Thomas A. Gover
University of Virginio Chorlottesviile
I
Diffusion 01 GCIS~S A physical chemistry experiment
T h e conventional method for measuring gas diffusion coefficients is with the Loschmidt apparatus. I n this method the diffusion tube is separated into two parts by a closable diaphragm and initially each part contains a different gas. The diaphram is opened to initiate the experiment and after an optimum time (I) the diaphragm is closed and the amount of gas which has diffused is determined by analyzing the contents of each half. From these data the diffusion coefficient, Dl2, is calculated. The Loschmidt apparatus has given excellent results; however, as a student experiment it presents difficulties. The use of a pinched rubber tube for a diaphragm is not satisfactory and the accuracy of the value obtained for D12depends on one analytical determination. The method outlined below was suggested by Onsager and used b?- Harned and co-workers (2) t o determine the diffusion coefficient,^ of aqueous electrolytes. They followed the progress of diiusion by measuring the change of conductivity with time. The inherent accuracy of conductivity measurements allowed them t o obtain diffusion coefficients reportedly accurate t o about 0.1%. Gas composition cannot he determined with equivalent accuracy but, using gas chromatographic analysis, we have obtained diffusion coefficients which seem t o he accurate to about 1%.
c=A
+
?,=m
B. exp n=1
A and B, are numbers and L is the length of the cell. This equation gives the concentration, c, a t any point x in the cell at any time, t. If one measures the concentration a t two points each a distance, a, from each cell end, one gets: CL = A
+ B, exp (-F) -- m s + Bz exp
cz= A
+ B, exp (--:;Dl) -
oos
Bn exp
2rz
+ . ..
2r(L - a)
+ . ..
cos
- a) +
r(L
ms
Note that: cos
(r - 2) L
= - COS
cos ( 2 r -
2ra
= cos
ra r,
2 n
Subtracting C1from C2 drops A and all the even terms and doubles all the odd terms. One obtains: CS - CI= ~ Bexp I
(- r'Dt LP
eos
na
+
ZBSexp
(I") 9raDt
em
3na
+ ...
If it is specified that the concentrations will be determined at exactly the total cell length from each end, i.e., a = L/6, then: 2Bn exp
where c is concentration, t is time, and x is the coordinate in the direction of mass flow. When this equation is solved for the case of a cell of finite length where concentration changes occur a t the ends of the cell, there is generated a Fourier series of the form:
(- 4rPDt I,)
(- 4+Dt T)
Theory
Fick's Second Law of Diffusion for the one dimensional case is:
[- (f)b,,t]eos nn-2
(1')
9n'Dt
cos
7 . Lg
37
=
0
Then if the time a t which the concentrations are determined is relatively long the fifth term in the series will he much less than the first and: CS- CI = (const)exp
(- 9)
Volume 44, Number 7, July 1967
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409
-SEPTUM W I B IN 0 D.
-
0.6 CM.
Figure 1. Approximoteiy half of the gar diffusion cell. The over-all length and sampling port distance from each end should b e as close to the ~peciRcd60.0 cm and 10.0 cm or possible. All other dimensions a m approximate.
The diffusion coefficient is obtained by plotting log C2 - C1 versus t . Then DI2 = (2.303) (L2)(slope)/r2. This is the treatment used by Harned and applied by him to electrolyte solutions. The result is general, however, and we have found that excellent results can be obtained with gases. The Experiment
Approximately one-half of the diffusion cell is shown in Figure 1. It consists of one inch id copper pipe with a 60-cm diffusion length. There are sampling ports 10 cm from each end and injection ports as near each end as can be managed. For flushing the cell with the host gas in. od copper tubing pierces the bottom and top plates. These flushing tubes are kept short and pinch clamps are used to close the rubber tubing connected t o these tubes as soon as the flushing is complete. The silicone rubber septa used in the various ports were the Barber-Colman (3) type and are standard for the injection ports of their gas chromatographs. The diffusion cell was wrapped with several layers of asbestos paper for insulation against air currents and heat from the student's hand during sampling. Students generally proceeded as follows: The cell was flushed thoroughly with the host gas and allowed ample time to reach room temperature. Then 2-4 cc of diffusing gas were injected by means of a syringe either into the top, if the diffusing gas had a lower density than the host gas, or into the bottom, if i t were more dense. With approximate values of Dl%, the student calculated when the mixing would be about 25% complete and after that time lapse began taking samples. Aliquots about 100 rl in size were drawn from the top and bottom sampling ports and injected into a gas chromatograph equipped with a flame ionization detector. Students generally used only one syringe and, with practice, could reduce the time lapse between top and bottom sampling to about 20 sec. If the samples from top and bottom were exactly the same size, then the peak areas on the chromatogram, being proportional to concentration, could be used directly in the plot of h(C1 - CI) versus time. However, students had difficulty in drawing samples which were duplicates to within 1%. This problem was solved in one of two ways. In the first, a host gas was chosen which would give a peak on the gas chromatograph. Since the concentration of the host gas is almost invariant throughout the run, the peak area from the host gas was proportional to the size of the injected sample. Thus, each sample gave two peaks, 410
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Journal o f Chemical Education
one from the diffusing gas and the other from the host gas. All peaks were normalized so that the area used for the calculations was that which would have appeared if all samples had been of the same size. This first solution to the sampling problem generally requires that the host gas be an organic compound to get a response from the flame detector. I n order that permanent gases could be used as hosts another procedure was followed. To the host permanent gas was added about 0.5% of an organic tracer. The tracer at this concentration did not influence D12but did give a peak which could be used for normalizing to a standard injected volume. The syringes used were Hamilton gas-tight types (4), though it should be possible to adapt other less expensive types. The gas chromatograph was a BarherColman 5000 series. The chart recorder was equipped with a Disc Instrument Co. disc integrator, a convenience but not a necessity. For the low molecular weight aliphatic hydrocarbons used in these experiments we found a 6 ft X in. column of activated alumina maintained at about 150°C to he convenient. Hydrocarbons were generally obtained from Phillips Petroleum Co. (5), some as pure grade, others as research grade. Most of the pure grade compounds contain detectable traces of impurities. One must always make sure that the host gas does not contain as impurity a significant amount of the diffusing gas. We found commercially available liquefied propane to he relatively free of methane, and that natural gas can become a good source of methane completely free of all of other hydrocarbons. A small U-tube of about 8 mm Pyrex tubing is filled with 13X molecular sieve (6) and cooled to -78°C with solid Coz. Of the aliphatic hydrocarbons only methane will move through this column a t a detectable speed. We have found that such treatment of commercial natural gas produces methane with no detectable hydrocarbon impurities.
I
30
70
110
150
TIME 1MIN.I Figure 2. Student data for two diffusing gar pairs. Each kind of point identifies a different student lab pair. Rervltr here ore within 2y0 o f the theoretisol valuer.
TIME (SECONDS X 1001 Figure 3. Student data for two runs where many points were taken to improve the line. Note the excellent line but poor correspondence to the predicted value of 0.076 cma/sec. Some mixing apparently occurs during sampling.
Results and Discussion
One of the virtues of the method outlined here is the wide variety of diffusing pairs which can be studied. Figure 2 shows some typical student data on the diffusion of the pairs methane-ethane and ethane-propane. To our knowledge Dl2 has not been determined for these systems before. Each kind of point identifies a different pair of students. Thus, as can be seen, it is possible for students who work carefully to reproduce each others work and, as will be seen, obtain values extremely close to that predicted by theory. Most of the sources of error will mix the gases faster, producing a high value of D I ~ . Figure 3 shows student results, a composite of two runs, where many points were taken to get a better line and presumably better results. However, although the line is quite good, the repeated samplings caused extra mixing and the values these students obtained for D I ~0.122 , cm2/sec, does not compare well with the expected value of 0.076 cmz/sec. Student results shown in Figure 2, where about four points were taken per run, gave excellent results. The value of Dl, obtained in these experiments for the methane-ethane system was 0.153 cm2/sec while theory predicts 0.150 cm2/sec. Similarly, the value obtained for the ethane-propane system, 0.078 cm2/sec, compares very well with the predicted value of 0.076 cm2/sec. Several other systems have been studied. In some cases the results were as good as those shown in Figure 2. I n other cases they were worse. The key factors seem t o he reasonable care in sampling and efficient use of the chromatograph. The peaks should be as nearly full scale as can be managed. Since differences in peak area are used a t a time when the
areas are approaching each other, significant figures are lost in the subtraction and effort should be expended t o start with the largest peaks possible. Since most of the available diffusion data in the literature have as one member of the pair a permanent gas which will not give a response in the flame detector, and since this method is simplest when both gases are flammable, students will be faced with data which cannot in many cases be checked against literature values. Fortunately, current theory is quite good and without difficulty the student will be able to calculate a predicted value which should be within 1% of the correct one. Students are referred to Hirschfelder, Curtiss, and Bird (7) for the theory. These authors have evaluated the collision integrals (8) for various molecules using a Lennard-Jones potential function. The transport properties they calculate agree very well with experiment. For example, student results for the diffusing pair ethanepropane gave an experimental DI2of 0.078 cmZ/sec while refined theory predicted 0.076 cm2/sec. Students need not understand the theory to use it in calculations. A sample calculation is shown in Reference (7) which any student can follow. In summary, an experimental method for measuring diffusion coefficients has been presented. The method seems superior to the widely used Loschmidt apparatus. Further, if the apparatus were refined t o incorporate analytical methods amenable t o accurate determination of concentration differences, the results should surpass in accuracy all other available methods for determining the diffusion coefficients of gases. One such refinement might be to use a diffusing gas which was radioactively tagged and to plot the log of the difference in counting rates versus time. No sampling would be required and the counting rates could be electrically subtracted. Acknowledgment
The author wishes to thank Mr. A. W. Nowelle who constructed the diffusion cell and the students W. Daniel, D. Gordon, J. Henry, J. Halbert, J. D. Lmehan, H. Lokey, E. A. Miller, D. Powelson, D. W. Van Zant, and D. Wallen whose data were used. Literature Cited (1) SHOEMAKER AND GARLAND,"Experiments in Physical Chemistry," McGraw Hill Book Co., New York, N. Y., 1962, p. 99. (2) HARNED. H. S.. C h a . Rev.., 40. 461 (1947). , , (3j Barber-dolmai Co., Industrial Instruments Division, Rockford, Ill. Order part no. 6-138. (4) Hamilton Co., P.O. Box 307, Whittier, Calif. 90608. Phillips Petroleum Co., Special Products Division, Bartlesville, OMa. (6)HIRSCHFELDER, J . 0 .,CWTISS,C. F., ANDBIRD.R.B.,"Moleaular Theory of Gases and Liquids," John Wiley & Sons, Inc., New York, N. Y., 1954, p. 579. (7) KAUZMANN, W., "Kinetic Theory of Gases," W. A. Benjamin, Inc., New York, N. Y., 1966, p. 232.
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Volume 44. Number 7. Julv 1967 1 A 1 1
