved for the three fugacities, from which all chemical fluxes can be deduced. This requires that D values be defined for flows of chemical in water, biomass solids, and air, for both degradation and surface volatilization. Clark et al. (1995) have described such a model, and Windows software and a BASIC program (STP) are available. The model is particularly useful not only for estimating the overall treatment efficiency but also the fraction of the chemical input that is volatilized, degraded, left in the sludge, or remains in the effluent.

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Figure 8.11 Transport and transformation processes in a typical activated sludge plant represented as three well mixed compartments. The relative chemical fluxes, e.g. (100), are illustrative.

8.11 8.11.1

INDOOR AIR MODELS

Introduction

We present here a very simple model of chemical fate in indoor air. Numerous studies have shown that humans are exposed to much higher concentrations of certain chemicals indoors than outdoors. Notable are radon, CO, CO2, formaldehyde, pesticides, and volatile solvents present in glues, paints, and a variety of consumer products. The key issue is that, whereas advective flow rates are large outdoors, they are constrained to much smaller values indoors. Attempts to reduce heating costs often result in reduced air exchange, leading to increased chemical “entrapment.” A nuclear submarine or a space vehicle is an extreme example of reduced advection. Fairly complicated models of chemical emission, sorption, reaction, and exhaust in multichamber buildings have been compiled [e.g., Nazaroff and Cass (1986, 1989) and Thompson et al. (1986)], but we treat here only the simple model developed by Mackay and Paterson (1983), which shows how D values can be used to estimate indoor concentrations caused by evaporating pools or spills of chemicals. An example of the effective use of fugacity for compiling mass balances indoors is the INPEST model, developed in Japan by Matoba et al. (1995, 1998). This model successfully describes the changing concentration of pyrethroid pesticides applied indoors in the hours and days following their application. Because of the reduced advection, there is a potential for high concentrations and exposures immediately following pesticide use, and it may be desirable to evacuate the room or building for a number of hours to allow the initial peak concentration in air to dissipate. The INPEST fugacity model, which is in the form of a spreadsheet, can be used to suggest effective strategies for avoiding excessive exposure. It can be used to compare pesticides and explore the effects of different application practices.

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Model of a Chemical Evaporating Indoors

We treat a situation in which a pool of chemical is evaporating into the basement air space of a two-room (basement plus ground level) building with air circulation. If the building were entirely sealed and the chemical were nonreactive, evaporation would continue until the fugacity throughout the entire building equalled that of the pool (fP). Of course, it is possible that the pool would have been completely evaporated by that time. The evaporation rate can be characterized by a D value D1 corresponding to kAZ, the product of the mass transfer coefficient, pool area, and air phase Z value. If the chemical were in solution, it would be necessary to invoke liquid and gas phase D values in series, i.e., the two-film theory as discussed in Chapter 7. The evaporated chemical may then be advected from one room to another with a D value D2 defined as GZ, the product of the air flow or exchange rate and the air phase Z value. From this second room, it may be advected to the outdoors with another D value D3. These advection rates are normally characterized as “air changes per hour” or ACH, which is the advection rate G divided by the room or building volume and is the reciprocal of the air residence time. Typical ACHs for houses range from 0.25 to 1.5 per hour. The outdoor air has a defined background fugacity fA. It is apparent that the chemical experiences three D values in series in its journey from spill to outdoors, thus the total D value will be given by 1/DT = 1/D1 + 1/D2 + 1/D3 and the flux N is DT(fP – fA) mol/h. Of interest are the intermediate fugacities in the rooms, which can be estimated from the equations. N = D1(fP – f1) = D2(f1 – f2) = D3(f2 – fA) This is essentially a “three-film” or “three-resistance” model. Degrading reactions could be included, leading to more complex, but still manageable, equations. Sorption to walls and floors could also be treated, but it is probably necessary to include these processes as differential equations. Worked Example 8.3 An example of a “spill” of a small quantity (1 g) of PCB over 0.01 m2 (e.g., from a fluorescent ballast) was considered by Mackay and Paterson (1983). The PCB fugacity was 0.12 Pa and the outdoor concentration was taken as 4 ng/m3 or 3.7 × 10–8 Pa. The three D values (expressed as reciprocals) are 1/D1 = 49000,

1/D2 = 30,

1/D3 = 15

Thus, DT is essentially D1, most resistance lying in the slow evaporation process from the small spill area. The molar mass is 260 g/mol, and the evaporation rate N is then

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N = DT(fP – fA) = 2.4 × 10–6 mol/h = 6.4 × 10–4 g/h The intermediate fugacities and concentrations are 3.6 × 10–5 Pa (3800 ng/m3) and 11 × 10–5 Pa (11500 ng/m3). The time for evaporation of 1 g of PCB will be 65 days. The amount of PCB in an air volume of 500 m3 would be on the order of 0.004 g, a small fraction of the small amount of PCB spilled. The significant conclusion is that, despite appreciable ventilation at an ACH of 0.5 h–1, the indoor air concentrations are over 1000 times those outdoors. Fortunately, the indoor fugacity is still very much lower than the pool fugacity. Similar behavior applies to other solvents, pesticides, and chemicals that may be used and released indoors. Although the amounts spilled or released are small, the restricted advective dilution results in concentrations that are much higher than are normally encountered outdoors. In many cases, this phenomenon is suspected to be the cause of the “sick building” problem in which residents complain repeatedly about headaches, nausea, and tiredness. The cure is to eliminate the source or increase the ventilation rate. Fugacity calculations can contribute to understanding such problems.

8.12

UPTAKE BY PLANTS

Uptake of chemicals by plants is one of the most important but still poorly understood processes. It is important because plants are at the base of food chains. Thus, a chemical such as a dioxin absorbed by grass can be transported to the cow, then to dairy and meat products, and thus to humans. Plants can be valuable monitors of the presence of chemicals in the environment, but they are only of quantitative value if the plant-environment partitioning phenomena are fully understood. Plants may also affect the overall environmental fate of a substance by removing it from the atmosphere or absorbing it from soils. An attractive “phyto-remediation” option is to use plants to reduce concentrations in contaminated sites. Ironically, although plants are much simpler organisms than animals on the scale of biological evolution, they are more difficult to model. A major difficulty is that plants grow quickly relative to animals, and their circulatory system is not as efficient as those of animals. There are flows in xylem and phloem channels, but these are not as well characterized as blood flows. Foliage, which is the primary contact area with the atmosphere, is very complex and variable from species to species. It consists of an external often waxy cuticle, but with access to the interior by stomata that are designed to ensure entry of CO2. The root, which is in contact with soil, presents a complex barrier to uptake of chemicals. It is not clear how the wood or bark of trees should be treated. Much of the mass of a tree is inaccessible to contaminants. Often, the consumed material is fruit, nuts, or seeds that form rapidly, but processes of chemical transport to them from foliage, roots, and stem are not yet well understood. These and other issues have been discussed in the texts by Nobel (1991) and Trapp and MacFarlane (1995). McLachlan (1999) has set out a framework for assessing uptake of chemicals by grasses from the atmosphere. Severinson and Jager (1998) have discussed the need for including plants in multimedia models. Actual fugacity or related models of uptake have been developed by Trapp et al. (1990),

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Paterson et al. (1991), and Hung and Mackay (1997). This topic is the focus of considerable research, and improved models will no doubt be developed in the near future.

8.13

PHARMACOKINETIC MODELS

Physiologically based pharmacokinetic models (PBPK models) treat an animal as a collection of connected boxes in which exchange occurs primarily in the blood, which circulates between all the boxes. The model can include uptake from air and food and possibly by dermal contact or injection. We then calculate the dynamics of the circulation of the chemical in venous and arterial blood, to and from various organs or tissue groups including adipose tissue, muscle, skin, brain, kidney, and liver. There may be losses by exhalation and metabolism, and in urine, feces and, sweat. In mammals, nursing mothers also lose chemical to their offspring in breast milk, and they lose tissue when giving birth. Analogous processes occur during egglaying in birds, amphibians, and reptiles. As in environmental models, partition coefficients or Z values can be deduced to quantify chemical equilibrium between air, blood, and various organs. Flows of blood to each organ can be expressed as D values. Metabolic rates can be expressed using rate constants, usually invoking Michaelis–Menten kinetics, as described in Chapter 6, and translated into D values. Mass balance equations then can be assembled, describing the constant conditions that develop following exposure to long-term constant concentrations or the dynamic conditions that follow a pulse input. Experiments are done, often with rodents, to follow the time course of chemical transport and transformation in the animal. The resulting data can be compared with model assertions to achieve a measure of validation. Much of the pharmacokinetic literature is devoted to assessment of the time course of the fate of therapeutic drugs within the human body. The aim is to supply a sufficient, but not too large and thus toxic, dose of drug to the target organ. Closer to environmental exposure conditions are PBPK models for occupational exposure to toxicants such as solvent or fuel vapors, which may be intermittent or continuous in nature. An example is the model of Ramsey and Andersen (1984), which was translated into fugacity terms by Paterson and Mackay (1986, 1987). Accounts of various aspects of pharmacokinetics and PBPK models and their contribution to environmental science are the works of Welling (1986), Parke (1982), Reitz and Gehring (1982), Tuey and Matthews (1980), Fisherova-Bergerova (1983), Menzel (1987), Nichols et al. (1996, 1998), and Wen et al. (1999). A fugacity model has been developed for whales by Hickie et al. (1999) and a rate constant model for birds by Clark et al. (1987). Figure 8.12, which is adapted from Paterson and Mackay (1987), illustrates the fugacity approach to modeling the fate of a chemical in the human body. In principle, it is possible to calculate steady- and unsteady-state fugacities, concentrations, amounts, fluxes, and response times, thus linking external environmental concentrations to internal tissue concentrations. Ultimately, from a human health viewpoint, it is likely that it will be possible to undertake these calculations and compare levels

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Figure 8.12 Transport and transformation in a multicompartment pharmacokinetic model as applied to humans.

of chemical contamination in vulnerable tissues with levels that are believed to cause adverse effects. There is clearly a need to link environmental and pharmacokinetic modeling efforts to build up a comprehensive capability of assessing the journey of the chemical from source to environment to organism and ultimately to the target site.

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8.14 8.14.1

HUMAN EXPOSURE TO CHEMICALS

Introduction

The multimedia environmental models described in this chapter lead to estimates of fugacities and concentrations in air, water, soil, and sediments. These abiotic fugacities can be used to deduce fugacities and concentrations in fish, and possibly in other animals and plants. The primary weakness is probably that they do not yet adequately quantify partitioning into the variety of vegetable matter that is consumed by animals and humans. For some compounds, such as the dioxins, the air-grasscow-milk-dairy product-human route of transfer is critical. In this chapter, we discuss briefly the principles by which these concentration data can be used to assess the impact of chemicals on humans and other organisms. The reader is directed to reviews such as that by Paustenbach (2000) for a detailed treatment of exposure and risk assessment. The first obvious use of these abiotic and biotic concentrations is to compare them with concentration levels that are believed to cause adverse effects. These levels are usually developed by regulatory agencies and published as guidelines, objectives, or effect-concentrations of various types. Target or objective concentrations can be defined for most media. For example, from considerations of toxicity or aesthetics, it may be possible to suggest that water concentrations should be maintained below 1 mg/m3, air below 1 µg/m3, and fish below 1 mg/kg. These concentrations can be compared as a ratio or quotient to the estimated environmental concentrations. A hypothetical example is given in Table 8.4, illustrating the quotient method. In this example, the primary concern is with air inhalation and fish ingestion. The proximities of the estimated prevailing concentrations to the targets are expressed as quotients, which can be regarded as safety factors. A large quotient implies a large safety factor and low risk. The high-risk situations correspond to low quotients. This quotient is also called a toxicity/exposure ratio or TER. The concentration level in fish may not be directly toxic to fish but may pose a threat to humans if the fish is consumed on a regular basis. The reciprocal ratio is also used in the form of a PEC/PNEC ratio, i.e., predicted environmental concentration/predicted no effect concentration. In this case, a high value implies high risk. Table 8.4 Comparison of Predicted or Measured Environmental Concentrations with Concentrations Producing a Specified Effect, or No Effect Concentrations Medium

Predicted Level

Air (µ/m3)

3

Effect Level 60

Quotient 20

Water (µg/L)

10

3000

300

Fish (µg/g)

2

10

5

Soil (µg/g)

1

100

100

Difficulties are encountered when suggesting target concentrations in soil and sediment, because these media are not normally consumed directly by organisms.

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Whereas simple lethality experiments can be designed using air, water, or food as vehicles for chemical exposure, it is not always clear how concentrations in the solid matrices of soils and sediments relate to exposure or intake of chemical by organisms. It is difficult to design meaningful bioassays involving interactions between organisms, soils, and sediments. One approach is to decree that whatever target fugacity is developed for water be applied to sediment. This effectively links the target concentrations by equilibrium partition coefficients. A second method is to use concentrations to estimate exposure or dosage in units such as µg/day of chemical to an organism, which for selfish reasons is usually a human. Individual and total dosages can be estimated to reveal the more important routes. This calculation of dose is enlightening, because it reveals which medium or route of exposure is of most importance. Presumably, steps can then be taken to reduce this route by, for example, restricting fish consumption. Table 8.5 lists representative exposure quantities for several human age classes. Table 8.5 Representative Exposure Rates for Four Human Age Classes Derived Principally from the EPA Exposure Factors Handbook Age Classes Route

<1

Air inhalation

4.5

Drinking water

300

1–5

6–19

20+

Units

6

10

13.5

m3/day

694

904

1500

mL/day

Foodstuffs fruit

135.6

145.1

161.6

166.9

g/day

61.9

108.0

209.1

271.3

g/day

grains

155.1

143.7

227.7

219.0

g/day

meat

26.1

59.1

113.8

123.3

g/day

fish

—

4

6

14

g/day

339.6

438.7

249.6

g/day

vegetables

dairy

570.9

human milk

775

0

0

0

g/day

—

0.11*

0.087

0.065

g/day

area

—

0.43

0.62

0.70

m2

duration

—

area

—

duration

—

Soil ingestion Dermal contact soil

1440

600

320

h/yr

water 0.65 82.4

1.25 82.4

1.81 82.4

m2 h/yr

*A pica child may ingest up to 10g/day

An average human inhales some 14 m3 of air per day. If the concentration in air is known in µg/m3, the amount of chemical inhaled in this air is readily calculated as the product with units of µg/day. Not all this chemical may be absorbed, but at least a maximum dosage can be deduced. The same human may consume 1.5 litres/day of water containing dissolved chemical, enabling this dosage to be esti-

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mated again in µg/day. Food, the other vehicle, is more difficult to estimate. A typical diet may consist of 1 kg/day of solids broken down as shown in Table 8.5. Fish concentrations can be estimated directly from water concentrations, but meat, vegetable, and dairy product concentrations are still poorly understood functions of the concentrations of chemical in air, water, soil, and animal feeds, and of agrochemical usage. Techniques are emerging for calculating food-environment concentration ratios, but at present the best approach is to analyze a typical purchased “food basket.” This issue is complicated by the fact that much food is grown at distant locations and imported. Beverages, food, and water may also be treated for chemical removal commercially or domestically by washing, peeling, or cooking. An example illustrates this method of estimating dose. Worked Example 8.4 A chemical of molar mass 181.5 g/mol has partitioned into the air, water, soil, and sediment resulting in the concentrations given in Table 8.6 below. As a result of contact with these abiotic media, fish, vegetation, and meat (and correspondingly, dairy products) are estimated to have the concentrations tabulated. Using these data and the exposure rates from Table 8.5 for an adult, calculate the dose to an adult human. Assume that the density of food is 1000 kg/m3. Table 8.6 Illustration of Calculation of Human Dose

Concentration C (mol/m3)

Intake Rate I (m3/day)

× 10–7 6.57 × 10–4 –6 Fish 2.75 × 10 1.40 × 10–5 –9 Meat 1.02 × 10 1.23 × 10–4 Dairy 3.23 × 10–10 2.50 × 10–4 –9 Water 1.16 × 10 1.50 × 10–3 –7 Soil 1.69 × 10 6.50 × 10–8 –11 Air 6.80 × 10 13.5 × 100 Total dose = 2.22 × 10–7 g/day = 0.222 µg/day Vegetation

4.02

Amount Consumed (C*I) (mol/day)

× 3.85 × 1.26 × 8.06 × 1.74 × 1.10 × 9.18 × 2.64

10–10 10

–11

10–13 10–14 10–12 10–14 10–10

Amount Consumed (C*I*181.5) (g/day)

× 10–8 6.99 × 10–9 2.28 × 10–11 1.46 × 10–11 3.16 × 10–10 1.99 × 10–12 1.67 × 10–7 4.79

In this case, inhalation in air causes 75% of the dose, and consumption of vegetation causes another 21%. Significant chemical exposure may also occur in occupational settings (e.g., factories), in institutional and commercial facilities (e.g., schools, stores, and cinemas), and at home, but these exposures vary greatly from individual to individual and depend on lifestyle. There emerges a profile of relative exposures by various routes from which the dominant route(s) can be identified. If desired, appropriate measures can be taken to reduce the largest exposures. The advantage of this approach is that it places the spectrum of exposure routes in perspective. There is little merit in striving to reduce an already small exposure.

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Exposure routes vary greatly in magnitude from chemical to chemical, depending on the substance’s physical chemical properties such as KAW and KOW , and it is not usually obvious which routes are most important. Data from the environmental fate models can provide a sound basis for estimating risk when used to assess quotients and to determine dominant exposure routes. If such information can be presented to the public, the individuals will be, at least in principle, able to choose or modify their lifestyles to minimize exposure and presumably risk. Individuals then have the freedom and information to judge and respond to acceptability of risk from exposure to this chemical compared to the other voluntary and involuntary risks to which they are subject.

8.15

THE PBT–LRT ATTRIBUTES

A regulatory issue in which evaluative mass balance models are playing an increasingly important role is in assessing the persistence, bioaccumulation, toxicity and long-range transport (PBT-LRT) attributes of chemicals. If chemicals that display these undesirable attributes can be identified, they can be considered for regulation, as has been done by UNEP for the “dirty dozen” high-priority substances discussed earlier. Monitoring data are usually too variable to enable them to be used directly in this priority setting task, and monitoring is impossible for chemicals not yet in use. Since there are many thousands of chemicals of commerce that require assessment, and (it is hoped) most are innocuous, there is an incentive to develop a tiered system in which there are minimal data demands initially and perhaps 90% of chemicals evaluated are rejected as of no concern. The remaining 10% of potential concern can be more fully evaluated in a second tier with a similar rejection ratio. A third tier may be needed to select the (perhaps) 100 top priority chemicals from a universe of 100,000 chemicals in a three-tier system. The challenge is to devise models or evaluation systems that will accomplish this task efficiently. Webster et al. (1998) have suggested using a Level III model similar to EQC, but with advection shut off, to evaluate persistence. Gouin et al. (2000) have described an even simpler Level II approach. This has the advantage that no “modeof-entry” information is required. Regardless of which model is used, it seems inevitable that models will play a key role in assessing persistence or residence time, since these quantities cannot be measured directly in the environment. Bioaccumulation can be evaluated most simply by calculating the equilibrium bioconcentration factors as the product of lipid content and KOW. For more detailed evaluation involving considerations of bioavailability, metabolism, and possible biomagnification from food uptake, the Fish model or a variant of it can be used. In some cases, a food web model may be required to determine if significant biomagnification occurs. Foodweb can be used for this purpose. Mackay and Fraser (2000) have suggested such a three-tiered approach for assessing the bioaccumulation potential of chemicals. Toxicity is not within our scope here. Long-range transport (LRT) presents an interesting challenge because, like persistence, LRT cannot be measured in the environment. It can only be estimated using

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models. Most interest is in LRT in the atmosphere but, in some cases, oceans and rivers can play a significant role. Even migrating biota can contribute to LRT. The most promising approach is to consider the fate of chemical in a parcel of Lagrangian air passing over soil or water and subject to degradation, deposition, and reevaporation. Such systems have been suggested by Van Pul (1998), Bennett et al. (1999), and Beyer et al. (2000). Beyer et al. (2000) showed that a LRT distance in air can be deduced from a simple Level III model as the product of the wind velocity, the overall persistence or residence time of the chemical, and the fraction of the chemical in the atmosphere. A model, TaPL3 (Transport and Pesistence Level 3), can be used for a Level III evalution of persistence and LRT. It is available on the website. It is expected that new models will be developed to assist in the evaluation of these attributes, especially in situations where there is no easy method of obtaining the required information from environmental monitoring data.

8.16

GLOBAL MODELS

The ultimate mass balance model of chemical fate is one that describes the dynamic behavior of the substance in the entire global environment. At present, only relatively simplistic treatments of chemical fate at this scale have been accomplished, but it is likely that more complex and accurate models will be produced in the future. Meteorologists can now describe the dynamic behavior of the atmosphere in some detail. Oceanographers are able to describe ocean currents. Ultimately, there may be linked meteorological/oceanographic/terrestrial models in which the ultimate fate of 100 kg of DDT applied in Mexico can be predicted over the decades in which it migrates globally. The obvious ethical implication is that a nation should not use a substance in such a way that other nations suffer significant exposure and adverse effects. These situations have already occurred with acid rain and Arctic and Antarctic contamination by persistent organic substances. The construction of global-scale models opens up many new and interesting prospects. It appears that there is a global fractionation phenomenon as a result of chemicals migrating at different rates and tending to condense at lower temperatures. Chemicals that do reach cold regions may be better preserved there because of the reduced degradation rates. Chemicals appear to be subject to “grasshopping” (or “kangarooing” in the Southern Hemisphere) as they journey, deposit, evaporate, and continue hopping from place to place until they are ultimately degraded as shown in Figure 8.13. Accounts of these phenomena, and models that attempt to quantify them, are given in a series of papers by Wania and Mackay (1993, 1996, 1999) and Wania et al. (1996). The most successful modeling to date has been of α-HCH, which was produced as an impurity in the insecticide, technical lindane (Wania and Mackay, 1999) but is no longer produced. An interesting insight from that study is an assertion that, despite α-HCH never having been used in the Arctic, about half the remaining mass on this planet now resides in the arctic oceans. This GloboPOP model is

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Figure 8.13 Schematic diagram of chemical “grasshopping” on a global scale.

APPLICATIONS OF FUGACITY MODELS

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available from the University of Toronto. A link is maintained from the Trent website, at which other models are available. As better models of global fate become available, they will provide an invaluable tool with which humanity can design, select, and use chemicals on our planet with no fear of adverse consequences. Whether we will be sufficiently enlightened to achieve this is a question only time will answer.

8.17

CLOSURE

Perhaps the task addressed by this book is best summarized by Figure 8.14, which depicts many of the environmental processes to which chemical contaminants are subject. The aim has been to develop methods of calculating partitioning, transport, and transformation in the wide range of media that constitute our environment. Ultimately of primary concern to the public, and thus to regulators, is the effect that these chemicals may have on human well-being. But there are sound practical and ethical reasons for protecting wildlife, and indeed all fellow organisms in our ecosystem. It is not yet clear how severe the effects of chemical contaminants are, nor is it likely that the full picture will become clear for some decades. Undoubtedly, there are chemical surprises or “time bombs” in store as analytical methods and toxicology improve and new chemicals of concern are identified. Regardless of the incentive nurtured by public fear of “toxics,” environmental science has a quite independent and noble objective of seeking, for its own sake, a fuller quantitative understanding of how the biotic and abiotic components of our multimedia ecosystem operate; how chemicals that enter this system are transported, transformed, and accumulate; and how they eventually reach organisms and affect their well-being. Modern society now depends on a wide variety of chemicals for producing materials, as components of fuels, for maintaining food production, for ensuring sanitary conditions and reducing the incidence of disease, for use in domestic and personal care products, and for use in medical and veterinary therapeutic drugs. We enjoy enormous benefits from these chemicals. Our industrial, municipal, and domestic activities also generate chemicals inadvertently by processes such as incineration and waste treatment. The challenge is to use chemicals wisely and prudently by reducing emissions or discharges to a level at which there is assurance that there are no adverse effects on the quality of life from chemicals, singly or in combination. It is hoped that the tools developed in these chapters can contribute to this process.

Figure 8.14 An illustration of a chemical’s sources, environmental fate, human exposure, and human pharmacokinetics.

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Index* A

bioaccumulation/bioconcentration 35, 59, 213, 229 biochemical oxygen demand 136 BIODEG 137 biodegradation 135 BIOLOG 137 biomagnification 217 bioturbation 156 biphenyls 50, 51 bulk Z values 107 burial from sediments 61, 199

absorption 94 adsorption 90 acids 51, 81 activity coefficients 80 advection 117, 123, 119 aerobic conditions 136 aerosol-air partition coefficient 97 activity 71 aerosols 57, 192 air changes per hour 222 air inhalation 227 air phase control 165 air-water exchange 191 Air-water model 194 air-water partitioning 49, 84, 86 alcohols 51 aldehydes 51 alkenes 50 anaerobic conditions 136 Antoine equation 76 aquatic biota 59 aquivalence 110 area factor 159 aromatics 50 arsenic compounds 53 atmosphere 55 atmospheric oxidation 139

C California EPA 190 CalTOX model 190 Canadian Environmental Modelling Centre 186 carcinogenicity 37 Carson, Rachel 1, 35, 51 characteristic times 27 ChemCAN model 190 chemical potential 71, 73 chemical properties 30, 40 chromophore 138 Clapeyron-Clausius equation 76 closed systems 11 Colborn, Theo 31 colloid 199 co-metabolites 127 concentration units 7 conductivity 147 critical-point temperature 76

B Beer-Lambert Law 138 * Specific mass balance models are in bold italic.

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D degradation 44 degradation products 117 density (units) 7 departure from equilibrium 147 deposition from atmosphere 57, 193 deposition from water 60, 199 deposition (dry) 58, 192 deposition (velocity) 60, 193 deposition (wet) 58, 192 dermal contact 227 diffusion 148, 150 diffusion in porous media 159 diffusive processes 25, 146 diffusivity (molecular) 146, 148, 153 diffusivity (eddy) 153 digestive processes 216 dimensions 9 dispersed phases 106 dissociation constant 41, 82 dissolved organic carbon 59 DDT 51 DNAPLs 64 dose 228 downloading models 186 drinking water 227 driving force 147 D values 122, 152, 189 E eddy diffusion 153, 154 egestion 214 enthalpy of phase transfer 86 entropy 72 enzymes 136 equilibrium 22, 69, 70, 73 error function 158 esters 52 estimation methods 30 ethers EQC model 190 Eulerian coordinates 20 evaluative environments 27 evaporation 153, 222

exposure 226, 233 exposure factors 227 exposure rates 227 F Fick’s laws 150, 156 fish 59, 229 Fish model 219 fish-water partition coefficients 96 fluorinated substances 51 fluxes 26 foliage-air partition coefficients 99 food 214, 227 food chains and webs 219, 220, 223, 229 Food-web model 229 force (units) 7 fugacity 2, 71, 73 fugacity capacity 78 fugacity forms 114, 115, 142, 235 fugacity ratio 75, 77 fulvic acids 59 G gas constant 8 gastrointestinal tract 217 Gaussian dispersion 156 Generic model 190 genotoxicity 37 gills 214 Globo-POP model 186, 230 groundwater 64 growth dilution 214 H half-time 26, 34 HCH 230 heat capacity analogy 103 Henderson-Hasselbalch 82 Henry’s law 80, 84, 86 humic acids 59 hydrocarbons 50 hydrolysis 137

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INDEX

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hydrophobicity 40, 88 hydrosphere 58 hydroxyl radicals 139 I indoor 233 indoor conditions 233 indoor air models 221 INPEST model 221 interfaces 90 interfacial fugacity 166 intermedia transport 145 ions 81 irreversible process 72 J Junge-Pankow equation 97 K ketones 51 L Lagrangian coordinates 20 Lake Ontario 203, 205 lakes 201 Lavoisier 11 leaching 196 Level I models 111, 113, 189, 190 Level II models 121, 128, 130, 141, 142, 143, 190 Level III models 175, 178, 182, 190 Level IV models 181 Lewis G.N. 73 linear additivity 180 lipids 96 lipid-water partition coefficients 96 lipophilicity 40 long-range transport 37, 229 M macropores 63 mass balances 11

mass transfer coefficients 151, 163 maximum fugacity 108 metabolism 214 metals 53 Michaelis Menten equation 127, 136, 224 Millington Quirk equation 160, 197 mineral matter-water partition coefficients 96 miscible 81 mixing ratio 7 models (downloading) 186 N NAPLs 64 nitrogen compounds 52 non-diffusive processes 25, 145 O Occam 11 occupational 233 octanol-air partition coefficient 49, 84, 88 octanol-water partition coefficients 44, 49, 84, 87 open systems 13 organic carbon-water partition coefficients 94 organic matter and carbon 59 Our Stolen Future 31 oxidation 140 oxides of nitrogen 139 ozone 139 P PAHs 50 Paracelsus 34 parallel D values 173 parallel resistances 174 particulate matter 58 partitioning phenomena 69 partition coefficients 70, 83, 85

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PBPK models 224 PCBs 51, 203, 205, 222 PEC/PNEC 226 persistence 26, 34, 41, 118, 132, 229 pharmacokinetics 233 pharmacokinetic models 224 pharmaceuticals 53 phenols 51, 81 phosphorus compounds 52 photolysis 137 photons 138 photosensitizing 139 phthalate esters 52 plant models 223 POPCYCLING-BALTIC model 186 porous media 159 POTWs 220 precautionary principle 39 prefices 8 pressure (units) 7 priority chemicals 31, 32 P-T diagram 75 pure substances (properties) 75 Q quantity 34 quantitative structure property relationships 91 quantum yield 138 quotient (risk) 226 QWASI models 201 QWASI multi-segment models 210 R rain 58, 192 Raoult’s law 80 Rate constant model 202 rate expressions 125 reactions 117, 125 reduction 140 reference fugacity 79 residence time 26, 34, 118, 132, 208 resistance 147

response time 135 resuspension from sediments 61, 199 retardation factor 64 reversible processes 72 risk quotient 226 rivers 61, 208 roll cells 154 run off 64 S scavenging 58 Schmidt number 153, 176 second-order reactions 126 sediments 60 sediment burial 124 Sediment model 201 sediment-water exchange 199 semipermeable membrane devices 82 sensitivity 187 series resistances 174 series D values 173 sewage treatment plants (STPs) 220 Silent Spring 1, 35, 51 snow 58 soil 63 Soil-Fug model 195 soil ingestion 227 Soil model 194 solubility 44 solubility in water 88 solubility in octanol 89 solubility of a substance in itself 89 solutes of negligible volatility 109 solution in the gas phase 78 solutions of ionizing substances 81 solution in liquid phases 79 solution in solid phases 82 steady-state conditions 11, 13, 22 Stokes-Einstein equation 160 STP model 220 stratosphere 124 Streeter-Phelps 210 sulfur compounds 52 supercooled liquid vapor pressure 75 superposition 180

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INDEX

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T TaPL3 model 230 TER (toxicity/exposure ratio) 226 teratogenicity 37 three solubility approach 85, 86 time (units) 7 tortuosity factor 159 toxicity 36, 229 toxic substances 1 transport 25 transport D values 167 transport parameters 178 triple-point temperature 76 TSP (total suspended particulates) 57, 192 turbulence 153 two-resistance concept 161, 162, 216 U UNEP 38 units 5

unsteady-state conditions 16, 133 unsteady-state diffusion 156 V vapor pressure 44, 75, 76 vegetation 64 volatile organic chemicals 40 volatilization 170, 196 W washout ratio 58, 193 water 58 water phase control 165 wet particle deposition 192 Whitman two resistance concept 161 Z zero-order expressions 127 Z values 112, 188

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