S. CHULIANG
910
Vol. 57
ON THE CALCULATION OF THERMAL TRANSPIRATION BY S. CHULIANG’ Division of Applied Chemistry, National Research Council, Ottawa, Canada Received April
4, le68
The em irical constants used for the calculation of thermal trans iration have been more accurately determined, and a new correl%ion between the wessure shifting factor and the gas coksional diameter becomes possible. Helium has been adopted as the new standard^toreplace nitrogen.
Introduction
Results and Discussion
When a temperature difference exists between the pressure measuring device and the,part of the system to be measured, such as in the cases of low temperature adsorption and vapor pressure measurements, the pressure measured differs from that desired because of thermal transpiration. For the calculation of this thermal transpiration effect R, an equation was introduced,2 where pl and p2 are
In the earlier works, nitrogen was chosen arbitrarily as the standard. It has now been found that the thermal transpiration effect is important in a much wider region than then anticipated, and the choice of standard cannot be made arbitrarily without causing inconvenience in the future. Helium has thus been decided upon as the new standard. Because helium has the smallest collisional diameter of all the gases, it has the largest thermal transpiration effect under the same conditions. Therefore, its effect can be measured most accurately. Furthermore, by redefining a new pressure shifting factor 4, eq. 1takes a new form more convenient for calculation
the true and measured pressures; a and p the empirical constants; X = p,d, d being the diameter of the connecting tube; R m = (T1/T2)lI2,TI and T2 being the temperatures (OK.) of the system to be measured and of the measuring device; and f the pressure shifting factor with f for nitrogen arbitrarily defined as unity; the empirical relationship
fdf,= 1
+ 4[(~1/rz)- 11
(2)
was suggested later on for the estimation of the R values of one gas from those of another, through the use of their collisional diameters rl and r2.a In the earlier communication, in addition to the obvious error that fi/fi appeared where f 2 / f 1 should, the condition rl/r2 3 1 was, unfortunately, left out. Without this condition, eq. 2 would require rl/r2to have a fixed value. The constants a, p and f’s have since been more accurately determined, and a more reliable correlation between r and f becomes possible. It is felt that the new information is useful enough to be reported.
Experimental The measurements were made by the “relative method” described in detail elsewhere.2 Briefly, the system to be measured was connected with two tubes of different diame$rs, one narrow and one wide, to two pressure gages. The measured pressure” pz was that measured through the narrow tube. The “true pressure” pl was represented by that measured through the wide tube of such a diameter d / h > l o 3 to 104, h being the mean free path of the gas, SO that the thermal transpiration effect became unim ortant. Thus, the larger the diameter of the wide tube, tge more accurate the measurements may be made at lower pressures. I n the present study, a 90 mm. i.d. tube was used as the wide tube. Two capillaries of 1.2 and 2.6 mm; i.d. were used as the narrow tubes. The pressure gage temperature was constant at 296 f 2’K. The cold bath temperature was at 195’K. (Dry Ice-acetone) and 77.5 f 0.1”K. for two sets of measurements. The gases measured included helium, neon, hydrogen, argon, nitrogen, krypton, xenon and ethylene. The ethylene was withdrawn from a commercial tank and urified by fractionation, the final gas was found to be at feast 99.9% pure by mass spectrometric analysis. All the other gases were obtained from Linde Air Products. (1) Researoh Laboratory, Dominion Tar & Chemical Co., 3547 Allard Straet, Ville LaSalle, P. Q., Canada. (2) 8. C. Liang, J . Applied Phys., 22, 148 (1961). JOUBNAL, 56,660 (1952). (3) 8. C. Liang, THIE
The relationship between f and 4 may be expressed, for a gas, dg = fHe/fg. In the present form, 4 has a minimum value of unity. The results obtained are summarized in Table I, where the previously published f values are also included and fHe/fg evaluated to compare with the newly determined values. Also included are the collisional diameters T of the various gases. It is of interest to note that, with the exception of nitrogen and krypton, a higher r is associated with a higher 4. TABLE I PRESSURE SHIFTINGFACTORS A N D COLLISIONAL DIAMETERS Gas Old factor f
He
Ne
Ha
A
Na
( j ~ t
= 1)
fHe/fg
2.80 2.0 2.20 1.25 1.0000 1.0000 1.40 1.27 2.24 2.80
New factor 0 (+He e 1)
Kr
Xe CnHi
,... ....
0.5 5.60
1.0000 1.41 1.52 2.93 3.28
3.90 6.87 7.2
2.58
3.60 4 . 1
Collisional diam. r,
A.
2.80 2.90 3.41 3.70
4.5
In Fig. 1 are plotted 4 us. r (in A.), both on a logarithmic scale. The straight line relationship 0.27 log 6
log T
- 0.41
(4)
is indicated. At present the 4 values can be determined only with 5% uncertainty, with the exception of that of helium which is defined as unity. The r values are not yet very well defined, and a 5% uncertainty is used for illustration. The cross on each point represents such uncertainties. It should be mentioned that the molecular diameters used are those calculated from the viscosity data by the Lennard-Jones 12 :6 potential method. It has been suggested to the author that the “equivalent elastic-sphere diameter^"^ given by Kennard should be used, because the krypton molecule under (4) E. H. Kennard, “Kinetic Theory of Gases,” McGraw-Hill Book Co., Ino., New York, N. Y., 1938, p. 149.
,
CALCULATION OF THERMAL TRANSPIRATION
Dee., 1953
such considerations, has a larger diameter than nitrogen. Since a universally accepted method for the calculation of molecular diameter is not available, and a discussion of the relative merit of different methods is out of the province of this communication, we shall leave this point out. This action may be justified because eq. 4 is only an empirical relationship (the true relationship is unknown) for the estimate of the thermal transpiration effect of a gas for which no experimental measurements have been made, and accuracy is unimportant, Accurate values, if desired, should be obtained experimentally, or calculated from eq. 3 if +value is obtainable by some other more accurate method. We wish to emphasize, however, although we are not convinced of the accuracy of the Kennard molecular diameter values, we agree that krypton should have a larger diameter than nitrogen. The thermal transpiration effect of nitrogen has been determined several times. The constants a and /3 for nitrogen obtained previously are compared with those obtained during the present study in Table 11. The difference is very small indeed. In fact, it coincides almost exactly with the results of Los and Fergusson.6 All these indicate that the existence of serious experimental errors is unlikely. In the case of krypton, our results agree well with those of Fergusson6a and also those of Holmes and Kington,6b and $Kr is unmistakably larger than ~ N Z , suggesting that krypton may have a collisional diameter larger than nitrogen, in fact, by some 6-7%. When krypton and nitrogen were used for surface area measurement by the adsorption method, it was always necessary to assign krypton a cross-sectional area some 15% higher than that of nitrogen, or 7.5% in terms of diameter.' The Kennard values indicate a 10% difference. TABLE I1 FOR NITROGEN CONSTANTS Cold end temp.,
a
OK.
Previous
195 77.3
22.76 29.5
New (u&+Nz')
27.1 27.1
Previous
5.00 12.5
'
New ('54N1)
4.79 12.4
In the earlier studies, the f values were thought to be a function of the cold end temperatures; this is (5) J. M. Los and R. R. Fergusson, Trans. Faraday SOC.,48, 730 (1952).
(6) (a) and (b) both unpublished. (7) R. A. Beebe, J. B. Beckwith and J. M. Honig, J . A h . Chsm. SOC.,67 1554 (1945).
0.7
t
91 1
