NOTES
3087
+
1
hfel jnle,x,[ lATelX,(zl) ~ e ~ ~ , Mel ( x ~ ) for the following binary melts: AgCl KC1, AgBr KBr, AgBr NaBr, AgBr LiBr, AgBr PbBrz, AgXOs LiKOa, PbClz 4- XC1. At the separation limit between the molten salts, no use of diaphragms was made. For the diffusion potential, we have taken as valid the relation, used for aqueous electrolytes,* of the form
+
+
+ +
e =
nF
JAB
(tl
d In al
+
+ tz d In a%)
+
(1)
By substituting the activity uzas a function of a],with the aid of the Gibbs-Duhem equation, there results
In virtue of the fact that the diffusion potential E is nil, we can write zzt, - %It2 = 0 (3) From relation 3 there results that the transport numbers of the two cations are variable and proportional to the molecular fractions of the corresponding salts. In a note publisheld in this journa1,j Berlin and his collaborators ascribe to us the statement that the mobilities or the transport numbers should remain constant all along the concentration range as used in the cell. This statement is not to be found in our works, and on the other hand the fact that the diffusion potential vanishes does not imply any direct conclusion as to the individual mobilities of the ions. (1) I. G. Murgulescu and I>. I. Marchidan, Z h . Fiz. Khtm., 34, 2534 (1 960). (2) I. G. Murgulescu and D. I. Marchidan, Bev. Chim. Acad. Rep. Populaire Roumaine, 5 , 17 (1960). (3) I. G. Murgulescu and 11. I. Marchidan, ibid., 5 , 299 (1960). (4) (a) K. Jellinek, “Lehrbuch der physikalischen Chemie,” I11 Band, Stuttgart, 1930, p. 780; (b) E. A. Guggenheim, “Thermodynamics,” 1957, p. 396. (5) A. Berlin, F. MBnBs, S. Forcheri, and C. Monfrini, J . Phya. Chem., 67, 2505 (1963).
are well known for flow in cylindrical tubes in the limit of low pressures (Knudsen flow) and of high pressures (Poiseuille flow) ; however, the intermediate (slip flow) regime is not well u n d e r ~ t o o d . ~In many of the recent theoretical treatments, the authors make use of Knudsen’s original data5 to relate their results to experiments. In this note, further experiments of this kind are reported; it will be seen that these data show features not present in previous work. Flow experiments were performed by measuring the time dependence of the pressure differences between two containers of known volume connected by twentyfive stainless steel tubes of length 10.000 cm. and i.d. 0.015 cm. The pressure differences were read to an accuracy of 3 X mm. by means of a differential capacitance nianonieter coupled to a chart recorder. Data have been obtained for helium, neon, and argon at a number of temperatures and a t pressures corresponding to a tube diameter to mean free path ratio ranging from 0.002 to 15. Flow rates are given in terms of the number of molecules passing through unit area of the tube in unit time, for unit gradient in the gas density. IF this quantity is denoted by D , one can calculate a dimensionless flow rate D* = D/av, where a = tube radius and V = average molecular speed. These rates are plotted in Fig. 1-3 as a function of a/X, where X = kinetic theory mean free path. It is most convenient to compute X from the viscosity of the gas
I I
2.4
D*
I
1 o THIS WORK
I
Fe Fe
I
H2 1288.5 IHe,Ne,t A L L
/
/ -
POlSEUl LL E
*8w KNUDSEN
Low Pressure Flow of Gases
0 0
by H. J. M. Hanley and W. A. Steele Whitmore Laboratory, Department of Chemistru, The Pennsylvania State University, University Park, Pennsylvania (Received June 4, 1964)
The isothermal flow of gases a t low pressures has been a subject of interest, both theoretically and experimentally, for some tinie.1-3 Theoretical equat,ions
I
I
I
I
5
10
15
20
oiA
Figure 1. Flow data for various workers. (1) W. G. Pollard and R. D. Present, Phys. Rev., 73, 762 (1948). (2) D. S. Scott and IF. A. L Dullien, A I.Ch.E. J . , 8,293 (1962). (3) 0. Gherman, Soviet P h y s . JETP,34, 1016 (1958). (4) S. A. Schaaf, “Handbuch der physik,” Band VIII/P, SpringerVerlag, Berlin, 1963. (5) M.Knudsen, Ann. Physik, 2 8 , 75 (1909).
Volume 58, Number 10 October, is64
NOTES
3088
.550+
where p is the average pressure of the flowing gas. In terms of the parameters defined here, the Knudsen equation leads to the result that lim D*
a/X+O
= z/3
I
+
He
0
Ne
(2)
and the Poiseuille equation becomes lim D* = ( 3 ~ / 6 4 ) ( a / X )
a/X-
(3)
m
In Fig. 1, the data obtained in the present investigation are compared with the predictions of eq. 2 and 3 and with some experimental results of other workers in this f ~ e l d . ~It, ~is seen that there is little difference between the various sets of data and, when viewed on a scale such as that used in Fig. 1, the flow rates seem to go rather smoothly from those characteristic of a Knudsen gas to those for a Poiseuille gas. However, Fig. 2 and 3 show some of the data plotted on an expanded scale. It can now be seen that the experimental data shown in the figures exhibit several interesting features: the minima in the flow rates are much sharper than those observed previously, particularly at higher temperatures and with the lower boiling gases (Le., the substances with the weaker intermolecular interactions) ; also, the limiting low pressure data do not agree with the prediction of the Knudsen equation, but vary with temperature and with the gas (this result is in qualitative agreement with the earlier work of Lund and Berman7). A theoretical explanation of these effects will necessarily involve a consideration of the dynamics of the niolecular collisions between molecules in the gas and between a gas molecule and the wall of the tube. Since the nature of these collisions is determined by the intermolecular forces, the observed differences in flow properties of the various gases are undoubtedly due to the differences in the gasgas and gas-surface interactions. However, a quantitative treatment of this effect is lacking a t present.
The Journ,al of Physical Chemistry
.400 I 0
,
I
1
I
.2
.4
I
.6
-8
a/ A
Figure 2. Flow rates a t 273°K.
r
I
1
.6
*a
1
.6OO.J/F li
,400 0
I
.2
.4 a/A
Figure 3. Flow rates at 386°K.
Acknowledgment. This work was supported by the National Science Foundation. ~~
~
( 6 ) H. ildzumi, Bull. Chern. SOC.Japan, 14, 343 (1939). (7) L. M. Lund and A. 8. Berman, J. Chem. Phye., 28, 363 (1958).
