Environ. Sci. Technol. 1985, 19, 869-870
(6) Peterson, T. W.; Moyers, J. L. Atmos. Enuiron. 1982,16, 1274.
Independence of D and E
Since I concur with the Peterson-Moyers responses (6), to Mage’s critique (3) regarding the assumed independence of D and E, and the boundedness of D, I need not add any more comment to their statements except for the discussion on sample averages. The correctly stated result for expectations in eq 4 (6) is due to the assumed independ, has nothing ence of D and E , i.e., E [ C ]= E [ D ] E [ E ]and to do with rollback or emission control. However, if the realizations of C and E (and derived D ) are those given by Mage (3) and the sample averages are = 5/2, E2 = 3/2, and D, = 3/2 (sample size is denoted by subscript n = 2), then it is true in general that the sample average for C does not equal the product of the sample averages for D and E, i.e.
Robert M. Ollver
Department of Industrial Engineering and Operations Research University of California Berkeley, California 94720
c2
e, z D,E,
finite n
(4)
even when D and E are independent and stationary. With independence the product relation is satisfied for expectation of sample averages, i.e.
E[c,] = E[D,]E[E,] = E [ D ] E [ E = ] E[C]
Comment on “Equilibrium Adsorption of Polycyclic Aromatic Hydrocarbons from Water onto Activated Carbon”
all n
(5)
SIR: In the June 1984 issue of ES&T, Walters and Luthy (1)describe their studies on the adsorption of polycyclic aromatic hydrocarbons from water onto activated carbon. The authors present two linearized forms of the Langmuir adsorption isotherm
and also, in the limit of large sample sizes, the equality holds in eq 4. I simply do not understand the confusion in ref 3 and 6 with regard to sample averages or the claims that one or two realizations of D yield an estimate p ( D ) of the theoretical probability density function, P(D). Summary
and
I appreciate the opportunity to respond to Dr. Mage’s critique as I am naturally interested in the reaction of a practicing professional on these scientific problems. On the basis of his critique I want to give more throught to the implications of mass-transfer rates Qf the pollutants and the underlying carrier (air) as well as some of the unsolved estimation questions which are truly necessary to validate any theory. I respectfully disagree with his claim of the inapplicability of conservation of expected flows or that the use of the log-normal distribution for receptor concentrations invalidates the derivation, application, or implications of a theoretical rollback model based on stationarity and conservation of expected flows. Naturally, where stationarity does not apply one should not force it. On the other hand, I would always encourage its use as a first step in model building in order to better understand order of magnitude effects of emission control policies. Finally, the implication that volume-transfer rates (V) a t a receptor must somehow be less than the volume-transfer rate at an emission source simply does not allow for the possibility of intermediate (nonsource) storage. Such an assumption is overly restrictive for purposes of studying the source-receptor relationship. It is not a question of the second law of thermodynamics; again, it hinges on the proper use of conservation of expected flow and volume-transfer rates. Literature Cited (1) Mage, D. T. Enuiron. Sci. Technol.,preceding paper in this issue. (2) Oliver, R. M. Enuiron. Sci. Technol. 1985, 19, 225-231. (3) Mage, D. T. Atmos. Enuiron. 1982, 16, 1273-1274. (4) Peterson, T. W.; Moyers, J. L. Atmos. Enuiron. 1980,14, 1439-1444. (5) Mage, D. T.; Ott, W. R. J . Air Pollut. Control Assoc. 1978, 28, 796-798. 00 13-936X/85/09 19-0869$01.50/0
(3) The authors make muchado as to the fact that one form fits their data better at low concentrations while the other form fits their data better at high concentrations. In reality, both equations, if properly treated statistically, should give identical estimates of q, and b. Equation 3 is obtained from eq 2 by divided by C,. These equations correspond to the linearizations of the Michaelis-Menten enzyme kinetics equation (4)
attributed to Woolf and to Lineweaver and Burk, respectively, i.e.
(5)
: (e)( A) v,,, 1
=
+
where the variables u and [SI and parameters Kmand Vm, in eq 4 can be identified with the variables qe and C, and parameters ( l / b ) and go in eq 1. Statisticians have demonstrated repeatedly that differences in parameter estimates obtained by using eq 5 and 6 to linearize eq 4 result from incorrect weighting of the experimental data. The same should hold true when eq 2 and 3 are used to linearize eq 1. I submit that the conclusions of Walters and Luthy concerning the relative merits of eq 2 and 3 are based on improper statistical analysis. All of the equations used by these authors, with the exception of Henry’s law, are nonlinear in their pa-
0 1985 American Chemical Society
Environ. Sci. Technol., Vol. 19, No. 9, 1985
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Environ. Sci. Technol. 1985, 19, 870-871
rameters and should be fitted by a properly weighted nonlinear regression process. This error casts doubts on the results and conclusions of this otherwise elegant study. I hope that the authors will be permitted the opportunity to reexamine their conclusions.
Literature Cited (1) Walters, R. W.; Luthy, R. G. Enuiron. Sci. Technol. 1984, 18, 395-403.
Table I. Comparison of Henry’s Law Estimates” compound
KH
NA ACY ACE FLE AN PY
2390 5270 16800 5640 6140 30500
linearized bqo bqo/KH 2690* 6990* 17300* 12900 6210* 51300
1.13 1.33 1.03 2.29 1.01 1.68
nonlinear bqO
b4OIKH
1330 1950 8820 5280* 5660 41300*
0.56 0.37 0.53 0.94 0.92 1.35
“ Asterisks indicate bqo value that best agrees with KH.
*Address correspondence to this author at Department 493, Abbott Laboratories, 14th and Sheridan, North Chicago, IL 60064. Table II. Comparison of Limiting Capacity Estimates
Willlam R. Porter* School of Pharmacy Center for Health Sciences University of Wisconsin-Madison Madison, Wisconsin 53706
SIR: Porter (1) indicates that (1)estimates of parameters of nonlinear equations should be obtained by fitting data using a properly weighted nonlinear regression process and (2) parameter estimation based on linearized forms results in errors that cast doubts on the results and conclusions of the study. These points raise important questions regarding generally accepted techniques for treatment of experimental data. First, we agree with Porter’s first claim; in general, estimates of parameters of nonlinear equations should be obtained by fitting data using a properly weighted nonlinear regression process. However, one difficulty here is in what constitutes a “properly weighted” nonlinear regression process. For the subject equation, the weighting scheme should properly reflect the way in which the variance of qe varies with qe, which can only be accurately known for data sets with a large number of points ( n > 50) (2). In our work, we are dealing with data sets of varying size n ranging from 11 to 48, which are not large enough to evaluate this variation with statistical certainty. This requires that assumptions be made about the variation, which introduces other issues for statistical criticism. On the other hand, there is no question that our approach of linearization where possible is a generally accepted one. Such classic environmental engineering texts as Weber (3) and Gaudy and Gaudy (4) prescribe the use of linearizations for this type of equation. Researchers continue t,o use these techniques in their analysis of data (5,6). Can there be any justification for the simplification provided by linearization? One way to answer this question is to compare parameter estimates from a nonlinear regression process to our data. Nonlinear estimates were determined by numerical optimization based on an efficient version of the steepest decent method; the observations were given equal weight, and the algorithm sought to minimize the sum of the squares af the errors. Results from this regression exercise are shown in Tables I and 11. Table I shows values for Henry’s law constants applicable to the low concentration region of the isotherm plots. Our prior graphical and “linearized Langmuir estimates of KH are shown with the nonlinear estimates. Asterisks denote the value of bqo (from either the linearized or nonlinear approach) which agrees most closely with the graphical value of KH. This shows that, for the six compounds for which a value of KH was clearly evident, the linearized estimate is in better 870
Environ. Sci. Technol., Vol. 19, No. 9, 1985
compound
sample size n
linearized q0
nonlinear q0
ratio (linearized/ nonlinear)
NA ACY ACE FLE PH AN FLA PY TRP BaA CHR
46 28 30 48 32 22 28 28 21 17 11
580 500 465 440 231 23.1 104 82.8 36.8 14.7 38.4
517 465 457 426 311 23.8 85.2 90.2 28.4 18.7 39.4
1.12 1.08 1.02 1.03 0.74 0.97 1.22 0.92 1.30 0.79 0.97
agreement than the nonlinear estimate. Table I1 shows linearized and nonlinear estimates of qo, from which it is apparent that there is very good agreement between the two values (agreement to within a difference of 30%). In view of this discussion, our use of linearized techniques certainly appears reasonable. Another reason for selecting linearized vs. nonlinear techniques pertains to the ultimate objective of the estimate. Clearly, if one is trying to evaluate apparent Henry’s law constants or limiting capacities from an isotherm, it makes sense to use a technique that weights data in either of these respective regions rather than to compromise the estimate by fitting the entire isotherm. Our approach follows this objective. In contrast, if one were trying to elucidate Langmuir parameters for a system that is expected to conform to the Langmuir model (e.g., monolayer coverage and homogeneous surface sites), it may be appropriate to consider the entire isotherm. While the Langmuir model is useful for describing isotherm data on an empirical basis, it is generally believed that few real adsorbing systems behave mechanistically according to the Langmuir model. It is apparent that there is a tradeoff to be made here. Nonlinear procedures give a more statistically accurate estimate of parameters at greater cost (a nonlinear regression routine and computer time must be available). The nonlinear approach has the added advantage of having only one set of parameters. Linearized procedures give results quickly, but the estimates may have statistical limitations. While having to keep track of which of the two sets of parameters to use from the latter technique is inconvenient, it is an effective means by which to remind users of the data of the experimental limits and ranges of applicability of these data. These points aside, we hope that the most important contributions of our work remain clear: (1)the general magnitude of adsorption onto activated carbon for this class of compounds and (2) the differences in trends in adsorption at low and high relative concentration.
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0 1985 American Chemical Society
