Environ. Sci. Technol. 1996, 30, 586-591
A Simple General Limiting Law for the Overall Decay of Organic Compounds with Global Pollution Potential U. MU ¨ LLER-HEROLD* Department of Environmental Sciences, Swiss Federal Institute of Technology, CH-8092 Zu ¨ rich, Switzerland
It is rigorously shown that the effective decay rate in the environment of a chemical is between the minimum decay rate in one of its possible compartments and an upper value, which is the weighted-average decay rates in all compartments. The weights are the compartments’ volumes and the equilibrium concentrations that would have occurred in the compartment due to transport alone, with no degradation. This upper value is approached, in the sense of a general limiting law, if degradation is much slower than transport. This limiting law, together with an estimate for the spatial range of a persistent chemical, could serve as a minimal base for exposure-based assessment of environmental risk. As a first illustration, the result is applied to DDT and hexachloroethane. A broader group of chemicals will be discussed elsewhere.
Introduction The half-life of organic pollutants in the environment is regarded as a key quantity in the assessment of ecological risks. This concerns persistent organic chemicals with low acute toxicity, since unlike degradable chemicals, they accumulate in the environment. If eventually serious noxious effects become evident, there is no way of removing these substances from the environment. Although this view has been advocated since the 1970s by scientists such as Stephenson (1), Korte (2), Klo¨pffer (3, 4), Tremolada (5), and Scheringer and Berg (6), the practical efforts of determining effective global decay rates have been hindered by serious difficulties. The difficulties come from the bewildering multitude of natural degradation mechanisms, from the complicated multiphase structure of the natural environment, and from the interplay of degradation and partitioning via multiple phase-transfer and mixing dynamics. The present paper tries to put this problem into a fresh perspective. In analogy to chemical thermodynamics, where rather complex systems are discussed on the basis of simple ideal limiting laws and successive refinements, * Present address: Laboratorium fu ¨ r Physikalische Chemie, ETHZentrum, CH-8092 Zu ¨ rich, Switzerland. Phone: +41-1-6324403; Fax: 41-1-6321021; e-mail address:
[email protected].
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it gives an exact general formula for the overall environmental decay rate in terms of local degradation rates and partitioning coefficients in the limit of rapid transport and phase exchange. This limiting case is close to the real situation with persistent chemicals where degradation is much slower than transport: On the global scale distribution mainly occurs by air and water transport. The atmospheric distribution in the troposphere happens on a time scale of less than 1 year. The global distribution via the oceans is 1-2 orders of magnitude slower, being on the order of decades. Globally persistent pollutants, on the other hand, can survive far longer. Transport of persistent chemicals with global pollution potential, such as DDT or the PCBs, in fact, is faster than degradation. The physical idea underlying the present treatment is an old one, and special forms of the result can be easily derived. For systems of two boxes, for example, the limiting behavior was derived by O’Connor (7), who determined analytical solutions to a system of two linear differential equationssby calculating the corresponding two eigenvaluessand then considered the case of rapid exchange (compare ref 7 Appendix I, eq 46). The analytical calculation of eigenvalues, however, is impossible for systems with many boxes. In these cases, it is sometimes possible to guess the smallest eigenvalue by stipulating instantaneous equilibrium between concentrations in the various boxes (compare ref 7, eqs 3, 4, and 10b). The present treatment, in contrast, is not restricted to low-dimensional systems and avoids ad-hoc approximations. It leads to the limiting law in its general form and gives a complete proof that the system really approaches the limiting behavior. It turns out that the law combines dynamic concepts of decay in the various phases with thermodynamic equilibrium properties of exchange between phases. On the one hand, this facilitates the application and avoids touchy involvement into the intricacies of environmental distribution kinetics. On the other hand, it makes the law a natural extension of wellestablished thermodynamical theories of partitioning such as the fugacity concept of Mackay (8-10) or related approaches. The necessary data for calculations may be available in standard data compilations of environmental chemicals (11-13). The result is derived for the case of linear dynamics of degradation and transport. In natural systems, a specific xenobiotic pollutant is generally present at a concentration that is sufficiently low not to affect the concentrations of its reaction partners in the environment in an essential way. This leads to a situation of general pseudo-first-order kinetics, i.e., of linear dynamics. The paper is organized as follows: The first section contains an explicit derivation of the limiting law in the most evident case of a two-box-model, and the second section gives the mathematical proof of in the general case. In third section, it is shown that the effective decay rate in real systems (with exchange of finite velocity) is between the minimum decay in one of the compartments and an upper value given by the limiting law. The next section presents hexachloroethane and DDT as examples of chemical interest, discussing the contributions of different factors to the final result and limits to the applicability of
0013-936X/96/0930-0586$12.00/0
1996 American Chemical Society
λ-(∞) )
k1/d1 + k2/d2 1/d1 + 1/d2
(6b)
λ-(∞) )
k1c1eq + k2c2eq c1eq + c2eq
(6c)
or
FIGURE 1. Structure of the two-box model showing exchange between two phases and associated degradation.
the law. The last section concludes with a short outlook on possible management use of the concept presented.
Introductory Example Let ci, i ) 1, 2, be the concentrations of a pollutant in a linear two-box model with corresponding effective decay constants ki and exchange parameters di (Figure 1). For boxes of equal volume, the dynamics is assumed to be given by
c˘ ) -d1c1 + d2c2 - k1c1 c˘ 2 ) d1c1 - d2c2 - k2c2
(1)
If K12 :) d1/d2 denotes the partition constant and if d :) d2 ) d1/K12 is the parameter controlling the speed of exchange between the boxes, the equations of motion are changed to c˘ ) -Lc with
L :)
(
(K12d + k1) -K12d
-d (d + k2)
)
(2)
The eigenvalues
q λ( ) + ( 2
x
q - k1k2 - k1d - k2K12d 4
(3)
are obtained as solutions of the characteristic equation
Det |L - λI| ) λ2 - λ{d + K12d + k1 + k2} + k1k2 + k1d + k2K12d ) 0 (4) with q ) d + K12d + k1 + k2. As the half-lives are related to the inverse of the smallest eigenvalue, one has to look at λ-. In the case of rapid exchange, d is large. (Note that the equilibrium with respect to exchangeswhich is fixed by the value of K12sremains unchanged by variation of d.) As q2 increases with d2 whereas the other entries under the radical grow only linearly with d, one gets as an asymptotic equality for large d
)
q k1k2 + k1d + k2K12d q λ-(d) ≈ + + ... 2 2 k1 + k2 + d + K12d
(5)
which in the limit of large exchange rates leads to
λ-(∞) :) lim λ-(d) ) df∞
k1 + k2K12 1 + K12
In the original parameters this reads
ci(t) ) ci(0+) e-λ(∞)t, t > 0, i ) 1, 2
(7)
where c1(0+) and c2(0+) are postinitial conditions, fulfilling c2(0+)/c1(0+) ) K12 after instantaneous equilibration due to infinitely rapid exchange. The question now arises as to how the physical content of eqs 6 can be extended to more general cases.
Limiting Behavior of a General N-Box Reaction-Distribution Model
2
(
where c1eq and c2eq are equilibrium concentrations with respect to transport. λ-(d) increases monotonously from λ-(0) ) k1 in the case of no transport to λ-(∞) in the case of rapid transport. Accordingly, λ-(d) has an upper and a lower bound regardless of the speed of material exchange between the boxes: k1 < λ-(d) < λ-(∞) for 0 < d < ∞. The limiting half-life τ(∞) ) ln 2/λmin(∞) exclusively depends on the decay constants of the two boxes and on the partition constant, so that indeed exchange solely enters via thermodynamics, in contrast to decay (Figure 2). One may ask how fast λ-, as a function of d, approaches its limiting value within 95%. An estimate starting from eq 5 shows that this occurs if d is larger than (k12 + k22K12)/ 0.05(k1 + k2K12)(1 + K12) which, in the case K12 ) 1 and k1 ) k2 = k, simplifies to d > 10k, suggesting that λ-(∞) is a good approximation to λ-(d) if exchange is of 1-2 orders of magnitude faster than degradation. After the limit d f ∞ the solutions for c1 and c2 show purely exponential decay
(6a)
In this section, it will be shown that the result literally carries over to a general N-box reaction-distribution system with linear dynamics, i.e., that the limiting overall decay constant is a kind of thermodynamic average of phase-related “local” degradation constants
λmin(∞) )
k1c1eq + k2c2eq + ... + kN cNeq c1eq + c2eq + ... + cNeq
(8a)
If, in addition, the boxes are admitted to have different volumes Vi the general result reads
λmin(∞) )
V1k1c1eq + V2k2c2eq + ... + VNkN cNeq V1c1eq + V2c2eq + ... + VN cNeq
(8b)
which means that the contribution of a particular box to the limiting value is weighted by its volume. In order to see this, let c˘ ) -Lc be the dynamics of an N-box model, where L :) K - D ˜ , cRN denotes the vector of concentrations, and K is the diagonal matrix of degradation constants. In order to establish D ˜ ) (d˜ij) as a matrix describing material exchange, whichsin the absence of degradationsleads to thermodynamic equilibrium between the boxes, it is assumed that D ˜ has nonnegative off-diagonal elements and negative diagonal elements such that (i) D ˜ has only non-positive eigenvalues, (ii) zero is a nonde-
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have the same eigenvectors andsup to a multiplication with dsthe same eigenvalues. In the case of rapid exchange, K/d is a small perturbation of -∆. Using RayleighSchro¨dinger perturbation theory for symmetric matrices (compare ref 15, for example, or any other textbook on quantum mechanics or quantum chemistry) with :) 1/d one gets for the eigenvalues of the perturbed system (eq 13b)
λi ) λi(0) + λi(1) + λi(2)2 + ... FIGURE 2. Behavior of the decay rates λ- and λ+ as functions of the exchange velocity d.
generate eigenvalue of D ˜ and the corresponding eigenvector ceq :) (c1eq, c2eq, ..., cNeq) can be chosen to have only positive entries, (iii) D ˜ fulfils a condition of detailed balance Vi d˜ijcjeq ) Vjd˜ji cieq with respect to this eigenvector. Conditions (i) - (iii) are standard in general first-order kinetics (14) and are related to the Perron-Frobenius theory of positive matrices (15). A parameter controlling the speed of exchange can be defined by D ˜ :) dD with d ) ||D ˜ || and D ) D ˜ /||D ˜ ||, where || || is any matrix norm. D then, by construction, is a matrix of norm l. (In the present case, d can be chosen to be the modulus of the largest negative eigenvalue.) The matrices D and D ˜ have the same equilibrium concentrations but differ with respect to the velocity of exchange between the boxes. In the limit of rapid exchange, the smallest eigenvalue yielding the half-life is then given by
λmin(∞) ) lim λmin(d) ) df∞
∑c
/∑c
ieqkiVi
i
jeqVj
(
xc1eq/V1
C) 0 0
0
0
xc2eq/V2
0
0
‚‚‚
xCNeq/VN
)
(10)
(15)
∑λ
(1) + lim λmin(∞) ) λmin
df∞ j>1
(1) min
1
d
j-1
(1) ) λmin
(16)
(1) The explicit form of the first-order correction λmin is taken from Rayleigh-Schro¨dinger perturbation theory and reads
(1) λmin )
〈γeq|Kγeq〉
(17)
〈γeq|γeq〉
where 〈 | 〉 denotes the usual scalar product in Rn. This leads to the final result
λmin(∞) )
∑c
jeqkjVj
/∑ c
ieqVi
(18a)
i
For ideal solutions, this can be equally expressed in terms of partition constants K1j = cjeq/c1eq
(
)/(
N
λmin(∞) ) k1 +
∑
kjK1jVj/V1
j)2
)
N
1+
∑K
1iVi/V1
i)2
(18b)
If all boxes have the same volume, eq 18b simplifies to
By the transformations c f C-1c = γ and ∆ = ∆ ˜ /d, accordingly, the equations of motion are changed into the equivalent system
(12)
with equilibrium vector γeq ) C-1ceq ) (γ1eq, γ2eq, ..., γNeq), where γieq ) {Vi cieq}1/2 and ∆ = ∆ ˜ /d. Second step: Determination of the smallest eigenvalue. The matrices Λ = C-1LC and Λ/d
9
}
and going to the limit of rapid exchange
j
δ˜ ij ) xVi/cieqd˜ijxcjeq/Vj ) xVj/cjeqd˜jixcieq/Vi ) δ˜ ji (11)
588
{
(1) 1 (2) 1 ... + λmin λmin(d) ) d 0 + λmin d d2
(9)
The similarity transformation C-1LC ) C-1KC + C-1D ˜ C then leaves K invariantsas K is diagonal one has C-1KC ) Ksand transforms D ˜ into a symmetrical matrix C-1D ˜ C :) ∆ ˜ ) (δ˜ ij)), δ˜ ij ) δ˜ ji, the symmetry being implied by the detailed-balance condition
γ˘ ) -(K - dδ)γ
where λi(0) is an eigenvalue of the unperturbed problem, i.e., of -∆. In particular this is true for the smallest (0) eigenvalue λmin ) 0. Multiplying with d gives the smallest eigenvalue of eq 13a
j
The proof is carried out in two steps. First step: Symmetrization of D ˜ . Let C be the diagonal matrix defined by eq 10
(14)
Λ ) K - dδ
(13a)
1 1 Λ) K-∆ d d
(13b)
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 30, NO. 2, 1996
(
λmin(∞) ) k1 +
N
∑k K
)/(
j 1j
j)2
N
1+
∑K i)2
1i
)
(18c)
Upper and Lower Bounds for the Overall Decay in the Case of Finite Exchange Velocity For practical applications, it would be desirable to establish a relation between the limiting behavior with d ) ∞ and a real situation. In real situations there might be some exchange between the boxes, i.e., d > 0; the velocity of exchange, however, is moderate: d < ∞. It turns out that the behavior of λ- in Figure 2 of the introductory example directly carries over to the N-dimensional case insofar as λ- grows monotonously to its limiting value. More precisely this means that, under the above condition, the smallest eigenvalue λmin(d) is bounded from below by the smallest of the decay constants ki and bounded from above by the limiting eigenvalue λmin(∞), so that indeed in real situations no other values of λmin outside a well-defined given interval can occur (!).
min ki e λmin(d) e λmin(∞)
(19)
TABLE 1
Half-Lives and Partition Constants of Hexachloroethane and DDTa hexachloroethane DDT
τ1/2 soil (h)
τ1/2 air (h)
τ1/2 water (h)
log KOW
KH (atm‚m3‚mol-1)
4.32 × (6 months) 1.4 × 105 (16 years)
6× (68 years) 177 (7.4 days)
4.32 × (6 months) 8.4 × 103 (350 days)
5.31
1.3 × 10-3
5.70
5.3 × 10-5
103
105
103
a The values are taken from the following: half-lives, ref 11; log K OW and KH of DDT, ref 12; log KOW of hexachloroethane, ref 13; KH of hexachloroethane, ref 17.
TABLE 2
Inverse Half-Lives and Dimensionless Partition Constants (Soil/Water and Air/Water) of Hexachloroethane and DDT As Calculated from Table 1 hexachloroethane DDT
1/τ1/2 soil (h-1)
1/τ1/2 air (h-1)
1/τ1/2 water (h-1)
KSW ) cs/cw
K′H) ca/cw
2.3 × 10-4 7.1 × 10-6
1.67 × 10-6 5.6 × 10-3
2.3 × 10-4 1.2 × 10-4
1.7 × 103 4.1 × 103
5.3 × 10-2 2.1 × 10-3
To understand this constraint, one starts from the definition of Λ(d) ) Κ - d∆ (eq 13a). Eigenvalues and eigenvectors depend smoothly on d. Now consider any eigenvalue λi(d) and the associated normalized eigenvector γi(d). As Λ(d) is a symmetric matrix, all eigenvalues λi(d) are real. The corresponding eigenvectors are orthogonal 〈γi(d)|γj(d)〉 for i * j and chosen to be normalized 〈γi(d)|γi(d)〉 ) 1. Taking the derivative with respect to d in the defining equation for the ith eigenvalue Λ(d)γi(d) ) λi(d)γi(d)
∂ ∂ Λ(d)γi(d) ) λi(d)γi(d) ∂d ∂d
(20)
and applying the product rule gives
Λ′(d)γi(d) + Λ(d)γ′i(d)γi(d) + λi(d)γ′i(d)
(21)
where the prime ′ indicates differentiation with respect to d. Multiplication by γi(d) from the left and taking scalar products yields
〈γi|Λ′γi〉 + 〈γi|Λγ′i〉 ) λ′i〈γi|γi〉 + λi〈γi/γ′i〉
(22)
Since first-order rate constants are related to half-lives through k/ln 2 ) 1/τ1/2, one has to take the inverse halflives in order to obtain the respective degradation constants (Table 2). The equilibrium between soil and water is defined through KSW ) cs/cw, which is proportional to the octanolwater partition constant KOW ) co/cw. (In what follows, cs, co, cw, and ca denote the pollutants concentration in soil, 1-octanol, water, and air.) If it is assumed that soil contains ∼2% organic material, and if it is further assumed that there is a factor of 0.41 between the partition constant for octanol and for the organic phase of soil (17), one has KSW ) 0.02 × 0.41 KOW ≈ 8 × 10-3KOW for nonpolar pollutants. The dimensionless Henry’s law constant K′H is related to KH through KH/RT = K′H ) ca/cw, where T ) 300 K and RT ) 24.6 atm‚L‚mol-1. The (relative) volumes of water (Vw), soil (Vs), and air (Va) are taken from previously published unit-world models (18, 19) and are assumed to behave as Vw:Vs:Va ) 250:1:2 × 105. Choosing water, soil, and atmosphere as phases 1, 2, and 3, respectively, and inserting V2/V1 ) 4 × 10-3 and V2/V1 ) 8 × 102 as well as KSW, K′H, 1/τ1/2 soil, 1/τ1/2air, and 1/τ1/2 water from Table 2 into eq 18b yields the final result:
hexachloroethane As Λ is real and symmetric (i.e., self-adjoint) one has 〈γi|Λγ′i〉 ) 〈Λγi|γ′i〉 ) λi〈γi|γ′i〉, which simplifies eq 22 to
∂ λ (d) ) 〈γi|Λ′γi〉 ) 〈γi| - ∆γi〉 g 0 ∂d i
(23)
The inequality is due to the fact that the numerical range of ∆swhich is spanned by the set of all eigenvalues of ∆sis nonpositive. Equation 23 means that all eigenvalues γj increase monotonously with d. As this is true in particular for the smallest eigenvalue λmin(d), this proves the proposition 19.
Examples DDT and hexachloroethane are chlorinated nonpolar organic chemicals with large octanol-water partition constants KOW and small Henry’s law constants KH. Conventional chemical wisdom, accordingly, suggests that their overall degradation should be governed, to a large extent, by their behavior in the organic phase. To show how the formalism works the data of Table 1 are used as a starting point.
λmin(∞) ) [2.3 × 10-4 + (2.3 × 10-4)(1.7 × 103)(4 × ln 2 10-3) + (1.67 × 10-6)(5.3 × 10-2)(8 × 102)]/[1 + 1.7 × 103 × 4 × 10-3 + 5.3 × 10-2 × 8 × 102] ≈ 3.7 × 10-5 h-1 τ(∞) ) ln 2/λmin(∞) ) 27 000 h ) 3 years
(24a)
DDT λmin(∞) ) [1.2 × 10-4 + (7.1 × 10-6)(4.1 × 103)(4 × ln 2 10-3) + (5.6 × 10-3)(2.1 × 10-3)(8 × 102)]/[1 + (4.1 × 103)(4 × 10-3) + (2.1 × 10-3)(8 × 102)] ≈ 5 × 10-4 h-1 τ(∞) ) ln 2/λmin(∞) ) 2000 h ) 83 days
(24b)
A closer look on the dominant contributions to these expressions shows that for hexachloroethane the overall behavior is essentially determined by high solubility and
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degradation constant and the subdominant second phase (in the denominator) modifies it to the final value of λmin(∞).
Concluding Remarks
FIGURE 3. Phase-related and global limiting lifetimes of DDT and hexachlorethane, based on a minimal 3-box unit-world model.
rapid degradation in soil (numerator) and by the large volume of the atmosphere (denominator). Taking the contributions of soil (numerator) and air (denominator) alone one gets
λmin(∞) VsKsw/τ12soil ≈ 3.7 × 10-5 h-1 ≈ ln 2 VaK′H
(25a)
corresponding to a global limiting lifetime of ∼3 years, which completely reproduces the result of eq 24a. For DDT things look different. Its overall behavior, as given by eq 18b, is dominated by rapid decay in the atmosphere on the one side and high solubility in soil on the other. Taking again these contributions alone leads to
λmin(∞) VsK′H/τ12soil ≈ 5.7 × 10-4 h-1 ≈ ln 2 VaKSW
(25b)
corresponding to 72 days which, again, is close to the complete result (eq 24b) of 83 days for DDT (Figure 3). The result of ∼100 days for the overall decay of DDT, however, is not realistic and shows that the law must not be applied blindly: The half-life of 7.4 days in the atmosphere is shorter than relevant transport processes such as atmospheric mixing times. The basic assumption that transport is faster than degradation is not fulfilled in this case, and the law does not apply. The limiting law, accordingly, should be applied exclusively to truly persistent chemicals where phase-related decay times are at least in the order of yearssfor which case the law was derived. What can be further learned from these examples? As can be seen from eq 18b, the limiting overall decay time of chemicals is a rational function of three types of quantities: phase-related half-lives, partition constants, and phase volumes. No simple, physically intuitive dependence of the limiting half-life τ(∞) on a few of them is to be expected in general. Such a picture may arise, however, for special classes of pollutants. The global decay of apolar, persistent, nonvolatile substances, for example, is governed by the interplay of soil (high concentration, low volume) and air (low concentration, large volume). Which one wins is decided by the degradation constants in the following way: The dominant phase (in the numerator) contributes its
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In the environmental risk assessment of organic chemicals, one may distinguish effect-based assessment (exposure taken only as a necessary condition for effect, assessment procedure focused on effect), and exposure-based assessment (evaluation in terms of exposure, assessment of effects not necessarily required). These two alternatives do not exclude but complement each other: An exposure-based assessment can always be followed by an effect-based assessment. There are several reasons to further develop the approach of exposure-based assessment (6): First, an exposure-based assessment makes it possible to reduce the complexity the assessment procedure has to deal with because less data have to be recorded, aggregated, and evaluated. Second, the logical order between exposure and effectssexposure logically and temporally comes before effectssmakes exposure a natural point of view for the regulatory needs of ecological prevention. One may argue that the minimal set of quantities required for exposure-based assessment are spatial range and persistence of a once-released xenobiotic chemical (6). To estimate these quantities in a transparent and reliable way, methods are needed that are at the same time simple, general, and theoretically wellfounded. This program as a whole, however, is still in its infancy. (Such a program was proposed at the 1995 meeting of the Society for Risk Analysis (Europe) and will be published elsewhere; Mu ¨ ller-Herold: U. Measures of Endangerment. In Risk Management for the Next Century; Kemp, R., Renn, O., Eds.; Kluwer: Dordrecht, The Netherlands, in press.) In particular there are no such estimates for spatial range. The limiting law as stated in eq 18 should contribute to this tool box. In the context of regulatory procedures, it seems conceivable that λmin(∞) alone could serve as a criterion for registration: As a necessarysbut not sufficientscondition for the registration of a new xenobiotic substance, one could require τ(∞) not to exceed a critical value. [A related principle was accepted in the Swiss guidelines for waste management: Every generation has to solve its own waste problemssultimately (20)]. In this case, noxious effects arising unexpectedly at later stages of use would cause no harm, except for a short time and within a restricted area.
Acknowledgments I thank Peter Baccini, Daniel Caderas, Gregor Nickel, Hans Primas, Martin Scheringer, and Andre´ Weidenhaupt for valuable discussions. Technical support by Pierre Funck and Florian Reitz as well as useful comments by three competent viewers are gratefully acknowledged.
Literature Cited (1) Stephenson, M. E. Ecotoxicol. Environ. Saf. 1977, 1, 39-48. (2) Korte, F. In Organische Verbindungen in der Umwelt; Aurand, K., et al., Eds.; de Gruyter: Berlin, 1978; pp 288 ff. (3) Klo¨pffer, W.; Rippen, G.; Frische, R. Ecotoxicol. Environ. Saf. 1982, 6, 294-301. (4) Frische, R.; Esser, G.; Scho¨nborn, W.; Klo¨pffer, W. Ecotoxicol. Environ. Saf. 1982, 6, 283-293. (5) Tremolada, P.; DiGuardo, A.; Calamari, D.; Davoli, E.; Fanelli, R. Chemosphere 1992, 24, 1473-1491.
(6) Scheringer, M.; Berg, M. Fresenius Environ. Bull. 1994, 3, 493498. (7) O’Connor, D. J. Environ. Eng. ASCE 1988, 114, 507-532. (8) Mackay, D. Environ. Sci. Technol. 1979, 13, 1218-1223. (9) Mackay, D.; Paterson, S. Environ. Sci. Technol. 1981, 15, 10061014. (10) Mackay, D.; Paterson, S. Environ. Sci. Technol. 1982, 16, 654A660A. (11) Howard, P. H., Ed. Environmental Degradation Rates; Lewis Publishers: Chelsea, MI, 1991. (12) Howard, P. H., Ed. Handbook of Environmental Fate and Exposure Data for Organic Materials; Lewis: Chelsea, MI, 1989; 3 vols. (13) Samiullah, Y. Prediction of the Environmental Fate of Chemicals; Elsevier: London, 1990. (14) Horn, F. Ber. Bunsen-Ges. Phys. Chem. 1971, 75, 1191-1201. (15) Gantmakher, F, R. The Theory of Matrices; Chelsea: New York, reprinted 1977; Vol. 2, Chapter 13 (translated from the Russian).
(16) Schiff, L. I. Quantum Mechanics, 3rd ed.; McGraw Hill: New York, 1969; Chapter 8. (17) Karickhoff, S. W. Chemosphere 1981, 22, 361-367. (18) Mackay, D.; Shiu, W. Y. J. Phys. Chem. Ref. Data 1981, 10, 11751199. (19) Mackay, D.; Paterson, S.; Shiu, W. Y. Chemosphere 1992, 6, 695717. (20) Baccini, P. Preface. In The Landfill. Reactor and Final Storage; Baccini, P., Ed.; Lecture Notes in Earth Sciences; Springer: Berlin, 1989.
Received for review May 22, 1995. Revised manuscript received September 11, 1995. Accepted September 13, 1995.X ES9503443 X
Abstract published in Advance ACS Abstracts, December 1, 1995.
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