Feb., 1963
EFFUSIOX OF GASESTHROUGH CONGAL OEIFICES
and 8.4 for the OH in 2,4,6-trichlorophenol, where its rotation should be hindered by intramolecular hydrogen bonding. The smaller size and electronic polarizability of the OH groupIs should result in a relaxation time smaller than that for the SH group rotation. Actually, r 2 for the OH group rotation in Table I11 is slightly larger than the value for the SH group, possibly, because the OH groups are more hindered in rotation by weak hydrogen bonding to the solvent molecules than are the SH groups. The apparent independence OF the group relaxation times of temperature should be attributed to the approximate character of the analysis of the known experimental data, and the shortness of the temperature range. The smaller contributions of group rotation in these two molecules are a little smaller than those in 2-methoxynaphthalene and 2-ethoxynaphthalene.8 The free energies AF*, heats AH*, and entropies AS* of activation for dielectric relaxation by over-all molecular rotation, calcu!ated from T~ in the usual manner,I9 are given in Table IV. The values in Table IV are similar in magnitude to the corresponding values for 1-chloronaphthalene and (15) C. P. Smyth, “Dielectric Behai ior and Structure,” McGraw-Hi11 Book C o . , New York, iY.Y , 1055, p. 409. (19) E. .J. Honnelly, W. &I. Heston, .Ji-., and C . P. Smyth, J. S m . Chem. SOL,70, 4102 (1948).
229
TABLE IV ACTIVATIONEXERGIES ( RCAL./MOLE) AKD ENTROPIES (E.c./ MOLE) FOR ROTATIOKAL ORIEWTATION OF MOLECULES Temp.. OC
AF*
AH*
AS*
20 40 60 20 40 60
2 9 3.0 3.2 3 0 3.1 3 2
0.7 .7
-7 5 -7 5 -7.5 -5 7 -5 8 -5.7
2-Naphthol
2-Naphthalenethiol
.7 1.3 1.3 1.3
1-bromonaphthalene in decalin.20 Due to inability to ascertain the temperature dependence of the small relaxation times T ~ it, has not been possible to obtain values of the heats and entropies of activation. The free energies of activation for the OH or SH group rotation are of the order of 1.6-2.0 kcal./mole, roughly two thirds of those for the over-all molecular rotation. The double-arc method has been successfully employed in the analyses of two 2-substituted naphthalenes where intramolecular mobility is suspected. It affords a more convenient, and, in some cases, a surer way to resolve data for two overlapping dispersion regions into two relaxation times than does the alternative method of chords. (201 E. L. Grubb and C. P. Smyth, zbzd ,83,4122(1961).
EFFUSIOS OF GASES THRdOUGH CONICAL ORIFICES BY RAYMOND P. ICZKOWSKI, Research Division, Allis-Chalmers Mfg. Co., Milwaukee 1, Wis.
Joas L. XLRGRAVE, Department of Chemistry, University of Wisconsin, Madison 6, Vis. ASD
STEPHES &I. ROBIXSOX
AVunzertcalAnalysis Laboratory, University of Wisconsin, Madison 6, W i s .
Received J u n e 18, 1968 The theory of Clausing for the effusion of rarefied gases through cylindrical orifices is generalized to apply t o conical orifices. Clausing factors are computed numerically for the flow in both directions through conical orifices of various lengths and angles. These results are aDplicahle to orifices commonly used in vapor pressure investigations by the Knudsen method. The results for the cylinder also are given to high precision as a degenerate case of the cone. The theoretical values compare well with the average values computed from the angular distribution data of Adams and Phipps.
Introduction Early analyses of the rate of effusion of gases through orifices were concerned with the “ideal” aperture1 shown in Fig. l a , an opening in a wall of infinitesimal thickness, with cylindrical tubes of lengths very much larger than their radii,2and with shorter right cylindrical orificesa such as shown in Fig. l b . These studies apply to gases which are sufficiently rarefied that the average mean free path of the molecules is larger than the diameter of the orifice. The rate of effusion of a gas through an orifice of a particular geometric shape may be taken as equal to the product of the rate of effusion through an “ideal” aperture having the same size and shape as the inner (1) M. Knudsen, Ann. Physik, 28, 999 (1909). ( 2 ) (a) M. yon Smoluchowski, t b t d , 33, l55Y (1910); (b) W. G. Pollard and R. D. Present, Phys. Rev., 73, 762 (1948). (3) P. Clausing, Ann. Physak, 12, 961 (1932).
face of the orifice times the Clansing factor for that orifice. Thus the Clausing factor for an “ideal” aperture would be unity. The Clausing factor for a long narrow tube has been shown2to be 8R/3L, where L is the length and R is the radius. The Clausing factor for the right circular cylinder has been given in tabular form.3 X very important application of the effusion of gases through orifices is the determination of vapor pressures by the Knudsen effusion method. However, in many vapor pressure investigations, “knife edge” orifices are used, which have the shape of a truncated right circular cone, such as that shown in Fig. IC, and the rate of effusion usually is assumed to be the same as that of the “ideal” case or corrected empirically. Failure to take into account the effect of the conical geometry limits the precision of such vapor pressure measurements. There have been several recent efforts to generalize
13. P. ICZKOWSKI, J. L. MAIAE~GRAVE, AKD S. A1. ROBIXSOK
230
Vol. 67
in Fig. 2a-d. A plane a t x or 4 forms a circular surface which will be referred to as s, and which has as its radius the radius of the cone at x or 4'. Two adjacent planes a t x and x dx or C; and E d[ inscribe a ring on the walls of the cone like those shown in Fig. 2b-d, which has the shape of a flat truncated cone of infinitesimal length and which will be referred to as r . By a generalization of Clausing's method, the following probabilities are defined subject to the assumptions given above. was (x,[) is the probability that a molecule passing through a surface located a t x and averaged over all points of that surface mill pass directly, without collision through another surface located at t . (See Fig. 2a.) w,,(z,t) dE is the probability that a molecule passing through a surface located at x and averaged over all points of that surface will strike the walls of the cone at a point within the boundaries of the ring located at E. (See Fig. 2b.) wrs (x,[) is the probability that a molecule leaving a point on the surface of the cone located within the ring a t 5 will pass through a surface located a t [. (See Fig. 2c.) wrr ($,E) d i is the probability that a molecule leaving a point on the surface of the cone located within the ring a t z mill strike the walls of the cone at a point within the boundaries of the ring located at [. (See Fig. 2d.) I n addition, w,, ([,x) and wsr (g,x) are defined in an analogous way, for a molecule passing through a surface at E and going to a surface and a ring, respectively, at
+
B
A
D
C Figure 1.
A
C
B
D
E Figure 2.
the Clausing factor for orifices of special shapes to get data which could be used for determining pumping rates in vacuum s y s t e r n ~for , ~ rarefied ~~ gas flow through orifices of special shapes16and for vapor pressure determinations by the Knudsen effusion technique.' The approach of Davis4s5uses a Monte Carlo calculation while those of Howard6 and of Balson7 involve the solution of equations similar to those developed by Clausing. I n the present paper, the theory of Clausing is extended to the cone and Clausing factors are computed for cones of various lengths and angles of the type shown in Fig. IC. The results may be utilized in a simple calculation to give the factor for the reverse cone shown in Fig. I d for which molecules enter the larger side and leave the smaller side. The Clausing factors for the right cylinder are given as the limiting case of a cone and are presented with higher precision than in previous studies. Clausing Factor Equations for the Cone.-The calculation is made under the following assumptions: (1) The scat teriiig amplitude for molecules striking the walls of the container follows the cosine law. (2) The angular distribution of the molecules entering the cone is given by the cosine law. ( 3 ) The mean free path of the molecules is greater than the cross dimensions of the cone. Therefore there are no interfering collisions of molecules with other molecules in the cone. The Clausing factor, W , may be defined as the probability that a molecule entering through the imaginary plane on one side of the cone eventually will escape through the imaginary plane which forms the opposite side, under the conditions assumed above. The molecule may pass through the opposite side directly, after entering the cone, or after making any number of collisions with the walls of the cone. The following probabilities are defined with reference to imaginary planes placed at points x and [ along and perpendicular to the centerline of the cone as is shown (4) D. H. Davis, J . A p p l . Phys., 81, 1169 (1960). ( 5 ) D. H. Davis, L. L. Levenson, and N. Milleion, in "Rarefied Gas Dynamics," Academic Press, Nex York, N. P., 1961, pp. 99-115. (6) W. M.Howard, Phys. Fluids, 4, 521 (1961). (7) E. W. Balson, J . Phys. Chem., 65, 1151 (1961).
+
2.
These probabilities are related to each other by the equations bwss (x,O
war
(.,E)
=
wrr
(x,E)
= -E
-6
at
awrs
(x,O
at
where E = -1 if E < x and E = +1 if > x. The first of these equations may be derived by noting from Fig. 2a-b that w,, ($,E) = w,, (x,t dt) wsr (x,E) dt. The other relations follow by analogous reasoning. I n addition, wrs (x,I) and wSr (E, x) are related by the equation
+ +
Wrs_( _ ~_ , t_ -)
As
(E)
(t,z)dx Ar (z) dx
Wsr
where &(E) is the area of the circular surface at [, and A , (x)dx is the infinitesimal area of the cone shaped ring a t x. This relation results from the fact that if the cone were homogeneously filled with molecules, then, in a given time, just as many molecules would go from the ring to the surface, per unit area, as from the surface to the ring, per unit area. The geometry of the cone is characterized in Fig. 2e by 01, the apex angle, Ro,the radius of the cone at its smaller truncation, and L, the distance between the two faces of the cone.
EFFUSIONOF GASESTHROUGH COKICAL ORIFICES
Feb., 1963
The formula for w,, (x,S) is known* to be given by the formula toss
(x,E) = P(,,E)
- m,o
where
231
w(x), which is defined as the probability that a molecule rebounding from the surface of the cone a t the point x according to the cosine law will eventually escape through the imaginary plane which forms the large end of the cone; it may do so directly, or after making any number of previous collisions with the walls of the cone. The probability of eventual escape from the point x is equal to the probability of passing from the ring a t x through the plane a t L Ro cot a plus the probability of going to some other point, E, times the probability of eventual escape from 4, summed over all points 4
+
and R ( x ) is the radius of the cone at the point x. The formula for w,. (E,z) is given by the same functions, except that the roles of x and E are reversed. From these formulas, the formulas for wss, w,,,w,,,and wrr can be derived. For the purpose of calculating W for large angles, a , it is convenient to use one coordinate system, while for small angles it is convenient to change to a second coordinate system. I n the first system, the zero point is taken to be a t the apex which would be formed by extending the walls of the truncated cone to their point of convergence, and in the second system the zero point is taken to be a t the smaller face of the cone. I n the first coordinate system, R(x) = x tan a and R(E) = E tan a . The smaller and larger faces of the Ro cot a. The cone lie at the points Ro cot a and L functions P and Q become
+
W ( Z ) = Wrs ( 2 , L
W
= wss
(Ro cot a , L
~ ) s s
( L -t Ro cot
L + Ro cot S R0 cot a
=
Wrr
From these probabilities it is possible to calculate ( 8 ) J. W.
T.Walsh. Proc. Phys.
Sac. (London), 32, 59 (1920).
a)
a
(L
wsr
CY,
Ro cot
+ Ro cot
CY,
+
CY)
E ) [l - w(()]dE (3)
+
+
(Ro+ E tan a ) 2 + 2(Ro + ( E - x)2 -t tan a ) ? 2(Ro t x tan a ) 2
'/2
IC
The probabilities w,,and w,, are given in the same functional form as those given previously, but in terrns of the new functions P , Q, T , and E. The probabilities w,,and w,,take the form wr.3
+ w - G) 1
+ Ro cot +
For the second coordinate system, R ( x ) = Ro x tan E tan a , the smaller and larger faces of the cone lie at the points 0 and L , and the functions P and Q become
P(Z,O
E(E,x)I
+
Since w(x) is the probability of passing out the larger end, 1 - w ( x ) is the probability of passing out the smaller end. The Clausing factor, W', for a molecule originating at the larger end and passing out the smaller end is given by
a and R(E) = Ro
The mobabilities can be expressed in terms of thefie
Ro cot a )
From w(x), the Clausing factor, W , can be calculated. The Clausing factor for a molecule going from the smaller face of the cone t o the larger face is equal to the probability of passing directly from the plane a t Ro cot a to theplane at L $.Rocot a , plus theprobabilityof going to some intermediate point, 5, times the probability of eventual escape from the point 5, summed over all points
W' = From these functions the following quantities are easily evaluated
$-
(%,E)
(x,E) =
cos 2
CY
= -- __
+ +
+ E tan a ) 2 + IC tan a )
(Bo (Ro
War
(E,x)
Ro E tan CY {2W(E,x) Ro x tan a tan a )( F - G) ] li;(E,x)]tan a (R, cos 2
CY
- __-
+
+
Expressions analogous to eq. 1, 2 , and 3 can be set up to determine values of w(x), W , and W ' , identical with those derived by means of the first coordinate system. The computation of W' from eq. 3 is reduncEant and is only used to assess the precision of the computations. W' can be calculated from W in a simpler way. Con-
R. P. ICZKOTI'SKI, J. L. R~ARGRAVE,ASD S. 14.ROBINSON
232
Vol. 67
sider two vessels containing the gas a t equal pressures, which are separated by a cone shaped orifice. The number of molecules moving from the first container to the second must be exactly balanced by the number moving from the second to the first
vAW' where v is the frequency a t which molecules collide with the walls per unit area and A > and A< are the areas of the larger and smaller faces of the cone, respectively. Therefore
L 4- Ro cot a
S
RQcot
A
