A. D. Kirk
of Victoria Victoria, B.C., Canada
The Range of Validity
University
of Graham's Law
I t is the purpose of this article to discuss effusion, diffusion, and some related processes and to outline the range of validity of Graham's Law. At the end some suggestions are offered regarding the presentation of these topics in introductory chemistry courses. Almost all introductory general chemistry and physical chemistry texts discuss diffusion, effusion, and Graham's Law to some degree in their treatment of kinetic theory of gases. Often the terms effusion and diffusion are used interchangeably or without definition and Graham's Law is applied to both processes. I n works on kinetic theory (1, 2, $), however,the two phenomena are carefully distinguished. Graham, in his original work (4), did recognize two situations for which the square root law was valid: effusive flow and the special case of diffusion through a porous plug across which no pressure differential exists (4). Nevertheless, in general, the two processes of effusion and diinsion are quite different and should be distinguished as such. The discussion of effusion begins with flow through apertures across which a pressure differential exists. Mason and Iironstadt, in their paper in THIS JOURNAL (4), discuss the case of equal pressure diffusive flow through a porous plug.
described by the Hagen-Poiseuille law for laminar flow. In between these extremes, that is when 10 > h/a > 0.01, there lies the transition region where the flow is changing character progressively from eff usive to hydrodynamic. This region is complex and may best be dealt with in a semiempirical manner. It will not be discussed except to comment that the flow is a mixture of hydrodynamic and effusive modes. Since, as will be shown, Graham's Law does not apply to hydrodynamic flow, one cannot make reliable predictions of relative flow rates in the transition region on the basis of that law. Readers interested in further detail of flow in this region are referred to the discussions in Dushman (8) or Carman (9). Effusion
It is a straightforward matter to show theoretically that Graham's Law applies to effusive flow through apertures. The detailed theory of effusion may he found in standard works on kinetic theory (10, 11) and only a simplified version will be presented here. Figure 1 depicts an idealized effusionsystem.
Flow through Apertures
By apertures is meant orifices in walls, or tubes and channels in porous solids or compacted powders. All of these systems have been used in effusion experiments (6fi,7). The object of this general discussion of flow through apertures is to outline the conditions under which true effusive flow occurs. Flow through apertures may always be classified into one of three broad categories depending on the value, for the particular system and conditions, of the Iinudsen number. For a cylindrical aperture or tube the Knudsen number is simply X/a, where X is the meau free path of the molecule under the conditions prevailing at the aperture and a is the radius of the aperture. When the mean free path is long, i.e., X/a > 10, true effusive flow occurs. That is, the molecules escape through the aperture completely independently of one another, no bulk gas flow toward the aperture develops, and the rate of gas loss is determined only by the velocity of the molecules and their collisions with the walls. As we shall show later, it is to this type of flow that Graham's Law applies. At the other extreme of short meau free path, when X/a < 0.01, hydrodynamic or viscous flow occurs, and the gaseous intermolecular collisions become the limiting factor in the flow rate. The gas flows as a viscous fluid and for long tubes and moderate flow rates may be
Figure 1. Depicts true molecvlor effusion in which molecules escape independently through o thin walled operture of diameter much less than the mean free path.
Assume that,: (1) The molecules of the gas me at constant temperature and possess a Maxwellian distribution of speeds. (2) The rtperture(s)isof sufficiently smallsize that no bulk flow of gas develops toward it but that molecules escape individually and randomly. (3) The concentration inmolecule cc-I. n. and thus theoressure.
into this treatment. (4) Every molecule entering the aperture in an outgoing direction succeeds in escaping. This a~sumptionrequires that the length of the aperture, that is the wall thickness of the container, be a. small fractionof its diameter. If this sssumptionis not made, a. constant factor (Clausing factor) may be incorporated into the Volume 44, Number 7 2, December 1967
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subsequent treatment to allow for the constant fraction of entering molecules that complete the passage through the aperture. The value of this Clsusing factor (6, I??, 13) depends on the geometry of the aperture.
It can readily be shown that for a Maxwellian gas the number of molecules crossing unit area of a plane in either direction in one second is 1/4 nE, where E is the average molecular speed. If the wall of the vessel contains an aperture of area S, then the number of mole cules escaping through the aperture, or the molecular rate of effusion, will be Assuming the gas obeys the ideal gas law, then rr = NPIRT
(2)
where N = Avogadro's Number. From kinetic theory,
where M = molecular weight of the gas. Substituting eqns. (3) and (2) into eqn. (I),
where k = R/N equals Boltzmann's constant and m is the mass in grams of the gas molecule. Equation (4) thus predicts that the rate of effusionis proportional to T-'I*, m-'/', S, and P. The T-'1' d o pendence is, a t first sight, surprising hut arises from the fact that at constant pressure the number of molecule cc-', n, decreases as 1/T while their velocity increases only as T"'. The rate of effusion at constant n, of course, increases as T"'. Since the gas flow is occurring in the region where gas phase intermolecular collisions are unimportant, the forward and reverse flow rates through the aperture are independent. Thus, where the pressure of the gas outside the aperture is not zero, one may replace P in eqn. (4) by Al' to allow for t,he rate of effusion in the reverse direction. Correspondingly, where dierent gases or gas mixtures are present on each side of the aperture, one may use eqn. (4) to calculate the independent rates of effusionfor each component in each direction from the partial pressure for each gas. The requirement for effusion of a small aperture (radius < 1/10 mean free path) and the experimental confirmation of eqn. (4) came first from the work of Iinudsen (14). Ihudsen used an aperture 0.025 mm in diameter in a platinum strip 0.0025 mm thick and found eqn. (4) to be obeyed by hydrogen, oxygen, and carbon dioxide at 22'C and 100°C over the pressure range 100-0.01 mm Hg. Graham's Law follows straightforwardly from eqn. (4). If one compares the separate rates of effusion of two different gases a t the same temperature and pressure through the same aperture, then
since all other quantities cancel, i.e., R.ri (molecule sec-I)
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Note that this inverse proportionality is only correct if the effusion rates are expressed in molecule or mole sec-'. If rates are measured in g sec-', as is common in physics, then one must multiply by the ratio of the molecular masses, is.,
For this reason Graham's Law may be best stated in the form: "The molar (volume) rate of effusion of a gas is inversely proportional to the square root of its molecular weight." Thus Graham's Law holds accurately for flow through apertures as long as the correct conditions for effusive flow are maintained. Before leaving the subject, another distinction between effusion and other flow processes will be noted. Because high velocity molecules have a greater prohability for escape than slow molecules, the gas through the aperture in a true effusion system does not possess a Maxwellian distribution of speeds but a distribution biased toward high speed molecules (15). This has two important consequences. The gas leaving the vessel has a higher effective temperature than the gas left b e hind and, secondly, the average energy of the escaping molecules is 1/2 kT higher than the value for molecules with a Maxwellian distribution. Thus, if the gas is monatomic, the average molecular energy is 2kT rather than 3/2kT, the usual value. Hydrodynamic or Viscous Flow
When the Ihudsen number X/a < 0.01, hydrodynamic or viscous flow occurs. For incompressible laminar (streamline) isothermal viscous flow through long cylindrical tubes, the rate of flow in P V units is given by the Hagen-Poiseuille law as
where is the coefficient of viscosity for the gas, 1 is t,he length of the tube, a is the tube radius, Pz is the entrance, and PI the exit pressure. Similar expressions may be derived for long tubes of cross-section other than circular (17, 18). The Hagen-Poiseuille law is valid in the form of eqn. (5) only if the flow is laminar, incompressible, and fully developed. The criteria for these conditions to he met are discussed by Dushman (16) and, under the conditions of flow usually met in laboratory experiments, flow ~ lhel incompressible and laminar but often not fully developed. This latter means that the gas must traverse a length of the tube after the entrance before the streamlines becon~eparallel to the walls. This effect is obviously most important when the tubes are short in relationship to their diameter and results in an increased pressure drop across the tube for a given flow rate. As a result, the rate of flow becomes a more complex function of the geometry and pressure drop across the tube than is implied by eqn. (5) (19). However, to allow for nondevelopment of the flow would complicate this discussion considerably and is unnecessary since the objective is merely to show that Graham's Law is invalid in this flow regime. Thus we will assume that molar rate of flow
a
a'
- (Pi
nl
- P,2)
(6)
From kinetic theory the viscosity coefficient of a gas is given by (20)
where pis the density of the gas. Now
and
where rn is the collision diameter of the molecule. This expression for the mean free path assumes a molecular model of rigid elastic spheres. More sophisticated molecular models lead to more complex, but analogous, expressions for the mean free path. Substituting eqns. (3), (S), and (9) into eqn. (7), then
Substituting eqn. (10) into eqn. (6) and incorporating any universal constants into the proportionality constant, then
If the separate rates of f l o of ~ two different gases for the same conditions of temperature and pressure are compared in the same apparatus, molar rate of flow of A molar rate of flow of B -
ss mme
oed~
This analysis clearly shows that, when isothermal, streamline viscous flow occurs, the relative molar flow rates have a stronger dependence on collision diameter than on molecular mass. Thus, for such flow Graham's Law is not valid. I t must, of course, be recognized that the molecular diameter does not vary over such a large range as the molecular mass [e.g., for He, 2.70 A and 4.003 amu; n CgHZO, 8.45 A and 128.4 amu; SnBr46.67 A and 438.3 amu (%)I. Asa result, the effect of molecular mass on flow rate in the viscous flow region may parallel that in the effusive flow region simply because the square of the collision diameter ratio is not as large as the square root of the molecular mass ratio. As a corollary of this, since in some cases the collision diameters of two gases of markedly different molecular mass will be virtually identical, Graham's Law may appear to be valid for systems in which true effusive flow is not occurring. Flow through Porous Solids
Since experiments on effusion frequently use porous plugs, as did Graham (4), it is appropriate to discuss in some detail the special aspects that may he of importance when a porous plug is used in place of individual apertures. It makes no difference to the subsequent discussion whether the medium through which the gas is flowing is a true porous solid or a compacted powder (or sinter). An idealized porous solid might consist of a number of parallel tubes. In reality, of course, such solids consist
of tubes which are neither parallel, of constant cross section, of same cross section and length, or straight. Many porous systems are markedly anisotropic in flow properties (81). However, despite these complications, it is possible to consider porous solids as assemblages of capillaries in series and in parallel. The effects of the bends and also of the "blind alleys" can he incorporated in a tortuosity factor which is a correction yielding an effective length for the capillaries. This tortuosity factor can he estimated by two methods. The first compares the specific electrical conductivity of the porous solid (which must be a nonconductor) impregnated with a salt solution and the specific conductivity of the salt solution in bulk, allowance being made for the fraction of the total cross-sectional area of the impregnated solid actually occupied by solution. The second compares the tme diffusion coefficient of water vapor in air with the apparent diffusion coefficient of water vapor through the air-filled pores of the solid. Either method yields an effective length for the capillaries in the solid. The problem of the distribution and variation of capillary cross sections can be met by measurements of the capillarity of the solid. Detail on all of these measure ments may be found in Carman ($2). This shows that a porous solid may legitimately be considered as an assemblage of parallel tubes of some effective length and effective cross-sectional area, both of these parameters being measurable. In the simplest cases, flow through such solids is analogous to flow through a single aperture or tube and falls into the same three categoriespreviously discusssed; true effusive flow when the Ihudsen number > 10, viscous flow when the Knudsen number < 0.01, and the transition region when the Knudsen number is intermediate in value. In this case the Iinudsen number may he taken as X/al, where a' is the effective capillary diameter. In the effusive flow regime, the flow rate is given by eqn. (4) with one important modification, incorporation of the Clausing factor. This correction is necessary b e cause the length of the aperture in a porous solid is much larger than its diameter, so that the fourth assumption in the treatment of effusion no longer pertains; only a fraction of the molecules independently entering the apertures succeed in escaping, many being reflected back by the aperture walls. The Clausing factor may be calculated by geometrical considerations and Dushman (23) presents results for a number of idealized tube cross-sectional shapes; for circular cross-section tubes the Clausing factor is a function of the 1ength:diameter ratio. It is clear that, for the present purpose of considering Grah:tm's Law, the Clausing factor, being a constant of the apparatus, is not of primary importance. Thus, when the conditions are correct for effusive flow, Graham's Law will apply rigorously to flow through porous plugs. Thus Ramsay separated helium and nitrogen by effusive flow through unglazed clay pipes and found a separation factor in agreement with Graham's Ltw ($4). (Since the molecular diameters are helium 2.70 A, nitrogen 3.681 A, any component of viscous flow should noticeably affect the separation factor, actually to enhance it, see eqn. (12).) Similar results were obtained by Farkas (25), Hertz (86), and Woolridge and Smythe ($7). Volume 44, Number 12, December 1967
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In applying Graham's Law to flow through porous solids, one must again recognize its limitation to effusion; once again fortuitous agreement may he obtained from experiments in the viscous flow regime on molecules with very similar molecular diameters. A possible example of this is to be found in elementary discussions of the gaseous diffusion technique for isotope separation, e.g., 238UF6 and 235UF6. These molecules are virtually identical in all molecular properties not involving directly the mass difference and have the same molecular diameter. The fact that Graham's Law applies to the process only establishes that the flow rate depends inversely on the square root of the molecular mass and does not prove, as is often implied, that the molecular mass is the only molecular property affecting the flow rate. Many porous solids are also adsorbents and this may have a new and extremely important influence on the flow rate. When the volume of an adsorbed gas film is small with respect to the pore volume, adsorption has no influence on the flow rate. When, however, measurable adsorption occurs, the flow rate of the adsorbed gas may be markedly increased either due to its surface mobility on the solid or to fluid flow through micropores. Carman quotes results illustrating this enhanced flow rate. Thus Wicke and Voigt (28) found that, for flow in the effusiveflow region through sintered glass, butane had a flow rate a t O°C that was 5.3y0 higher than calculated; at -35"C, 9.8y0 higher thancalculated; and a t -80°C, 12% higher than calculated. Similarly, they found that for effusive flow through active charcoal at O°C, COzwas 5.3% higher and SFs, 12.6% higher than calculated. Some very interesting work by Haul (29)indicates that surface diffusion effects can be significant even for isotopic molecules. He found that the separation factor for an '60-'80/160-160 mixture flowing effusively thnn~gh:I 4ica ping at 77°C \\.as +ignificantlygreater tli:~:~ r:il(:uiatcd frun~Gr:~h:im'sIAW,while :IT r(mni ternperature the separation factor was as expected from Graham's Law. This enhanced separation was ascribed to the greater surface mobility of the lighter isotopic molecule; 37% of the flow occurred by surface diffusion. Similar results were obtained a t liquid nitrogen temperature for a6A/@Amixtures. Diffusion of gases through nonporous solids may also be a process of some importance. Such flow occurs by diffusion of the individual gas molecules through the solid. One very important factor determining the relative rates for different gases is their molecular diameter (30). I n summary, then, Graham's Law only applies for true effusive flow through porous solids where surface adsorption effects are known to be unimp0rtant.l
until the composition gradient is eliminated and the system is homogeneous. This process of diffusion, unlike effusion and related flow processes, depends upon the existence of a composition gradient, not a pressure gradient. This gaseous interdiffusion at constant pressure and temperature is the process operative in demonstrations, such as the ammonia/hydrochloric acid experiment, which purport to illustrate Graham's Law. For simplicity consider only isothermal, constant pressure systems of two gases in which a composition gradient occurs in only one dimension. Such a system is shown diagramatically in Figure 2, where the gradient of composition is chosen to exist in the x direction. For
Figure 2. The curves lobelled n* and ns depict the voriatian with distance x of the cancentrotions of A and 0 molecules. Since n, the t o l d number oi molecule CC-', is con.tant, J A , ~ = -Je,*.
a mixture of ideal gases, the requirement of constant pressure dictates that the total number of molecule cc-', n, is a constant, independent of x, i.e., n
=
n* f
n8 =
constant
where n~ and n~ are respectively the number of molecules of A and B per cc. Since n is everywhere constant, then
Because of the concentration gradients of each component gas. they will be flowing in opposite directions across any arbitrary yz plane. By Fick's First Law of Diffusion these rates of flow are proportional to the concentration gradients so that rate of flow of gas A (molecule em-a seerL)=
and rate of flow of gas B (molecule em-* see-') =
Diffusion
Gaseous diffusion or interdiffusion is the process, quite different from those discussed up to this point, to which Graham's Law is perhaps most frequently incorrectly applied. When two or more gases are present in a system and a composition gradient exists, diffusive mixing will occur But see Mason and Kronshdt, (4).
where DAtB and DB+Aare respectivelj the diffusion coefficients of A into B and B into A. However, these rates of flow must he equal and opposite if n is to remain everywhere constant (31, 32) i.e.,
From eqns. (13) and (14) it is clear that D A - ~is equal to DB-A, showing that the diffusion of both gases is described by one diusion coefficient, DAB. Thus the rate of diffusionof a gas is a function not only of its own molecular parameters but also of those of the gas into which it diffuses. Experimentally, it is convenient to have an expression for the time rate of change of concentration in any volume of the system. Thus for the volume element of F i p r e 2 hounded by the planes x and x dx and of unit cross section, the net change in number of A molecules per sec is,
of considerable complexity (33). The resulting expressions are also complex and depend upon the particular model chosen to represent the gas. Fortunately, the more complex results are necessary only to predict diffusion coefficients to an accuracy better than a few percent and to predict correctly their composition dependence (34). For the simplest molecular model of rigid elastic spheres the first approximation to the diusion coefficient becomes Langevin's Formula,
+
Here uABis the effective collision diameter for collisions between molecules of A and B and may reliably be taken as equal to (OA Q)/2 (36). The other symbols have their usual meanings. Langevin's Formula may be relied upon to yield values of the diffusioncoefficients
+
But
+-ddz'h * . d z (2). ("2) * + d. =
:.
Jl*, - J < .+ dr) = D
Table 1. Comparison of Experimental Diffusion Coefficients with Values from Langevin's Formula
dh* (=) .dz
--
Dividing this by the volume of the element, dx,gives the rate of change of concentration per sec. thus,
Gas
Gas
A
B
(A)
HI CO On
GO, Cog
2.968 3.590 3.433 3.882 2.968 3.433 3.996 3.882
CH, HI
C0z CO, a!r
0s
s!r
CIL
am
COI
.*a
a!r
mc
(1) 3.886 3.996 3.996 3.996 3.617 3.617 3.017 3.617
D A B ~ (erpt) (em* aec-I)
0.550 0.137 0.139 0.153 0.611 0.178 0.138 0.190
DAB((calol (om( reo-1)
70 Error
0.500 0.141 0.141 0.156 0.564 0.178 0.139 0.186
Moleoulsr diameters from viaooaity mesaurements, refereone (SKI. b V~hlueaa t 0% from reference (34). ' Calculated nt O'C from Langevin's Formula.
a
which is Fick's Second Law. Thus the time rates of diffusion of A and B are given by
for gases within the uncertainty of most of the experimental data. This is illustrated in Table 1 where a comparison is made of some experimental and calculated diffusion coefficients at O°C. It will be seen that, except for H2, good agreement is obtained. In these calculations aaB was taken to he (aA m)/2 and the collision diameters, quoted in (36), were obtained from viscosity data. On the basis of this agreement, Langevin's Formula can be used to compare the relative rates of diffusion of pairs of gases, A and B, into a third gas C. DACand DBCwill both be given by Langevin's Formula, so that,
+
and - = D m dhs
dl
again equal rates.
It follows from this that under comparable conditions of diffusion the relative rates of diusion of two gases into a third is given by their relative diffusion coefficients into the third gas, viz.
The derivation from kinetic theory of a precise expression for the diffusion coefficient of a gasis a problem Table 2. Test of Graham's Law versus Langevin's Formula
DAC Entry
Gas A
Gas B
Gas C
=
a
=
b
cox co* coz co1 C01 cox air a?r a!c a!r
a!r am
=
c
abe
% Error Graham's Law
+- 07 . 25
+18 - 7.0 -21 +26 16 +j 4 +15 - 9.3 9.7
+
+
+28 167
a
obc gives ratio
of theoretical
% Error Langevin
-11 -12 0.0 - 1.0 -12 - 0.9 - 5.5 - 9.5 - 7.0 - 3.9 - 3.2 - 2.7 69
-
diffusionrates according to Langevin's Formula. Volume 44, Number 12, December 1967
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749
This expression differs from Graham's Law by the first two factors and only if they cancel will it be apparently obeyed. That this expression is better than Graham's Law in predicting relative rates of inter-diffusion is shown in Table 2 which compares, by pairs, the gases in Table 1. Again, of course, hydrogen gives anomalous results, interestingly in very good agreement with Graham's Law for the first two entries. For the remaining gases, however, Langevin's Formula gives much better agreement than Graham's Law. Remembering that both Langevin's Formula and the experimental data ignore a composition dependence that may be as high as 8-loyo, the agreement is as good as can be expected. The last row of Table 2, showing the total percentage error for Langevin's Formula and Graham's Law, shows that, even including the cases involving hydrogen, Langevin's Formula is far superior.
have been individually tabulated so that the reader may see the magnitudes of these factors, often partially cancelling each other, hut not always (see entries 3, 6, and 12). Table 2 also shows clearly that the relative rates of diffusion of two gases into a third is dependent on the nature of the third, e.g., compare entries 1 and 7, 5 and 11, and 6 and 12, both theoretically and experimentally. This is shown further in Table 3. Thus it is inTable 3. Relative Rates of Diffusion of Various Gas Mixtures Hydrogen and Oxygeu
Hydrogen and Air into
Methane and Carbon Dioxide into
The ratios in this table have been calculated from the experiment,al diffusioncoefficients tabulated in reference (36).
correct to apply Graham's Law to gaseous inter-diiusion since it ignores both the effect of molecular diameter and the dependence of diffusion rate of a diiusing gas on the properties of the gas into which it is diffusing. Graham's Law in Freshman Chemistry
I t is now appropriate to turn to the question of what might be said on the topic of Graham's Law in the freshman chemistry course. F i s t , we might ask why this topic is discussed at all. The answer to this seems to be that a discussion of effusion illustrates molecular motion very well and that a t least one phenomenon depends solely on the molecular velocity. It further shows that the speed of molecules is inversely proportional to the square root of their mass and thus provides a very nice confirmation a t this level of the statement that all mole cules at the same temperature have the same average kinetic energy. This leads naturally to discussion of the gaseous "diffusion" process for isotope separation. The name is
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unfortunate. In discussion of this process it might be worth indicating that Graham's Law applies only if flow is occurring under the correct conditions and that, since isotopic molecules have the same molecular diameter, effects due to this parameter may disappear. This w a r n of the possible dangers of extrapolation from UFs to SFs. Finally, one should not say that the rate of gas diffusion in air obeys Graham's Law. A freshman student can appreciate that the rate of diiusion of ammonia, say, through nitrogen depends not only on the mass and diameter of the ammonia molecule but also on the mass and diameter of the nitrogen molecule. In conclusion, wherever intermolecular collisions are important in determining a gas property, the molecular diameter will be an important. parameter and cannot be overlooked. Literature Cited ( 1 ) KENN~RD. E. H., "Kinetic Theory of Gases," (1st ed.), McGraw-Hill Book Co., New York, 1938. ( 2 ) JEANS,J. H . , "The Dynamical Theory of Gases," (4th ed.), Dover Publications. New York. 1954. ( 3 ) LOEB,L. B., "The ~ i n e t i c~ h e b r yof Gases," (3rd ed.), Dover Publications, New York, 1961. (4) MASON,E. A., AND KBONSTADT, B., J. CHEM.EDUC.,44, 740 (1967). J. R., J. CHEM.EDUC.,39, 23 (1962). ( 5 ) HOLLAHAN, (. 6.) WEINER.S.. AND JOHNSON, . D.,. J. CHEM.EDUC.. . 26.. 599 (1949j. ( 7 ) LEROY,L. F., J. CHEM.EDUC.,25, 215 (1948). (8) DUSHMAN, S., "Scientific Foundations of Vacuum Technique," (Revised by LAFFERTY, J. M.), John Wiley & Sons, Inc., New Yark, 1962, Chapter 2. P. C., T I O W of Gases through Porous Media," ( 9 ) CARMAN, (1st ed.), Butterworths Scientific Publications, London, 1956, Chapter 3. E. H . , op. cil., p. 60. (10) KENNARD, (11) Loss. L. B.. on. cd.. D. 301. (13) DUSHMAN. S., o p . cit., p. 90. M., Ann. Phyaik., 28,75 (1909), quoted by KEN(14) KNUDSEN, NARD, E. H.. o p . cd., p. 65. E. H., op. cd., p. 62. (15) KENNARD, S., op. cd., p. 85. (16) DUSHMAN, P. C., op. cd., p. 11. (17) CARMAN, S.. on. c*.. D. 86. (18) DUSHMAN. ( i g j bid., p. si. (20) CHAFMAN,S., AND COWLING, T. G., "The Ml~thematioal Theory of Nan-Uniform Gases," (8md ed.), Cambridge University Press, London, 1952, p. 218. P. C., o p . eit., p. 5 and p. 52. (21) CARMAN, (22) Ibid., Chapter 3. S., o p . eit., p. 94. (23) DUSHMAN, (24) Quoted by KENNARD, E. H., op. cit., p. 65. (25) FARKAS, A,, "Light and Heavy Hydrogen," Cambridge University Press, London, 1935, p. 120. (26) HERTZ,G., Z . Phy~ik.,7 9 , 108 (1932). D. E., AND SMYTHE, W. R., Phy8. Rev., 50,233 (27) WOOIDRIDGE, (1936). (28) WICKE,R. E., AND VOIGHT,U., Angew. Chemie, B19, 94
..
.l"',,,,A,, \
..
/.
(29) HAUL,R. A. W., N a t ~ r w i e a e n a c h a f41.255 t~ (1954). JOST. W.. "Diffusion in Solids. Liouids. Gases." (rev. ed.). (30) , ~ & d e & i cPress, New York, '1966, CLpter 7 : E. H., op. Cit., p. 184. (31) KENNARD, S., AND COWLING, T. G., o p . Cit., p. 123. (32) CHAPMAN, (33) Ibid., Chapters 7 , 8, 9 . (34)Ibid., p. 244. (35) HIRBCHFEIDER, J. O., BIRD,R. B., AND SPOTZ,E. L., Chem. Rev., 44, 205 (1949). (36) "American Institute of Physics Handbook," (end ed.), McGraw-Hill Book Co., New Yark, 1963, p. 235.
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