length of the sampling interval is critical because small deposits of material on the collection surface may cause increased collection efficiency through interception effects. I n addition to the problems mentioned, variations of the basic flow pattern because of boundary-layer growth and changing velocity profiles in the impactor jets might be expected a t reduced pressures. These phenomena are particularly evident when making pressure drop measurements because the relative pressure differential across a nozzle must be increased as the absolute pressure is decreased to maintain a constant-volume flow rate. For the range of pressures studied and the impactor configurations tested, however, examination of the data has not revealed any significant variations in collection efficiencies not already taken into account by the inertial parameter. This problem, nevertheless, should be considered in any atteimpt to use published data as a basis of predicting the performance of impactors with small jet dimensions a t pressures less than those reported here. Nomenclature
C = Cunningham slip correction, dimensionless D, = diameter of round jet or width of rectangular jet, cm. D, = diameter of aerosol particle, cm. L
= length of rectangular jet, cm.
S V
= impactor jet-to-slide distance, cm = velocity of aerosol particle in jet, cm./sec.
Z ij
= mean molecular velocity, cm./sec. = total collection efficiency, %
h
= mean free path length,
P
= = = =
2/J =, cm. PV
p p,
+
viscosity, grams/(cm.) (sec.) air density, grams/cc. particle density, grams/cc. inertial parameter, dimensionless
literature Cited
(1) Davies, C. N., Aylward, M., Proc. Phys. SOC. (London) B64, 889 (1951). (2) Einbinder, H., Battelle Memorial Inst., Tech. Rept. BMI 2409-1,Contract DA-18-064-CML-2569, March 1955. (3) May, K. R., J. Sci. Znstr. 22, 187 (1945). (4) Millikan, R. A.,Phys. Rev. 22, 1 (1923). (5) Mitchell, R. I., Pilcher, J. M., IND. ENG.CHEM.51, 1039 (1959). (6)’ Ranz, W. E., Wong, J. B., Ibid., 44, 1371 (1952). (7) Schadt, C., Cadle, R. D., Anal. Chem. 29, 864 (1957). (8) Thomas, D. G., Lapple, C. E., A . I. Ch. E. J . 7, 203 (1961). RECEIVED for review September 11, 1961 ACCEPTED September 12, 1962 Research sponsored by Division of Biology and Medicine, U. S. Atomic Energy Commission under Contract A T 11-1-401.
METHOD FOR RAPID DETERMINATION OF DIFFUSION COEFFICIENTS Theory and Application J .
CALVIN
GlDDlNGS AND SPENCER
L. SEAGER
Department of Chemistry, Lrniniversily o f Utah, Salt Lake City 12, Utah
A method similar in operation to chromatographic techniques has been theoretically and experimentally extended for measuring a wide range of diffusion coefficients. Design of the experimental apparatus is guided b y chromatographic theory. For very slow diffusion processes, such as those occurring in liquid systems, it is possible to magnify the over-all diffusion effect so that it is measurable after a very short period of time. The experimental work deals with gaseous diffusion coefficients measured a t various flow velocities, concentration levels, etc. Where data are available for comparison, agreement with other methods is satisfactory. Although the potential speed of the method has not yet been developed, it appears that diffusional ancilysis b y this method is already much more rapid than b y most conventional methods of similar accuracy.
method for mearsuring gas phase diffusion coefficients (76). T h e apparatus used consisted of a commercially available gas chromatography unit, with an empty tube replacing the packed column. The theory applying to the system is a special case (73) of a more general theory (7, 70, 7 7, 74) of chromatography, although Taylor (26) first derived the special case used here in 1953. The extension of the method to a wide range of diffusion measurements, involving both gas and liquid phases, is highly feasible in view of the aforementioned general theory. NEW
A was proposed in am earlier communication
Further extensions of the method are discussed here and additional experimental results on gas phase systems are reported. The principal feature of the present method is the speed with which diffusion measurements can be made. Although this paper reports the more general characteristics of the method, and no attempt has been made to increase speed, it has nonetheless been found possible to take and interpret the data for 200 separate determinations of the diffusion coefficient ( D ) in 36 hours. Consequently, the compilation of extensive tables of diffusion data should be much more feasible than before. VOL 1
NO. 4
NOVEMBER 1 9 6 2
277
The need for extensive diffusion coefficientmeasurements has been pointed out by several investigators (2, 27, 33, 34). T h e current methods (7, 4, 34) used to determine diffusion coefficients employ a number of individual measurements for a single evaluation of the coefficient. Each evaluation is timeconsuming. Boardman (7) and Coward and Georgeson (4) indicate that a minimum run time of 1 hour was necessary to obtain the data for the one determination of the diffusion coefficient when a Loschmidt-type apparatus was used. In the case of the point source method of U’alker and Westenberg (30-32, 34, the time for a determination is not estimated but it appears that it would be shorter than methods mentioned above but longer than the method described here. Some other determinations seem faster than the abovementioned methods but less precise (27-29). The precision of current methods ranges from 0.4% for the best Loschmidttype determinations to 570for other types. The precision reported here is on the order of 1% for individual determinations. This could he improved through the use of longer columns. Theory
Recent developments in the theory of chromatography have made it possible to predict in a n exact manner certain contributions to the dispersion of a flowing gas pulse as a function of the packing geometry within the flow tube, kinetics of adsorption and desorption, and the diffusion coefficients in the bulk phase or phases contained within the tube (7>70, 77, 73, 74). If sufficient information is available concerning the contents of the tube, the equations can be inverted to yield diffusion cozfficients as a function of pulse dispersion. This cannot he done satisfactorily with the usual granular media used in packed columns (except perhaps by a comparative method), since the geometry is not well defined. Suggestions to this effect have, however, been made by Keulemans (20) and Carberry and Bretton (3). I t is the object of this discussion to indicate the possibilities of obtaining accurate diffusion coefficients from peak dispersion data, and to deal with a special case that has proved highly successful (76). The height equivalent to a theoretical plate in a typical gas chromatographic column can he written as
where X and y are constants of order unity, u is the average carrier gas velocity, d, is the average diameter of the contained particles, and D , is the binary diffusion coefficient for the sample and the carrier. The plate height, H, is obtained experimentally as L?/t2. where L is the column length, 7 is the standard deviation of the eluted peak in time units, and t is the retention time of the peak measured to its center. The constants C,, C L ,and Ck are nonequilibrium terms (representing gaseous diffusion, liquid diffusion, and kinetic processes) which can he calculated exactly using the theory of chromatography. These terms originate in the nonuniform flow velocity existing over the tube cross-section. A given sample molecule a t a particular instant may be located in a fast-moving streamline (near the center of a channel), in a slow-moving streamline, or it may he stationary because of a sorption process. In any case the existence of unequal velocities introduces a group of elementary displacement steps resembling a random walk ( 9 ) . The length of a step (directly influencing the peak dispersion as reflected in H ) is related to the equilibration time, or the time required for a sample molecule to jump from one velocity regime to another. This is often diffusion-controlled (although for adsorption-desorption processes it is reaction-controlled), 278
l&EC FUNDAMENTALS
and hence the C, and Cl terms are related to the respective diffusion coefficients. The rapid analysis found possible here and the very high potential speed of the method are due, first, to the requirement, in addition to the calibration curves, of only a single measurement for each experimental value, and second, to the short path over which diffusion must occur for equilibration. The pa.th length is roughly the distance from the inside to the outside of a flow tube, or the distance through a liquid-filled pore, both of which can he made very short (in some cases, microscopic). In principle it is possible to magnify these short diffusion paths, so that the dispersion of a zone is far greater than that to which axial diffusion by itself would lead. Consequently a diffusion path or group of paths which are too small in extent for direct measurement are magnified in effect and easily measured. I n principle there is no limit to the magnification and thus the speed of the method, although practical limitations exist. Previous methods have depended almost entirely on obtaining a direct diffusional displacement large enough for a concentration profile measurement. This \\ ould obviously require more time, especially with liquids. The potential role of the magnification effect can he indicated by a very crude quantitative argument. The length, 1, of the displacement step mentioned above is the difference in velocity between a fast and slow velocity regime, approximated by the mean flow velocity, u. multiplied by the time, t,, needed for transit between the two regimes a distance 6 apart. 1 = ut,
(2)
The transit time itself can be approximated by t , = 6*/2D. Substitution of this leads to
I = -v6
-
6
20
(3)
which, as the ratio of two elementary displacement distances, gives the magnification of zone dipersion. This can be made very large by employing high velocities. For liquids, in which D is very small, the above ratio is large (on the order of l o 4 ) under most flow conditions. The degree of magnification is, of course, expected to he highly important for liquid diffusion measurements. I n the case of gaseous and vapor systems, diffusion is often sufficiently rapid to preclude the need for magnification. This applies to the measurements reported in this paper. Magnification is most easily obtained where it is most needed, because of the inverse dependence of 1,’6 on D. The assumptions basic to Equation 1 are: t >> 7: a negligible axial diffusion occurring Lvithin the liquid phase, and a competition between lateral gas phase diffusion and “eddy” diffusion as outlined in the theory by Giddings (73)>and leading to the first term in Equation 1. The simplest case to which Equation 1 can be applied is an empty tube of circular cross-section. This is the system employed in the experimental discussion. I t is assumed that adsorption a t the wall is negligible, so that Cs = 0. Since no liquid is contained in the tube, C L = 0. The quantity 2Xd, goes to infinity because there are no mixing stages in the tube (73). Furthermore, for this precise geometry, y = 1. Thus
(4) where the correct expression (73: 26) for a circular tube, r J / 2 4 D , , replaces Cg. An equivalent form of this expression was first derived by Taylor (26). The binary diffusion coefficient in the gas phase, Do,can he obtained in terms of the measured value of H as
(5)
This expression is the basis of the experimental work reported below. Next, consider a column packed with glass beads over which a small amount of liquid has been deposited. After equilibration, the liquid accumulates (by capillary action) around the contact points of adjacent beads. The term Ci.representing the diffusion of sample molecules into and out of this liquid, has been derived (7) as Ci
=
R(l
- R)d2/12Dl
(6)
where R represents the fraction of sample molecules within the interstitial gas space a t equilibrium (the remaining molecules are partitioned in the liquid). a quantity obtained as the ratio of the retention time of an inert peak and the sample peak. The distance, d, is measured from the point of bead contact to the liquid meniscus. The diffusion coefficient for sample molecules in the liquid is D E . The operation of such a column can be carried out in the following way. The pressure and the flow velocity of the carrier gas are varied from run to run: such that the product of the two is constant. The terms 2Xd,, C p , and 2yD,/u remain is inversely proportional constant throughout these changes (Do to pressure and C, is inversely proportional to D o ) . The term Ckis zero if adsorption effects are properly eliminated. Thus the prescribed changes in pressure and velocity are reflected only in a change in the term Clv of Equation 1. A plot of H us. u yields a straight line of slope C l . With the use of Equation 4 a D l value can be obtained from this plot. Numerous other examples could be cited, where it is possible to determine D . All that is required is a well defined geometry for which theoretical expressions can be derived. The actual packing of the glass beads in the above example does not require a precise configuration, since effects arising from the packing configuration rvmain constant and are subtracted out. Only the contained liquid requires definite boundaries. A system which has been used extensively for chromatographic purposes is the so-called capillary column, a fine tube of about 0.01-inch diameter which presumably has a thin film (-1 micron) of liquid deposited on the inner wall. The theory and practice of such columns were first worked out by Golay (78). While these columns seem to provide all the necessary requirements for liquid diffusion measurements, some doubt exists about the uniformity of the liquid film ( 8 ) . If capillary condensation forces by themselves determine the distribution of liquid, a uniform film on the inside wall of a tube is thermodynamically unstable. Adsorption forces beyond a few monolayers are very weak and probably d o not suffice to hold the film. Coupling this with the fact that fine tubes are never perfectly uniform, it is doubtful if the conditions set u p in the model can be achieved. This is borne out by the anomalous results obtained with such columns ( 5 , 25).
meters. T h e length of the short piece was measured conveniently with a 2-meter scale and found to be 104.10 0.05 cm. This short tube was placed in the instrument and the time for air samples to travel through the tube, a t various carrier gas (helium) flow rates, was determined. These times and the carrier flow rate made it possible to calculate the volume of the tube i n the instrument. This tube volume was equal to the volume determined from the time-flow measurements minus the instrument volume. T h e instrument volume was previously determined by means of air peak flowtime data with a capillary tube i n the instrument. The volume of the capillary had been determined by weighing the capillary both empty and filled with water. The volume of the short ‘,14-inch tube could then be used to obtain the average radius of the tube, since the length was known. The long piece of tubing was then put into the instrument and the air peak flow-time data were again taken. These data were used to calculate the volume and hence the length of the diffusion tube. I t was assumed that the radius in the short and long tubes was the same. T h e length of the long tube was 14.27 rt 0.05 meters. Gas sampling and injection were done using either a Hamilton No. 1001 I-ml. capacity gas-tight syringe or a 1-ml. capacity tuberculin syringe. Both had scales which could be read to the nearest 0.01 ml. T h e plunger of the tuberculin syringe was coated with light silicone oil, to ensure a gas-tight fit. The gas samples were obtained by piercing a rubber tube which was connected to the sample source and contained sample gas under slight pressure. All gases used were of no less than 99.6% purity. T o correct for end effects and for diffusion occurring in the instrument dead volume, all data were taken with both the long and the short tube. T h e data of the short tube were then subtracted from those of the long tube. T h e equation for H f r o m which D was obtained is
*
(7) where subscripts d and G refer to the long diffusion tube and the short correction tube, respectively. The information obtained consisted of gaussian-type sample peaks, the time for diffusion, and the carrier gas flow rate. T was determined for the sample peaks as w/4, where w is the peak width a t the base line. Tangents were drawn a t the inflection points of each peak and extended to the base line. The distance between the intersections of these tangents with the base line was taken as w . T h e sample sizes ranged from 0.02 to 0.30 cc. The sample was, except in a few rare cases, no larger than 0.08 cc. Table I shows the results of a calculation used to check the mean peak concentration relative to carrier gas concentration of sample gas as the sample traverses the length of the diffusion tube. In this particular calculation, sample spreading due to instrument volume or detector was ignored. I t is seen from the table that the sample can be considered to be present as a trace, particularly in view of the very slight concentration dependence reported below.
Experimental
A Perkin Elmer Model 154 C gas chromatographic unit was used for this work. This instrument has a thermistor katharometer as the dletection device. The flow rate of carrier gas was measured with a soap film flowmeter and pressure measurements were made with a mercury manometer. The flow rate and inlet pressure of the carrier gas were controlled by mrans of a Cartesian diver manostat, needle valves, and/or lengths of thermometer capillary tubing. T h e diffusion tube was made from standard ‘/d-inch outside diameter copper refrigeration tubing. A 50-foot coil was cut into two pieces with approximate lengths of 1 and 14
Table 1.
Relative Concentration of Sample as a Function of Migration Distance
Distance Traveled, Cm. 100.00 500.00
.G,rnple
0.03-CC.
sample 0.017
n
008
0.005
1000.00 1322.79
0.005
VOL. 1
NO. 4
0.70-cc. sample 0.088 0,041 0,030 0.026
NOVEMBER 1 9 6 2
279
v (cm/sc)
Figure 1.
Nature of two roots of diffusion equation as a function of flow velocity
Results
Equation 5, used to evaluate the binary gaseous diffusion coefficient, D , or D I , Zyields two values of D I ,for ~ each measured value of H. The nature of the two solutions is best indicated by choosing a n arbitrary D1,2value (0.500), calculating H a t various velocities from Equation 4, then plotting the two D1,2 values obtained from the calculated H a n d Equation 5 . The results are shown in Figure 1.
fi
U p to velocity u, = D I , z / ~the o positive root of Equation 3 is to be taken. Beyond velocity u, the negative root is valid. The two values are equal when u = u,. This also corresponds to the minimum value of the plate height, H. (Another method for choosing the correct roots of Equation 5 comes from noting that as u -+ m the first term of Equation 4 is negligible and H + m , and thus the correct value, D, = &/24 H, is acquired only by using the negative root. At the other extreme, u 0, the second term is negligible and H+ m , and thus the correct value, D , = Hu/2, is obtained only with the positive root. T h e change-over must occur a t u = u,, since only here are the two roots equal. The foregoing method of choosing the correct solution is the conventional one of choosing the particular root which leads to a physically meaningful solution. The nature of the roots, however, is best illustrated graphically as discussed above.) The eventual upswinging of the positive value of D 1 , 2 is no doubt largely responsible for the
-
~
~~~~~~~~
Table II. Exptl.
Temp., "C. 23.0 25.0 24.0 24.0 26.5 23.9 24.0 25.6 22.6 24.6 23.7 24.2
280
Press., Cm. Hg 63.92 64.25 64.91 64.91 64.22 64.01 64.22 64.46 64.64 64.49 64.70 64.70
cc.
I&EC FUNDAMENTALS
~
~~
Experimental Results for Diffusion Coefficient of Binary Systems
Sample Size, 0.02 0.08 0.04 0.04 0.04 0.04 0.10 0.10 0.04 0.03 0.05 0.03
anomalous increase in the diffusion coefficient reported earlier (76). If the correct root is taken, there is apparently no velocity effect except that which would occur a t the onset of turbulent flow. Reynolds numbers as high as 150 have been successfully used. Nonetheless, the best precision is obtained a t low flow velocities, and most measurements have been made at velocities less than uc. Binary coefficients have been determined for the systems N2-CO2, Nz-02, He-H2, He-A, N r H 2 , COz-Hz, He-NH3, COZ-NZ, Nz-He, He-Nz, OZ-He, He-02, COZ-He, and HeCO2, where the first mentioned of any pair is the carrier gas and the second the trace sample. In addition to the binary coefficients, concentration effects have been determined for the He-CO2 and He-NZ systems, and the three-component system He-C02-H2 has been studied. Before any coefficients were obtained, the detector of the instrument was checked as to linearity of response by varying the sample size. The detector was found to be linear over the concentration employed. The correction technique was also investigated by taking data for seven different sample sizes on both the long and short columns. The ? of the short column was subtracted from that of the long column and the difference was found to be reasonably constant within the experimental error of the data. The effects on coefficient measurements due to diffusion tube
Experimental CoeJicient , System He-A HeHz Nz-Hz N2-H2 COz-Hz HeNHa Nz-C02 c02-N~ N2-He HeN2 02-He Heon
q . Cm./Sec. 0.866 i= 0.009 1 ,339 f 0.046 0.914 f 0.019 0.912 f 0.008 0.793 f 0.003 0.923 f 0.007 0.192 f 0.001 0,214 f 0.003 0.833 f 0.007 0.808 f 0.006 0.860 f 0.005 0.840 f 0.011
Carrier Velocity Range, Cm./Sec. 2.53- 8.32 3.39- 8 . 4 5 3.38-16.60 Constant Constant Constant Constant Constant Constant Constant Constant Constant
Co1r. Temp., ' C. 23.0 25.0 0.00 0.00 25.0 ...
25.0 25.0 25.0 25 .O 25.0 25 .O
Corr. Pressure, Cm. Hg 76.00 76.00 76.00 76.00 76.00 76.00 76.00 76.00 76.00 76.00 76.00 76.00
Corr.
CoeBcimt 0.729 1.132 0,687 0.686 0.665 0.163 0.181 0.717 0.687 0.737 0.718
Lit. Values 0.725 (37),0.733 (24) 1 . 4 1 (77) 0.691 ( 7 ) 0.691 ( 7 ) 0.638 (7), 0.646 (2) 0,167 (34) 0.167 (34) 0.730 (34) 0,688 (34)
...
...
1.000
a
0, (pos. !;quare mot term)
0
13. (Neg. square root term)
/
-2 c
-I 0.500
0
v (cm/sec)
Figure 2. Experimental values of diffusion coefficient as a function of flow velocity for N2-C02system Equation 5 used t o e v a l u a t e d a t a
bending and variation in tube radius were investigated and found to be negligible (72, 75). Reynolds numbers were calculated to check the carrier gas stream for laminar flow. The Nz-COz and N2-02 systems were investigated to check the shape of the experimental D1,2us. u curve against that obtained theoretically. These results are plotted in Figures 2 and 3. T h e experimental curves can be satisfactorily interpreted using the previous theory. The value of D l , % obtained from the horizontal section of the Nz-CO2 curve is 0.195 :k 0.004 sq. cm. per second a t 20.0"C. and 64.78 cm. of H g (atmospheric pressure at this location). This value, when corrected to standard pressure (DIP1= DoPo), becomes 0.163 sq. cm. per second, a quantity which agrees exactly with the value obtained by Westenberg (30) using the point source technique. The value of Dl,z obtained from the horizontal section of the N2-02 curve is 0.171 =k 0.008 sq. cm. per second a t 25' C. and standard pressure.
2.0c
0
D, (FQs.
o
D- (Nag. square root term)
T h e data for the h T 2 - 0 2 system were not taken a t a velocity less than u,, because the thermal conductivities of the two gases are similar and it was necessary to operate the instrument a t high sensitivity. The thermal conductivity detection cell, especially when in series with an empty tube, is very sensitive to normal laboratory pressure and flow fluctuations. This effect was more noticeable a t low flow rates; hence the experimental values were taken above u,. The experimental results of the remaining systems studied are tabulated in Table 11. The precision indicated is the average deviation of the experimental values. All values represent the average deviation of the experimental values. All values represent the average of a minimum of nine measurements. All temperature corrections were made using the "2 power law. The error of this correction is small, since the temperature intervals are small. In the cases in which the carrier gas and the trace sample gas were interchanged, a difference was found for the experimental coefficients. The two values bracket the value obtained with a conventional Loschmidt apparatus. This is in agreement with resultsfound by Walker and Westenberg (37,34),although the values do not agree completely in all cases with those of the point source technique. Reference to Table I1 shows that the precision of measurements made a t constant carrier velocity is better than of those made over a velocity range. This is probably due in part to incomplete flow stabilization in cases of varying flow velocity. For measurements made at various flow velocities the values of taken a t or near uc were difficult to obtain and were subject to large errors. For maximum accuracy the measurements made a t constant flow velocity employed a velocity appreciably different from u,.
Concentration Effects The effect of concentration on the diffusion coefficient was studied by injecting sample gas into a gas stream made up of a mixture of the carrier and the sample gas. The gas flow and composition were controlled by pieces of thermometer capillary
/
square root term)
8 0
\
(Y
E
0
n
1.00
0
5.0
15.0
10.0
20.0
v (cm/sec)
Figure 3.
Experimental plot for NZ-02system VQL 1
NO. 4 N O V E M B E R 1 9 6 2
281
~~
Table 111.
Mole Fract.
He
coz
He
coz He GO2 He c02
Table IV.
He in Carrier 0.000 0.182 0.182 0.321 0.321 0.392 0.392 0.492 0.492 0,582 0.582 0.632 0.632 0.719 0.719 0,821 0.821 0.874 0.874 1.000
NZ He N2 He Nz He Nz He Nz
Exptl. Tfmp., C. 25.9 26.7 26.7 26.0 26.0 26.4 26.4 27.0 27.0 26.0 26.0 26.7 26.7 26.8 26.8 26.6 26.6 26.0 26.0 26.5
Exptl. Press., Cm. Hg 64.36 64.62 64.62 65.01 65.01 65.05 65.05 65.03 65.03 64.42 64.42 64.23 64.23 64.16 64.16 64.25 64.25 64.20 64.20 64.34
Coeflcient, Sq. Cm./Sec.
0.694 f 0.002 0.688 f 0.004 0.739 f 0.003 0.634 f 0.007 0.685 f 0.007 0.676 f 0.005 0.725 f 0.009 0,666 Z!= 0.008 0.731 f 0.004 0.691 f 0.006 0.756 f 0.005 0,698 f 0.006 0.761 f 0.008 0.688 f 0.005 0.775 f 0.008 0.696 f 0.008 0.789 f 0.009 0.705 f 0.009 0.726 f 0.009 0.723 f 0.003
Experimental Results for Concentration Studies on He-NQ System
Mole Fract. Sample He
~
Experimental Results for Concentration Studies on H e 4 0 2 System
Exptl. Temp.,
He 0.000 0.178 0.178 0.342 0.342 0,490 0.490 0.644 0.644 1 .ooo
O
c.
22.6 29.8 29.8 30.0 30.0 29.2 29.2 30.5 30.5 24.6
Exptl. Press.,
Cm. Hg 64.64 64.32 64.32 64.50 64.50 64.50 64.50 64.32 64.32 64.49
Coeficient, Sq. Cm./Sec. 0.833 f 0.007 0.859 f 0.004 0.852 Z!= 0.003 0.849 f 0.003 0.834 f 0.007 0.843 f 0.007 0.830 zk 0.003 0.859 Z!= 0.003 0.835 f 0.006 0,808 f 0.006
through which the two gases were passed a t known rates from the source tanks. The two gas combinations studied were COZ-He and Nz-He. In Tables I11 and I V each value is the average of a minimum of nine separate determinations. Sample sizes were comparable to those used for the one-component carrier gas measurements. The results shown in Tables I11 and IV, corrected to standard pressure and 25.0' C., have been plotted in Figures 4 and 5 . For the He-COZ study a n increase in the mole fraction of helium tends to increase the value of the diffusion coefficient. Lonius (22) used a Loschmidt technique to study the Nz-Hz pair and found a similar increase in 01,~ with increasing mole fraction of the less dense component. In the case of the He-Nz pair, an increase in the mole fraction of He tends to decrease the value of the coefficient. Lonius (22) and Schafer, Corte, and Moesta (23) used a Loschmidt technique on the systems H2-COz, He-A, COZ-Nz, and Ns-H2, and found a decrease in D with increase in the mole fraction of the less dense component. A three-component system was studied in an attempt t o verify the work of Hirschfelder and Curtiss (79) and of Fairbanks and Wilke (6) experimentally. According to them, the equation for the diffusion coefficient of a trace sample of one gas into a mixture made up of two other gases is
where x2 and x3 are mole fractions of the two gases making u p the mixture and Hz-1 and 0 3 - 1 are the diffusion coefficients of the sample gas in each of the mixture gases. In the system studied, H Bwas used as the trace sample in mixtures of He and COz. The experimental results (Table V) agree well with those predicted theoretically. Walker and Westenberg (32) have also been able to obtain good results for this type of system using the point source technique. Conclusion
The method discussed has been successful in rapidly obtaining experimental gaseous diffusion data under a wide range of conditions. While the object of this research has been the exploration of the range of applicability, little work was done in improving the apparatus beyond what is available commercially. Although such apparatus is adequate for many
1.00. Dmixt, He
0
o
0.1
a2
a3
a4
0.5
0.6
0.7
0.8
Mole Fraction He in Cmrier Gas
Figure 4. 282
l&EC FUNDAMENTALS
Effect of concentration in He-C02 system
0.9
Table V.
Mole Fract. He 0.222 0,360 0.513 0.720 0.750
Experimental and Theoretical Results for Three-Component Study
Exptl. Temp.,
Exptl. Press., Cm. Hg 64.70 64.91 64.90 64.90 64.68
c.
29.8 28.7 28.9 29.0 29.8
Sample
Size,
Cc. 0.03 0.03 0.04 0.06 0.06
0.838 =!= 0.003 0.885 =I=0.003 0.938 k 0.007 0.999 i 0.006 1.046 =!= 0.005
D at Std. P and 25.0” C. Exptl. Calcd. 0.697 0.714 0.742 0.749 0.785 0.792 0,836 0.857 0.869 0,868
I
n -
070
060
1
3 t
purposes, significant imxovements are possible which should both increase accuracy and reduce time. Both would be useful for the compilation of extensive diffusion data. The first step in improving the apparatus would involve a significant reduction in the dead volume of the injection and detection devices. This would eliminate the need for the short tube corrections, and permit greater accuracy. Increased tube length would also improve accuracy. The feasibility of such improvements is now under active invee8tigation. Literature Cited
(1) Boardman, L. E., Wiild, N. E., Proc. Roy. Soc. (London) A1629 511 (1937). (2) Boyd, C. A,, Stein, N., Steingrimssom, Y., Rumpel, W. F., J . Chem. Phys. 19,548 (1951). (3) Carberry, J. J., Bretton, R. H., A . I. Ch. E . J . 4, 367 (1958). (4) Coward, H. F., Georgeson, E. H. M., J . Chem. Soc. 1937, 1085. (5) Desty, D. H., Goldup, .4.,“Gas Chromatography,” R. P. W. Scott, ed., p. 162, Butterworths, Washington, 1960. (6) Fairbanks, D. F., Miilke, C. R., IND. ENG. CHEM.42, 471 (1950). \ - - - -, -
(7) Giddings, J. C., Anal. Chenz. 33,962 (1961). (8) Zbid., 34,458 (1962). (9) Giddings, J. C., J . Ch(!m.Educ. 35, 588 (1958). (IO) Giddings, J. C., J . C,hem. Phys. 31, 1462 (1959) (11) Giddinzs. J. C.. J.C,hromatoe. 3. 44:3 (1960). , (12) Ibid., p-520. ’ (13) Giddings, J. C., Ibid., 5 , 46, 61 (19Cil).
(14) Giddings, J. C., .%‘ature 188,847 (1960). (15) Giddings, J. C., unpublished work. (16) Giddings, J. C., Seager, S. L., J . Chem. Phys. 33, 1579 (1960); (17) Gilliland, E. R., IND.ENG.CHEM.26, 681 (1934). (18) Golay, M. J. E., “Gas Chromatography,” D. H. Desty, ed., p. 36, Butterworths, London, 1958. (19) Hirschfelder, J. O., Curtiss, C. F., “Third Symposium on Combustion, Flame and Explosion Phenomena,” p. 124, Williams and Wilkins, Baltimore, Md., 1949. (20) Keulemans, A. I. M., “Gas Chromatography,” Reinhold, New York, 1959. (21) Kimpton, D. D., Wall, F. T , J . Phys. Chem. 56,715 (1952). (22) Lonius, A , , Ann. Phys. 29,664 (1909). (23) Schafer, K., Corte, H., Moesta, H., Z. Elektrochem. 55, 662 (1951). (24) Schmidt, R., Ann. Phys. 14,801 (1904). (25) Scott, R. P. W., Hazeldean, G. S. F., “Gas Chromatography,” R. P. W. Scott, ed., p. 144, Butterworths, Washington, 1960. (26) Taylor, G., Proc. Roy. Soc. (London) A219,186 (1953). (27) Waldmann, L., Naturwissenschaften 32,222 (1944). (28) Waldmann, L., 2. Naturforsch. 59, 327 (1950). (29) Waldmann, L., Z. Physik 124,2 (1947). (30) Walker, R. E., Westenberg, A. A , , J. Chem. Phys. 26, 1544 (1957). (31) Ibid., 31,519 (1959). (32) Ibid., 32,1314 (1960). (33) Wall, F. T., Kidder, G. A., J . Phys. Chem. 50,235 (1946). (34) Westenberg, A. A., Walker, R. E., J . Chem. Phys. 29, 1139 (1958).
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RECEIVED for review November 2, 1961 ACCEPTEDJune 4, 1962
VOL. 1
NO. 4
NOVEMBER 1 9 6 2
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