I John H. Smith Butler University Indianapolis, Indiana 46207
I (
Determination of Molecular Diameters
by the Use of the Crookes Radiometer An undergraduate experiment
The Crookes radiometer, often displayed in chemistry laboratories as a curiosity and sold commerically as a toy, affords a visually gratifying method for studying the kinetic theory of gases, thermal transpiration, and mean free path properties of gases. A commonly seen vane radiometer, depicted in Figure 1, consists of aluminum vanes blackened on one side and mounted on an axis free to rotate in a partially evacuated bulb. The velocity of revolution of the vanes, and thus the radiometric forces being applied, may he investigated as a function of pressure for different gases a t a constant temperature of irradiation.
In 1919, West (10) gave a qualitative account of the change in temperature gradients surrounding a vane as a function of pressure. First, it is understood that an irradiated vane blackened on one side and polished on the other will absorb more heat on the blackened side and thus have a slightly higher temperature than the polished side. Second, the peripheral portions of the vane will he a t a slightly lower temperature than the interior due to the increased dissipation of heat at the edges. The body of gas coming into thermal contact with the vane will reflect the temperature of the vane as depicted by the isothermals of Figures 2 and 3 for the low and high pressure cases, respectively. I t is seen
'iv VACUUM SYSTEM
-m PIWNI GAUGE
Figure 2.
Low pressure isotherrnols.
Figure 3.
High prrrwrc isothermalr.
_B
-A
Figure 1. Radiometer apparatus. A, Constant temperature enclosure; B. 2 5 0 4 boiling flask; C, epoxy cement; 0, stmight pin; E, glorr tube; F, aluminum vanes blackened on one side.
Brief History and Theory of Operation
Fresenl (1) first observed the repulsion of an irradiated body suspended in a gas. W. Crookes (Z) constructed several devices illustrating this repulsion phenomena and showed that the radiometer forces were a function of the gas pressure and intensity of radiation. Briiche and Littwin (3) have shown that the force was a maximum for a certain optimum pressure (later to be seen in Fig. 5), implying that two separate theories are required to explain the high and low pressure cases. Several review articles are available (4-9) which attempt to explain the radiometric phenomena for these two conditions but no coherent pattern of behavior has yet emerged. 590 / Journol of Chemical Education
that when the pressure is low and the mean free path between molecular collisions, and thus energy transfer, is large, the molecules are more efficient in transferring the heat of the vane a distance into the body of the gas. The isothermals for a supposed temperature difference of 10"for the two sides (Fig. 2) thus reflect the temperature gradients on the surface and are more widely spaced and less extreme than in Figure 3. In the latter diagram, the pressure is higher and the mean free path is lower, with the result that temperature transfer is immediate and the isothermals are more extreme and closely spaced around the vane. The temperature gradients on the surface of the vane are no longer apparent in this case. Radiometric forces exerted on the vanes in general arise from a streaming of gas molecules from the colder
regions of the vane to the warmer regions. This streaming, called thermal transpiration (11), is caused by a greater dissipation of tangential momentum of the warm gas as compared to the cooler gas. The blackened (charcoal) surface is more efficient in transferring momentum from the particles to the vane, i.e., the accommodation coefficient, a, the fraction of translational energy transferred on a collision @), is higher for the dark surface thau for the polished. This steady drain of tangential momentum from the warm gas sets the neighboring colder gas into tangential motion and a thermal creep fiom cold to warm regions occurs. The direction of this streaming is indicated by the dotted arrows in Figures 2 and 3 for the low and high pressure cases, respectively. These two cases can now be considered separately. As seen from Figure 2, in the low pressure case (usually below 20 p ) the thermal streaming of the gas occurs over the entire surface of the vane rather than a t the edges as in Figure 3. The radiometric force on the vane in this low pressure case is due to the greater normal momentum transfer of the warmer gas to the blackened side of the vane as compared to the cooler gas on the polished side. The warmer gas molecules on the average strike the blackened side of the vane with increased velocity and have their momentum transferred with a greater efficiency than the polished side. The body of the gas is not thinned out by the rebounding of high velocity molecules from the warmer side of the vane because cooler molecules are constantly replenishing the supply due to the thermal streaming effect. This thermal contact of gas molecules with the warm and cool side of the vane is the basis for the Icnudsen effect. The radiometric effect is further complicated, however, by the fact that this contact is effected by the process of thermal transpiration rather than the proximetry of the two surfaces. By calculating the normal momentum transfer of gas molecules as they collide with the surface of the vane, Knudsen (IS) and more recently Wu (14) have determined the resulting radiometric force in the low pressure case to be proportional to
The quantities TI, To and Poare respectively, the temperature of the vane and the temperature and pressure of the body of the gas. The blackened side of the vane (with higher accommodation coefficient al) has a greater efficiency of normal momentum transfer thau the polished side (of lower coefficient G),and the force is seen to be directly proportional to the pressure for this low pressure case. The high pressure case is seen from Figure 2 to be largely an edge effect, i.e., the streaming of molecules must occur primarily around the edges. Figure 4 depicts this edge in greater detail, showing the direction of molecular streaming and the force of reaction to this streaming on the vane. Several authors (15-18) have obtained expressions for this latter force by d i e r e n t methods but largely agree that it is proportional to the quantity
where L is the mean free path, and dT/dX is the temperature gradient on the edge of the vane. Expressed in another form, the force is proportional to T dT - PddX
(3)
where u is the collision diameter. Thus the radiometric force is seen to be inversely proportional to the pressure for this high pressure case. By combining eqns. (1) and (3) by use of the empirically determined constants a and b (19), the radiometric force becomes, a t constant temperature
When pressure is high
and when it is low
The constant b can therefore be associated with (T/04)(dT/dX) from eqn. (3) and the constant a with
from eqn. (1). By differentiation of eqn. (4) the maximum force is seen to be
and thus F"""
=
K
2
a t constant T. The parameter K may be determined empirically. In a mathematically rigorous (but perhaps physically less elucidating) treatment, AIason and Block (9) unify the above separate pressure theories into one equation quite similar to eqn. (4)
I n this expression a1 and Q are terms involving, among other quantities, the thermal diffusion behavior of the gas. Thermal diffusion is a term usually used to describe the behavior of binary gas mixtures in a temperature gradient. I t is found that such a mixture tends to separate into its two components with the heavier gas molecules tending to accumulate in the low-tempera-
UNIFORM TEMPERATURE
COLD
4.
Radiometric edge effects at high prewure.
Volume 47, Number 8, August 1970
/
591
ture region. Starting, therefore, with the ChapmanEnskog one-dimensional diffusion equation for a binary mixture, Mason and Block associate the heavier component of the "mixture" with the stationary vanes of the radiometer and the lighter component with the actual gas under consideration. The vanes thus "diffuse" to the colder regions of the gas surrounding the polished surface by the same mechanism as that of thermal diffusion (20),i.e., due to increased momentum transfer or momentum drain of the warm gas molecules to the vane (heavier gas molecules) than vice versa. Now, thermal diffusion in which the heavier component of the gas mixture is held stationary (the vanes), describes the thermal streaming of the lighter gas molecules from the low to the high temperature regions, corresponding to the polished and black sides of the vane, respectively. This is the transpiration phenomenon described previously which prevented the thinning out of the body of the gas due to the rebounding of warm gas molecules. The transpiration process tends to beimportant a t low gas pressure (Fig. 2) but tends to fall off a t higher pressure (Fig. 3) thus leading to a maximum radiometric force. I t is understandable therefore that eqn. (4) should be largely equivalent to eqn. (5). Further, since the minor bracketed term added in eqn. (5) is not a function of pressure, the maximum radiometric force expression remains unaltered. Experimental and Results
The vane radiometer is constructed as shown in Figure 1. The aluminum vanes are blackened on one side by adhering charcoal granules to a thin layer of epoxy cement on the vane. Four of these vanes are then cemented to a small glass tube sealed on one end and the assembly is positioned on a pin cemented to the bottom of a glass flask. Constant temperature of irradiation during the laboratory period is assured by encasing the flask in a larger glass tube and allowing equality of temperature to obtain inside and outside the flask. The pressures of various gases admitted through the vacuum system are determined with a Pirani gauge and the time required for 50 revolutions, as sighted through a cathetometer telescope, is measured with a timer. A small vertical line painted on the glass tube and flask is helpful in this latter respect. The flask can then be steadily irradiated by an infared lamp (250 W) clamped a t a constant distance from the flask. The temperature fluctuations are usually less than half a degree throughout the laboratory period. Since we are only interested in the relative maximum radiometric forces for various gases, the force is assumed proportional to the reciprocal of the time required for a certain number of revolutions. I t is further assumed that the accommodation coefficients for various gases are the same for the same vane surfaces in eqn. (1). This is largely the case for heavier gases that are appreciably absorbed and not true for Hz as will he seen below. The results for 5 gases taken during a laboratory period are shown in Figure 5. As seen, the maximum radiometric forces occur between 5 and 50 p pressure a n d are tabulated as the reciprocal of the minimum quantity of seconds per 10 revolutions (from Fig. 5) in the table. The parameter k was found to be 1.20 X 10P, obtained by averaging this quantity for the 592
/
Journal of Chemical Education
Figure 5. Time required for ten revolutions of vans ossombly for voriour go9er or o function of pressure.
gases 0 2 , N20, and SOz. A large discrepancy between the calculated and literature value of u for H, has presumably occurred because of the large difference hetween the accommodation coefficients of this compound and the other gases for the same type of surface ($1). A smaller discrepancy is seen to occur for the molecule C8H8,which is probably the result of inelastic collisions of this bulkier molecule with the sides of the vanes. Calculated values of u for the other gases, however, are seen to compare quite favorably to the accepted literature results. Literature Cited (1) FREBNEL, q., Ann. Chirn. Phya.. 29, 57. 107 (1825). (2) C n o o x ~ sW., , Phil. Trans., 164,501 (1874); 166,340 (1876). W., Zeds f. Phydk;,52, 318 (1028). (3) BBUCn%E.. AND *WIN, E. H., Kinetic Theory of Gases. MoGraw-Hill Book Co., (4) KENNARD, New Ymk. 1938.11.3:t7. (5) LOEB, L. B., ' ' K i k e Theory of Gases." (let ed.). MoGraw-Hill Book Co.. New York, 1927, p. 205. . S., GLASSTONE, S., "A Treatise on Physical Chemistry," (6) T ~ r m s H. (3rd ed.), Lanoaster Press, Lanoaster. Pa., 1951, Vol. 2, p. 144. (7) Dnsaum. S.. LArren~r. J. M.. "Scientific Foundations of Vacuum ~eehnique,"(2nd ed.), John ~ i i e v& Sona. Inr., New York, 1962, p. 277. . . R o s e w a u w , P., A N D LA Mmn, V. K., Phys. Reo., 70,385 (1945). M*son, E. A.. *wo B ~ o o xB., , Ann. Phya. (N.Y.), 37(1). 7 (10.66). WEST,G. D., Pmo. P h y ~SDC. . (London).32,168, 222 (1019). KENNARD, E. H., "Kinetic Theow of Gases, (1st ed.) MoGraa Book Co., New Yark. 1938, D. 322. KADZM*WW. W., "Kinetic Theory of Gaaes," (let d l . W . A. Benjr Ino.. New York. 1966.11.205. KNODBEN. M., Ann. d. Physik, 32, 800 (,1910); 34, 823 (1911): 6 ,,n.zm ,Av"",.
(14) (15) (16) (17)
W a , Y., Ann d. Physit. 19, 144 (1067). EINBTEIN, A,, Z d t 8 . f . Phydk, 27. 1 (1924). EP~TEIN, P. S., 2 e i t . f . Physik. 54,537 (1920). C z ~ r w r M.. , AND HEITNER.G.. 2eits.f. Physik. 30, 258 (1024).
Relative Maximum Radiometric Force for Various Gases (from Fig. 5 ) Relative radiometric force
a
D~Rocco,A,,
(1962).
BENESCH,
AND
Calculated c X lo8
Literature e X lo8
HOOVER,J., J . Chem. Phys., 36, 916
W .,AND ELDER, T., Phya. Rev., 91, 308 (1952).
(18) Loes. L. B.. "Kinetic Theory of Gases," (let ed.) MoGraw-Hill Book Co.. New York. 1 9 2 7 , ~ 300. . (19) Bnucnn, E., A N D Lm~wrn,W., Zeila. I. Phydk, 52,318,334 (1928).
(20) F ~ A N X B8. L, P.. P h y i Rm..57,661 (1940). (21) Loes, L. B., "Kinetic Theory of Gaaes:' (1st ed.), MeGraw-Hill Book CO.,New York. 1 9 2 7 , ~ 275. .
Volume 47, Number 8, August 1970
/
593
