CORRESPONDENCE
relation Equation 4. If the mixture is formed by adding a fraction 7 of parcel 1 and a fraction (1 - 7) of parcel 2, then
SIR: Calvert ( I ) has suggested an explanation for the apparent deviations of atmospheric ozone and nitrogen oxides concentrations from the photostationary state relationship in photochemical smog. We wish to demonstrate in this correspondence that Calvert’s suggestion does, in fact, provide an explanation for the observed deviations. The principal reactions governing the relationship between the oxides of nitrogen and ozone in photochemical smog are
0
+ O2 + M3
2
O3 + NO-NO2
+
O3 M
(2)
+ O2
(3)
At 25 “C and typical sunlight intensities, k l varies between 0 and 0.6 min-l (I),kz = 2.0 X 10-5ppm-1 min-l (with M = lo6 ppm) (Z),and k 3 = 25.2 ppm-l min-l(Z). Reactions 1-3 are generally fast relative to other reactions involving oxides of nitrogen and ozone in the urban atmosphere. Consequently, Reactions 1-3 achieve a steady state in which the rates of the three reactions are equal. In this steady state, called the photostationary state ( 3 ) , (4)
A number of investigators have reported deviations of ambient data from Equation 4. Eschenroeder et al. ( 4 ) reported that Los Angeles monitoring data did not obey Equation 4. In an analysis of the LARPP data (5),Calvert ( I ) found that the measured values of the left-hand side of Equation 4 generally exceed the value of the right-hand side based on solar irradiance measurements of k l and the known value of k 3 . He suggested that the discrepancy arises because the sampled air mass is heterogeneous, consisting of pockets of NO-rich air originating from fresh emissions mixed with pockets of relatively 03-rich background air. Thus, for the most part, Equation 4 is valid locally in the atmosphere; however, when spatially-averaged data (or equivalently, time-averaged data) are used to test Equation 4, the existence of heterogeneity of the sample gives rise to an apparent deviation of the data from satisfying Equation 4. We wish to demonstrate here the feasibility of this hypothesis. Consider an initial mixture of NO2, NO, and O3 of concentrations [NO&,, [Nolo, and [03]0. At equilibrium, or photostationary state, the concentrations of the three species are
where
We can consider the initial concentrations, [NOzlo, [Nolo, and [O3lO,to be the result of the mixing of two separate air parcels, labeled as 1 and 2, one with concentrations, [NOzll, [N0]1, and [03]1 and the other with concentrations, [N02]2, [N0]2, and [03]2. Note that the concentrations in air parcels 1 and 2 are assumed to satisfy the photostationary state 1218
EnvironmentalScience & Technology
[No210 = ~ [ N 0 2 1+ i (1- 1)[NO2I2
(9)
+ (1 - T ) [ N O ] ~
(10)
[Nolo = o[NO]i [0310
= do311-k (1 - )1)[0312
(11)
Equations 5-11 give the local concentration of an air parcel (in photostationary equilibrium) that results from the mixing of two other air parcels (each in photostationary equilibrium). The concentrations depend on the mixing fraction 7 as well as on the concentrations in the two air parcels before mixing. Reported ambient concentrations usually represent an average of instantaneous, or local, concentrations over a specified sampling interval. Let us suppose that over the sampling interval the air sampled is characterized by a distribution of values of the mixing fraction 7. In fact, the mixing fraction 7 a t any point can be characterized by a probability density function p,,(q). Then the average concentration of each species, NO2, NO, and 03,over a sample is simply,given by, e.g.,
with similar relations for NO and 03. The problem of interest is then whether the photostationary state relation ( 4 ) written in terms of the average concentrations is obeyed. Specifically, we seek to determine if the quantity
is equal to unity. Because 0 depends nonlinearly on the mixing fraction 7, it is, of course, evident that the average of the photostationary state relation,
is not equal to the photostationary state relation based on the averages, Le., Equation 13. In fact, ( 0 ) = 1, since Equation 4 is assumed to hold locally at every point. We wish to calculate 0 from Equation 13 for a simple choice of p,,(v).To illustrate the elements of the phenomenon, we assume that TJ is normally distributed with mean i j and standard deviation a,
Since 7 is a fictitious parameter, our choice of its probability density function should not bear too strongly on the physical plausibility of our arguments. The normal density is perhaps an obvious choice, although a more detailed study would involve a more thorough analysis of p,,(~).Finally, note that because 7 varies only from 0 to 1, it is necessary to normalize p vby the value of its integral from 0 to 1. We are interested in examining 8 as a function of the following variables: [NOzll, [N0]1; [NO2]2, [NO]2;i j and a. Figure 1 shows 0 as a function of u for k l = 0.4 mind’, i j = 0.5 and [NOz]l = 0.01 ppm, and [NO11 = 0.10 ppm for two sets of values of [NO212 and [NOIz. The concentrations, [NO211and [N0]1, reflect those that might exist in a parcel of “sourcerich” air, whereas [NO212 and [NO12 are typical of ambient polluted atmospheric concentrations. We selected the value of the mean mixing fraction i j as 0.5, representing, perhaps, early morning conditions when deviations from Equation 4 have been most marked (1). The behavior of 0 is governed strongly by the standard deviation a of the fluctuations in TJ. At [NO212 = 0.1 ppm and [NO12 = 0.01 ppm, 0 attains a value
Thus, for liquids x = l/y, since f R = f L , and for solids x = In terms of I.L mol/L, the correlating parameters used by Chiou can be related to x; hence, y for dilute solutions as: (fs/fR)/?w.
S = 55.5 x x lo6 pmol/L or
S = 55.5 X 106/y, (liquids) or 55.5 X
lo6 ( f s / f R ) / y , (solids)
The partition coefficient K in units of g or mol/L ratio can be obtained from the fugacity equation written as Figure 1. Photostationary state group as a function of standard deviation u of mixing fraction 7 between source-rich and background urban
air of 1.78 at u = 0.5, whereas a t [NO212 = 0.2 ppm and [NO12 = 0.02 ppm, 0 reaches 1.43 at u = 0.5. The effect of the intensity of the fluctuations in the mixing fraction 7 on the ratio 0 is understandable because this level of fluctuations determines how greatly 0 differs from ( O ) , Le., unity.
Literature Cited (1) Calvert, J. G., Enuiron. Sci. Technol., 10,248 (1976). (2) Hampson, R. F., Jr., Garvin, D., Eds., Chemical Kinetic and Photochemical Data for Modelling Atmospheric Chemistry”, NBS Tech. Note 866,1975. (3) O’Brien, R. J., Enuiron. Sci. Technol., 8,579 (1974). (4) Eschenroeder, A. Q., Martinez, J. R., Nordsieck, R. A., “Evaluation of a Diffusion Model for Photochemical Smog Simulation”, Final Rep. of General Research Corp., Santa Barbara, Calif., to EPA Contract No. 68-02-0336, 1972. ( 5 ) Eschenroeder, A. Q., “Los Angeles Reactive Pollutant Program (LARPP) Summary of Data Management Activities”, Environmental Research and Tech. Inc. Document P-165613, 1976.
John H. Seinfeld Chemical Engineering California Institute of Technology Pasadena, Calif. 91 125
f
= x,y,fR
where subscripts w and o refer to the water and octanol phases. The reference fugacities f R cancel, regardless of whether the solute is solid or liquid at the system temperature. Now C, is x, 55.5 mol/L, and C, is 6.36 x,. Since the density of octanol is 827 g/L and its molecular weight is 130, 1L of octanol contains 6.36 mol. Thus,
K = C,/C,
= 6.36 x,/55.5 x,
f
= xy,fR
where x is the mole fraction of the compound in water, y, is its activity coefficient in aqueous solution, and f R is the reference fugacity, i.e., the fugacity of the pure liquid compound at the system temperature. For liquid compounds f R is simply the liquid vapor pressure f L . For solids f R is the extrapolated liquid fugacity (below the triple point), not the solid fugacity (or vapor pressure) f S . However, the ratio f R / f S can be estimated by standard techniques (2).
= 0.115 y,/yo
A correlation between log K and log S is thus a correlation between (log y, - log yo - 0.94) and (-log y, log ( f s / f R ) 7.74). Since y, varies from a value of about lo3for chloroform to about IO9 for 2452‘4‘5’ PCB, its variation dominates the correlation. If yo and ( f S / f R )were constant, the correlation illustrated in Figure 1 by Chiou would be a line of slope -1. This is nearly true for liquids, as can be seen by the clustering of the points for liquid compounds at the right of Figure 1 about a slope of -1. In practice, for solid compounds as molecular weight and melting point increase ( f s / f R ) becomes smaller and yo increases as the compound exhibits more positive nonideality in the octanol phase, due in part to the increase in molar volume difference. By inserting the above definitions of K and S in Chiou’s correlation and rearranging it, it can be shown that
+
+
(fs/fR)Yo=
SIR: Chiou et al. recently obtained an empirical correlation between n -octanol/water partition coefficients and aqueous solubilities for a wide variety of compounds ( I ) . Since partition coefficient or solubility correlates with biomagnification, there is justification for using these purely physical-chemical properties as estimators of environmental and toxicological behavior. The authors omitted to mention that there is a physical-chemical basis for their correlation; indeed, both the partition coefficient and aqueous solubility are to a large extent functions of the same property, the aqueous phase activity coefficient of the compound, and as shown below the correlation is partly between that property and its reciprocal. The aqueous solubility of a hydrophobic compound in water can be expressed in terms of the fundamental equation in its fugacity f , using the Raoult’s law convention,
= X,7,fR
0.0158
This is an empirical relationship (equivalent to the Chiou correlation) which states that as molecular weight, melting point, and hydrophobicity increase, the fugacity ratio is observed to decrease and yo is observed to increase in proportion to the cube root of the decreasing solubility. If data were available for yo, melting point, entropy of fusion, and solid and liquid heat capacities, then the precise relationship between K and S could be calculated. Chiou’s work is entirely valid, and it will be useful as a means of overcoming a lack of physical-chemical data needed for interpretation of biomagnification and toxicity studies and for checking the “reasonableness” of K and S values. It is, however, noteworthy that part of the success of the correlation is attributable to correlating a quantity against its reciprocal. Finally, the quoted value for the solubility of benzene (820 ppm) is substantially lower than the more recent values which are about 1780 ppm (3).
Literature Cited (1) Chiou, C. T., Freed, V. H., Schmedding, D. W., Kohnert, R. L., Enuiron. Sci. Technol., 11 ( 5 ) ,475 (1977). (2) Prausnitz, J. M., “Molecular Thermodynamics of Fluid Phase Equilibria”, Prentice-Hall, Englewood Cliffs, N.J., 1969. (3) McAuliffe, C., J . Phys. Chem., 70,1267 (1966).
Donald Mackay Department of Chemical Engineering and Applied Chemistry University of Toronto Toronto, Canada M5S 1A4
Volume 11, Number 13, December 1977 1219
