Anal. Chem. 1989, 61, 843-847
emissions stopped, reduced, or vented. A study of cigarette patterns indoors could also prove useful in discussing secondary exposure to cigarette smoke. Row order data entry produces easily interpretable eigenchromatograms. These abstract chromatograms often correspond to chromatograms of volatile organics from a particular source. With further refinement and establishment of a library of volatile patterns from common organic sources, eigenchromatograms could be matched to chromatograms of the indexed sources. This approach could be invaluable in off-site monitoring of industrial atmospheric effluents in areas where there are several major sources of atmospheric organics.
ACKNOWLEDGMENT The intricacies of the original computer program were explained by Mark Dale. His program was the backbone of the program used in this study. Timely suggestions for improved data analysis were provided by Dr. J. A. Llewellyn. LITERATURE CITED (1) Cobb, 0. P.; Hua, L. M.; Braman, R. S. Anal. Chem. 1986, 58(11), 2213-2217. (2) Cobb. 0. P.; Braman, R. S. Measurement of Toxic Ah Poilutants; USEPAIAPCA: Raleigh, NC. 1986; pp 314-325.
a43
(3) Harman, H. H. Modern Factor Analysis; The Unlversity of Chicago Press: Chicago, IL, 1976. (4) Selvers, R. Measuremnt of Toxic AC Pollutants; USEPAIAPCA: Raleigh, NC, 1986; pp 287-303. (5) Amlck, D. A.; Walberg, H. J. Introduction to Muhlvarlent Analysis; McCutchan Publishing Co.: Berkley, CA, 1975. (6) Malanowskl. E. R.; Howery, D. G. Factor Analysis in Chemisfty;John Wiley & Sons, Inc.: New York, 1980. (7) Coomans. D.;Broeckaert, I. Potenfhl Pattern RecognlNon ln Chemlml and Medichal Decision Maklng ; John Wiley & Sons, Inc.: New York, 1986. (8) Windig, W.; Jakab, E. Anal. Chem. 1987, 59(2), 317-323. (9) Gilbert, R. A.; Llewellyn, J. A.; Swartz, W. E.;Palmer, J. W. Appl. Spectrosc. 1982, 36(4), 428-430. (IO) Gllbert, R. A.; Llewellyn, J. A.; Swartz, W. E. Appi. Spectrosc. 1985, 39(2), 316-320. (11) Jurs, P. C. Science 1988, 232, 1219-1224. (12) McConnell, M. L.; Rhodes, G.; Watson, U.; Novotny, M. J . ChromatOgr. 1979, 762, 495-506. (13) Lavine, 6.; Carlson, D. Anal. Chem. 1987, 59(6), 468A-470A. (14) Lohniger, H.; Varmuza, K. Anal. Chem. 1987, 59(2), 236-244. (15) Ramos, L. S. Anal. Chem. 1986, 58(5), 294R-315R. (16) Delaney, M. F. Anal. chem. 1984, 56(5), 261R-277R. (17) Ildlko, E. F.; Kowalski, B. R. Anal. Chem. 1982, 54(5), 232R-243R. (18) Kowalski, 8. R. Anal. Chem. 1980, 54(5), 112R-122R. (19) Uguagliati, P.; Benedetti, A. Compuf. Chem. 1984. 8(3). 161-168.
RECEIVED for review August 25,1988. Accepted January 19, 1989.
Fundamental Factors in the Performance of Diffusive Samplers Daniel P. Adley
Liberty Mutual Insurance Company, 600 Grant Street, Pittsburgh, Pennsylvania 15219 Dwight W. Underhill*
School of Public Health, University of South Carolina, Columbia, South Carolina 29208
The performance of dlffuslve samplers can be determined from dlmensionless parameters characterizing the sampling time and the geometry of the dlffuslve sampler. Should the sorptlon isotherm contaln adJustableparameters, the number of dhmshbs parameters increases accordingly. For Hnear and lrreverdble Isotherms, sampler performance was caiculated In closed form, whereas for the Dublnln-Radushkevich Isotherm (which describes the uptake of many compounds on activated carbon), sampler performance was determined by using the Crank-Nlcobn implicit numerlcal procedure. For all three Isotherms, generailzed plots of sampling efflclency were developed as a function of sampllng tlme and sampler geometry. With the linear isotherm, a very rapld loss of sampilng efficiency occurs, whereas for both the DubinlnRadushkevich and the lrreverdble isotherms, hlgh sampling efficiencies can be malntained until a slzabie fractlon of the adsorbent is saturated. These calculatlons show the performance that can be expected from a dmusive sampler and, In particular, explain the hlgh pedomance seen with diffusive samplers containing an actlvated carbon adsorbent.
INTRODUCTION Figure 1 gives a schematic diagram of a common type of diffusive sampler. While this sampler is in use, the analyte diffuses across an internal air gap to a sorbent, which serves to retain it. The windshield at the front of the sampler serves 0003-2700/89/0361-0843$01.50/0
as a barrier to convective air movement, which otherwise would disrupt the rate-limiting aspect of the diffusion across the air gap in the diffusive sampler. The impervious barrier supporting the sorbent prevents loss of analyte through the back of the sampler. Ideally a diffusive sampler would, until the sorbent is exhausted, have an uptake rate exactly proportional to the instantaneous ambient concentration of analyte. This ideal response is usually only approximated because (1) for many analytes there is no useful sorbent for which the sorption process is irreversible and therefore loss of some sample by desorption and reverse diffusion occurs and (2) as the more exposed layers of sorbent become saturated, the distance that the analyte must travel before reaching fresh sorbent increases, decreasing the sampling rate of the analyte. Any realistic model of a passive sampler must include both effects.
THEORY Equations for Mass Transfer in a Diffusive Sampler. The model developed here includes these factors: (1) the external concentration of analyk, (2) the width of the air gap, (3) the diffusion coefficient of the analyte in air, (4) the porosity, (5) tortuosity, and (6) thickness of the sorbent pad, and (7) the sorption isotherm. In the following analysis, these factors are combined into a minimum of two dimensionless parameters, i.e. one dimensionless parameter describing the sampling time and another the geometry of the sampler and, should the sorption isotherm have adjustable parameters, additional nondimensional parameters for each of these. 0 1989 American Chemical Society
844
ANALYTICAL CHEMISTRY, VOL. 61, NO. 8, APRIL 15, 1989
Concentration gradient 1 Initially. 2 After sampling for an appreciable period. UEquilibrium concentration of
prediction of diffusion of sorbate across a sorption bed. Nondimensional Parameters. If the following nondimensional parameters are defined as reduced distance, x x
=x/w
(6)
c/c,
(7)
reduced concentration, c
sorbate in bulk adsorbent
c=
reduced partition coefficient,
k
k = K/Ko KO= value of K a t C = Co Concentration of sorbate in interparticle space
t
Air gap
Adsorbent
Ambient Concentration
mass transfer resistance, a
t
reduced time, r
Backing
Figure 1. Schematic diagram of a diffusive sampler.
r=
The partial differential equations for mass transfer in this system are as follows: Boundary Condition for Exposed Surface of Sorbent. The flux of analyte across the air gap to the sorbent pad is
D
F1 = -(Co - C(X=O)) L
where F2 is the flux of analyte into sorbent pad, g/(m2s), y is the tortuosity factor for interparticle diffusion, dimensionless, and e is the fractional interparticle void volume, dimensionless. In eq 2 the factors y and e account for the tortuous nature of the diffusion paths in the sorbent pad and the fact that only a fraction of the volume of the sorbent pad is available for interparticle diffusion. The buildup of analyte in the air gap of a diffusive sampler is negligible unless the air gap is exceptionally wide (1-3). Then assuming that the flux of the analyte entering the sorbent pad equals the flux of analyte to the sorbent pad (Fl = F&,the boundary condition for mass transfer at the exposed surface of the sorbent is
(3)
Boundary Condition ut the Unexposed Surface of the Sorbent. If no effluent flux of analyte is permitted at the unexposed surface of the sorbent, then
dc/ax(,=,
=
o
tD (1 - 4LWKO
and remembering that in practical diffusive samplers, the partition coefficient is always so high that (1- €)KO >> 1,then eq 3-5 can be rewritten nondimensionally as
(1)
where Fl is the flux of analyte across air gap, g/(m2 s), L is the width of air gap, m, D is the molecular diffusion coefficient for analyte in air, m2/s, Co is the ambient concentration of analyte, g/m3, C(xm0)is the concentration of analyte at the exposed surface of sorbent, g/m3, and X is the distance into sorbent pad, m. The flux of analyte into the sorbent pad is
rrLaclax(x=+o,= co - C ( X 4 )
(8)
(4)
where W is the thickness of the sorbent pad.
Mass Transfer within the Sorbent Pad. Mass transfer by diffusion within the sorbent pad is given by
(5)
(13) Solutions for Common Isotherms. The sampling efficiency, E , is defined as the ratio of the uptake of analyte to the uptake of analyte that would occur over the same period of time were there no mass transfer resistance and no reverse diffusion of desorbed analyte. In terms of the dimensionless variables given above, the sampling efficiency is equal to
E = 1-
(14)
dT/T
The sampling efficiency, which is the averaged overall sampling rate divided by the initial (and therefore most rapid) sampling rate, is not equivalent to the sampling accuracy. The latter depends on how well we determine the time weighted concentration from uptake, and in some cases this can be accurately done even after the sampling efficiency has dropped to a low value. Once the reduced partition coefficient (k)is known as a function of concentration, the above equations can be solved for the sampling efficiency as a function of the two dimensionlew parameters, a and T (plus any additional parameters introduced by the sorption isotherm/partition coefficient). We now give solutions calculated for three important sorption isotherms. Linear Isotherm. The most commonly considered isotherm is the linear isotherm, in which the equilibrium quantity of sorbed analyte is proportional to the partial pressure of the analyte. For this isotherm the reduced partition coefficient, K , equals 1(see Figure 2). Solving eq 11-13 for the sampling efficiency, E , for this case gives (6, 7) 1
where C is the concentration of analyte in the interparticle void of the sorbent pad, g/m3, and K is the partition coefficient for analyte-sorbent, defied as (g of analyte/m3 of compacted sorbent)/(g of analyte/m3 of carrier gas). References 4 and 5 with &Kr and 133Xeshow that eq 5 can give a very good
LTc(x=+
m
E = 7- - C
2a2 exp(-q,%/a)
n=1Q;(qn2
+ a2 +
(15) ff)T
where E is the sampling efficiency, qn are the positive roots of qn tan (4") = a, and cy is the mass transfer resistance, dimensionless.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 8, APRIL 15, 1989 845
0
E 0.9 ISOTHERM 8 o,8 - AE = LINEAR IRREVERSIELE ADSORPTION '.O
-C
= =
DUEININ-RADUSHKEVICH ISOTHERM
0.6 -
E 0.5 8 0.4 fi 0.30 3 0.2 D
I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
I
I
I
I
I
I
I
I
I
lj
o'oO.O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
FRACTIONAL SATURATION OF ADSORBENT Flgure 2. Reduced partition Coefficient as a function of analyte uptake: ( X axis, fractional saturation of sorbent; Y axis, l/(reduced partition
REDUCED TIME Figure 4. Sampling efficiency, assuming irreversible sorption: X axis, reduced time, 7;Y axis, sampling efficiency, E .
coefficient). In order to place the results on a linear scale, the reciprocal of the reduced partition coefficient, i.e. I l k , was plotted as a function of analyte uptake. Thus the Y axis gives the ratio l / k = [(concentrationin gas phase)/(concentrationin gas phase at saturation)]/ [(concentrationin sorbent)/(concentrationin sorbent at saturation)]. Note that the reduced partition coefficient calculated from the linear isotherm remains constant and equal to 1 for all analyte concentrations, whereas for the case of irreversible sorption, it is infinite (with l l k = 0) until saturation occurs. The reduced partition coefficient calculated from the Dubinin-Radushkevich equation is intermediate between these two extremes.
coefficient corresponding to this important case. The vapor-phase concentration remains at zero until the sorbed (or reacted material) reaches a fixed amount. At this point the chemical reactant-were such a material present and responsible for the irreversible uptake-is spent and the vapor-phase concentration increases sharply without any corresponding increase in the concentration of sorbed material. The sampling efficiency calculated for the irreversible sorption is (see Appendix)
E
+
= ( ( 2 ~ 1)0.5- I)/w
E = 1 / ~for
1
T
for
T
< a/2 + 1
> a/2 + 1
(17)
An abrupt change in the slope of the sampling efficiency occurs at T = a12 1, because at this reduced time the sorbent reaches saturation and further uptake of analyte ceases. Assuming 7 < a12 1, the following equation can be used to calculate the ambient concentration, Co, from the uptake:
+
+
co = a(M + bW)/t 0.1 '.?JL
d.1 d 2 0!3 014 0)s 0)6 017 018 019 IlO 1)1 1.12 lj3 11!
REDUCED TIME Flgure 3. Sampling efficiency, assuming a linear isotherm: X axis, reduced time, T ; Y axis, Sampling efficiency, E . In this and the following figures, a is the mass transfer resistance.
Figure 3 gives the sampling efficiency as a function of the two variables in eq 15, the reduced sampling time, T , and the mass transfer resistance, a. This figure shows that the reversibility of the sorption process causes significant loss of sampling efficiency even in the absence of mass transfer resistance (Le. a = 0). For this latter case (equivalent to eq 17 of Coutant et al.) 1 - exp(-T) E= (16) 7
Accordingly, by the time 20% of the sorbent capacity is spent, the sampling efficiency is less than 90%, and thus a significant loss of sampling efficiency occurs by the time a small fraction of the sorption capacity is spent. The same Laplace transform technique used to derive eq 15 also gives a general solution (8) directly applicable to a diffusive sampler with multiple layers of different sorbent materials. But this procedure can be used only if the sorption isotherm(s) is (are) linear. Irreversible Sorption. If the sorbent interacts chemically with the analyte-possibly through the interaction of an impregnant added to the sorbent-then the sorption may be irreversible. Line B in Figure 2 shows the reduced partition
(18)
where a and b are constants, M is uptake of analyte, and t is sampling time. This calculation is currently used for strain length diffusive samplers in which the analyte diffuses into a tube containing silica gel (or some other suitable support) impregnated with a chemical having an irreversible reaction with the analyte (9).The stain length corresponds to the quantity of sampled analyte. The calibration scale for such samplers usually involves a square root relationship consistent with eq 18 given above. This simple relationship between uptake and the time weighted average concentration does not imply that sampling can be carried out as long as desired without increasing the analytical error. For as the sampling time increases, the uptake rate decreases and the calculated concentration becomes increasingly sensitive to analytical error. Figure 4 gives the sampling efficiency, calculated assuming an irreversible sorption, as a function of the reduced sampling time, 7,and the mass transfer resistance, a. Note that for small values of a (i.e. for low mass transfer resistances), the sampling efficiency remains high until the sorbent capacity is entirely spent. To be specific, if the mass transfer resistance, a, is 0.5 or less, the sampling efficiency is at least 0.8 as long as the sorbent is not completely used up. It is instructive to compare Figures 3 and 4 in order to see the difference between sampling with a sorbent having a linear isotherm and sampling with a sorbent having an irreversible isotherm. With a linear sorption isotherm, a sampling efficiency greater than 0.8 is not possible after the sorbent reaches 38% of saturation. An irreversible isotherm does not guarantee high sampling efficiency. Sampler geometry is also important. Note that
848
ANALYTICAL CHEMISTRY, VOL. 61, NO. 8, APRIL 15, 1989
Table I." Error in Estimating Concentration from Uptake as a Function of the Form C = a ( M + bM2)/T
b= T er
0.0 0.9960 0.0128 0.0052
0.1 1.0016 0.0629 0.0010
0.2 1.0031 0.1177 0.0005
0.5 1.0057 0.2775 0.0008
1.0 1.0061 0.5409 0.0010
2.0 1.0055 1.0614 0.0010
5.0 0.9967 2.6367 0.0007
10.0 0.976 5.353 0.0005
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.9972 0.9985 0.9998 1.0010 1.0022 1.0030 1.0026 0.9990 0.9865
1.0005 1.0000 0.9998 0.9998 0.9999 1.0000 1.0003 1.0006 0.9972
1.0011 1.0001 0.9997 0.9995 0.9996 0.9998 1.0000 1.0004 1.0006
1.0017 1.0001 0.9995 0.9994 0.9995 0.9997 1.o001 1.0005 1.0010
1.0021 1.0000 0.9993 0.9992 0.9994 0.9997 1.0002 1.0007 1.0012
1.0022 0.9999 0.9992 0.9992 0.9994 0.9998 1.0002 1.0006 1.0011
1.0016 0.9997 0.9994 0.9995 0.9997 0.9999 1.0002 1.0005 1.0007
1.0010 0.9998 0.9996 0.9997 0.9999 0.9999 1.0002 1.0003 1.0004
a= a=
"The first four rows give, respectively, the mass transfer resistance, a,the coefficients a and b in the correlation equation, and the root mean square error (er) in the estimates which are given in each column. The reduced time, T , for each estimate is given in the first column on the left. The data used are the same as those plotted in Figure 5, where it was assumed that the sorption isotherm follows the Dubinin-Radushkevich equation. In the model used here the hypothetical concentration was 1.0 (in reduced units), so the difference between the calculated concentration and unity represents the experimental error. The worse estimate (0.9865) was seen at T = 0.9, with a mass transfer resistance, a = 0. Under these hypothesized conditions, the sampler was approaching saturation. if the mass transfer resistance is greater than 2, the sampling efficiency is largely controlled by mass transfer resistance and the sampling efficiency may be less than with a well designed diffusive sampler containing a sorbent having a linear sorption isotherm for the analyte. Dubinin-Radushkeuich Equation. Diffusive samplers containing activated carbon are very important to the industrial hygienist for the determination of solvent vapors. Calculation of the sampling efficiency of such samplers requires, as it did for the other cases examined here, assumptions regarding the reduced partition coefficient for the analyte between the sorbed and gaseous phases. Very often sorption on microporous sorbents such as activated carbon leads to a Dubinin-Radushkevich equation (10, 11). The reduced partition coefficient derived from this equation is exp(-A ln2 (rc)) k= (19) c exp(-A ln2 ( r ) ) where A = B R 2 P ,B is structural constant for the sorbent, mo12/J2,R is gas constant, 8.3143 J/(mol K), T is absolute temperature, K, r = (ambient vapor pressure of analyte)/ (vapor pressure of pure analyte), and c is reduced concentration. Figure 2 gives the reduced partition coefficient for the sorption of benzene on activated carbon, as calculated over the range 0-100 ppm for benzene at 25 "C, using a value of 8.5 X 10-lo mo12/J2for the structural constant, B (ref 12). The reduced partition coefficient calculated from the Dubinin-Radushkevich equation is seen in Figure 2 to be intermediate between the reduced partition coefficients calculated from the linear and irreversible isotherms. Note that at low reduced concentrations, k , the reduced partition coefficient calculated from the Dubinin-Radushkevich equation is closer to the irreversible isotherm than to the linear isotherm. Then, a t least initially when the concentration of analyte is low, the Dubinin-Radushkevich isotherm may lead to results that are close to what is seen from an irreversible isotherm. Our calculations must be based on numerical procedures because analytical methods are too limited to give closed-form solutions for eq 11-13 if the reduced partition coefficient, k , is calculated from the Dubinin-Radushkevich equation. The implicit numerical procedure of Crank and Nicolson was chosen because of its high stability (13). In this analysis, the sorbent pad was divided into a hundred laminae, each having a thickness, 6x. Diffusion took place in successivestages, each having a reduced time increment of 67. The basic assumption was that we knew the concentration in the lamina at the reduced time, 7, and wish to calculate the corresponding
2 ,
I
I
I
,
I
I
I
I
I
I
I
I
,I
o'oO.O0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 REDUCED TIME
Figure 5. Sampling efficiency, assuming the Dubinin-Radushkevich function: X axis, reduced time, r; Y axis, sampling efficiency, E .
concentrations at the reduced time, 7 + 67. The equations containing these unknown concentrations form a tridiagonal system that can be solved rapidly (the number of calculations being proportional to the number of laminae) using a wellknown algorithm (14). The computer program for these calculations was written in BASIC and run on a personal computer. Figure 5 shows the results of these calculations. With a sufficiently low mass transfer resistance ("a"),sampling efficiency remains high until the capacity of the sorbent is almost entirely spent. In fact with a = 0, the sampling efficiency is 0.947 at a reduced sampling time, r = 1. For the irreversible isotherm under the same conditions, the sampling efficiency would remain at 1.0 until r = 1.0 and then drop off to zero. The sampling efficiencies given in Figures 4 and 5 are even closer at higher values of the mass transfer resistance, a. This closeness between the sampling efficiency calculated from these two isotherms is the result of the similarity between these isotherms, as seen earlier in Figure 2. These results give a theoretical explanation for the very high sampling efficiencies often seen with microporous sorbents-with such sorbents the sampling efficiency can, under a wide range of conditions, approach that would be observed if the sorption were irreversible. If the sampling efficiencies calculated from the irreversible isotherm and the Dubinin-Radushkevich equation are so similar, might not eq 17 also be applicable if the sorption isotherm followed the Dubinin-Radushkevich equation. We have examined this theoretically from the goodness of fit of eq 17 to the sampling curves shown in Figure 5. For each
ANALYTICAL CHEMISTRY, VOL. 61, NO. 8, APRIL 15, 1989
curve, it was found that the average error was less than 1% over the range of reduced times, 7 = 0.1 to 7 = 0.9. The results, obtained by regression analysis, are given in Table I. From this excellent fit, it would appear that eq 17 could be a powerful tool in the analysis of data taken by diffusive samplers containing activated carbon; this has in fact been found to be true (15). DISCUSSION One would expect diffusive samplers containing a thin layer of sorbent, and therefore having essentially no mass transfer resistance from diffusion of analyte through the sorbent, to show at any given reduced time, the highest possible sampling efficiency. In fact, such high sampling efficiencies are seen in Figures 3-5, where the curves labeled a! = 0 correspond to diffusive sampling with no mass transfer resistance. This case was discussed earlier (16). In reality, diffusive samplers cannot contain an infinitely thin layer of sorbent, and somehow this fact must be taken into account. The results of these calculations are shown in Figures 3-5. With these results given in dimensionless variables, they are presented in a generalized form applicable to the widest possible combination of factors. These calculations show the underlying similarity in diffusive samplers-for despite great differences in geometry, sorbent, and application, all diffusive samplers rely on diffusive processes that can be described in terms of a limited number of dimensionless variables. The theoretical considerationsgiven here must be tempered with an understanding of all the variables that can affect diffusive sampling. A well-designed diffusive sampler may still suffer real-world problems such as contamination and loss of analyte by reverse diffusion after its intended use. A sampler in which the analyte is irreversibly bound can be contaminated during transport. But at the very least, we can determine in advance of the actual use of diffusive samplers what designs are appropriate for which particular use. ACKNOWLEDGMENT This paper is based on the Master’s Essay, “Mathematical Models for Diffusive Samplers”,submitted to the faculty of the Graduate School of Public Health, University of Pittsburgh, Pittsburgh, PA, 1984, by Daniel P.Adley in partial fulfillment of the requirements for the degree of Master of Science (in hygiene). APPENDIX Performance of a Passive Sampler Assuming Rapid Irreversible Sorption of t h e Analyte. If there is rapid irreversible sorption of the analyte upon contact with unreacted sorbate, the instantaneous rate of uptake is equal to the concentration of analyte in the ambient atmosphere, multiplied by the exposed surface area of the sampler, and divided by the sum of the mass transfer resistances of the air gap and the reacted portion of the sorbent pad. By use of the nomenclature given in the text, this is equivalent to
dM -_ dt
Cd L
WM
(1)
E+=
where A is exposed surface area of diffusive sampler, m2, Co is ambient concentration of analyte, g/m3, D is molecular diffusion coefficient for analyte in air, m2/s, L is width of
847
air gap, m, M is uptake of analyte at a time, t , g, Mo is maximum uptake of analyte, g, W is thickness of the sorbent pad, m, e is fractional interparticle void volume, dimensionless, and y is tortuosity factor for interparticle diffusion, dimensionless. Note that the mass transfer resistance in the sorbent pad is equal to the thickness of the reacted zone of sorbent in the pad ( WM/Mo),divided by the fractional interparticle void volume, by the tortuosity factor for interparticle diffusion, and by the molecular diffusion coefficient for the analyte in air. Next let a! = W / y € L dT =
A C a dt
LMO w = M/Mo
Then a7
dw = 1+ a w Integration of eq 2 gives
w
+ 0.5aw2 =
7
(3)
If, as before the sampling efficiency, E, is defined as WIT, eq 3 leads directly to eq 17 of the text. Registry No. C, 7440-44-0. LITERATURE CITED (1) Hearl, Frank J.; Manning, Michael P. Am. Ind. Hyg. Assoc. J . 1980, 41, 998-983. (2) Underhill, Dwight W. Am. Ind. Hyg. Assoc. J . 1985, 4 4 , 237-239. (3) Bartley, David L.; Doemeny. Laurence J.; Taylor, DavM G. Am. I n d . Hyg. ASSOC.J . 1983, 4 4 , 241-247. (4) Curzio, Giorgio G.; Gentili, Albert0 F. Anal. Chem. 1972, 4 4 , 1544-1 555. (5) Underhill, Dwight W. Nucl. Scl. Eng. 1977, 63,133-142. (6) Crank, John The Mathemetics of Dlffuslon; Clarendon Press: Oxford, 1975. Coutant, R. W.; Lewis, R. 0.; Mullk, J. Anel. Chem. 1985, 57, 219-223. Spencer, H. Garth; Barrie, James A. J. Appl. Pokm. Scl. 1980, 25, 2807-2814. McKee, Elmer, S.; McConnaughey, Paul, W. Am. Ind. Hyg. Assoc. J . 1985, 46, 407-410. Smisek, Milan; Cerny, Slavoj Active Carbon; Elsevier: New York, 1980; pp 103-130. Goiovoy, A., and Braslaw, J.; J. Ak Pollut. Control Assoc. 1981, 3 7 , 881-885. Sansone, Eric B.; Jonas, Leonard A. Am. In d. Hyg. Assoc. J . 1981, 42, 688-69 1. Press, William H.; Flannery. Brian P.; Teukolsky, Saul A.; Vetterllng, William T. Numerical Recipes-The Art of Scienfffb Computing; Cambridge University Press: New York, 1986; pp 637-645. Westlake, Joan R. A Handbook of Numerical MaWx Inversion and Solution of Linear Equations; John Wiley: New York, 1968. Tidwell, Cecilia J.; Underhill, Dwight W. Anal. Chem. 1989. 67, 917-921. Underhill, Dwight W. Am. Ind. Hyg. Assoc. J . 1984, 45, 306-310.
RECEIVED for review July 8,1988. Accepted January 4,1989. Financial support for the analysis of the data in this paper was provided to D.W.U. by the Center for Environmental Epidemiology, University of Pittsburgh. Thus although the information in this document was funded in part by the United States Environmental Protection Agency under Assistance Agreement CR 812761-01 to the Center for Environmental Epidemiology,University of Pittsburgh, it does not necessarily reflect the views of the Agency and no official endorsement should be inferred.
