Environ. Sci. Technol. 1999, 33, 503-509
General Formulation of Characteristic Time for Persistent Chemicals in a Multimedia Environment D E B O R A H H . B E N N E T T , †,‡ W I L L I A M E . K A S T E N B E R G , * ,§ A N D T H O M A S E . M C K O N E ‡,⊥ Department of Mechanical Engineering, University of California at Berkeley, Berkeley, California 94720, Environmental Energy Technologies Division, Lawrence Berkeley Laboratory, Berkeley, California 94720, Department of Nuclear Engineering, University of California at Berkeley, Berkeley, California 94720-1730, and School of Public Health, University of California, Berkeley, California 94720
A simple yet representative method for determining the characteristic time a persistent organic pollutant remains in a multimedia environment is presented. The characteristic time is an important attribute for assessing long-term health and ecological impacts of a chemical. Calculating the characteristic time requires information on decay rates in multiple environmental media as well as the proportion of mass in each environmental medium. We explore the premise that using a steady-state distribution of the mass in the environment provides a means to calculate a representative estimate of the characteristic time while maintaining a simple formulation. Calculating the steadystate mass distribution incorporates the effect of advective transport and nonequilibrium effects resulting from the source terms. Using several chemicals, we calculate and compare the characteristic time in a representative multimedia environment for dynamic, steady-state, and equilibrium multimedia models, and also for a single medium model. We demonstrate that formulating the characteristic time based on the steady-state mass distribution in the environment closely approximates the dynamic characteristic time for a range of chemicals and thus can be used in decisions regarding chemical use in the environment.
Introduction Persistence is an important attribute for determining the overall human health and ecological impact of a chemical release to the environment. Since the late 1970s, investigators have looked at different measures of persistence with varying levels of complexity (1-8). Persistent organic pollutants originate from a range of activities including combustion for energy production and transportation, industrial processes, and agriculture. Persistent pollutants pose a greater potential * Corresponding author phone: (510)643-0574; fax: (510)6439685; e-mail:
[email protected]. † Department of Mechanical Engineering, University of California at Berkeley. ‡ Lawrence Berkeley Laboratory. § Department of Nuclear Engineering, University of California at Berkeley. ⊥ School of Public Health, University of California. 10.1021/es980556a CCC: $18.00 Published on Web 12/16/1998
1999 American Chemical Society
concern per unit release because they cannot be rapidly removed from the environment if adverse health or ecological effects are later discovered. A methodology to determine a measure of persistence in the environment for chemical pollutants is needed to classify the chemical as one with short-term or long-term impacts. From a policy perspective, this method needs to be straightforward to calculate, yet be representative of the complex dynamic environment. Potential applications for this measure include risk assessment, pollution prevention assessment, health effects studies, pollutant mass balance studies, life cycle analyses, and sustainability and regulatory impact studies. An appropriate measure of persistence is the characteristic time a chemical remains in the environment. The characteristic time can theoretically be determined by finding the overall decay rate of a pollutant in a closed, defined landscape system. Determining the characteristic time in the environment requires knowing both the mass distribution among environmental media and the media-specific half-lives. Both sets of information are needed because the decay rates in each environmental media can differ significantly for a given pollutant. Recently, Mu ¨ ller-Herold (1, 2) proposed a limiting law bounding the multimedia decay rate between the slowest media specific decay rate and the decay rate based on equilibrium partitioning of mass in the environment. The equilibrium limit was derived from the eigenvalues of a simplified transfer matrix (the source-free or homogeneous solution) for a two-compartment model (1, 2). Mu ¨ller-Herold points out that the equilibrium partitioning limit is only a good approximation of the actual decay rate when decay rates are orders of magnitude slower than the transfer rates (1, 2). Hence, this approach is not applicable for an important set of environmental conditions and a subset of compounds potentially released to the environment. Scheringer (3, 4) has developed an appropriate representation of the characteristic time for pulse inputs. While the calculation methods are concise, the approach uses multiple spatial boxes and time steps, requiring a numerical simulator to complete the calculation. In this paper, we derive an exact analytical solution for pollutant concentrations in a dynamic two-compartment system with a distributed source (the nonhomogeneous case). From this solution, we derive three alternate formulations for the characteristic time: equilibrium; source weighted (the chemical does not transfer out of the compartment to which it was released); and steady-state (a continuous source term in at least one compartment that can cause a displacement from equilibrium where the concentrations do not change with time (9)). Last, we compare the results using these three approaches with the exact solution over a range of input values. We complete a case study, considering transformations in air, water, and soil as well as diffusive and advective transport between compartments. We propose that the characteristic time calculated from the steady-state mass distribution is a reliable approximation of the characteristic time in a dynamic system. It is more representative than an equilibrium model because it accounts for the shift from equilibrium resulting from the source and advective phase-transfer processes (active transport processes such as rainfall, wet and dry particle deposition, and runoff may transfer mass from a region of lower fugacity, to a region of higher fugacity). Moreover, the steady-state approach retains sufficient simplicity to complete calculations in a tractable form, such as a spreadsheet, which is VOL. 33, NO. 3, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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solution. The transient overall decay rate and characteristic time for the system illustrated in Figure 1a are
kT(t) )
N1(t)k1 + N2(t)k2 N1(t) + N2(t)
τT(t) ) 1/kT(t)
(4a) (4b)
The time-dependent masses are derived from the following rate equations and initial conditions:
FIGURE 1. (a) Diagram of the two-compartment system used to calculate characteristic time and (b) the configuration of the evaluation unit used in the case studies. useful if the output of the analysis is to be utilized as a factor in decision making or subjected to an uncertainty analysis.
Methods We examine a closed two-compartment system, illustrated in Figure 1a, and relate the characteristic time and decay rates by deriving the most general characteristic time formulation (i.e., time dependent). We then generalize to the multicompartment system illustrated in Figure 1b. Calculating the Characteristic Time for a Two-Compartment Model. The instantaneous mean life or average life expectancy of a molecule in an environmental compartment, τ, is defined as the inverse of the decay rate in the compartment (10). We will refer to this as the characteristic time:
τ ) 1/k
(1)
where k is the decay rate, representing radioactive decay or chemical reactions that irreversibly remove the chemical from the system. In a two-compartment system, the effective decay rate is mass averaged between the two compartments, leading to the following instantaneous overall decay rate:
koverall )
N1k1 + N2k2 N1 + N2
(2)
where Ni is the mass in compartment i (kg). Having defined the overall decay rate, we define the overall instantaneous characteristic time for the entire closed system as
τoverall ) 1/koverall
(3)
Since we can derive an analytical solution for N1(t) and N2(t) (mass in compartments 1 and 2 as a function of time), we define an “instantaneous overall decay rate” and corresponding “characteristic time”, based on a fully dynamic 504
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dN1/dt ) -(k1 + T12)N1 + T21N2 + S1
(5a)
dN2/dt ) T12N1 - (k2 + T21)N2 + S2
(5b)
N1(0) ) N10
(5c)
N2(0) ) N20
(5d)
where Tij is the mass-based transfer factor from i to j (day-1), ki is the decay rate in compartments i (day-1), and Si is the source term to compartments i (kg/day). Equations 5a and 5b can be written in matrix form and an analytical solution can be obtained for a constant source by determining the eigenvalues, eigenvectors, adjoint vectors, normalized modal matrix, and inverse modal matrix for the system. These intermediate steps and results can be found in the Supporting Information. The final time dependent solution can be written as
N1(t) ) b1 + b2eλ1t + b3eλ2t
(6a)
N2(t) ) f1 + f2eλ1t + f3eλ2t
(6b)
where the constants (bi, fi) and eigenvalues (λi) are defined in the Supporting Information. These time dependent masses are used in eq 4a,b to define the most general formulation for the characteristic time. We make three simplifying assumptions to calculate koverall. First, we assume an equilibrium mass distribution in calculating the overall decay rate and characteristic time. This is the approach proposed as a limit by Mu¨ller-Herold (2), and in this case k and τ are defined as
N1,EQk1 + N2,EQk2 N1,EQ + N2,EQ
(7a)
τEQ ) 1/kEQ
(7b)
kEQ )
where N1,EQ and N2,EQ are the equilibrium masses in compartments 1 and 2 (the normalized equilibrium mass in a compartment is with Zi and Vi representing the fugacity capacity and volume in compartment i, respectively (9)). Second, we can base the overall decay rate and characteristic time on the steady-state mass distribution, defined as the steady-state decay rate and characteristic time:
kSS )
N1,SSk1 + N2,SSk2 N1,SS + N2,SS
(8a)
τSS ) 1/kSS
(8b)
where N1,SS and N2,SS are the steady-state masses in compartments 1 and 2, calculated by solving eqs 5a and 5b with the time derivatives equaling zero. In the absence of multimedia transfers, the chemical would remain in the compartment to which it was released. This third alternative is the extreme assumption of no transfer across boundaries and is representative of decay rates for
FIGURE 2. Source, equilibrium, transient, and steady-state characteristic times are plotted using: k1 ) 0.1 day-1, k2 ) 0.01 day-1, S1 ) 2 kg/day, S2 ) 1 kg/day, T12 ) T21 ) 0.1 day-1. The transient characteristic time initially equals the source characteristic time, and approaches the steady-state characteristic time. short times (short relative to the characteristic time). We define this alternative as the source weighted decay rate and characteristic time:
kST )
S1k1 + S2k2 S1 + S2
τST ) 1/kST
(9a) (9b)
Having defined four characteristic time expressions, we consider what occurs when a chemical is released into a chemical-free closed system. When mass is first released into a compartment (i.e., time equals zero), the initial instantaneous decay rate of the system is the decay rate of the pollutant in that compartment as all the mass is in that compartment. As time progresses, some mass is transported to the other compartment, and the system moves toward, and eventually reaches, steady state. Thus for short times, τT(t) is close to τST, and for longer times, it moves toward τSS. Hence these are the limits for τT(t). One example is shown in Figure 2. This particular example illustrates that τSS and τEQ are not necessarily equal. In this case, the decay and transport rates are on the same order of magnitude because there is a continuous source. The various characteristic times, τT(t), τEQ, τST, and τSS, were calculated for a broad range of decay and transport rates and are plotted in Figure 3, parts a and b. When one of the decay rates is on the same order of magnitude as the transport rates, τSS and τEQ can differ, as in case 1 of Figure 3a. As the transport rates become faster than the decay rates, as in cases 2 and 3, τSS approaches τEQ in the absence of advective transport processes. When decay is 1-2 orders of magnitude slower than transport, as in case 3, transport and decay can be decoupled, and the equilibrium limit is approached (1). When advective processes are considered, τSS does not approach τEQ, even if transport is much faster than decay. In Figure 3b, the transport increases relative to the decay rate in each subsequent case, with half the transport resulting from advective processes. Case 2 converges to case 3 in the limit that transport is faster than decay. However, because of the advective processes, τSS is notably different than τEQ. If the time to steady state is short compared to the characteristic time, the transient model converges to the steady-state model before a significant quantity of the chemical has decayed. Thus it is quite appropriate to use a steady-state model to approximate the characteristic time. If the chemical is not steadily released into the environment, but is released as a series of pulses, we need to consider when it is appropriate to use a steady-state model as opposed to a dynamic pulse input model. If the time between releases is less than the characteristic time in the environment, it is appropriate to use a steady-state assumption. If the time
FIGURE 3. (a) Source term, equilibrium, and all transient and steadystate characteristic times are plotted using k1 ) 0.1 day-1, k2 ) 0.01 day-1, S1 ) 4 kg/day, S2 ) 1 kg/day. Case specific transport factors are (slow) T12 ) T21 ) 0.01 day-1, (medium) T12 ) T21 ) 0.1 day-1, and (fast) T12 ) T21 ) 100 day-1. (b) Source, equilibrium, and all transient and steady-state characteristic times are plotted using k1 ) 0.1 day-1, k2 ) 0.01 day-1, S1 ) 4 kg/day, and S2 ) 1 kg/day. All transfer factors from compartment 1 to 2 include half advective flow and half diffusive flow. Flow from compartment 2 to 1 is diffusive. Transport values are (slow) T12 ) T21 ) 0.01 day-1, (medium) T12 ) T21 ) 1 day-1, and (fast) T12 ) T21 ) 100 day-1. between releases is longer than the characteristic time, one needs to determine if transport is faster than decay or vice versa. If transport is faster than decay, the dynamic, steadystate, and equilibrium models all yield the same result. If transport is slower than decay, a dynamic model is most appropriate. However, a steady-state approximation is more representative than an equilibrium approximation in this case. Case Study. To illustrate the use of characteristic time for persistent pollutants, characteristic times were calculated for lindane, dieldrin, hexachlorobenzene (HCB), carbon tetrachloride, cyclohexane, and dioxin using the four approaches outlined above. These chemicals were selected because they were, with the exception of dioxin, evaluated by both Scheringer and Mu ¨ ller-Herold (1-3), and thus their findings can be compared to ours. Lindane, dieldrin, and hexachlorobenzene (HCB) are all found in pesticides (HCB is often used as an inert ingredient) (11). Lindane and HCB are either injected into ground or sprayed on the ground or foliage (11). Dioxin was selected because it is primarily released to the air but strongly interacts with the soil. The evaluation unit presented in Figure 1b was used in the calculations and approximates the distribution of environmental media found across the earth (12). To examine the difference in the characteristic time as the source term is moved from the soil to the air, characteristic times for VOL. 33, NO. 3, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. Fugacity Capacities and Transfer Coefficients Used in Case Studies Air
Fugacity Capacity Zair ) 1/RT
Water
Zwater ) 1/KH
Soil
Zsoil ) foc0.41Kow/KH
air to watersdiffusive
Transfer Factors T diff aw ) uwua1/(uw + ua1Zair/Zwater)(Aaw/Va)
water to airsdiffusive
diff T diff wa ) T aw (Zair/Zwater)(Va/Vw)
air to soilsdiffusive
T diff as ) [ua2(us1 + us2Zwater/Zair)(Aas/Va)]/[(ua2 + us1 + us2Zwater/Zair)]
soil to airsdiffusive
diff T diff sa ) T as (Zair/Zsoil)(Va/Vs)
air to soilsrain washout
rain T rain (Zwater/Zair)(Aas/Va) as ) u
air to watersrain washout
rain T rain (Zwater/Zair)(Aaw/Va) aw ) u
air to soilsparticle deposition
dry T dep + uwet)(Aas/Va) as ) (u
air to watersparticle deposition
dry T dep + uwet)(Aaw/Va) aw ) (u
soil to water
Tsw ) (urunoff (Zwater/Zsoil) + urunoff )(Aas/Vs) w s
air to watersoverall
rain dep Taw ) (T dif aw + T aw )(1 - Φ) + T aw Φ
air to soilsoverall
rain dep Tas ) (T dif as + T as )(1 - Φ) + T as Φ
soil to air
Tsa ) 0.001T dep as Φ
TABLE 2. Parameter Values Assumed in Case Studies parameter
symbol
value
units
ref
transfer velocity (water) transfer velocity (air over water) transfer velocity (air over soil) transfer velocity (air in soil)a transfer velocity (water in soil)a dry deposition velocity wet deposition velocityb rain rate water runoff rate soil runoff rate volume soil volume water volume air interfacial area air/soil interfacial area air/water universal gas constant temperature fraction organic carbon
uw ua1 ua2 us1 us2 udry uwet urain urunoff w urunoff s Vs Vw Va Aas Aaw R T foc
0.72 72 24 0.16 6.2E-05 260 460 2.3E-3 9.4E-4 5.5E-7 1 233 2.0E+5 10 23.3 8.314 288 0.02
m/day m/day m/day m/day m/day m/day m/day m/day m/day m/day m3 m3 m3 m2 m2 Pa m3/mol K K unitless
13 13 13 4 4 13 4 13 13 13 scaled from 12 scaled from 12 scaled from 12 scaled from 12 scaled from 12
a Derived from diffusivities defined by Jury (17) for soil that is 30% water and 20% air by volume fraction using a diffusion path length in soil of 0.05 m as suggested by Mackay (13). b Based on rain rate (9.7 × 10-5 m/h) (13) times scavenging ratio (2 × 105) (18).
TABLE 3. Chemical Properties Assumed in Case Studies chemical name
ks (day-1)
kw (day-1)
ka (day-1)
Kow
KH (Pa m3 mol-1)
Φa
VP (Pa)
Tm (K)
lindane dieldrin hexachlorobenzene carbon tetrachloride cyclohexane dioxin
5.46E-3 1.10E-3 4.53E-4 2.53E-3 6.66E-3 9.50E-5
5.46E-3 1.10E-3 4.53E-4 2.53E-3 6.66E-3 1.61E-3
3.27E-1 7.48E-1 8.06E-4 1.89E-4 3.48E-1 6.93E-2
4.07E+3 2.09E+4 2.04E+5 6.76E+2 2.75E+3 5.70E+6
0.296 5.88 132 3080 19500 3.75
3.4E-4 1.2E-3 6.6E-5 6.9E-10 2.4E-9 2.1E-1
7.4E-3 5.0E-4 2.5E-3 1.5E+4 1.3E+4 9.4E-8
386 448 501 250 279 577
a Φ indicates the particle bound fraction in air and is estimated by the Junge equation (18), Φ ) (0.17 × SA)/[VP[exp[-6.81[1 - (T /T)]]] m + 0.17 × SA], where SA is the surface area of particles, SA ) 1.5 × 10-4 m2/m3.
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lindane and carbon tetrachloride were calculated with emissions to both soil and air. Defining the Model. We calculate the mass distribution in the evaluation unit with a fugacity model, a common approach for describing partitioning in multimedia systems (9, 13). The fugacity capacities (the chemical concentration per unit chemical fugacity) are defined for each compartment and are listed in Table 1. The chemical exchange is based on overall mass transfer coefficients, “T” values. The algorithms used in this model have been proposed by other researchers (4, 13) and are listed in Table 1. The values for the landscape properties are listed in Table 2. The dimensions and properties of the evaluation unit have been used by other researchers, including Scheringer and Mu ¨ ller-Herold (1, 4, 12). Representative chemical property values used in the calculations can be found in Table 3 (1, 3, 14). The steady-state concentrations in each environmental compartment are determined from the interactions between the three environmental compartments and the decay rate in each compartment. The equations needed to define the mass distribution can be found in the Supporting Information. Because an analytic solution to the three-compartment system is difficult to obtain, we approximated the dynamic system with the partially dynamic model in Figure 1b. In the cases we have presented, there are significant differences between the decay rates in soil and air while the decay rates in soil and water are similar. Therefore, the portion of mass in air has a controlling effect on the characteristic time, particularly if the decay rate in air is considerably faster than in soil. Thus, we can model air and soil in a dynamic relation with each other, each in a steady-state relationship with surface water for emissions to soil. The equations for determining the concentration in each compartment based on a steady input to soil can be found in the Supporting Information.
Results The source weighted, steady-state, transient, and equilibrium characteristic times were calculated and plotted in Figure 4, parts a-h. The pulse input, multicompartment characteristic time calculated by Scheringer and the values reported by Mu ¨ ller-Herold were also plotted (1-3). In all of the cases presented, the value of τSS lies between τST and τEQ. This occurs because under steady-state conditions, the source causes a shift from equilibrium. The pulse input model calculated by Scheringer generally results in a longer characteristic time value than the steady-state model. A larger proportion of the chemical decays in the soil in a pulse input model because all of the mass is initially in the soil and the soil generally has a slower decay rate. The steadystate solution is not expected to yield the same solution as the pulse input solution as the two systems are different both mathematically and physically. For lindane, τT(t) rapidly approaches τSS as seen in Figure 4a. The time it takes the system to reach steady state is short compared to τSS and thus we consider the steady-state model representative of the environment. A significant portion of the lindane remains in the soil in the steady state when the source term is to soil. Therefore, the net transfer rate between soil and air, resulting from both diffusive and advective processes, creates a constant displacement from equilibrium at steady state, resulting in a higher fugacity in soil than air. Forcing the system to an equilibrium state increases the mass in air by 10%, which causes a 75% increase in the portion of mass decayed in the air, leading to a significant change in the characteristic time. Figure 5a illustrates the importance of this slight shift in the mass fraction in air. If lindane is released to air, more of the chemical will be in the air than if it were released to soil. The mass distribution
is much closer to an equilibrium state and thus τSS is much closer to τEQ, as seen in Figure 4a. The τSS differs when the source is released to soil versus air because of the shift in the mass distribution from soil to air coupled with the difference in decay rates between these two mediums. This indicates that it is important to properly characterize the emissions. For dieldrin, plotted in Figure 4b, the difference between the half-life in air and in soil is greater than for lindane, thus magnifying the differences between τEQ and τSS. Equilibrium partitioning yields more dieldrin in the atmosphere than the steady-state mass distribution does. The characteristic times for HCB released to the soil are plotted in Figure 4c. For HCB, the steady-state and equilibrium results are similar because the decay rates in air and soil are comparable and thus the difference in the mass distribution is less influential. For HCB, the portion of mass in each compartment is closely related to the mass decayed in that compartment, as illustrated in Figure 5b. This should be viewed in contrast to the importance of the mass distribution on the portion decayed in each compartment for lindane, as seen in Figure 5a. The characteristic time for carbon tetrachloride was calculated for both emissions to soil and to air and the results are plotted in Figure 4d. The time to steady state is relatively long, thus it might be desirable to use the transient formulation of the problem. Carbon tetrachloride decays much more slowly in air than in soil. Almost all of the chemical will eventually be found in the air, regardless of whether the emission was to air or soil. Thus for carbon tetrachloride, it is less important to characterize the emissions than it was for lindane. Cyclohexane is rapidly transported from soil to air, where it undergoes rapid degredation. The transport rate exceeds the soil decay rate, and thus steady-state and equilibrium partitioning are similar and either can be used to calculate the characteristic time, as seen in Figure 4e. The characteristic times for steady dioxin emissions to air are plotted in Figure 4f. The air concentration at steady state is higher than at equilibrium, thus the steady-state characteristic time is less than the equilibrium characteristic time. A significant portion is decayed in air, consistent with the findings of the dynamic mass balance completed by Eisenberg et al. (15).
Discussion In any effort to assess the potential adverse health and environmental effects of a chemical, the characteristic time (representing a measure of persistence) is an important component of the analysis. The duration over which impacts are assessed should be on the same order of magnitude as the characteristic time for that chemical. Characteristic time together with spatial range (18) provides insight on the overall impact of a chemical in the environment. Several assumptions must be made in determining the characteristic time based on a single evaluation unit steadystate mass distribution. The characteristics of the evaluation unit are globally representative, thus it is assumed that either the spatial range is large enough or the use pattern regular enough to enable the chemical to be spread globally. Additionally, it is assumed all the properties can be spatially averaged. If the uncertainties associated with the characteristic time (based on the above assumptions) are smaller than the uncertainties associated with other physical quantities being used in a decision, the above assumptions may be quite appropriate. We demonstrated simple methods for making preliminary yet reliable estimates of the characteristic time. The steadystate mass distribution can be calculated analytically for any system and is useful if the output is to be used for environmental decision making. As we learn more about VOL. 33, NO. 3, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 4. (a) For lindane, τT(t), τST, and τSS for a source to soil, τST and τSS for a source to air, and τEQ are plotted with the values calculated by Scheringer and Mu1 ller-Herold. (b) For Dieldrin released to soil, τT(t), τST, τEQ, and τSS are plotted with the values calculated by Scheringer and Mu1 ller-Herold. (c) For HCB released to soil, τT(t), τST, τEQ, and τSS are plotted with the values calculated by Scheringer and Mu1 llerHerold. (d) For carbon tetrachloride, τT(t), τST, and τSS for a source to soil, τST and τSS for a source to air, and τEQ are plotted with the values calculated by Scheringer and Mu1 ller-Herold. (e) For cyclohexane released to soil, τT(t), τST, τEQ, and τSS are plotted with the values calculated by Scheringer and Mu1 ller-Herold. (f) For dioxin released to air τST, τEQ, and τSS are plotted.
FIGURE 5. The bars indicate both the percentage of mass in and decayed in the air compartment under equilibrium and steady-state conditions. Part a indicates the sensitivity to the proportion of mass in the air because of the rapid decay rate in air. Part b shows a contrast because in this case, the mass decayed is more dependent on the proportion of mass in the compartment. 508
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multimedia systems, different transport algorithms or additional compartments can be incorporated. We demonstrated over a broad range of chemicals that the steady-state characteristic time is a reliable approximation of the dynamics that occur for a steady or pulse input. This approach is more representative of actual environmental conditions than an equilibrium model over a wider range of chemical pollutants and environmental conditions. Thus, we recommend steady-state calculation methods when making decisions regarding chemical impacts in the environment.
Supporting Information Available Equations for the solution of the transient two-component model, steady-state unit world, and two box transient system (5 pages). Ordering information is given on any current masthead page.
Literature Cited (1) Mu ¨ ller-Herold, U.; Caderas, D.; Funck, P. Environ. Sci. Technol. 1997, 31, 3511-3515. (2) Mu ¨ ller-Herold, U. Environ. Sci. Technol. 1996, 30, 586-591. (3) Scheringer, M. Environ. Sci. Technol. 1997, 31, 2891-2897. (4) Scheringer, M. Environ. Sci. Technol. 1996, 30, 1652-1659. (5) Tremolada, P.; Diguardo, A.; Calamari, D.; Davoli, E.; Fanelli, R. Chemosphere 1992, 24, 1473-1491. (6) Ballschmiter, K. Environ. Carcinog. Ecotoxicol. Rev. 1991, C9, 1-46.
(7) Frische, R.; Esser, G.; Schonborn, W.; Klopffer, W. Ecotoxicol. Environ. Saf. 1982, 6, 283-293. (8) Korte, F. In The Evaluation of Toxicological Data for the Protection of Public Health; Hunter, W. J., Smeets, J., Eds.; Pergamon Press: Oxford, 1977; pp 235-246. (9) Mackay, D. Multimedia Environmental Models, the Fugacity Approach; Lewis Publishers: Chelsea, MI, 1991. (10) Lamarsh, J. R. Introduction to Nuclear Engineering, 2nd ed.; Addison-Wesley: Reading, MA, 1983. (11) USEPA. Pesticide Fact Handbook; USEPA: Park Ridge, NJ, 1988. (12) Klein, A. In Handbook of Environmental Chemistry; Hutzinger, O., Ed.; Springer: Berlin, 1985; pp 1-28. (13) Mackay, D.; Paterson, S. Environ. Sci. Technol. 1991, 25, 427436. (14) Mackay, D.; Shiu, W. Y.; Ma, K. C. Illustrated Handbook of Physical-Chemical Properties and Environmental Fate for Organic Chemicals; Lewis Publishers: Boca Raton, 1995. (15) Eisenberg, J. N. S.; Bennett, D. H.; McKone, T. E. Environ. Sci. Technol. 1998, 32, 115-123. (16) Bennett, D. H.; Matthies, M.; McKone, T. E.; Kastenberg, W. E. Environ. Sci. Technol. 1998, 32, 4023-4030. (17) Jury, W.; Spencer, W.; Farmer, W. J. Environ. Qual. 1983, 12, 558-564. (18) Bidleman, T. F. Environ. Sci. Technol. 1988, 22, 361-367.
Received for review June 1, 1998. Revised manuscript received October 13, 1998. Accepted October 28, 1998. ES980556A
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