ES&T FEATURE
Fugacity revisited The fugacity approach to environmental
Donald Mackay Sally Paterson Department of Chemical Engineering and Applied Chemistry University of Toronto Toronto, Ontario, Canada In previous articles we have described the fugacity approach to environmental modeling in general ( / ) and the simpler Level I and II calculations in more detail (2). Here we introduce the concepts of transfer between environmental compartments and unsteady state behavior as Level III and IV calculations to bring the model predictions a step closer to re-
ality. This feature article is a summary presentation of a more detailed report (3). These more advanced calculations yield more information about a chemical's likely "behavior profile" in the environment. In addition, we suggest methods of displaying these profiles pictorially rather than numerically. To recapitulate, fugacity is a thermodynamic quantity related to chemical potential or activity that characterizes the escaping tendency from a phase. At equilibrium, fugacities (which have units of pressure) are equal. The fugacity calculations can be applied to an actual environment of defined volumes but it is more convenient to use a hypothetical or evalua-
FIGURE 1
Tank analogy of levels I-IV fugacity calculation conditions =0
=g
HL·? pS^ psj Equilibrium steady-state no-flow system Level I
Equilibrium steady-state flow system Level II
Nonequilibrium steady-state flow system Level III
^4 t ^ t^i Before emissions
Approaching Level III Recovering after emissions steady state cease Nonequilibrium nonsteady-state flow system Level IV
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Environ. Sci. Technol., Vol. 16, No. 12, 1982
transport
tive environment or "unit world" consisting of volumes of homogeneous air, water, soil, bottom sediment, suspended sediment, and biota as discussed by Neely and Mackay (4). Each phase is assigned a reasonable volume and properties, and the distribution of the compound is then calculated from a knowledge of its properties to yield a behavior profile. The behavior and concentrations in the real environment will differ from those calculated, but the dominant behavior characteristics should be mimicked. In Level I, the equilibrium partitioning of a fixed amount of a nonreacting compound is calculated using fugacity capacities that are calculated from physical chemical data and partition coefficients. The equilibrium ratios of concentrations and the relative amounts in each phase can be calculated. In Level II, the steady-state equilibrium concentrations of the compound are calculated for fixed emissions that are balanced by reactions of various types in each phase. This provides information on the compound's overall persistence and what proportion reacts in each phase. Figure 1 illustrates these calculation approaches by analogy with water flow into tanks in which the level in the tank is equivalent to fugacity. Level I corresponds to a fixed amount of water (chemical). Level II corresponds to fixed inflow (emission) balanced by the sum of various leakages (reactions). In both cases connections between the tanks are sufficient to ensure equal height of water (interphase transport is rapid). Figure 2 illustrates the unit world and Table 1 shows the properties of a hypothetical organic chemical similar to naphthalene. Figure 3 gives the Level I and II data in pictorial form as
0013-936X/82/0916-0654A$01.25/0
© 1982 American Chemical Society
pie charts of mass distribution and reaction and as concentration on a log scale. Since the absolute concentra tions have no significance (unless the volumes and amounts of chemical are known), the scale numbers may be eliminated and the information con veyed by only relative position on the scale. The aim of the diagram is to in dicate where the substance goes, where it reacts, and what relative concen trations it adopts at a common fu gacity. The most severe limitation of these calculations is that they assume that the chemical achieves equilibrium between all phases. In practice, there are transfer resistances that limit the transfer from phase to phase and thus tend to "contain" the chemical. These resistances are analogous to valves between the tanks in Figure 1. In cluding valves or interphase transport resistances brings us to Level III nonequilibrium steady state, in which each phase may have a different fugacity. Level IV introduces the concept of inflows (emissions) that change with time, causing levels in the tanks to change, i.e., conditions are nonequilibrium, unsteady state. If emissions are constant, the levels will approach the Level III steady-state condition at long times. The task here is to provide a simple but physically realistic method of cal culating these interphase resistances. In principle, it is application to the environment of the two-resistance model concept originally devised by Whitman in 1923 for chemical pro cessing (5). This model successfully describes diffusive transport processes that are driven by fugacity or concentration differences, i.e., in a given phase, a chemical will diffuse from regions of high to low concentration or fugacity. The diffusion process is merely a manifestation of mixing, which tends to eliminate concentration gradients. Within one phase, concentration is an adequate descriptor of diffusion "driving force" (as in Fick's law) but between phases, concentration fails because a chemical may diffuse from low to high concentration across a phase boundary. The correct driving force is then fugacity since diffusion always proceeds from high to low fu gacity. It is usually assumed that the fugacities immediately adjacent to the phase boundary are equal in the two phases. A second method of interphase transport occurs when a quantity of material moves from one phase to an other, "piggy backing" chemical with it. Examples are deposition of SO2
from air in rainfall or dustfall and sediment deposition and resuspension. These material transport processes are not driven by fugacity differences and must be treated separately.
FIGURE 2
Unit w o r l d v o l u m e s (V m 3 ) a n d a r e a s (A m 2 )
Diffusive transport W e consider two compartments in which the solute has respective con centrations, fugacities, and fugacity capacities, C\ and C2,f\ a n d / 2 , and Z ] and Ζ2 separated by an area A m 2 through which the solute is diffusing at a constant rate TV defined as
Ν = Ζ>12(/Ί -f2)
Air V = 6.0 χ 10 9 A = 7.0 χ Soil V = 4.5 χ 10"
mol/h.
D is thus a transport coefficient with dimensions of m o l / P a h, which is in a very convenient form for fugacity calculations. The two-resistance model is merely the application of Fick's first law of diffusion to each phase in series; thus Ν = KtAACi =
=
105£jj
Water V = 7.0 χ 10 6
_Ju— ^~-Λ
Biota V = 7.0
Suspended solids V = 35.0
K2AAC2
KiAAd
Emissions > Transfer Reaction (mol/h)
CT>
10
0.942 ,1
Emissions
n.
(0 0
Soil f = 4.90 x 10~ 8 C = 6.42 χ 10" 1 m = .029 fc = .016
Water
Sediment-
f = 2.52 χ 1 0 " s C = 8.41 χ 10 ' m = 5 8 . 9 k = .016
•10" 5
Water -
τ
6.51 χ 1 0 " 5
lo.o Suspended Solids f = 2.52 χ 10~ 5 C = 6.61 x 10~ 4 m = .023 k = 0.0
ΙΟ"4 Suspended solids —Water Biota
Biota f = 1.76 χ 10~ 5 C = 9.30 χ 1 0 - " m = .0065 /r = .01
Air —Sediment 10"6 Soilr-10r
, 4 . 8 6 χ 10 Mass d i s t r i b u t i o n
— Soil
Sediment f » 1.77 X 10~ e C = 4 . 6 3 χ 10^ : m = .972 k = .005
10- 8
f = fugacity (Pa) Air C = concentration (mol/m 3 ) m = amount (mol) k = reaction rate constant (h~ 1 )
10"
L
Reaction d i s t r i b u t i o n
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Environ. Sci. Technol., Vol. 16, No. 12, 1982
Concentrations (mol/m 3 )
10" Fugacity (Pa)
ties differ as shown on the log scale. Comparison of Figures 3 and 5 shows clearly that since emissions are into water and there is a resistance to transfer from water, there are higher water concentrations and fugacities, and water degradation processes be come more significant. Biota and sus pended solids in intimate contact with water have fugacities almost identical to water. Level IV The Level IV model describes the response of the system to emissions that change with time. This model is useful in determining (/) the time re quired for a contaminant to accumu late to given concentrations in various phases after emissions start and con tinue at a defined level, and (ii) the time it will subsequently take for the system to "recover" to lower concen trations when emissions are discon tinued or reduced. An incremental change in fugacity is calculated for each compartment over a time increment At. Considering nonadvection conditions, the differ ential equation for compartment i is ViAQ/At
= Ej - ViQki - ZjDijtfi
-fj).
The terms that include Z),y may be positive or negative depending on the direction of the transport, which is determined by the relative values of/} and fj. Replacing the C terms by Zf and rearranging the equation gives the simple finite difference form Afi = (At/ViZi)
( £ , - VxZifikt - VjDijifi -fj))
Values of Af are calculated for all six compartments, and the new fuga cities are calculated as
f^f.
+ Af.
The tendency is for f to increase until the group {E — VjZj...) above reaches a zero value such that emis sions are exactly balanced by reaction and interphase transport. The values off which are then reached are iden tical to those obtained in the Level III steady-state calculation. Advection may be included by considering addi tional input in the form of inflow GJCBÎ and associated loss from outflow GjCj or GiZjfi. Adding the difference of these two terms to the equation gives: Af = ( Δ ± / Κ , Ζ , ) ( £ , + GiiCsi - Zift) - ViZJikt - VjDijU
-fj)).
Figure 6 illustrates the increase of fugacity with time for initial concen
TABLE 3
Level III equations and solution Basic equations Ει = fiVi - f 2 D 1 2 -
f3D13
£2 ~ f2V2 ~ /1Ο12 -~ /4D24 "~ feDzS "** Τβ&2β E3 = /3V3 - f t D 1 3
Ο = ΫV4 — f2D2t 0= fsVsf2D2S 0= fevB— f2D2& where v, = V-,k,Z, + D 1 2 + £>13
v2 = V2fc2Z2 + D 12 + D 24 + D 2 5 + D M v3 = v"3k3Z3 + D 1 3 V4 = V ^ f e A + D24
vs = V 5 k 5 Z 5 + θ26 Ve = VekeZ6+ D2% Solution h = (BDn + AE2)l[Av2 - D?2 - A(Df«/V4 + Df 6 /v 5 + £&/v 6 )] where Λ — v-t — D?3/ v3 and Β = Ei + £3013/ v3 f, = (f 2 D 12 + B)/A f3 = (E3 + fiD13)/va U = feD24/v4 fs = f2D2S/vs f$ = t2D26/v6
trations of zero. In this case, the system approaches the Level III concentra tions by 200-300 h. When the emis sions cease at 670 h, the system re covers rapidly—reaching 90% of the original zero concentrations by 1000 h. Discussion The Level III and IV calculations yield new information about the chemical's behavior and contribute to a more detailed behavior profile. Im portant interphase transport processes can be identified. The sensitivity of the results to changes in D values can be explored, and often it can be concluded that only a very approximate estimate of D is necessary. If D is large and transport fast, the phases may be close to equilibrium; thus the results are insensitive to the actual value of D. If D is small and transport slow, the process may be in significant and again the behavior is unaffected by moderate changes in D. There may be an intermediate range of D values in which the value is impor tant. Before devoting effort to measuring D, it is useful to have advance infor mation about how important the value is likely to be. Knowing which phase resistance dominates is also useful because there may be no need to know the other phase resistance. Further, the transport behavior of one compound can be deduced from another if both are controlled by the same resistance.
The most common case is the estima tion of volatilization rates of volatile organics from water using oxygen transfer correlations. The Level IV output gives inter esting insights into behavior. A phase of low accessibility, or high volume and low reactivity, experiences a rather slow concentration buildup; but, hav ing established that concentration, it may persist for a longer time than in other phases. In Figure 6, the sediment concentration increases slowly and recovers slowly; thus, viewing the unit world at different times results in quite different distributions. Perhaps models of this type can help to interpret envi ronmental observations and detect or provide warning of undesirable situa tions in which we realize too late that we have created a long-term contam ination problem. These calculations help to place environmental science in perspective. Estimation of Ζ values arises from equilibrium-phase partitioning mea surements and correlations. The study of environmental reactions yields k estimates. Diverse activities such as study and correlation of volatilization rates from water and soil, wet and dry deposition from the atmosphere, sedi ment-water exchange processes, up take and release kinetics from biota, and suspended matter all contribute to establishing reliable D values. Unfor tunately, there remains considerable doubt about the rates of many of these processes, notably those involving Environ. Sci. Technol., Vol. 16, No. 12, 1982
659A
tion, and transport processes that in teract to determine the exposure to which organisms are subjected. At least, such models can provide com parative exposure information about chemicals and assist priority setting. It is encouraging to find that this general approach is being followed as is described, for example, in the recent account by Schmidt-Bleek et al. (6) of Organisation for Economic Co-oper ation and Development procedures for environmental hazard assessment and ranking of new chemicals.
FIGURE 6
Level IV or unsteady-state behavior with emissions into water of 1 mol/h ceasing after 670 h 10" :
-
Water, suspended solids s^~~~~ Biota
. 10s ~
\
\
(/^
'a
j ο CO
if
10"
6
/
"
ΙΟ' 7 ~
Air ^^~
1 '/
Sediment
/ 10
D
Acknowledgment
\ \ \
Emissions cease 200
400
The authors gratefully acknowledge the financial support of Environment Canada and the scientific contribution of Dr. W. M. J. Strachan to this work. Before publication, this feature article was read and commented on for technical accuracy by W. Brock Neely, Environ mental Sciences Research Laboratory, Dow Chemical U.S.A., Midland, Mich. 48640 and Seong T. Hwang, U.S. Envi ronmental Protection Agency, Washing ton, D.C. 20460.
^
Soil
/
8
\
600
800
\ \ 1000
1200
Time (h)
Computer programs Computer programs are available from the authors upon request. References
sediments in which the assumption of a well-mixed volume is clearly inade quate. Model Validation Critics of models are rarely slow to demand evidence of validation, the assertion being that the results from an unvalidated model cannot be used with confidence. Some validation is possible and is, of course, highly desirable by conducting microcosm or pond ex periments. Instead of using a micro cosm directly as a regulatory tool for elucidating environmental behavior (as has been suggested), an appealing approach is to regard it as a device to validate or calibrate a model that is then used as the regulatory tool. A model that faithfully describes the microcosm behavior of a solute such as anthracene can probably be used (with qualifications) for other compounds of the same class. If environmental models can be de veloped for large-scale situations the output can be juxtaposed with envi ronmental monitoring data, inferences drawn about the level of agreement (or disagreement), and hence some vali dating (or invalidating) evidence pro vided. It is often forgotten that inval idation is as valuable in making sci entific progress as is validation. Vali dation using actual concentration data is difficult since there is a wide spec 660A
trum of concentrations in the envi ronment even in one phase. Perhaps environmental models will eventually yield spectra of concentrations with a mean and variance instead of a single value. Solutes of similar characteristics may yield similar variances. The reg ulator would then be able to make statements such as: "The mean con centration should be 10 but in 5% of the volume we expect values in excess of 50 and in 1% in excess of 80." But perhaps concentration match ing is not the best validating criterion. When molecules are discharged into the environment, they experience a cohort fate profile at various times, i.e., a running mass balance or budget. This budget actually exists, the prob lem being that its experimental deter mination is difficult. If environmental analyses could establish a budget that agrees well with the model budget then the problem of environmental fate prediction and validation would be largely solved, without comparing concentrations. The aim would be to match partitioning properties and residence times in specific environ ments, i.e., to match the cohort fates. The environmental behavior of chemicals is sufficiently complex that the human mind must be assisted by some form of systematic and physi cally reasonable mathematical model that describes the partitioning, reac
Environ. Sci. Technol., Vol. 16, No. 12, 1982
(1) Mackay, D. Environ. Sci. Technol. 1979, 13, 1218. (2) Mackay, D.; Paterson, S. Environ. Sci. Technol. 1981, 15, 1006. (3) Mackay, D.; Paterson, S. "Fugacity Models for Predicting the Environmental Behaviour of Chemicals," report prepared for Environ ment Canada, 1982. (4) Neely, W. B.; Mackay, D. "Evaluative Model for Estimating Environmental Fate" in "Modeling the Fate of Chemicals in the Aquatic Environment"; Dickson, K. L.; Maki, A. W.; Cairns, J., Eds.; Ann Arbor Science: Ann Arbor, Mich., 1982; Chapter 8, p. 127. (5) Whitman, W. G. Chem. Metal. Eng. 1923, 29, 146. (6) Schmidt-Bleek, F.; Haberland, W.; Klein, A. W.; Caroli, S. Chemsphere 1982, / / (4), 383.
Donald Mackay is α professor and Sally Paterson is α research associate in the Department of Chemical Engineering and Applied Chemistry at the University of Toronto. Their research interests are the environmental fate and effects of toxic substances, especially modeling, studies of volatilization of organic contaminants from water, and measurement of physical chemical properties.
