Environ. Sci. Technol. 1984, 18, 756-764
George L. Maddrea, Jr., associated with the Cessna aircraft measurements; C. Gerald Clendenin, Jr., Thomas L. Owens, Bordie D. Poole, Jr., and Otto Youngbluth, Jr., associated with the tethered balloon measurements. We thank Jason K. S. Ching, NOAA-EPA, Research Triangle Park, John Salop, Commonwealth of Virginia, State Air Pollution Control Board, Gerald M. Gregory, NASA Langley Research Center, and James E. Smith, Department of Geophysical Sciences, Old Dominion University, for their cooperation and assistance during the case studies. Registry No. Ozone, 10028-15-6.
Literature Cited (1) Browell, E. V.; Carter, A. F.; Shipley, S. T.; Allen, R. J.;
Butler, C. F.; Mayo, M. N.; Siviter, J. H., Jr.; Hall, W. M. Appl. Opt. 1983,22, 522. (2) Vaughan, W. M.; Chan, M.; Cantrell, B.; Pooler, F., Jr. Bull. Am. Meteorol. SOC.1982, 63, 258. Browell, E. V.; Ismail, S.; Shipley, S. T. (submitted for publication in Appl. Opt.). Browell, E. V.; Shipley, S. T.; Butler, C. F.; Ismail, S. “Airborne Lidar Measurements of Aerosols, Mixed Layer Heights, and Ozone During the 1980 PEPE/NEROS Summer Field Experiment”; PEPE/NEROS Archive, CAPITA Special Studies Data Center: Washineton University, St. Louis, MO, 1983. Lyons, W. A.; Dooley, J. C., Jr.; Whitby, K. T. Atmos. Environ. 1978, 12, 621. Burke, H. K. IEEE Trans. Geosci. Remote Sens. 1982, GE-20, 154. Norton, C . C.; Mosher, F. R.; Hinton, B.; Martin, D. W.; Santek, D*;Kuhlow, W*J*APPl. MeteoroL 1980, 199 633. Lyons, W. A.; Calby, R. H. 1983, Final Report on EPA Contract 68-02-3740. Haagenson, P.; Shapiro, M. A. NCAR Technical Note, 1979, NCAR/TN-l49+STR. 1
I
Ching, J. K. S., “The Role of Convective Clouds in Venting Ozone from the Mixed Layer”. Third Conference on Applications of Air Pollution Meteorology, AMs,San Antonio, TX, Jan 12-15, 1982. Wakimoto, R. M. Mon. Weather Rev. 1982, 110, 1060. Sisterson, D. L.; Shannon, J. D.; Hales, J. M. J . Appl. Meteorol. 1979,18, 1421. Austin, J. M.; Fleischer, A. J. Meteorol. 1948, 5, 240. Vukovitch, F. M.; Erlich, D. P.; Clark, T. U.S. Environmental Protection Agency, 1981, EPA 68-02-3428. Orndorff, B. L. MS Thesis, Old Dominion University, Norfolk, VA, 1983. Durham, J. L.; Whelpdale, D. M. (co-chairmen) “United States-Canada Memorandum of Intent on Transboundary Air Pollution”. Atmospheric Sciences Review Sub Group, Work Group 2, 1982, Report 2F-A. Csanady, G. T. “Turbulent Diffusion in the Environment”; Reidel: Dordrecht, The Netherlands, 1973. Uthe, E. E.; Ludwig, F. L.; Pooler, F., Jr. J. Air Pollut. Control. Assoc. 1980, 30, 889. Clarke, J. F.; Ching, J. K. S.; Godowitch, J. M. “Lagrangian and Eulerian Time Scale Relationships and Plume Dispersion from the Tennessee Plume Study”; Sixth Symposium on Turbulence and Diffusion, AMs, Boston, MA, March 22-25, 1983. Danielsen, E. F. Arch. Geophys. Bioklimatol., Ser. A 1959, AI?
ni 1 ,
nnQ
OW.
(21) McRae, G. J.; Shair,F. H.; Seinfeld, J. H. J. Appl. Meteorol. 1981,20, 1312. (22) Salop, J.; Maier, G. F. J . Air Pollut. Control Assoc. 1978, 28, 1217. (23) Pasceri, R.; Predale, R.; Perritt, A. J . Air Pollut. Control Assoc. 1979, 29, 639. (24) &toque, M. A. J . Atmos. Sei. 1962, 19, 244.
Received for review September 15, 1983. Revised manuscript received April 6, 1984. Accepted April 19, 1984.
Transformation and Fate of Organic Esters in Layered-Flow Systems: The Role of Trace Metal Catalysis Marianna Plastourgou and Michael R. Hoffmann*
Environmental Engineering Science, W.M. Keck Laboratories, California Institute of Technology, Pasadena, California 9 1 125
rn A simple mass transport model with chemical and biochemical reactions has been developed to predict the relative degree of degradation of organic esters in a layered-flow or density-stratified system. A numerical method was used to solve a system differential equations involved when metal-catalyzed hydrolysis, biodegradation, and adsorption were assumed to be major pathways for ester transformation. The relative importance of each pathway for the time-dependent concentration profiles of an organic ester was examined by use of dimensionless parameters given a known initial concentration profile across the depth of the water column for a catalyst, for a microbial population, for particles, and for a hypothetical ester. Both the ester and the catalyst were allowed to interdiffuse and to react. Results show that the characteristic times for metal-catalyzed ester hydrolyses alone are in the range of 6-700 days. These values depend strongly on initial concentration profiles, the magnitude of either the diffusion or dispersion coefficients, and the magnitude of the hydrolysis rate constant. Limitations of model applicability are discussed. Introduction Current research in the fate of pollutants in aquatic environments has been focused on pathways for pollutant 756
Environ. Sci. Technol., Vol. 18, No. 10, 1984
transformation. Important pathways that have been identified include biological degradation, photolysis, autoxidation, adsorption, and hydrolysis. A primary pathway for the transformation of organic esters in aquatic environments is hydrolysis. Hydrolysis reactions are normally sensitive to a variety of catalytic influences that include specific acid or base catalysis, metal oxide surface catalysis, general acid or base catalysis, and metal ion catalysis. Positively charged metal ions can function effectively as Lewis acids in these reactions and can be expected to catalyze reactions which are similarly catalyzed by Bronsted acids. In this paper, a model is presented that predicts mass transport effects in two-phase flow systems for a given aquatic system in which a catalyst and a substrate interdiffuse and react to yield products. Emphasis has been placed on metal-catalyzed hydrolysis of organic esters as a decomposition pathway although the mathematical formalism presented is equally applicable to biodegradation and adsorption as transformation pathways. Organic esters such as amino acid esters, carbamate esters, organophosphate esters, phthalate esters, and thiophosphate esters have physicochemical properties similar to other trace organic pollutants. In particular, they have low water solubility, they form organic microlayers or micelles, they readily adsorb to particle surfaces, and
0013-936X/84/0918-0756$01.50/0
0 1984 American Chemical Society
they are subject to degradation by chemical and biochemical pathways. For example, Epstein (1)has shown that isopropyl methylphosphonofluoridate (Sarin) hydrolyzes rapidly in seawater to methylphosphonic acid, a relatively nontoxic product.
i
0
II I F
t H20
(CH3,&CHO--P--CH~
+
ICH&CHO-P-CH3
I
directly polarizes the substrate and enhances displacement reactions: case I (Lewis acidity)
HF (1)
OH
Epstein (I)was able to show that, in the absence of specific catalysis, half-life of Sarin in water a t pH 6.5 and 25 "C was 175 h. However, with the addition of 1ppm of Cu2* at the same pH and temperature, the half-life for the above reaction was reduced to 2 h. Similarly, in seawater, at pH 8.2 where the primary catalysts were shown to be the hydroxy complexes of Ca2+and Mg2+,the half-lives were reduced to a few minutes. During the latter part of 1969 the fate of Sarin in seawater was a highly topical question because the U.S. Army had chosen the Atlantic Ocean off of New Jersey as a disposal site for leaking, concrete-encasedwarheads loaded with Sarin. After disposal the following question was posed: "What is the persistence and potential impact of the chemical warfare agent, Sarin, that is being slowly released to the marine environments?" The answer to this question as shown by Epstein (I)was that the lifetime of Sarin in seawater was remarkably short due to the combined effects of metal- and base-catalyzed hydrolysis such that the potential for a catastrophe in the immediate marine environment was minimal. Similar questions can be posed about the fate and persistence of other organic esters that may be introduced to marine environments either through wastewater discharges or through atmospheric deposition (2, 3). Literature Review Hydrolysis (i.e., solvolysis) is an important pathway for the transformation of organic esters in aquatic environments. Most often hydrolytic reactions in natural systems are sensitive to specific acid or base catalysis. In acidcatalyzed ester hydrolysis, the role of the proton according to Bender and Brubacher (4) is to provide a reaction pathway of lower energy by withdrawing electrons and therefore weakening the bond to be broken. A mechanism for acid catalysis has been suggested by Bender and Brubacher ( 4 ) and is as follows: 'OH
0
II R-C-OR'
+
II
Ht $ R-C-OR'
'0H
I1
R-C-OR'
t H20
I I OH:
R-C-OR'
(2) OH
OH
I
$ R-C-W'
4
I 1 OH H
(3)
OH
I + I 1 OH H
R-C-OR
RC02H f R'OH t H t
(4)
where eq 4 represents a summation of two steps involving proton transfer in addition to bond breaking. A metal ion can mediate or facilitate hydrolysis or nucleophilic displacement in two ways: either by direct polarization of the substrate and subsequent external attack by the nucleophile (i.e., OH-/H20) or by the in situ generation of a reactive basic reagent such as MOH+ (5). In the fiist case, the metal ion functions as a Lewis acid that
where N: is a nucleophile. Polarization can also be achieved by coordination to the leaving group as in the case of phosphate ester substrates.
I
OH-/H,O
\Mn2+/
)6 ' \p/
RO\p/
0 '
R 0 'I
HOJI'
Direct polarization of the carbonyl or phosphoryl function results in a more electrophiliccarbon center. The ability of a particular metal to exert this effect will depend on its overall charge, size, coordination number, and lability. The rate expression predicted for a direct polarization mechanism would be as follows: k&[ zn],[ RCOY] [N ] v = (5) 1 + K[RCOY] where N is an appropriate nucleophile, kN is a secondorder rate constant, and K is a pre-rate-determining association constant. In the case of Co(II1) complexes, k N is accelerated by a factor of 104-10s for carbonyl-bound esters. In the case of the Co(II1)-catalyzedhydrolysis of ,&alanine esters, Buckingham et al. (6) found a distinct effect of chelate ring size. A six-membered ester chelate was accelerated by a factor of lo6 while a comparable five-membered chelate was accelerated by a factor of 10'. Bender and Brubacher ( 4 ) have reported that Cu(H20)62+is an effective catalyst for the hydrolysis glycine ethyl ester. At 25.0 OC k N was reported to be -lo6 M-l s-l as compared to kOH of 1M-l s-l. They (4) proposed that the rapid hydrolysis of a-amino acid esters involves the direct complexation of Cu2+by the ester group. In general, the effectiveness of a metal ion catalyst in case I hydrolysis appears to parallel the respective formation constant (Le., kN K ) . Buckingham (5) has pointed out that divalent metal ions increase the rate of hydrolysis by a factor of -lo4 when a direct polarization mechanism is operative. In the case of Co(II1)-promoted reactions, laO-labelingexperiments indicate that the rate-limiting step in ester hydrolysis is elimination of the alcohol function from the tetrahedral intermediate (4). In the second case, the aquated metal ion hydrolyzes to give a coordinated reactive nucleophile. The bimolecular nucleophilic displacement can be represented as follows: case I1
'.'+'i"' I
I
R
L Y'
I
Ho:-7
:OH
O
-
I
Y
O
M(H20),0Hn+ + RCOY M(n+l)++ RC02- + Y- (6) Equation 6 is applicable for labile metals-the case for Environ. Sci. Technol., Vol. 18, No. 10, 1984
767
most of the other first-row transition metals. Buckingham and Clark (7) have reported that both exchange-labile and nonlabile (substitution inert) metal hydroxide species readily promote the hydrolyses of 2,4dinitrophenyl acetate (Dnpa) and 4-nitrophenyl acetate (Npa). Product analysis for these reactions was consistent with a case I1 mechanism. However, because Cu2+is a labile metal, no conclusive evidence for case I or I1 could be presented, although a linear free energy relationship (LFER) was shown to exist between log kN and pK,, for metal hydrolysis. This suggests in part that the mechanism was similar for all metal complexes studied (i.e., case 11). However, the MOH-promoted reactions were not overly sensitive to the pK, of the bound water molecule, The slope of the LFER was relatively small. The relative insensitivity of kN to pK,, indicates that the metal-bound nucleophile (OH-) is superior kinetically to the free nucleophile in a pH range where the latter is essentially protonated. Similar conclusions were drawn by Buckingham and Clark (7) for the catalytic hydrolysis of acetyl phenyl phosphate by Co(III), Ca2+,and Mg2+. Martell (8) has argued that distinctions between case I and case I1 catalysis for labile metals are meaningless, particularly in the case of fluorophosphate ester hydrolysis, since a number of pre-rate-determining equilibria may occur that effectively introconvert the various intermediates to give a single rate-limiting intermediate. Wolfe et al. (9), Faust and Gomaa (IO),and Weber (11) have studied experimentally the relative hydrolysis rates of thiophosphoric acid esters and carbamic acid esters. Woife et al. (9) studied with the alkaline hydrolysis of carbaryl, propham, and chloropropham at pH 9. Carbaryl was hydrolyzed rapidly with a calculated half-life of 0.15 day at 27 “C. Chloropropham and propham were more resistant to degradation. Their half-lives were reported to be lo4 days for each at the same temperature and pH. Rate constants for the neutral and acid catalysis terms were not determined because of very slow reaction rates at low temperatures. Basically, all carbamate esters that were studied by Wolfe et al. (9) were found to be resistant to degradation in the pH range 3.0-7.0. Faust and Gomaa (10) studied the rate of hydrolysis of parathion and its oxidation product paraoxon at pH 7.4 and 20 “C. They found that parathion has a half-life of 108 days and paraoxon has a half-life of 144 days. For the carbamate esters baygon and carbaryl, they calculated hydrolysis half-lives of 1.6 days and 2.5 h, respectively, using the data of Aly and El-Dib (12)for pH 9. However, they concluded after an extensive review of kinetic results reported by others that the thiophosphoric acid and carbamic acid esters are more resistant to hydrolytic degradation in the pH range (5.5-8.5) of natural waters than previously suspected. Of additional concern are the hydrolysis products such as p-nitrophenol, which is also toxic, although its inherent toxicity is somewhat less than its parent compound. Weber (11) studied the hydrolysis rate of parathion in sterilized and filtered marine and estuarine water samples at elevated temperatures. A t approximately pH 8 and at 70 OC the experimentaly determined half-lives for distilled water, estuarine water, and seawater were 22.7,30, and 17 h, respectively. No attempt was made by Weber to minimize trace metal catalysis in his distilled water samples with added buffer salts. Also potential contributions due to general base catalysis were not considered. However, Weber reported that a significant fraction of parathion (0,O-diethyl 0-(p-nitrophenyl) thiophosphate) was hydrolyzed by an alternative pathway involving dealkylation 758
Envlron. Scl. Technol., Vol. 18,No. IO, 1984
to give a secondary ester of phosphoric acid which still contained the p-nitrophenol moiety. Deethylation is thought to be limited to reactions catalyzed by complexation with heavy metal ions (13). This hypothesis was confirmed by earlier work. Ketelaar et al. (14) were able to show that Cu2+catalyzed the hydrolysis of parathion efficiently. With concentrations of M (0.01 pM or 0.64 ppb) Cu2+and M parathion at pH 8.5, the catalytic effect produced a rate that was 20 times more rapid than the control. Ethylenediaminetetraaceticacid (EDTA) was used in the control flasks to minimize the contribution of nonspecific trace metal catalysis. As pointed out above, in case I hydrolysis a critical aspect of trace metal catalysis involves the formation of metal substrate complexes in a rapid equilibrium step before nucleophilic displacement by either OH- or H20. Green et al. (15) have reported that the hydrolysis of 0-aminothio esters are catalyzed by Cu(I1) and Ni(I1). They isolated and characterized the 1:l complexes of Cu(I1) and Ni(1I) with various thiol esters ({i))as shown in eq 7 and 8. The R
(i) + H 2 0
-
R
A
R‘
0
R2N(CH2I2SH+ R’C02H
(8)
catalytic effect was most pronounced for Cu(I1) although Ni(II), Hg(II), Pb(II), and Ag(1) also exhibited a positive catalytic influence. Barca and Freiser (16) studied the influence of metal ions on the rate of hydrolysis of 8acetoxyquinoline in water and at 25 OC. Again the proposed mechanism involves an initial coordination to the catalytic metal followed by a nucleophilic displacement in the rate-determining step. Ma+ + E (ME)”+ + OH-
(ME)”+
k
(9)
MO,(”-l) + CH3C02H (10)
slow
-
The catalytic order observed for the divalent metal ions was Cu(I1) > Zn(I1) > Pb(I1) > Cd(I1) > Ni(I1). The relative catalytic activity seems to parallel the ability of each metal to selectively form a seven-membered,chelated activated complex as illustrated.
$AJ
\Mn+
0
‘c=o
I
I
CH3
Similar results were reported by Wells et al. (17) in their study of the hydrolysis of methyl esters of substituted imidazoles in which a bound metal hydroxide acts as nucleophile in an intramolecular hydrolysis. Significant catalytic activity was observed for Co(11) and Ni(II), and Co(I1) being somewhat superior as a catalyst. Catalytic activity was most pronounced in the pH range 6.5-8.5. Total metal concentrations were in the range 1-10 pM. Finally, Fife and Przystas (18) have reported that Ni(II), Co(II), and Zn(I1) are active catalysts for the hydrolysis of 2-pyridylmethyl hydrogen phthalate. First-order de-
-
9
7
V
-
Table I. Symbols Used in Development of Interdiffusion Model for the Fate of Esters in Density-Stratified Systems
IL
Flguro 1. Basin of a hypothetical, density-stratified lake in which the shape 1s approximated by a series of horirontal layers of arbtkary size.
pendencies were observed for these metals with no apparent saturation effects. They previously reported rate enhancements up to lo9 for the metal-catalyzed hydrolyses of certain phenolic esters (19,20) and benzaldehyde methyl-8-quinolyl acetals (21). Decarboxylation reactions are also catalyzed by a variety of metal ions. Prue (22)investigated the metal ion catalyzed decarboxylation of acetonedicarboxylic acid (eq 11) HO&CH&(=O)CH2C02H CH&(=O)CHa + 2C02 (11) over a wide pH range and found that the deprotonated acid, A2-, was most strongly catalyzed by metal ions. He found that a linear free energy relationship (LFER)existed between the rate constant for the catalytic pathway and the dissociation constant for the corresponding metalmalonate complex. In this case the activated complex was postulated to be similar to the bidentate ligand malonate. The order of catalytic activity was reported as: Cu2+> Ni2+ > Zn2+ > Co2+B Mn2+ > Cd2+. Similar catalytic effects were reported by Gelles and Salama (23)for the decarboxylation of oxaloacetic acid for which the following order of catalytic activity was observed Cu2+> Zn2+> Co2+> Ni2+> Mn2+> Ca2+. Metal ions catalyze other hydrolysis reactions (4)of potential environmental significance such as the hydrolysis of tripolyphosphate.
DZ X
L D cz,o
+
Dz' D'
+
P3010-5 + H20
-
HP209
+ HP04-2
dimensionless concentTation of the catalyst at any given time (C, = Cl/Cl,o) dimensionless concentlration of the reactant at any given time (Cz = CZ/C2,o) initial catalyst concentration (however, when initial values for the catalyst concentration are different in the various layers, Clp is the highest concentration) dimensionless time for the catalyst (rl = Dlt/L2) dimensionless time for the reactant (Q = Dzt/L2) molecular diffusion coefficient for the catalyst, om2 s-l molecular diffusion coefficient for the reactant, cm2 s-l dimensionless distance along the x axis where the origin of axes is in the bottom layer characteristic length of the problem which is equal to the depth of all layers dimensionless parameter (C,okmL2/Dz) initial substrate concentration (however, when initial values are different in the various layers, C2,0is the highest concentration) D z / ( R 1) where R is the Freundlich adsorption constant dimensionless parameter (when adsorption is included) (Cl,&mL2/D,') dimensionless parameter (when biodegradation is included) ( L 2 F , C ~ / ( D , l K ~ ) ) maximum velocity of uptake, mg mg-' h-' concentration of bacterial dry mass, mg L-' half-saturation constant, mg L-'
B
of arbitrary size as shown in Figure 1. The existence of density-stratified layers and their dampening effects on advective forces allows the contribution of advective transport to be ignored, and it allows the use of realistic initial boundary conditions. The problem of interdiffusion with simultaneous chemical transformation due to hydrolysis can be represented mathematically in terms of the following equations: the first one for diffusion of the catalyst as a conservative species
(12)
Catalytic activity is again correlated to the ability of a particular metal to form a complex with the substrate. In aqueous solution at pH 7-9 the tripolyphosphate anion is extremely stable; however, with the addition of Cu2+,Zn2+, or Pb2+,the hydrolysis reaction is measurable accelerated (24). In order to predict the spatial and temporal distribution of organic esters in surface waters influenced by specific discharges, a mass transport model with chemical or biochemical reactions is needed. A simple mathematical formalism to predict the time-dependent fate of esters in layered-flow (i.e., density-stratified) systems is presented below. Model Development The conceptual framework for this development involves the basic problem of a reductant (pollutant) introduced in an upper layer of a body of water while the catalyst resides in a lower density-stratified layer. Hence, the mathematical treatment is concerned with the transport of reactant and catalyst from opposite ends with simultaneous chemical reaction. In this simple layered-flow model, the shape of the water basin can be approximated in terms of horizontal layers
or, in dimensionless terms, as
el
where = Cl/Cl,o, r1 = Dlt/L2,and X = x / L (definitions of the appropriate symbols are given in Table I); the second one for interdiffusion with transformation of the reactant
or, in dimensionless terms, as
e2
where = C2/C2,0and r2 = D2 t / L 2 . A general rate law for a substrate undergoing hydrolysis can be written as follows: -d [substrate] = Iz,b,d[substrate] (17) dt where koM is a pseudo-first-order rate constant containing multiple terms. Environ. Scl. Technol., Vol. 18, No. 10, 1984
759
kob,d
kl[H2O]
+ k2[H+] + ka[OH-] + Clzi[HBJ + i Ckj[Bj] I
+ Ckm[Mn+] (18) m
The klterm reflects the rate of hydrolysis due to a neutral water molecule alone; the k2 and k3 terms account for specific acid and base catalysis, respectively; the kiand k. terms reflect the rate contribution made by general acid and general base catalysis, respectively; in the last term, k, accounts for the contribution to the overall rate of hyrolysis made by specific metal catalysis. In the region of pH 6-8, where the k1 term dominates, the hydrolysis rate is predicted to be independent of pH. This is the pH range in which metal catalysis is expected to have the greatest influence on the overall rate of hydrolysis. A basic assumption in this model is that hydrolysis of a specific ester such as parathion occurs primarily via a metal-catalyzed pathway over the pH range 6-8. In the particular case of parathion -d[parathion] = kObsd[parathion] (19) dt where parathion (0,O-diethyl 0-(p-nitrophenyl) phosphorothionate) has the structure S
and where
k, is a general second-order rate constant for hydrolysis by a specific metal ion catalyst, Mn+,and the summation term reflects the contribution of each specific catalyst to the overall rate of hydrolysis. k, appears as a kinetic coefficient in eq 15 and 16. In reference to Figure 1, fundamental chemical and physical considerations in model development are as follows: initially two or three horizontal layers of arbitrary size are considered; discontinuous discharges of reactants are allowed in the vertical direction whereas in the horizontal direction there is assumed to be uniformity at any time, t ; inputs are allowed into any layer; however, at t = 0 inputs are considered to be uniformly distributed across any specific layer; transfer between layers occurs by diffusion alone; influxes of reactants and catalysts are expressed in terms of concentration, and volume variations are approximated by changing the number and size of the layers; calculations are made on a sequential time basis; rate constants, pH, and diffusion coefficients are assumed to be constant with depth; temperature and pH effects can be implicitly included by variations of specific rate constants. Equation 16 can be modified to simultaneouslytreat the problems of interdiffusion of catalyst and substrate with chemical and biochemical transformation in dimensionless terms as follows:
for which the fundamental considerationsand assumptions are that biodegradation can be treated as another parameter (i.e., the dimensionless parameter, B = L2V,C,/ (D2K,), which indicates the relative importance of biodegradation as compared to diffusion as a loss process, that pollutant transformation for a given bacterial concentration is first order with respect to substrate, given a constant 760
Envlron. Scl. Technol., Vol. 18, No. 10, 1984
ratio of enzyme activity to biomass, i.e., ester biodegradation is expressed as
and for C2
