July, 1961
SOMEASPECTSOF MOLECULAR EFFUSIOX
1151
SOME ASPECTS OF MOLECULAR EFFUSIOK BY E. W. BALSON Department of Chemistry, Universit2/ of Southampton, Southampton, England Received December 1 7 , 1.960
Methods used originally by Clausing t o compute the chance of outflow a t low pressure through cylindrical outlets have been extended and generalized in order to deal with the chance of outflow through a conical orifice, the question of molecular flow through a sharp pipe bend, and the over-all chance that a molecule shall effuse in a typical set-up for measuring vapor pressures by the Knudsen effusion-weight loss method. Values of the chance of outflow in some typical instances are listed.
Although the Knudsen efiusion method for measuring lo^ vapor pressures has been much used in the past, it is only comparatively recently that particular attention has been focussed on the consistent errors with which it is always associated. Under ideal conditions, the saturated vapor in an enclosure effuses out through a perfect hole into a space where the pressure of vapor is zero, the vapor pressure then being calculated from the area of the effusion oui,let and the rate of weight loss. I n practice, the effusion outlet usually takes the form of a short (cylinder, though occasionally outlets of conical form have been employed. Values for the chance of outflow for cylindrical holes of various 1enp;th to radius ratios have been computed by Clausing,’ but the case of the conical outlet does not appear to have been treated. This paper describes a generalization of the methods used by Clausing which can be used to deal completely with the problem of effusion from a cylindrical container with a small axial effusion hole, a matter which has received attention recently from Whit man, who describes an approximate treatment of the problem which is most likely to be in error as the area of the effusion hole is made smaller, relatively, and the vapor within approaches saturation. A somewhat similar treatment has been described hy M ~ t z f e l d t ,which ~ includes a practical solution to the problem. It has proved possible to deal not only with the above prohlem, but also to treat the problem of molecular ~’Jow through a conical outlet, and to examine the flow through a sharp pipe bend. Matters will be dealt with in the reverse order to that given above, the question of the pipe bend being discussed initially, since in this instance the operations can be dealt with in a concise way. Preliminary Discussion.-As in the treatment of Clausing, the function W S S ( z ) is defined as the chance that a molecule passes through a circular disc of radius R and enters another similar disc a t distance z, without collision with other molecules or surfaces. Figure 1 ,3howsthe coaxial discs S enclosed by the spherical surface ABCDEF. The chance WSX(z) is given directly hy the ratio of the spherical surface ABC to that of .4BCDF provided that the cosine law for emission of molecules applies, for when this is so, molecules emitted through the plane FD from surface DEI;’ land evenly over the remainder of the spherical surface. In terms of the dimensions 5 , and R ( I ) P.Clausing, A n n Physzk, 11, 961 (1932) ( 2 ) C . I. Whrtrnan, J Chem P h y s , 10, 161 (1962) (3) I;> r and can be evaluated graphically. For example, when r/R = 0.2, the three terms in (21) are 0.96426, 0.00315, and 0.00299, respectively. Table VI shows values of the over-all chance W O computed in this way, together with those calculated from the relation developed by Whitman; the procedure of Motzfeldt leads to the same result. TABLE VI THEOVER-ALLCHANCEOF OCTFLOWW o FROM A CYLINDRXAL EFFUSION VESSEL( R = 1; L = 2 ) WITH SMALL AXIALEFFUSION HOLE( T ) Hole radius (r) 0 1 0 2 0 3 0 4 0 5 1 0 IVo ( E W R ) 0 99530 0 97040 0 93480 0 88906 0 83540 0 51318 Wo ( W , M) 0 9906 0 9635 0 9215 0 8685 0 8083 ..
It is considered that the uncertainty in the computed values of Wois somewhat less than the probable deviation shown in Table V, since the constant a is the major contributor to WOwhen R>> I .
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of Wo thus differ by little more than 0.1% even with the comparatively large value of l / r chosen, so that with this and shorter outlets the resultant chance is quite adequately represented by WOWI,. The Influence of the Evaporation Coefficient.Consider a plane surface of evaporating material placed across the entrance to the containing cylinder (Fig. S), from which surface the emission rate is n molecules see.-’ cm.-2 and the outflow rate through the ideal hole is nr2nWomolecules set.-'. If the maximum evaporation rate is nomolecules see.-’ and the evaporation coefficient is a, the emission rate of nR2n from S is composed of nR2noamolecules actually evaporating, together with those which arrive from the walls and from and fail to condense. The arrival rate a t S is
s
1
WSS(L) (\VSS(L)
+ JorA (2/IZ) (n,/n)WIIS(z)ds 11
The case when R = r = 1 corresponds to the This me write as rR2nG,for brevity, where G can open tube discussed by Clausing, who reports a be computed from the relative dimensions and the value of 0.5136 for W0, in close agreement with that calculated values of n,/n. The outflow rate is now given above. Furthermore, the theory shows that in this instance the ratio n,/n a t half distance _ _7rT*noo1 __ along the cylinder should be exactly 0.5 (cf. the 1 - G(1 - CY) pipe bend mentioned earlier). The values of a, b and c listed in Table V give a and leads to the over-all chance of outflow WO,,, where IVo,,/Wo is given by value of 0.50004 for n,/n a t half distance. Q The Outflow through a Short CyIindrical Effusion ll~0,Ullvo= Hole,--The extension of the computations to the case when effusion takes place through a short Now lVO,,+Wo as G 4 1 and 50 becomes independcylindrical outlet can be made with some certainty ent of a. For example, when rjR = 0.1, the value if the molecules which enter this outlet do SO uni- of a has little iiifluence 011 W O , , . formly over the whole of the entrance area. This In this instance nx,/nis almost unity along the is very nearly the case when r / R = 0.1, for here entire length of the container, so that it is conventhe minimum value of n,/n along the container is ient to work in terms of (1 - n,,’n) in the relation 0.9934, so that conditions within are virtually those for G, which then becomes which obtain when the effusion vessel is part of a completely closed system and, when this is so, the G = 1 - JTSs(L) - JoL (1 - nJn)\VSlt(L - z)dz nature and amount of flow from left to right is the L same as that in the opposite direction. i P - r2 The resultant chance for an effusion vessel with (1 - n x / ? ~ ) i l W I t S ( ~ ) d(23) ~ such an outlct will be WolVl, to a first approximation, where Wlr is the chance of outflow for the out- The last two terms iii (23) are comparatively small let of length I and radius r. This is, however, an and ran he evaluated graphically, when it is found underestimate, for some of the molecules which col- that when r / R is 0.1, G is 0.9968, and has fallen to lide with the walls of the outlet return to the walls 0.9531 when r/R has risen to 0.2. Table VI1 shows the influence of the value of a of the container and so augment the arrival rate the evaporation coeffjcient on the ratio W~,~/ll.‘o in nx. With a short outlet, this increase will be small these two instances. The table shows the slight influence of the value and leads to a, further term Wo(1 - Wi,)WRs(z:) 011 the right-hand side of equation 19. Revised con- of the evaporation coefficient on the over-all efstants a’, b’ and c‘ then lead to a more exact value fusion rate; reduction of a from unity to 0.5 reduces the over-all rate by only 0.3% when r / R is 0.1, and for Wo’. When this cyclic approximation is applied to the by only 370 when a is as low as 0.1. When r / R is case when R = 1, L = 2, with r = 0.1 and E/r = raised to 0.2, the influence of the magnitude of ct is 0.5 (a value greater than would normally be used in much more pronounced. The exprwsion deduced practice), for which TVl, is 0.8013, and W O = by hIotzfeldt to cover the influence of a on the 0.99530, it is found that WQ’= 0.99673, and WO” chance of outflow has been used to compute values of lVo,,/lVo shown in columns (M), for comparison. = 0.99670, for two cycles. The successive values TV-P
mm
SOMEASPECTSOF MOLECULAR EFFUSION
July, 1961
TABLE VI1 THEINFLUENCE OF THE EVAPORATION COEFFICIENT ON
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THE
EFFUSIONRATE FROM A CYLINDRICAL CONTAINER WITH SMALL EFFUSION HOLE (Cont,ainer: R = 1, L = 2) Evaporation coeftirirnt (a)
0.9 .8 .7 .6 .5 .4
.8 .2 .1 .01
IvO,U/IvQ r/R = 0.1 (EWB) (M)
0.9996 ,9992 .9986 .9979 .9968 .9952 .9926 ,9874 .9720 .7463
0.9990 .9975 .9955 .9933 ,9900 ,9850 .9770 .9615 .9095 .5050
14’0, a/Il‘o
r/R (E’S’B)
0.9948 ,9884 .9803 .9697
.9552 .9342 .go13 .8420
.7031 .1767
=
0.2
(M)
0,9960 .9905 .9840 ,9750 .9625 .9460 .9180 .8660 .7425 .2076
When considering the influence of the evaporation coefficient it must, be borne in mind that the evaporating surface, if that of a solid, will seldom approximate to a plane surface, and will usually consist of a layer of small crystals, loosely packed, with small irregular spaces between them. Conditions within these “irregular effusion vessels” may well be similar to those already discussed, so that the effective rate of eL-aporation from the “surface” will be greater than that indicated by the evaporation coefficient. At the same time it is clear that the practice of filling the effusiori vessel full of chips, crystals, or turnings, eto., in a n effort to assist the saturation of the vapor within, may in some circumstances have the opposite effect to that desired, for in the limiting case, with the effusion vessel filled completely up to the effusion hole, the value of Wo,,/Wo = a for G is then zero, and the evaporation coefficient is exerting its maximum effect. This effect has been noted by Stern and who report a marked diminution in the effusion rate for iodine when the effusion vessel was filled to within about 1 mm. or less from the top; substantially constant effusion rates were recorded when less iodine was used, provided that there was sufficient to cover the base of the cylindrical effusion vessel. The Vapor Path to the Cold Trap.--It remains finally to estimate the chance that a molecule shall reach- the . cold trap having emerged from the effusion hole. Since the effusion rate from the cylindrical vessel in Fig. 6 is d n W o molecules sei.-’, while that of entry is rR2?i:,then the chance WLR that a molecule travels right through from left to right is given by WLR = ( r / R ) W o
Furthermore, if the cylinder shown is part of SL closed system, equal numbers of molecules must travel in the opposite direction, ie. R ~ W L R= T ~ W R L
so Wo = WRL
where WRLis the chance of motion from right to left. ( 5 ) J. H. Stern a n d N. W. Gregory, J . Phus. Chem.. 61, 1226 (1957).
I
i
i-‘ /I
Consequently, for the experimental arrangement depicted schematically in Fig. 7, the chance that an emergent molecule shall reach the cold trap is WL‘R’ and the over-all chance W for the whole system is given by W
=
WOWR’L’W~I
(24)
where Wo refers to the effusion vessel, W R ~ to L ~the path to the cold trap, and Wrl to the effusion outlet. Of the three terms in equation 24, Wrl is likely to be the smallest, for with r = 0.1, R = 1, and L = 2, then I, the length of the outlet, will not be much smaller than 0.01, so that Wrl will be about 0.95, while W O= 0.9953, and W R ~will L ~ be about 0.99 to 0.995 with a compact experimental arrangement. It is not always possible to ascertain the value of W R ~for L ~in many cases the path from outlet hole to cold trap is less direct than that shown, and it is desirable that it should be so in order that there is no direct path between the cold surface and the effusion vessel; radiation of heat from vessel to trap is then prevented. However, a value for W R ~ can L ~ be inferred from the ratio of the straightened length of the outlet path to that of its radius, for in the first section it was shown that the introduction of a bend into the path has little effect on the over-all chance of outflow. In such cases an extrapolation method similar to that used by the author,6 in the case of spherical effusion vessels where the effusion recoil force was measured, could be used. In this, a value of the vapor pressure, a t a constant temperature, duly corrected for the imperfection of the effusion hole, is found for each of a series of effusion vessels with outlets of varying size, in an apparatus where all of the geometry save that of the outlets is kept constant; the plot of this corrected pressure against the (6) E. W. Baleon, “Surface Phenomena in Chemistry and Biology,” Pergamon Prese, New York, N. Y..1958, pp. 117-132.
1158
REMOLO CIOLAASD ROBERT L. BURWELL, JR.
area of the effusion hole, is approximately linear, for both W o and W R ~ are L ~ approximately linear functions of the hole area when the latter is small, and the intercept a t zero area gives the saturation vapor pressure. For an effusion vessel with R = 1and L = 2, and with 12' = 2 and L' = 4, then when r = 0.1, Jti, = 0.9953, while W R ~will L ~ be even closer to unity, a t ea. 0.!399, so the combined probability will be a little more 1han 0.994, provided th2t the evaporation coefficient is not far from unity. When r is raised to 0.2, the combined probability has fallen to about 0.922, indicating that results for this orifice should lie about 5yclower than those for the former, but should the evaporation coefficient fall seriously short of unity, the difference is even greater. -411 extreme case has been reported by Stern and Gregory (ref. .?), who find a very low value for the evap-
T'ol. 65
oration coefficient of iodine. Several effusion vessels were used with values of r R bctn-een 0.01 and 0.1. Despite these low value', values for the effusion rate per unit area of hole ~vcreabout twice as large using the smaller holes as with the larger. I t may be concluded that an effusion cell with dimensions near to R = 1, L = 3, and r = 0.1, should be most satisfactory, for all of t h p correction terms, save perhaps that for the path to the cold trap, differ but little from unity. The use of much smaller holes is of doubtful advantage, since it is then difficult to keep the ratio of length t o radius small, and to be certain of a truly cylindrical profile; the use of divergent conical holes would seem t o offer some advantage here, since the value of TV, remains high even when the length to entrance radius ratio is quite large, and t h e outlrt need not be formed in very thin material.
HYDROGESSTIOY OF 3,3-DISIETHYL-1,4-PESTADIESE O S SICKEL CAT,ALYSTS. A TEST OF T H E DEGREE OF DIFFUSIOSAL CONTROL I N CATALYTIC HYDROGENATIOSS BY REMOLO CIOLAASD ROBERT L. BURWELL, JR.? Department of Chemzstry, Sorthuestern Cnzz'ers?ty, ELanston, Illznozs Recerved December 19, 1960
On nickel wire and on nickel-silica catalysts in the vapor phase and on nickel-silica in the liquid phase, the tn-o double bonds of 3,3-dimethyl-1,4-pentadieneappear to hydrogenate independently. In the absence of diffusional control, 3,3dimethylpentene is the only initial product. If substantial concentration gradients in the catalyst pores lead to diffusional control, 3,3-dirnethylpentane appears as an apparent initial product and its initial ratio to dimethylpentene constitutes a measure of the degree of diffusional control introduced by hydrocarbon concentration gradients. Under the conditione employed in the vapor phase reaction, such gradients xere small or negligible for oxidized nickel nirr, large on 40-60 mesh nickel-silica and still larger on 20-40 mesh nickel-silica. In the liquid phase a t room temperatures on 100-200 mesh nickelsilica, the concentration gradients of olefin %ere small but those of hydrogen appeared to introduce a serious degree of diffusional limitation. In the liquid phase, the ratio of the rate constants of hydrogenation of the diene and of the mono-ene was about 9 and the ratio of 3,3-dimethylbutene and 3,3-dimethylpentene m-as 1.6. On the assumption that the kinetic forms for the rates of hydrogenation of diene and mono-ene are identical, a kinetic treatment of the course of hydrogenation is derived as a function of the initial ratio of mono-ene to alkane.
I n any heterogeneous catalytic reaction, reactants must diffuse t o the catalyst surface and products niust diffuse from it. With usual porous catalysts, two stages of diffusion may be characterized: diffusion between the bulk fluid phase of the reactant stream and the exterior surface of the catalyst particle; diffusion between the exterior surface and the interior surface through the pore structure of the catalyst particle. I n gas phase reactions, the first type of diffusion is usually rapid but the seco:nd type frequently will affect over-all rates and I n practice, t'wo methods of assessing the degree of diffusional control have been most commonly employed. I n one, one compares reaction rates obtained wit,h equal weights of different part'icle sizes of the same catalyst. The degree to which reaction rate increases with decreasing size of (1) This work was supported by the Bir Force Office of Scientific Research and Development. Presented at the New York meeting of the American Chemical Society, September 14, 1960. (2) T o whom inquiries about this paper should be sent. (3) For reviews of this subject see: (a) A . Wheeler. "Catalysis," edited by P. H. Emmett, Vol. Il., Chap. 2, Reinhold Publ. Corp., New York, N. Y . , 1955; (b) A. Wheeler, Advances in Cdtalyeisi 8, 250 (1950); (e) E. Wicke, Z.Elektrochsm., 60, 774 (1956);
catalyst particles measures the degree of diffusional control. The other involves a theoretical treatment of diffusional control often cia the dimensionless parameter, h, of Wheeler which is a function of catalyst particle geometry, the pore volume and surface area of the catalyst, diffusion coefficients mid the rate constant of the Quantitative application of either method requires a knowledge of the exact kinetics of the reaction. The first method is applicable only to catalysts in reproducible steady state conditions. The second method always involves some degree of uncertainty in its quantitative application because of the simplifying assumptions ahout pore geometry made in its derivation. The question of the presence or absence of diffusional control continually befogs attempts a t mechanistic interpretation of studies of heterogeneous catalytic reactions. I t would be useful to have a simple experimental means of assessing the degree of diffusional control which was independent of particular assumptions about the nature of pore structure and which did not require determination of exact kinetics. The present;
