Radioactive Tracer Technique for Molecular Diffusion Coefficients in Granular Media E. R. Bennett‘ and W. E. Bolch2 Sanitary Engineering Research Laboratory, University of California, Berkeley, Calif.
A radioactive tracer technique for the measurement of gaseous molecular diffusion coefficients in granular media or in open columns is described. A sealed glass ampoule of radioactive trace component gas was introduced into one end of a column containing granular media and counting gas in the pore volume. The mathematical expression for the arrival of the trace component at the other end of the column is given. The arrival was monitored with a radiation detector. Using this technique, the effect of pore opening obstruction on the molecular diffusion coefficient in granular media was investigated. Gaseous molecular diffusion coefficients in randomly packed, uniform, spherical, media were found to be reduced to a factor of 0.59 i 0.02. The effect of nonuniformity and nonspherical shape of the granular media was shown to further reduce the molecular diffusion coefficient. The extreme case of a flat plate media was found to produce a diffusion obstruction factor of 0.197. Temperature and pressure correlations with the molecular diffusion coefficient were also made. MOLECULAR DIFFUSION of a conservative solute from a point source in a three-dimensional medium can be expressed as: d- C_ dt
-
DIn,dV2C
When the boundary conditions for a particular problem are used, the surface concentration at any location and time can be calculated if the effective media diffusion coefficient, Dme+ is known. The effective media diffusion coefficient is composed of the molecular diffusion coefficient and selected media parameters. The molecular diffusion coefficient of a trace component of gas diffusing in another pure gas can be calculated by any of several equations ( I ) relating diffusive transport to the thermodynamic properties (2-4) of the gases. (A trace component is a constituent that is present in concentrations that can be measured but are so small as to not change the physical and chemical properties of the carrier gas. In this experiment the radionuclide concentration was approximately 10-7~4.) In this study a radioactive technique was used. The inert carrier gas required a small percentage of a n alkane to serve as the quench gas in the radiation detector. The inert carPresent address, Department of Civil and Environmental Engineering, University of Colorado, Boulder, Colo. Present address, Department of Environmental Engineering, University of Florida, Gainesville, Fla. (1) R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena,” Wiley, New York, N. Y., 1960, p. 505. (2) “Handbook of Chemistry and Physics, Western Edition, Chemical Rubber Company, Cleveland, Ohio, 1970. (3) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N. Y., 1954, p 539. (4) S. Chapman and T. C. Cowling, Mathematical Theory of
Non-Uniform Gases, 2nd Ed., Cambridge University Press, New York N. Y.,1951.
rier, the quench gas, and the radionuclide therefore produced a three-component diffusion system. Wilke ( 5 ) has shown that the molecular diffusion coefficient of a multicomponent system can be calculated from the harmonic sum of the ratio of the mole fraction t o the molecular diffusion coefficient for each component. When diffusion takes place in the void area of a packed bed or column, the molecular diffusion coefficient may be modified by a number of media effects. F o r granular media larger than 200 microns, the Knudsen effect can be considered as negligible. The major effect is that due to obstructed diffusion. This causes a reduction in the molecular diffusion coefficient due to the extra length of travel of the diffusing molecules o n the tortuous path around the media granules (6) and due t o the change in mass flux in the ever changing crosssection size and shape from point to point in the media pores (7). The obstructive factor of molecular diffusion can be defined as a parameter expressing the ratio of the diffusion coefficient in the pore area of granular media t o the diffusivity in a n open space. From the values presented in Table I (8-22), it can be noted that the obstructive factor for randomly packed, uniform spheres has been found t o be in the range of 0.5 t o 0.8. EXPERIMENTAL METHOD AND MATERIALS
Packed columns were used to provide a one-dimensional regime in which gas diffusivities and obstructive factors could be measured under varied conditions. The pressure within the column could be maintained over a range of values and measured by means of a manometer. The column assembly was located in a constant temperature and radiological enclosure with a maximum temperature variation less than ( 5 ) C. R. Wilke, Chem. Eng. Progr., 46,95 (1950). (6) P. C. Riest, Enuiron. Sci. Technol., 1,567 (1967). (7) E. E. Peterson, A.I.Ch.E.J., 4,343 (1958). ( 8 ) L. J. Klinkenberg, Bull. Geol. SOC.Amer., 62, 559 (1951). (9) M. R. T. Wyllie, and A. R. Gregory, Ind. Eng. Chem., 41, 1379 (1955). (10) J. M. Coulson, Trans. Inst. Chem. Eng. (1949). (11) R. R. Sullivan and K. L. Hertel, Aduan. Colloid Sci.,1, 37 (1952). (12) P. C. Carman, J. SOC.Chem. Ind., 57,225 (1938). (13) J. H. Knox and L. McLaren, ANAL.CHEM.,36,1477 (1964). (14) R. J. Blackwell,J. R. Rayne, and W. M. Terry, Trans. A.I.M.E., 216, l(1959). (15) W. E. Brigham, P. W. Reed, and J. N. Dew, SOC.Petrol. Ena. J.. 1, 1 (1961). (16) I. Hoogachogen, Ind. Eng. Chem., 47,906 (1955). (17) J. C. Sternberg and R. E. Poulson, ANAL.CHEM.,36, 1492 (1964). (18) E. Glueckauf, “Gas Chromatography 1958,” K. H. Desty, Ed. Butterworths, London, 1958, p 33. (19) E. V. Evans and C. N. Kenney, Trans. Ins:. Chem. Eng. 44, 189 (1966). (20) S. J. Hawkes and S. P. Steel, J. Chromatogr. Sci., 8, 256 (1970). (21) J. A. Currie, Brit. J. Appl. Phys, 11, 314 (1960). (22) H. L. Penman, J. Agr. Sci., 30,437 (1940).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971
55
Medium Glass spheres Glass spheres Glass spheres Spheres Spheres Spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres .Glass spheres Glass spheres Glass spheres Lead spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres Glass spheres
Table I. Obstructive Factors in Uniform, Spherical Media Obstructive Porosity Mean size, m m factor 0.435 8 0.71 0.368 3 0.72 0.402 3 0.71 0.40 ... 0.59 0.39 ... 0.59 ... 0.575 0.36 0.38 0.50 0.62 0.38 0.50 0.58 0.38 0.275 0.62 ... 0.50 0.80 ... 0.04 0.71 0.35 0.10 0.71 ... 1 vol 1.0-1.25 1 vol 3.26 0.66 ... 1 vol 1.0-1.25 4 vol 3.26 0.63 0,086 0.64 0: 395 0.21 0.74 0.403 0.42 0.66 0.418 0.21 0.73 0.474 0.42 0.79 0.529 0.42 0.76 0.643 1.15 0.75 ... 1.20 0.70 0.36 0.81 0.70 0.374 1.96 0.70 ... 0.21 0.73 0.405 0.75 0.67 0.396 0.75 0.674 0.381 0.75 0 672 0.395 0.38 0.67 0.375 5-6 0.682 0.376 5-6 0.67 0.397 3 0.803 0.364 0.5 0.774 I
Table 11. Physical Properties of the Media Mass mean Uniformity Solid density, size, microns coefficient gram/cm3 Media type Glass beads A 6080 1.07 3.00 1.15 2.485 Glass beads B 380 2.485 Glass beads C 220 1.20 350 1.26 2.64 Sand I 1.32 2.63 480 Sand I1 480 2.78 2.64 Sand IIA 2.63 730 1.24 Sand I11 1.20 2.62 Sand IV 2640 2.77 800 x 800 Microplates X 30
MANOMETER-
GLASS COLUMN A N D MEDIUM
COLUMN 8
U
Figure 1. Diffusion column + l OC. A schematic drawing of the system is shown in Figure 1 and more complete descriptions are given elsewhere (23,24).
(23) E. R. Bennett and W. J. Kaufman, SERL Rep. No. 67-8, University of California, Berkeley, Calif., July 1967. (24) W. E. Bolch, R. E. Selleck, and W. J. Kaufman, SERL Rep. No. 67-10, University of California, Berkeley, Nov. 1967. 56
A radioactive trace component gas was released a t the sample chamber and the rate of approach to equilibrium at the opposite face was recorded with a n internal flow proportional radiation detector. The column voids contained one of three stagnant gas solutions: 99% helium and 1% isobutane (Q gas); 90% argon and 10% methane (Plo); or a standard air composition. The trace component radioactive gases used were krypton-85, tritiated hydrogen, and xenon133. The properties of the different column packing are given in Table 11. Three sizes of uniform, nonporous, glass spheres were selected so as to be similar in shape and size distribution. The two smaller sizes were “3M Superbrite” glass beads, number 100 and 080. The larger size was six-millimeter diameter, laboratory “boiling” beads. Washed, graded, kiln dried, Monterey sands were used as raw material for the second classification of media. Additional grading and separation led to series (1, 11, 111, 1V) of media having the primary variable of particle size. Sand 1was selected to yield a natural medium of approximately the same mean size as
ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971
07 06
,IJ
i
01
00 002
considered as the superimposed effect of the molecules diffusing from the source and those “reflected” at x = L , and “reflected” again at x = 0 and returning to the detector at x = L, plus molecules that have “traveled” multiples of the two L “reflection lengths” This analogy results in the form of Equation 3.
1 I
01 D!, L‘
with f having positive integer values from zero. The equation has been normalized in terms of a completely mixed concentration.
.o
~
,c ,,
Figure 2. Theoretical molecular diffusion curve
(4)
A plot of Equation 3 is shown in Figure 2. The curve rises rapidly until the maximum of the first term is reached and continues to rise until the completely mixed concentration is approached. The time required to reach a specific fraction of the function Ct,, is therefore, indicative of the molecular diffusion coefficient. The parameter D , is replaced by Dmedfor the analysis of a packed co:umn.
Glass beads B. Sand I I A was prepared to provide a larger uniformity coefficient while retaining the same mean size as Sand 11. A third classification consisted of a media from microsized aluminum plates. For each diffusion measurement, an ampoule containing the radioactive trace component was positioned in the sample chamber, the column was flushed with the appropriate carrier gas, then sealed at the desired static pressure and held at a constant temperature. The ampoule was shattered with the screw plunger and the detection and recording instruments were simultaneously activated. The accumulative total count from the scaler was recorded at time intervals and these data were reduced to a plot of counting rate as a function of elapsed time. The detector was also coupled to a rate meter and a continuous curve was obtained from the chart recorder. Diffusional transport in an infinitely long, packed column can be considered on the basis of Equation 1. The solution of this equation for diffusion from a plane source in a onedimensional, single-directional, regime of infinite length is
DISCUSSION AND RESULTS
A “reflection” analogy (25) is used to express the effect o’f closure at some distance L away from the source. Therefore, for a finite column of length L, with a source at one end and the detector at the other, the concentration history can be (25) J. Crank, “Mathematics of Diffusion,” Oxford University
Press, New York, N. Y. 1956.
M eAL
= -
Figure 3 is a plot of the counting rate or concentration as a function of time curves for the gas pairs studied and for each of the media and column lengths. Since the detection volume was the same for all portions of the study, the counting rate values can be used to express the concentration terms. The counting rate-time curves were fitted to Equation 3 using Figure 2 to determine the diffusion coefficient. This procedure eliminated the need for a precise value of the total amount of trace component injected or the exact efficiency of the detector. A summary of the diffusion test results is presented in Table 111. Note that runs 1, 9, 19, 23, 24, and 25 were without any media in the column and therefore determined experimental values for the molecular diffusion coefficients for the various tracer-gas solutions. Experimental diffusion coefficients were determined for variations in pressure from 1 atmosphere to 1.72 atmospheres and Dmed was found to be an inverse function of pressure, as would be expected. A temperature range from 15 to 35” C was studied and Dmedwas found to vary with absolute temperature raised to the 1.55 power.
I .o
0.9
0.8 0.7 0.6
0.5
Figure 3. Molecular diffusion measurements
0.4 0.3 0.2 0.I
0.0 IO
20
30
50
100
200
500
1000
2000
T I M E (MINUTES)
ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971
57
Run No. 1 2 3
Trace gas 3H 3H 3H 3H 3H 3H 3H 3H 85Kr 85Kr 85Kr 85Kr 85Kr
4
5 6 7 8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 He
Q
=
Carrier gas Hea He He He He He He He He He He He He 85Kr He 85Kr He 85Kr He 86Kr He 85Kr He 86Kr Argon 86Kr Argon 86Kr Argon 85Kr Argon 85Kr Air 133Xe He 133Xe Air 99% helium 1
+
Table 111. Diffusion Coefficients in Granular Media Media Column Temp, Press, Media dia, length, "C atm type mm Porosity cm 27 1 .oo None ... 1.Ooo 134 27 1.oo GlassA 6.00 0.423 134 27 1.oo GlassB 0.380 0.372 134 27 1 .oo Glass C 0.220 0.384 134 27 1.386 GlassC 0.220 0.384 134 27 1.724 GlassC 0.220 0.384 134 15.5 1 .oo GlassC 0.220 0.384 134 35 1.00 GlassC 0,220 0.384 134 27 1.OO None ... 1.ooo 123 27 1.00 GlassA 6.00 0.423 134 27 1 .oo GlassB 0.380 0.372 134 GlassC 0.220 0.384 134 27 1.oo 27 1 .oo Sand I 0.350 0,403 62 27 1 .oo Sand11 0.480 0.398 62 27 1 .oo Sand IIA 0.480 0.375 62 27 1 .oo Sand I11 0.730 0.386 62 27 1.00 SandIV 2.640 0.350 62 27 1 .oo Microplate ... 0.565 247 27 1.oo None ... 1 .Ooo 134 27 1 .oo GlassA 6.00 0.423 134 GlassB 0.380 0.372 134 27 1.oo 27 1.oo GlassC 0.220 0.384 134 None ... 1 .Ooo 62 27 1.oo 27 1 .oo None ... 1.Ooo 62 27 1.oo None ... 1.Ooo 62 isobutane. Argon = 90% argon 10% methane.
D,,
D, calcd 1.580 ...
Dmed
obsd 1.380 0.840 0.820 0.800 0.552 0.444 0.750
.
I
Obstructive factor, Y
... 0.61 0.59 0.58
.
...
...
...
...
... ...
0.840
...
0.533 0.396 0.362 0.365 0.291 0.293 0.264 0.293 0.302 0.104 0.162 0.100 0.094 0.094 0.157 0.450 0.136
0.590 ... ... ...
... ... ... 0.65 0.59 0.60 0.55 0.56
... ... ...
0.50
...
...
0.56 0.57 0.197
... ... ...
0.62 0.59 0.59
0.158 0.550 0.125
... ...
... 0.154
. . I
...
+
0.0
0.6
0.4 E
0
0.2
1
0.0 0.0
X
/
, 0.2
0.4
220 u
,
O :!K :
0.6
0.0
, 1.0
1
220 u OB6 2.44
Figure 4. Effect of pressure on the molecular diffusion coefficient
1
-246 log T
2.40
2.50
(OK)
Figure 5. Effect of temperature on the molecular diffusion coefficient Plots of these results are shown in Figures 4 and 5. Values of the molecular diffusion coefficient calculated from critical constants are also given in Table 111. Porosity determinations are also shown. The media diffusion coefficient, Dmed, determined for molecular transport within each of the granular media, was lower than that for a n open tube without media because of the tortuous path and changing size and shape of the void areas. The change can be expressed as Dmedia= yD,. The value of the obstructive factor, y , was determined for each size medium and gas pair. The observed values of obstructive factors for the glass spheres are similar to those found by previous investigators. For randomly packed, spherical media, such as the 220micron and 380-micron sizes, the obstructive factor was found to be 0.59 =t0.02. With the six-millimeter beads, the column diameter to particle diameter ratio was only 8.4, and edge effects may have caused the increase in porosity and the 6z increase in obstructive factor. The uniform sands (I, 11, 111, and IV) have nearly identical 58
obstructive factors. The values of y for these irregularly shaped particles is about 6z smaller than the value observed for spherical particles of comparable size The uniformity of the medium can produce a substantial change in the obstructive diffusion as seen from the comparison of Sand I1 and Sand IIA and the extreme case illustrated by the aluminum microplates. SYMBOLS
Cross-sectional area of the column interior Concentration of solute molecules per unit void volume Ct+- = Concentration a t the completely mixed condition D, = Molecular diffusion coefficient without media in column Dmed = Molecular diffusion coefficient with media in column
A
C
ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971
= =
f L
5 T t X
V
= Integration function, integers from zero
Length of the sample column = Total gas system pressure in atmospheres = Absolute temperature, OK = Time after injection = Distance from the source = Three-dimensional derivatives =
t
=
y
=
Porosity Obstructive factor, equal to Dmed/DllL
RECEIVEDfor review April 8, 1970. Accepted October 5 , 1970. This work was part of grant AT(l1-1)34 Project 30, USAEC completed in the Sanitary Engineering Research Laboratories of the University of California, Berkeley.
Spectrophotometric Analysis of Carbonyl Compounds in the Presence of Carbohydrates without Prior Separation Edward B. Sanders and Jack Schubert Department of Radiation Health, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, Pa. 15213
A simple, rapid method is described whereby normal carbonyl compounds can be spectrophotometrically determined in the presence of carbohydrates, without prior separation, using the reagent 2,4-dinitrophenylhydrazine. The method exploits a technique which suppresses formation of carbohydrate 2,4-dinitrophenylhydrazones. The effect of such parameters as pH, temperature, and time of heating on the condensation of both carbonyl compounds and sugars with 2,4dinitrophenylhydrazine to form 2,4-dinitrophenylhydrazones is discussed. The fraction of a-dicarbonyl compounds present can also be determined. THE REAGENT 2,4-DINITROPHENYLHYDRAZINE (2,4-DNP) i S widely used for the analysis of carbonyl compounds, especially as applied by Lappin and Clark ( I ) . In their method, carbonyl compounds are reacted in an acidic media with 2,4-DNP to give a 2,4-dinitrophenylhydrazone(2,4-DNPH) which, after addition of methanolic potassium hydroxide to give the anion, is determined spectrophotometrically. This method has been used, with slight variations, for a number of diverse applications, such as analysis of total carbonyl content in irradiated amino acids (2, 3), ethylene glycol ( 4 ) , and cyclohexane (5); analysis of carbonyl functions in polymers (6); and the determination of total carbonyl content in foods (7). However, the method of Lappin and Clark cannot be directly applied for the analysis of the total carbonyl content when carbohydrates are present, especially in complex systems such as food, DNA, and cellular material. Consequently, analyses for simple aldehydes and ketones in the presence of sugars are carried out only after their prior separation from sugars, usually by volatilization or extraction into an organic phase. Such separations exclude high boiling carbonyl compounds, or highly polar, water soluble compounds such as glyceraldehyde or glycolaldehyde, and in some cases, lead to the chemical loss, destruction, or transformation of certain carbonyls. In our continuing investigations on the chemical changes produced (1) G. Lappin and L. C. Clark, ANAL.CHEM., 23,541 (1951). (2) B. M. Weeks, S . A. Cole, and W. M. Garrison, J. Phys. Chem., 69,4131 (1965). (3) J. W. Harlan, F. Leo Kauffrnan, and F. Heiligman, “Radiation Preservation of Foods,” American Chemical Society, Washington, D. C., 1967, p 39. (4) M. Ahmud, M. H. Awan, and D. Muhammad, J. Chem. SOC. ( B ) , 1968,945. (5) A. J. Bailey, S. A. Barker, R. H. Moore, and M. Stacy, J . Chem. SOC.,1961,4086. (6) J. Belisle, Anal. Chim. Acta, 43, 515 (1968). (7) H. P. Fleming, W. Y. Cobb, J. L. Etchells, and T. A. Bell, J. Food Sci., 33,572 (1968).
upon gamma-irradiation of carbohydrates and other food components (a), it is necessary to determine the total carbonyls known (9) to be produced upon irradiation. During our attempts to apply the method of Lappin and Clark (1) to irradiated sugar solutions, we developed a rapid and reliable spectrophotometric assay, described here, for total carbonyls in the presence of carbohydrates without requiring the prior separation of the carbonyls. We are able to eliminate or minimize the interference of sugars in the 2,4-DNP assay of carbonyls by exploiting the fact that carbohydrates, in general, exist in aqueous solution as cyclic hemiacetals rather than open chain aldehydes. Consequently, they react much more slowly with nucleophilic reagents than do carbonyl compounds whose structures either preclude hemiacetal formation or where the hemiacetal-open chain aldehyde equilibrium favors the open chain aldehyde. When we applied the method of Lappin and Clark to aqueous solutions of glucose and fructose, we noted that the visible absorption due to the sugars was considerably less than would be expected for typical carbonyl compounds. This reflected either a lower molar absorptivity for these sugar 2,4-DNPH anions or incompleteness of reaction under these conditions. Examination of the visible spectra of the anions of the 2,4DNPH’s of synthetic glucose and fructose ruled out the former possibility, indicating that proper choice of conditions for the analysis might, and in fact did, result in formation of 2,4DNPH derivatives of normal carbonyls but not those of the sugars. Further, as described later, since the a-dicarbonyl compounds such as glyoxal necessarily form osazones with a different absorption spectrum and relatively very high molar absorptivity from that of the 2,4-DNPH anions, we are also able to determine the fraction of a-dicarbonyl present in a mixture of carbonyl compounds. EXPERIMENTAL
Reagents. CARBONYL-FREE METHANOL.To about 500 ml of reagent grade methanol (Fisher), add 5 grams of 2,4D N P and a few drops of concentrated hydrochloric acid and reflux for 2 hr. The methanol is then distilled through a Vigreux column. When kept tightly stoppered, the methanol remains suitable for use for several months. 2,4-DINITROPHENYLHYDRAZINEREAGENT.2,4-DNP (Eastman) is recrystallized from carbonyl-free methanol. The recrystallized material melts at 199-200 “C and is pure ac(8) J. Schubert, Bull. W.H.O., 41,873 (1969). (9) G. 0. Phillips, Aduatz. Carbohydrate Chem., 16,13 (1961).
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