Anal. Chem. 1991, 63, 1011-1013
salt and its gaseous precursors (11). Under controlled conditions, this sublimation equilibrium could be used in a similar way as with ammonium chloride to provide a much needed measurement standard for nitric acid. Ammonium halides sublime with the rate of sublimation, increasing in the order NH41< NH4Br < NHICl < NH4F (21),thus providing sources of their respective acids. Ammonium formate and ammonium acetate also excert a vapor pressure of acid over the salt as recognized from the pungent odor of these salts. Ammonium hydrogen sulfide decomposes into hydrogen sulfide and ammonia at room temperature. Trace concentrations of many of these gases can be useful as measurement standards.
ACKNOWLEDGMENT I thank P. K. Dasgupta and K. Irgum for valuable discussions and M. Sharp for linguistic revision of the text. Registry No. HCl, 7647-01-0; NH,+.Cl-, 12125-02-9.
1011
(5) Denyszyn. R. E.; Sassaman, T. ASTM Spec. Tech. Pub/. 1987, 957, 101-109. (6) Namiesnik; J. J. Chrom. 1984. 300, 79-108. (7) Jockel, W.; W c e k , C.; Eollmacher, H. Staub-Reinhelt. Luft 1887. 47, 280-282. (8) Scarano, E.; Calcagno, C.; Cignoli, L. Anal. Chlm. Acta 1979, 170, 95-106. (9) Jockel, W. Staub-Reinha#. Luft 1880, 40, 145-150. (10) Pio, C. A.; Harrison, R. M. Atmos. Envlfon. 1887, 2 1 , 1243-1246. (11) Allen, A. G.; Harrison, R. M.; Erisman, J.-W. Atmos. Envlron. 1988, 23, 1591-1599. (12) Plo, C. A.; Harrison, R. M. Atmos. Environ. 1987, 2 1 , 2711-2715. (13) Lindgren, P. F.; Dasgupta, P. K. Anal. Chem. 1988, 61, 19-24. (14) Genfa, 2.; Dasgupta, P. K. Anal. Chem. 1989, 61, 408-412. (15) Wagner, H.; Neumann, K. 2.Phys. Chem. 1981, 2 8 , 51-70. (16) Clementi, E.; Gayies, J. N. J. Chem. Phys. 1987, 47, 3837-3841. (17) Stephenson, C. C. J. Chem. Phys. 1944, 12, 316-319. (18) de Kruif, C. G. J. Chem. Phys. 1882, 77, 6247-6250. (19) Goldfinger, P.; Verhaegen, G. J. Chem. Phys. 1989, 50, 1467-1471. (20) Hradil, J.; Svec, F.; Kalal, J.; Eelyakova. L. D.; Kiselev, A. V.; Platonova, N. P.; Shevchenko, T. I. React. Polym. 1982, 1 , 59-65. (21) Chaiken, R. F.; Sibbet, D. J.; Sutherland, J. E.; Van De Mark, D. K.; Wheeler, A. J. Chem. Phys. 1962, 37, 2311-2318.
LITERATURE CITED (1) Goldan, P. D.; Kuster, W. C.; Albritton, D. L. Atmos. Envlron. 1986, 2 0 , 1203-1209. (2) Stellmack, M. L.; Street, K. W. Am. Lab. 1982, dec., 25-33. (3) Mlguei. A. H.; Natusch, D. F. S. Anal. Chem. 1975, 4 7 , 1705-1707. (4) Dasgupta. P. K.; Dong, S. Atmos. Envlron. 1988, 2 0 , 565-570.
RECEIVED for review October 12,1990. Revised manuscript received February 6,1991. Accepted February 11,1991. This work was supported by the Swedish Environmental Protection Agency.
Boundary Layer Effect in Diffusive Monitoring Dwight W. Underhill* and Charles E. Feigley Department of Environmental Health Sciences, School of Public Health, University of South Carolina, Columbia, South Carolina 29208
I n diffusive sampling, It is usually assumed that the rate of uptake is directly proportional to the diffusion coefficients of the analytes in air. But this may not be correct, because the anaiytes must pass through a boundary layer of air before reaching the sampler. I t was found, in agreement with boundary layer theory, that the resistance to mass transfer across this layer Is a nonlinear function of the diffusion coefficients of the anaiytes in air. This becomes important in calibrating a diffusive sampier for a wide range of analytes. the same phenomenon is found in absorption towers, and it is suggested that the calibration of diffusive samplers include a factor similar to the Schmidt number used in the analysis of the performance of absorption towers.
INTRODUCTION Much of the current data regarding human exposures to toxic gases and vapors in the workplace as well as in the home has been obtained through the use of diffusive samplers. These devices use diffusion through a quiescent layer to obtain an uptake that is proportional to both the time of exposure and to the arithmetic mean airborne analyte concentration. One common assumption underlying diffusive samplingthat the uptake is directly proportional to the diffusion coefficient of the analyte-is in general not true and can lead to significant error in otherwise carefully controlled work. A nonlinear relationship occurs because, between the air in the workplace, which can always be considered to be in turbulent motion, and the quiescent space within the diffusive sampler there must be a boundary layer through which the analytes 0003-2700/91/0363-1011$02.50/0
pass. In such boundary layers, the transport of analyte is proportional to a fractional power of the diffusion coefficient. Various conceptual models of the physical processes within a fluid boundary layer have led to different predictions of the dependence of the mass flux on the molecular diffusion coefficient, D. Higbie’s unsteady molecular diffusion (UMD) model (1)assumes that fluid elements do not remain a t the boundary long enough to achieve equilibrium and that mass transfer into these elements is controlled by molecular diffusion. Applying the usual diffusion equation
where D is the diffusion coefficient for analyte in air, cm2/s; C is the concentration of analyte in air, g/cm3; t is the time, s; and x is the distance, cm; to the mass transfer by diffusion into a surface element of lifetime, to,gives the average flux into that surface element over its lifetime as
J=2 C 0 d z where to is the lifetime of the surface element, s; Co is the concentration of analyte in the bulk of the fluid, g/cm3; and J is the flux of analyte, g/cm2/s. The UMD model predicts that the rate of mass transfer is a function of the square root of the molecular diffusion coefficient. Different surface elements may have different lifetimes, but it can be shown that, regardless of the age distribution of these surface elements, eq 2 always yields a flux proportional to (2). This model was originally developed to characterize gas and liquid-phase 0 1991 American Chemical Society
1012
ANALYTICAL CHEMISTRY, VOL. 63, NO. 10, MAY 15, 1991
Table I. Diffusion Coefficient ( D ) ,Sampling Rate (Q& and Boundary Layer Conductance (Qbl) for the 3M Model 3500 Diffusive Sampler
D,cm2/s Qo, cm3/s
compd
Figure 1. 3 M Model 3500 organic vapor monitor.
mass-transfer resistances in absorption columns and is still important in that application (3). A similar dependence has been predicted for mass transport into diffusive samplers (4). There are other hydrodynamic models for mass transfer that rely on the similarity between heat, mass, and momentum transfer. The Reynolds analogy, and the Coburn-Chilton (5) analogy developed from it, assume that the rate of mass transfer is controlled by turbulence rather than by molecular diffusion. In particular, the Coburn-Chilton analogy predicts that when the ratio of the kinematic viscosity to the diffusion coefficient (i.e., the Schmidt number) is near unity, as is to be expected for airborne analytes, the mass flux is proportional to D2/3(2). However, these models have been important primarily for mass transfer to or from liquids in contact either with solids or with much more viscous liquids. These latter models would seem applicable to diffusive samplers were the analyte sampled from a solution rather than from air.
BASIS OF THE ANALYSIS The sampling rate (Bo, in cm3/s) for an analyte depends on the conductance across two regimes placed in series, i.e., across the external boundary layer, and the sampler’s internal air gap. The sampling rate is the harmonic mean of the conductances across these two regimes. Thus, 1 1 1 (3)
_-
80- Si
+ -Qbi
where Qo is the sampling rate, cm3/s; Qi is the internal conductance of the sampler, cm3/s; and Qbl is the conductance of the boundary layer, cm3/s. Rearranging eq 3 gives Qbl
=
QiQo/(Qi
- 80)
(4)
which permits the conductance of the boundary layer, Qbl, to be calculated from Qi and Qo, which are both determinable parameters. The overall sampling rate, Qo, is equal to
Qo = m / t c where m is the mass of analyte sampled, g; t is the sampling time, s; and c is the atmospheric analyte concentration, g/cm3. The internal conductance, Qi, is calculated from
= AD/L (5) where D is the diffusion coefficient of analyte in air, cm2/s; A is the internal cross-sectional area, cm2;and L is the air gap thickness, cm. Qi
DEVELOPMENT OF THE ANALYSIS Figure 1 shows the 3M Model 3500 diffusive sampler for organic vapors. The sampling rates for 56 organic analytes, as reported by 3M (6)for the 3M Model 3500 diffusive sampler are given in Table I. The factors A and L were determined by direct measurement. For the 3M Model 3500 diffusive sampler, the ratio of AIL is 8.54 cm (7).Diffusion coefficients for 56 of the compounds were estimated by 3M using the
acetone amyl acetate benzene bromochloromethane bromoethane butanol 2-butanone 2-butox yethano sec-butyl acetate 4-tert-butyltoluene carbon tetrachloride chlorobenzene cumene cyclohexane cyclohexanol cyclohexanone cyclohexene 1,2-dibromoethane 1,2-dichlorobenzene 1,2-dichloroethane 1,2-dichloroethylene dichloromethane 1,2-dichloropropane 2,6-dimethyl-4-heptanone 2-ethoxyethanol 2-ethoxyethyl acetate ethyl acetate heptane hexane 2-hexanone 4-hydroxy-l-methyl2-pentanone isophorone mesitylene 2-methoxyethanol 2-methoxyethyl acetate methyl acetate methylcyclohexane 2-methyl-1-propanol 2-methyl-1-propyl acetate 4-methyl-2-pentanone a-methylstyrene nonane octane pentane 1-pentanol 2-pentanol 2-pentanone 1-propanol propyl acetate styrene tetrachloroethylene toluene lI1,2-trichloroethane l,l,l-trichloroethane trichloroethylene xylene
Qbi,
Cm3/s
0.1096 0.0668 0.0947 0.1005 0.1013 0.0879 0.0943 0.0681 0.0728 0.0599 0.0857 0.0812 0.0690 0.0851 0.0760 0.0802 0.0876 0.0824 0.0732 0.0973 0.0992 0.1102 0.0833 0.0606 0.0820 0.0682 0.0883 0.0721 0.0796 0.0756 0.0707
0.668 0.433 0.592 0.573 0.607 0.572 0.605 0.470 0.477 0.345 0.503 0.488 0.408 0.540 0.492 0.482 0.538 0.493 0.463 0.553 0.587 0.632 0.510 0.410 0.540 0.443 0.575 0.482 0.533 0.495 0.470
2.34 1.80 2.20 1.73 2.03 2.40 2.43 2.45 2.04 1.06 1.61 1.65 1.33 2.10 2.03 1.62 1.92 1.65 1.79 1.66 1.91 1.92 1.80 1.97 2.36 1.86 2.42 2.21 2.48 2.12 2.12
0.0635 0.0660 0.0911 0.0740 0.1009 0.0769 0.0908 0.0793 0.0761 0.0700 0.0617 0.0664 0.0864 0.0787 0.0790 0.0838 0.1004 0.0793 0.0764 0.0786 0.0827 0.0836 0.0855 0.0874 0.0748
0.362 0.438 0.605 0.483 0.617 0.482 0.598 0.517 0.500 0.417 0.410 0.443 0.575 0.520 0.538 0.550 0.662 0.502 0.447 0.472 0.523 0.495 0.515 0.518 0.455
1.09 1.97 2.72 2.06 2.17 1.81 2.62 2.18 2.17 1.38 1.85 2.03 2.61 2.30 2.66 2.38 2.90 1.94 1.42 1.59 2.02 1.61 1.75 1.70 1.58
Hirschfelder equation, as modified by Wilke and Lee (8). These values are given in Table I. This table also gives the conductance across the boundary layer, as calculated by using eqs 4 and 5. Because the boundary layer conductance was determined from the difference between two larger values, the experimental error is relatively high. Nevertheless, the large number of data pairs makes it possible to make meaningful deductions. Of particular interest is how the boundary layer conductance varies as a function of the diffusion coefficient of the analyte in air. The quiescent air film model, the Coburn-Chilton analogy, and the U M D model predict relationships of the form In
Qbl
=a
+ b In D
(6)
ANALYTICAL CHEMISTRY, VOL. 63, NO. 10, MAY 15, 1991
between the boundary layer conductance and the diffusion coefficient of the analyte in air. These models differ by predicting values of 1.0, 0.67, and 0.5, respectively, for the coefficient, b, in eq 6. By using the values in Table I, linear regression of the natural logarithm of the boundary layer conductance vs the natural logarithm of the diffusion coefficient for the analyte in air gives In Qbl = 2.086
+ 0.563 In D
(7) with the values of a and b being equal to 2.086 and 0.563, respectively. The coefficient of correlation, r, for this regression is 0.39. Furthermore, the "t" statistic (9) for the slope of the regression line is 5.57(0.563 - B), with 54 degress of freedom, where B is a possible value of the slope. This result is inconsistent with the quiescent air film model (in which it is assumed that B = 1.0) because the t test gives a probability of 0.02 for B 11.0. However, neither the UMD model nor the Coburn-Chilton analogy can be excluded. DISCUSSION AND CONCLUSIONS The role of the boundary layer mass-transfer resistance must be understood if high accuracy is to be obtained with diffusive samplers. For example, the short internal air gaps used in some current diffusive samplers give high sampling rates, but in these same diffusive samplers, the boundary layer mass-transfer resistance is a significant part of the overall mass-transfer resistance. For the diffusive sampler investigated here, the boundary layer contributed about 30% of the total mass-transfer resistance. The dependence of the sampling rate on the diffusion coefficient can be rewritten by using dimensionlessparameters, e.g., q = (1
+ ruSmfl)-'
(8)
where q is the nondimensionalized sampling rate = Q&/AD; Sm is the Schmidt number, dimensionless = p/pD; p is the viscosity of air, P (0.OOO 185P at 25 O C , 1atm); p is the density of air, g/cm3 (0.001 185 g/cm3 at 25 "C, 1atm); and a and ,9 are coefficients for the Schmidt number. The coefficient, a, is expected to be a function of the sampler geometry and the air flow. A similar factor containing the Schmidt number is commonly used in the design of absorption towers, where the UMD model has found widespread application (3). For the diffusive sampler examined here, the factors CY and ,9 are 0.471 and -0.437, respectively. Equation 8 can be rewritten to give the sampling rate for a compound having a known diffusion coefficient; e.g.,
80=
AD(1
+ crSm@)-l L
(9)
and this result can be applied equally well to a single diffusive sampler simultaneously sampling a number of analytes or to a set of diffusive samplers independently sampling different analytes under similar conditions. Such calculations will clearly show that, as is common with boundary layers, if the diffusion coefficient increases, the relative contribution of the boundary layer to the overall mass transfer also increases.
1019
Finally, these findings explain an earlier puzzling result. The quiescent film model predicts that the sampling rate is directly proportional to the diffusion coefficient for the analyte in air. In this model, a linear regression of the sampling rate vs diffusion coefficient in the form
Qo = a'+ b'D
(10) should yield a statistically insignificant value for a ' because the regression line should pass through the origin. In fact, this test was carried out earlier (7), and positive values for a' were found. Applying this same analysis to the data given in Table I gives values of 0.068 and 5.45 for a' and b', respectively. The value for a'is significantly different from zero (based on a one "tail" test for significance, a < 0.01). This result is difficult to explain except in terms of the theory presented above.
ACKNOWLEDGMENT We thank the 3M Occupational Health and Environmental Safety Division for providing the diffusion monitor illustration. Registry No. Acetone, 67-64-1; amyl acetate, 628-63-7; benzene, 71-43-2; bromochloromethane, 74-97-5; bromoethane, 74-96-4;butanol, 71-36-3; 2-butanone, 78-93-3; 2-butoxyethanol, 111-76-2;sec-butylacetate, 105-46-4;4-tert-butyltoluene,9851-1; carbon tetrachloride,56-23-5; chlorobenzene, 108-90-7;cumene, 98-82-8; cyclohexane, 110-82-7; cyclohexanol, 108-93-0; cyclohexanone, 108-94-1; cyclohexene, 110-83-8;1,2-dibromoethane, 106-93-4; 1,2-dichlorobenzene,95-50-1; 1,2-dichloroethane,10706-2; 1,2-dichloroethylene,540-59-0; dichloromethane,75-09-2; 10883-8; 1,2-dichloropropane,78-87-5; 2,6-dimethyl-4-heptanone, 2-ethoxyethanol, 110-80-5;2-ethoxyethyl acetate, 111-15-9;ethyl acetate, 141-786;heptane, 142-82-5;hexane, 110-54-3;2-hexanone, 123-42-2;isophrone, 591-78-6;4-hydroxy-4-methyl-2-pentanone, 78-59-1; mesitylene, 108-67-8; 2-methoxyethanol, 109-86-4; 2methoxyethyl acetate, 110-49-6; methyl acetate, 79-20-9; methylcyclohexane, 108-87-2; 2-methyl-1-propanol, 78-83-1; 2methyl-1-propoyl acetate, 110-19-0;4-methyl-2-pentanone,10810-1;ahmethylstyrene, 98-83-9;nonane, 111-84-2; octane, 111-65-9; pentane, 109-66-0; 1-pentanol, 71-41-0; 2-pentanol, 6032-29-7; 2-pentanone, 107-87-9;1-propanol,71-23-8; propyl acetate, 10960-4; styrene, 100-42-5; tetrachloroethylene, 127-18-4;toluene, 108-88-3; 1,1,2-trichloroethane, 79-00-5; l,l,l-trichloroethane, 71-55-6; trichloroethylene, 79-01-6; xylene, 1330-20-7. LITERATURE CITED (1) Higbie, R. Trans. Am. Inst. Chem. Eng. 1935, 31, 365-389. (2) Astarita, G. Mass Transfer with Chemical Reaction; Ekevier Publishing Co.: New York, 1967; Chapter 1. (3) Perry, R. H.; Green, D. W.; Maloney, J. 0. Peny's Chemical Engineers' Handbook. 6th 4.;McGraw-Hill Book Co.: New York, 1984. (4) Zurlo, N.; Andreoletti, F. Effect of Air Turbulence on Dlffuslve Sampling. In Dlffusive Sampling: An Altemattve Approach to Workplece Ak Monltorlng; Berlin, A., Brown, R. H., Saunders, K. J., Eds.; The Royal Society of Chemistry: Burlington House: London, 1987. (5) Perry, R. H.; Green, D. W.; Maloney. J. 0. Perty's chemical Engineers' Handbook, 6th ed.;McGraw-Hill Book Co.: New York. 1984. (6) 3M. 3M Orgenic Vapor Monltor Sampling Guide for Grganic Vapor Monnor #3513510, Organic Vapor Monitor with Back-Up SeCMon #3520 Pub. No. R35SGA (1221)R2, St. Paul, MN, 1982. (7) Felgley, C. E.; Lee. B. M. Am. Ind. Hyg. Assoc. J . 1987, 49, 266-269. (8) Wllke, C. R.; Lee, C. Y. Ind. Eng. Chem. 1055, 47, 1253-1257. (9) Spiegel, Murray R. StaNStics;McGraw-Hill Book Co.: New York, 1961.
RECEIVED for review October 15, 1990. Accepted February 19, 1991.
