Environ. Sci. Technol. 1997, 31, 3511-3515
Validity of Global Lifetime Estimates by a Simple General Limiting Law for the Decay of Organic Compounds with Long-Range Pollution Potential U. MU ¨ LLER-HEROLD,* D. CADERAS, AND P. FUNCK Department of Environmental Sciences, ETH-Zentrum, Swiss Federal Institute of Technology, CH-8092 Zu ¨ rich, Switzerland
In order to assess the persistence of organic compounds with global pollution potential on the basis of a minimal set of compound-specific parameters, Mu¨ ller-Herold has proposed a simple general law giving a lower bound τ(∞) for the overall lifetime τ of a chemical (Environ. Sci. Technol. 1996, 30, 586-591). τ(∞) is the limiting value of τ in case of infinitely rapid exchange between the compartments of the environment. This paper investigates the accuracy of τ(∞) as an estimate for the overall lifetime τ if degradation is much slower than transport. First, a criterion is given for degradation to be slower than transport. Then τ and τ(∞) are calculated from a three-box unit-world model for a group of 29 relevant chemicals. It turns out that, for the chemicals considered, (i) τ(∞) is indeed a perfect estimate if degradation is slower than transport, i.e., under the conditions the method has been designed for; (ii) τ(∞) should not be used as an approximation if the pollutant is short-lived and its geographical range is only local; (iii) in all other cases, τ(∞) has at least the same order of magnitude as τ.
Introduction Lifetimes and spatial ranges are the minimal set of quantities for an exposure-based assessment of xenobiotic chemicals (1, 2). In order to estimate these quantities in a transparent and reliable way, methods are needed that are at the same time simple, general, and theoretically well-founded. In a recently published paper, Mu ¨ller-Herold has derived a general limiting law for the overall decay of persistent organic compounds (3). It was rigorously shown that the overall degradation rate constant λmin of a chemical is limited by a lower bound, which is the degradation rate constant kmin in the compartment where degradation is slowest, and an upper bound λmin(∞), which is the weighted average of the degradation rate constants ki of all the compartments (see eq 1). The weights are the volumes Vi of the compartments multiplied by the equilibrium concentrations ci eq that would occur in the compartments if there were only transport and no degradation:
λmin(∞) )
V1k1c1eq + V2k2c2eq + V3k3c3eq + ... V1c1eq + V2c2eq + V3c3eq + ...
(1)
λmin(∞) approaches λmin, in the sense of a general limiting law, if degradation is much slower than transport. Furthermore, * Author to whom correspondence should be addressed. Phone: +41-1-6324403; fax: +41-1-6331136; e-mail: mueller-herold@ umnw.ethz.ch.
S0013-936X(97)00257-5 CCC: $14.00
1997 American Chemical Society
FIGURE 1. Unit-world dynamics.
TABLE 1. Phase Volumes and Interfacial Areas in Relative Units,a According to Ref 8 phase
volume
interface
area
soil water air
Vs ) 1 m 3 Vw ) 233 m3 Va ) 2 × 105 m3
air/soil air/water
Ags ) 10 m2 Agw ) 23.3 m2
a The numerical value of the soil/water interface is not required for the model calculations.
TABLE 2. Transport Parameters, According to Refs 9 and 10 phase water air over water air over soil air in soil water in soil
transfer velocity
uw ) 0.03 m h-1 ua1) 3 m h-1 ua2 ) 1 m h-1 us1 ) 6.7 × 10-3 m h-1 us2 ) 2.6 × 10-6 m h-1
it can be seen that the limiting rate constant λmin(∞) is identical with the overall rate constant obtained under the assumption of instantaneous equilibrium between the compartments. As a first illustration, the result was applied to DDT and hexachloroethane. For practical applications, it is an important question as to how much faster than degradation transport has to be for the limiting rate constant λmin(∞) to be a good estimate for λmin, rather than a mere upper bound. Degradation being much slower than transport means that degradation and transport occur on different time scales. In dynamics with different time scales, it is usually possible to decouple the dynamics of the respective dynamical layers and to obtain closed (i.e., autonomous) equations for the lower-level dynamics alone. Experience from various fields suggest that asymptotic decoupling gives good results if the time scales differ at least by 1 or 2 orders of magnitude. A well-understood example is the decoupling of molecular dynamics into electronic, vibrational, and rotational degrees of freedom. In the present study, the limiting law is applied to a broader group of environmental chemicals in order to clarify the range of its validity as an estimate (and not only as a bound). To this end, overall degradation rate constants λmin calculated
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TABLE 3. Rate Constants and Partition Coefficients, Following Ref 5a compound
ks (s-1)
kw (s-1)
ka (s-1)
log Kow
KH (Pa m3 mol-1)
1 CCl3F 2 CH3CClF2 3 Carbon tetrachloride 4 tetrachloroethylene 5 1-butanol 6 dioxane 7 cyclohexane 8 benzene 9 pyridine 10 acetone 11 formaldehyde 12 aldrin 13 chlordane 14 dieldrin 15 endosulfan 16 hexachlorobenzene 17 lindane 18 mirex 19 hexachlorobiphenyl 20 chlorobenzene 21 1,4-dichlorobenzene 22 1,2,4-trichlorobenzene 23 toluene 24 p-cresol 25 nitrobenzene 26 malathion 27 methyl-parathion 28 disulfoton 29 atrazine
2.93 × 10-8 2.93 × 10-8 2.93 × 10-8 2.93 × 10-8 2.00 × 10-6 7.71 × 10-8 7.71 × 10-8 7.64 × 10-7 2.00 × 10-6 2.00 × 10-6 2.00 × 10-6 2.62 × 10-8 9.88 × 10-9 1.27 × 10-8 1.74 × 10-6 5.24 × 10-9 6.32 × 10-8 1.80 × 10-9 4.17 × 10-9 7.36 × 10-8 7.71 × 10-8 7.71 × 10-8 6.17 × 10-7 2.27 × 10-5 7.63 × 10-8 1.61 × 10-6 4.34 × 10-8 6.69 × 10-7 6.06 × 10-8
2.93 × 10-8 2.93 × 10-8 2.93 × 10-8 2.93 × 10-8 2.00 × 10-6 7.71 × 10-8 7.71 × 10-8 7.64 × 10-7 2.00 × 10-6 2.00 × 10-6 2.00 × 10-6 2.62 × 10-8 9.88 × 10-9 1.27 × 10-8 1.74 × 10-6 5.24 × 10-9 6.32 × 10-8 5.36 × 10-8 4.17 × 10-9 7.36 × 10-8 7.71 × 10-8 7.71 × 10-8 6.17 × 10-7 2.27 × 10-5 7.63 × 10-8 2.88 × 10-7 3.48 × 10-7 6.69 × 10-7 1.07 × 10-6
2.72 × 10-10 1.40 × 10-9 2.19 × 10-9 9.11 × 10-8 3.99 × 10-6 4.32 × 10-6 4.03 × 10-6 6.99 × 10-7 2.73 × 10-7 1.25 × 10-7 5.31 × 10-5 3.85 × 10-5 6.77 × 10-6 8.66 × 10-6 1.41 × 10-5 9.33 × 10-9 3.78 × 10-6 1.76 × 10-10 1.16 × 10-7 4.80 × 10-7 1.75 × 10-7 2.73 × 10-8 3.38 × 10-6 2.33 × 10-5 6.44 × 10-5 3.56 × 10-5 3.34 × 10-5 7.27 × 10-5 1.01 × 10-5
2.53 1.60 2.83 3.40 0.88 -0.27 3.44 2.13 0.65 -0.24 0.35 6.50 5.54 4.32 3.83 5.31 3.61 6.89 6.80 2.84 3.52 4.02 2.73 1.94 1.85 2.36 2.86 4.02 2.75
9.83 × 103 2.42 × 104 3.08 × 103 1.51 × 103 5.64 × 10-1 4.94 × 10-1 1.95 × 104 5.50 × 102 7.09 × 102 3.72 × 100 3.31 × 10-2 5.03 × 101 4.91 5.88 1.13 1.32 × 102 2.96 × 10-1 7.10 × 101 3.00 × 101 3.49 × 102 1.52 × 102 1.44 × 102 6.01 × 102 9.79 × 10-2 2.47 × 100 2.03 × 10-3 1.01 × 10-2 4.04 × 10-1 2.66 × 10-4
a Most of the material is taken from refs 12 and 13. Exceptions: k of 2, ref 14; k of 18, ref 15; k of 18, ref 16; log K a s w ow of 18, ref 9; 19, values according to refs 9 and 17; degradation rate constants of 29, ref 14.
TABLE 4. Quality of the Limiting Lifetime τ(∞) as an Estimate for the Overall Lifetime τ a compound
log[δ2/kmax]
log [min(dij + dji)/kmax]
τ (days)
τ(∞)/τ
1 CCl3F 2 CH3CClF2 3 carbon tetrachloride 4 tetrachloroethylene 5 1-butanol 6 dioxane 7 cyclohexane 8 benzene 9 pyridine 10 acetone 11 formaldehyde 12 aldrin 13 chlordane 14 dieldrin 15 endosulfan 16 hexachlorobenzene 17 lindane 18 mirex 19 hexachlorobiphenyl 20 chlorbenzene 21 1,4-dichlorobenzene 22 1,2,4-trichlorobenze 23 toluene 24 p-cresol 25 nitrobenzene 26 malathion 27 methyl-parathion 28 disulfoton 29 atrazine
0. 688 0. 688 0. 689 1. 29 -0. 646 -0. 630 -1. 46 47. 9 -2. 54 -0. 985 -0. 365 -0. 179 -0. 204 -0. 229 -0. 216 -0. 879 -0. 245 -0. 365 -0. 322 4. 22 -1. 61 -0. 749 -1. 60 -0. 652 -0. 316 -0. 216 -0. 214 -0. 176 -0. 190
0. 688 0. 688 0. 689 1. 29 -0. 647 -0. 630 -1. 46 47. 9 -2. 54 -0. 985 -0. 366 -0. 180 -0. 211 -0. 231 -0. 222 -0. 880 -0. 266 -0. 367 -0. 326 4. 22 -1. 61 -0. 749 -1. 60 -0. 652 -0. 316 -0. 286 -0. 268 -0. 191 -0. 348
28600 5720 3620 88.2 3.45 11.5 1.99 11.5 28.7 8.50 3.11 1.54 10.3 1.71 1.70 921 21.7 4360 877 16.7 46.4 30.0 2.38 0.344 0.270 24.8 17.7 1.09 7.63
0. 9990 0. 9999 0. 9996 0. 9999 0. 8677 0. 1334 0. 2257 0. 9999 0. 9830 0. 2167 0. 7757 0. 0050 0. 0127 0. 0028 0. 3694 0. 6676 0. 1719 0. 9820 0. 4562 0. 9993 0. 8727 0. 3347 0. 4205 0. 9975 0. 0040 0. 8913 0. 0975 0. 0910 0. 0577
i*j
a τ(∞) and τ are calculated from a three-box model. The closer τ(∞)/τ approaches unity, the better τ(∞) approximates τ. In columns 1 and 2, the numerical values of criterion eq 2 and eq 2′ for the comparison of degradation rates and transport velocities are listed. If the log is positive, transport is said to be faster than degradation.
from a three-box unit-world-type model are tested against limiting rate constants λmin(∞) of the same model. To make the comparison more transparent, λmin and λmin(∞) are then converted into the corresponding lifetimes. The paper is
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organized as follows: The first section states the limiting law in the context of N-box models. In the next section, the threebox unit-world model is specified. The third section presents the results for a selection of 29 relevant chemicals, and the
FIGURE 2. Quality of the limiting lifetime τ(∞) as an estimate for the overall lifetime τ, with τ(∞)/τ as a measure of accuracy (see Table 4). For comparison of degradation and transport, the ratio δ2/kmax of the slowest transport velocity and the largest degradation rate constant kmax is used (criterion eq 2). In agreement with theory, the estimate is perfect [i.e., τ(∞)/τ ≈ 1] if transport is faster the degradation: δ2 > kmax, i.e., if log[δ2/kmax] > 0 (substances 1-4, 8, and 20). fourth gives a simplified criterion for the comparison of degradation rates and transport velocities.
Global Limiting Lifetimes in a Class of N-Box Models Let c˙ ) -Lc be the dynamics of a linear N-box model, where c ) (c1, c2, ..., cN )T denotes the vector of the concentrations (given in mol L-1) of a pollutant in the respective compartments. The N × N matrix L (all entries of which are given in s-1) can be written as:
(
k1
L)
0 k2 ·· ·
kN
0
)
+D
where ki denotes the degradation rate constant of compartment i, and D is a matrix describing material exchange, which in the absence of degradation would lead to thermodynamic equilibrium. In technical terms, this is guaranteed by a detailed balance condition for D (see ref 3, p 588). Physically, this means that one-sided pathways of phase transfer such as wet and dry deposition or soil runoff are not considered here. By comparison of Table 4 with the work of Scheringer on the same model (4), it can be seen that, within the physical range of their magnitude, these mechanisms hardly affect the overall lifetimes. Within the model, the overall lifetime is defined by τ ) ln 2/λmin, with λmin being the smallest eigenvalue of L. The limiting lifetime of the model, in turn, is given by τ(∞) ) ln 2/λmin(∞), with N
k1 +
∑k K
j 1jVj/V1
j)2
λmin(∞) )
(1′)
N
1+
appropriately as wholes. This has to be done carefully. The rate constants ki are non-negative numbers and have the direct physical meaning of inverse lifetimes. The same, however, does not apply to the transport coefficients dij of D. Indeed, the relevant relaxation times with respect to transport are not the entries of D, but rather its eigenvalues, which are non-negative as well. If these eigenvalues are larger than all the degradation rate constants ki, it seems reasonable to say that transport is faster than degradation. There is one difficulty, however: due to mass conservation in transport, the smallest eigenvalue δ1 of D is always zero. The eigenvalue relevant for the comparison of decay and transport is the smallest non-zero eigenvalue δ2 of D. Transport, accordingly, is said to be faster than degradation if the smallest non-zero eigenvalue δ2 is larger than the largest degradation rate constant, i.e., if
0 ) δ1 e k1 e k2 e ... e kN < δ2 e δ3 e ... e δN where the rate constants ki and the eigenvalues δj of D are ordered with respect to their magnitude.
Specification of the Model Parameters In order to test the range of validity of the limiting law as an estimate, we use a simplified three-box version (see Figure 1) of the more extended unit-world model recently discussed by Scheringer in this journal (5) and in his doctoral thesis (4). Let c ) (ca, cw, cs)T be the vector of concentrations in a unit world, where ca, cw, and cs denote the concentrations of a pollutant in air, water, and soil, respectively. If Va, Vw, and Vs are the volumes of the respective compartments and if ka, kw, and ks denote the corresponding degradation rate constants, the matrix L in the rate equation c˙ ) -Lc is given by
(
∑K
which is equivalent to eq 1. Here the Kij are the partition coefficients for thermodynamic equilibrium between compartment i and j without degradation (for details see ref 3). Since λmin(∞) is an upper bound for λmin, the limiting lifetime τ(∞) is a general lower bound for τ, i.e., τ(∞) e τ (for details, see ref 3). As there are several degradation rate constants and transport coefficients relevant for the system, one has to specify a method for comparing degradation and transport
)
ka 0 0 L ) 0 kw 0 + D 0 0 ks
1iVi/V1
i)2
with
(
daw + das Va daw Vw Va - das Vs
D) -
Vw dwa Va
dwa + dws V - wdws Vs
Vs dsa Va Vs - dsw Vw
-
dsa + dsw
(3)
)
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FIGURE 3. Quality of the limiting lifetime τ(∞) as an estimate for overall lifetime τ, with τ(∞)/τ as a measure of accuracy (see Table 4). Here a simpler criterion (eq 2′) is taken for comparison of degradation and transport: transport is said to be faster than degradation if the smallest sum of non-zero conjugate transport coefficients exceeds the largest degradation rate constant, i.e., if log [min{dij + dji}/kmax ] > 0. Since there is only one visible difference (substance 29), eq 2′ does as well as the more involved criterion eq 2 (see Figure 2). Following Scheringer (4, 5), the phase-transfer dynamics is modeled along the lines of Mackay and Paterson (6, 7). Restricting the model dynamics to diffusive phase transfer between soil and air as well as between water and air, one gets the following transport coefficients dij:
Va dwa ) daw Kaw Vw
daw )
Aaw uwua1 uw + ua1Kaw Va
(5a)
Va dsa ) das Kas Vs
das )
ua2(us1 + us2Kaw) Aas ua2 + us1us2Kaw Va
(5b)
dsw ) 0
dws ) 0
(5c)
where Kaw and Kas are the air/water and air/soil partition coefficients, respectively. The meaning and the numerical values of the other parameters appearing in eqs 5 can be found in Tables 1 and 2. The partition coefficients are obtained from Henry’s law constants KH (given in Pa m3 mol-1) and octanol/water partition coefficients Kow by
Kaw ) KH/RT Ksw ) focKoc ) foc × 0.41Kow The fraction foc of organic carbon in soil is set to 2%. The factor 0.41 converts the octanol/water partition coefficient Kow into the organic matter/water partition coefficient Koc (11). The values of these substance-related parameters are listed in Table 3.
Results The results are given in columns 1, 3, and 4 of Table 4 and in Figure 2. It can be seen that (i) The limiting lifetime τ(∞) is a perfect estimate for the overall lifetime τ if transport is faster than degradation, i.e., if log (δ2/kmax) g 0. This applies to a group of six substances (1-4, 8, and 20), and there is no exception. As all these substances have a geographical range of 10 000 km and moresif released from a point source (18)sthey represent the type of environmental pollutants with global pollution potential the limiting law was designed for (3). (ii) Even if transport is slower than degradation, i.e., if log δ2/kmax is negative, τ(∞) can fairly well approximate overall lifetimes, i.e., τ(∞)/τ ≈ 1, or give at least the correct order of magnitude, i.e., τ(∞)/τ g 0.1. This concerns 14 other substances (5-7, 9-11, 15-19, and 21-24);
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(iii) There are seven substances (12-14, 25, and 27-29) for which the limiting law does not give the overall lifetime within the correct order of magnitude, i.e., where τ(∞)/τ < 0.1. In the model environment of the three-box system, they all have lifetimes of 20 days or less. Furthermore, their geographical ranges are less than 2000 km (18). They are neither persistent, accordingly, nor do they have long-range pollution potential. Summing up, it can be said that eq 1′ gives reasonable or even exact lifetime estimates for all chemicals that are either persistent or have a long-range pollution potential. Within the group under consideration, the limiting law accordingly yields reliable results for all environmental chemicals except those giving rise to ecological concern predominantly on the basis of the excessive quantities released into the environment. For practical purposes τ(∞), being a lower bound, can always be used as a reliable, i.e., conservative estimate for the assessment of the environmental endangerment by a chemical through its longevity. As τ(∞) e τ, the real situation may possibly be worse. Since, in addition, this is achieved on the basis of only a few measured chemical quantities inserted into the simple analytical expression (1′), τ(∞) has the ideal properties of a simple and effective screening factor for an exposure-based assessment of new and existing chemicals.
A Simpler Method for Comparing Degradation Rates and Transport Velocities In practical cases, criterion eq 2 for the comparison of degradation rates and transport velocities is not particularly convenient: in fact, one has to know all the entries of D in order to calculate its second eigenvalue, which is just the same expenditure as calculating the lifetimes right away from L. For practitioners, it may appear more natural to compare decay constants and transport coefficients directly. In a twobox model analogous to eqs 3-5, this turns out to be possible since the second eigenvalue δ2 is given by
δ2) d12 + d21
(6)
In order to replace criterion eq 2, this observation is generalized to the following simpler requirement: Transport is said to be faster than decay if
min{dij + dji} > kmax i*j dij*0
(2′)
where the minimum is taken over all non-zero pairs of conjugate transport coefficients. Criterion eq 2′ is much easier
to apply than criterion eq 2 as it amounts to a direct comparison of transport coefficients and degradation rate constants, without any calculation of eigenvalues. The corresponding results are given in columns 2 and 4 of Table 4 and in Figure 3. It turns out that in the cases considered criterion eq 2′ does exactly as well as criterion eq 2. In fact, with the single exception of atrazine (substance 29), there are not even visible differences between Figure 2 and Figure 3. It suggests the perspective of deriving a general inequality
δ2 g min{dij + dji} > kmax i*j dij*0
(7)
for the second eigenvalue of the type of so-called Laplacian matrices that D represents. Such an inequality would be useful as a justification for the application of the more convenient criterion eq 2′ beyond the case of three-box systems.
Literature Cited (1) Scheringer, M.; Berg, M. Fresenius Environ. Bull. 1994, 3, 493498. (2) Mu ¨ ller-Herold, U. Geneva Pap. Risk Insurance 1996, 21, 383392. (3) Mu ¨ ller-Herold, U. Environ. Sci. Technol. 1996, 30, 586-591. (4) Scheringer, M. Environ. Sci. Technol. 1996, 30, 1652-1659. (5) Scheringer, M. Dissertation, ETH Zu ¨ rich, 1996. (6) Mackay, D.; Paterson, S. Environ. Sci. Technol. 1982, 16, 654A660A.
(7) Mackay, D.; Paterson, S.; Shiu, W. Y. Chemosphere 1992, 24, 695-717. (8) Klein, A. W. In Handbook of Environmental Chemistry; Hutzinger, O., Ed.; Springer: Berlin, 1985; pp 1-28. (9) Mackay, D.; Paterson, S. Environ. Sci. Technol. 1991, 25, 427436. (10) Jury, W. A.; Spencer, W. F.; Farmer, W. J. J. Environ. Qual. 1983, 12, 558-564. (11) Karickhoff, S. W. Chemosphere 1981, 10, 833-846. (12) Howard, P. H.; Boethling, R. S.; Jarvis, W. F.; Meylan, W. M.; Michalenko, E. M. Handbook of Environmental Degradation Rates; Lewis: Chelsea, MI, 1991. (13) Howard, P. H. Handbook of Environmental Fate and Exposure Data for Organic Chemicals. Vol. I (Large Production and Priority Pollutants), Vol. II (Solvents), Vol. III (Pesticides); Lewis: Chelsea, MI, 1991. (14) Nimitz, J. S.; Skaggs, S. R. Environ. Sci. Technol. 1992, 26, 739744. (15) National Research Council. Kepone/Mirex/Hexachlorocyclopentadiene: An Environmental Assesssment; National Academy of Sciences: Washington, DC, 1978. (16) Mackay, D.; Paterson, S.; Cheung, B.; Neely, W. B. Chemosphere 1985, 14, 335-374. (17) Shiu, W. Y.; Mackay, D. J. Phys. Chem. Ref. Data 1986, 15, 911929. (18) Mu ¨ ller-Herold, U.; Nickel, G. A Simple Formula to Estimate the Isotropic Geographical Range of Environmental Chemicals. Presented at the Annual Meeting of the Society for Risk AnalysisEurope, Stockholm, Sweden, June 1997.
Received for review March 20, 1997. Revised manuscript received August 26, 1997. Accepted September 5, 1997.X ES9702576 X
Abstract published in Advance ACS Abstracts, October 15, 1997.
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