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Diffusional Processes in Knudsen Cells
by N. A. Gokcen Chemical T h e r m d y a m i c s Seclwn, Laboratories Division, Aerospace Corporation, El Segundo, California (Received M a y 4, 1966)
The volume and surface diffusions in Knudsen cells have been analyzed and their erroneous effects on vapor pressure measurements have been discussed. It is found conclusively that the volume diffusion may be large in some cases. A useful equation has been derived from the relationships of Motzfeldt and of Winterbottom and Hirth, leading to a simple graphical method capable of showing measurably large contributions of surface diffusion to effusion. It is shown that in favorable cases the Knudsen cells may be used for obtaining reasonable values of the root mean square of surface diffusion distance, E-l. Suggestions have been made to minimize the diffusional errors and to make calculational corrections in the absence of data on E.
Introduction Measurements of equilibrium vapor pressures with Knudsen cells require the elimination of numerous sources of experimental errors (see, for example, Carlsonl). The errors from diffusional processes, however, have been generally ignored because it is usually assumed that a choice of appropriate cell materials would eliminate them. The volume diffusion in graphite cells has been observed and evaluated by Fujishiro and Gokcen2and the surface diffusion through the cell orifice has been critically examined and elegantly formulated by Winterbottom and Hirth.3 The latter formulation3has been verified experimentally by Boyer and Meadowcroft4for Ag(1) in Mo cells. The purpose of this paper is (a) to show that in some cases the volume diffusion may be large, (b) to derive useful equations from the relationships of Motzfeldt5 and of Winterbottom and Hirth which lead to simple graphical methods capable of showing contributions from surface diffusion, and (c) to show that in favorable cases the Knudsen cells may be used for obtaining reasonable values of the root-mean-square diffusion distance. Suggestions have been made to minimize or eliminate the diffusional errors and to make calculational corrections.
Volume Diffusion A number of investigators have used a refractory nonmetallic crucible with an attached metallic lid which is easy to drill and grind in order to obtain D, knife-edge The Journal of Physical Chemistry
orifice of desired dimensions. Since the coefficients of expansion of the two cell components are not the same, this procedure should be avoided whenever possible in order to eliminate unexpected leakage through the lid joint. The cell should therefore be made of one single material whenever possible. The vapor pressure of silver has been determined by this method but we shall take the cell as constructed entirely of n i ~ k e l . Silver ~ and nickel are immiscible in the solid and the liquid states but there is an estimated solid solubility of roughly 1 to 3% by weight of silver in solid nickel6 at 1300°K. We shall adopt a value of 2% silver by weight. The diffusivity, D,of silver in nickel is not known but it may be estimated as 10-lo cm.2/sec., which is about the same as that of copper in nickel or gold in nickel. The flux J D of silver in grams per square centimeter through a cell wall is given by Fick’s first law
(1) K. D. Carlson, “Molecular and Viscous Effusion of Saturated Vapors,” Argonne National Laboratory, ANL-6156, Argonne, Ill., 1960. (2) S. Fujishiro and N. A. Gokcen, J . Phys. Chem., 65, 161 (1961). (3) W. L. Winterbottom and J. P. Hirth, J. Chem. Phys., 37, 784 (1962). (4) A. J. Boyer and T. R. Meadowcroft, Trans. A I M E , 233, 388 (1965). (5) K.Motzfeldt, J. Phys. Chem., 59, 139 (1955). (6) M. Hansen and K. Ankerko, “Constitution of Binary Alloys,” McGraw-Hill Book Go., Inc.. New York, N. Y., 1958.
DIFFUSIONAL PROCESSES IN KNUDSEN CELLS
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where c is the concentration in grams per cubic centimeter and dc/bx is the concentration gradient. Assuming that (a) the steady-state diffusion prevails and bc/bx = Ac/Az, (b) the wall thickness is 0.02 em., or Ax = 0.02, and (c) the concentration of silver on the surface of the Knudsen cell is zero, then the concentration gradient is Ac/Ax = - 9 g . / ~ m . ~ The . amount of silver lost per second through the entire cell wall is the surface area, A , times JD. The value of A for an average size cell is about 10 cma2;hence, the loss of Ag is 0.9 X g./sec. The rate of weight loss, W in grams per second by effusion through a knife-edge orifice is obtained from
P (dynes/cm.2) = 2.2856 X 104'4E a
(2)
Substitution of a = cm. for an unusually small but not uncommon4orifice, T = 13OO0K.,M = 107.9, and P = 12*52 dynes/cm*2from the equilibrium data yields = 1.58 x lo-' g./sec* It is therefore Seen that the volume diffusion may be large in exceptional cases. The foregoing may be in error by a factor Of more than l o because (a> cannot be estimated by comparison with D for other systems, (b) the solubilitg of silver in nickel is not known with a sufficient degree of accuracy, and (e) bc/dx may be much smaller than the assumed value because the Concentration on the surface may be large and c os. x may not be linear. Increasing both the wall thickness and the orifice dimension within limits and selecting nondissolving crucible materials would minimize or entirely eliminate the volume diffusion. However, in the case when the cell material itself is one of the reactants or the products, as graphite cells containing dissociating carbides,2 the choice of another material is not possible. Fujishiro and Gokcen2 showed that in the cells made of the densest graphite available at that time, a substantial loss of weight occurred by diffusion through graphite and possibly through a small number of connected pores, as measured by means of cells without orifice. Fortunately, it is possible to eliminate this error entirely by using a torsion effusion cell symmetrical in shape with respect to the axis of suspension. The loss of diffusion generally increases faster than the increase in pressure inside a given cell with increasing temperature because the activation energy for diffusion is generally greater than the standard free energy for vaporization but not necessarily for dissociation,2 and the solubility of a sparingly soluble substance exponentiauy with temperature and thus sets up a greater concentration gradient.
Surface Diffusion Equations. Fick's law for surface diffusion is
(3) where J , is the flux in molecules per second per centimeter of length perpendicular to the direction of diffusion, D, is the surface diffusivity in square centimeters per second, and n is the surface concentration of diffusing substance in molecules per square centimeter. When an effusing gas is adsorbed on the inner surface of a cell, it tends to diffuse out through the orifice and reach the external surface where it is desorbed under a high vacuum. This process may contribute considerably to the observed weight loss in Knudsen cells. The use of a torsion cell in this case cannot eliminate the error. Winterbottom and Hirth3 derived a number of useful equations by solving eq. 3 for n and J, in terms of measurable quantities by using the prevailing boundary conditions for Ni and Mo cells containing Ag. They also obtained an expression for the ratio of surface diffusion current,W , in grams per second to the effusion current Weff,ie., P = We/Weff,in terms of the usual quantities encountered in surface diffusion phenomena, By their eq. 14, 21, and 27, we derive the useful relationship
(4) In this equation, n' is the surface concentration of the effusing substance at the top of the circular cylindrical orifice where the gas leaves the cell; ne is the same quantity at the bottom or inner part of the orifice, hence the equilibrium value of n with Ag; y is the transmission coefficient which closely approaches the Clawing' factor with decreasing ratio of the length of the cylindrical orifice, L, to its radius, R; 1 / E is the root mean square of diffusion distance; and K1 and KO are the modified Bessel functions of the second kind of first and zeroth order, respectively, both being functions of the variable ER. The quantity E is given by
E2D, = u exp(-AGodes/kT) (5) where u is the frequency factor and AGode, is the free energy of desorption. The values of the quantities in eq. 5 for Ag(g) adsorbed on Mo cells are 2 X lo9 < u < 4 X l o l l , AGodes = 2.2 i 0.2 e.v., and D. = cm.2/sec. as obtained by Goeler and Peacock.aJ The value of D, varies somewhat with temperature but the (7) P. Clausing, Ann. Phusik, 12, 961 (1932). (8) E. Von Goeler and R. M. Peacock, J . Chem. Phys., 39, 169 (1963).
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activation energy for Da is probably small, and therefore, in view of the uncertainties involved in other quantities in eq. 5, the assumption that De is constant3 is reasonable. The values of E computed by taking u = 1O1O sec.-I and by substituting the preceding quantities in eq. 3 are 29, 540, and 1100 cm.-’ at 1000, 1300, and 1400”K., respectively. We shall now take advantage of a very useful property9 of the ratio &/KO in eq. 4. This ratio is close to unity when ER is in excess of 2, corresponding to p = 1; ie., the surface diffusion current is about equal to the effusion current. In the experiments of Boyer and Meadowcroft,*ER ranged from about 4 to 10 for the cells with knife-edged Mo lids at their lowest temperature. Computations from the appropriate equations of Winterbottom and Hirth show that n’/(yne) is very nearly equal to unity for L / R about 0.2 and smaller. For a knife-edge orifice (L/R = 0) this ratio is identically unity. Therefore, eq. 4 assumes the remarkably simple form
p = 2 -B ER-R
For large values of p, e.g., p = 5 corresponding to ER = 0.65, eq. 4 may still be expressed by eq. 6 for a fivefold range in ER, with an accuracy of better than 15%, but an appropriate numerical coefficient is necessary to relate B to E in B = 2/E. We shall limit our discussion to the range where eq. 6 is valid, keeping in mind that extending the range of ER to lower values requires simple appropriate adjustments in B. The pressure prevailing in the cell decreases with increasing €2 according to a useful equation derived by Motzf eldt
+ 2
4~
where P o b s d and Pesare the observed and the equilibrium pressures, respectively, cy: is the accommodation coefficient, and A is the surface area of vaporizing phase inside the cell. Although the degree of precision of this equation has been questioned1 in view of the assumptions used in its derivation, the significant aspects of its usefulness for our purposes are as follows: (1) a plot of 1/Pobsd VS. R2 is h e a r , and the intercept is l/Pes,and (2) the slope is a positive quantity, rather small for large values of a! and Pep. The total observed rate of weight loss, W, from a cell, after correction for volume diffusion when necessary aa obtained from a cell whose orifice may be sealed after evacuation, consists of the sum of the effusionloss, Wefi and the surface diffusion loss, we. The quantity from The Joumal of Physical Che?nistrv
6
6
I/PR
\-----0 J -----
n 1.4 (*L. \ N
G 1.2
-
1.00
1.2 1.6 2.0 I/PR Figure 1. Plots of 102/Pobsd in cm.2/dyne vs. 1/(PobsdR) in cm./dyne. All lines are calculated, and points 0 represent data4 for solid lines, except two points10 corresponding to Pept 01 1/(PobsdR) = 0.
0.4
0.8
which Pobad is computed by means of eq. 2 is therefore W - Wal or since p = B / R = Ws/Weff,then W*ff =
WR B+R
~
Substitution of this quantity in eq. 2 and then in eq. 7 gives the important relationship
- -1- + -- -1Pobsd
(7)
4
peq
rR2y aAPeq
B PobadR
(8)
where P o b a d is calculated by using W in eq. 2. We shall next deal with the experiments in which y is kept constant or the knife-edge orifices in which y = 1, while R is varied from one measurement to another in order to reduce the number of independent variables. Applications. Equation 8 shows that while in the absence of surface diffusion, ie., when B = 0, a plot of 1/Pobsd vs. R2 is linear with a positive slope, this is not true when the surface diffusion is present. As a matter of fact, for a! = 1, A = 3 for Ag in Mo cells with R = 0.05 cm., the second term is smaller than the absolute value of the third term by 1500 times at 1000°K. and 15 times at 1400°K. We may argue this point in a more rigorous manner that we are interested (9) E. Jahnke and F.Emde, of Functions,,, Dover Publications, New York, N. Y . , 1945,p. 236.
DIFFUSIONAL PROCESSES IN KNUDSEN CELLS
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in measurably large values of 6, such as p 20.1, for this, P o b s d is not greatly different from P,, and therefore the second term is only one-tenth to one-thousandth as large as the third term for the orifice radii from 0.1 to 0.01 cm., respectively, Consequently, the second term in eq. 8 may be ignored so that
- -1_ _-_ 1_ _ Pobsd
A plot of
peq
B PobsdR
(9)
1/Pobsd vs. 1/(Pobsd R) is a straight line whose slope is -B. The solid lines in Figure 1 show eq. 9 corresponding to Ag in molybdenum cells. The broken portions of the straight lines are where R is so large that even the requirement that R be only one-twentieth as large as the mean free path is not met; however, they must intercept at l/Peqand 1/R = 0 where Peqwas obtained from McCabe and Birchenall.’O The experimental data of Boyer and Meadowcroft4are represented by the appropriate points. The agreement is fair and the discrepancy may possibly be attributed to the errors in optical temperature measurements and possible leakage at the lid joint in the cell. The upper lines represent eq. 7 on the same coordinates and show that the slope is zero; ;.e., there is no surface diffusion. It is evident that in the absence of any data on E, appropriate experiments may be carried out and the results may be plotted as shown in Figure 1 to obtain B
from which fair values of E can be computed. For this purpose, it is essential to use small enough orifices and low enough temperatures so that p may be measurably large. The existence of surface diffusion is readily detected in a plot of 1 / P o b s d vs. R2. It would show that 1/ Pobad always remains below l/Peq, and the relationship is not linear. The surface diffusion can be minimized or eliminated by using different cell materials if this is experimentally permissible. For a given cell, the best that can be done is to carry out the measurements at high enough temperatures so that the effusion contribution is greater than the diffusional contribution, or the slopes of the solid lines in Figure 1 are small and therefore the resulting corrections constitute a small fraction of the observed pressure. In the absence of experimental data on E, even two points with each cell material are adequate to compute B from which P,, can be obtained and compared with similar sets of data for other cell materials in order to obtain consistent values of P e q for a given vaporizing substance. (10) C. L. McCabe and C. E. Birchenall, Trans. A I M E , 197, 707 (1953); their data are in complete agreement with P. Grievson, G. W. Hooper, and C. B. Alcock in “Physical Chemistry of Process Metallurgy,” G. R. St. Pierre, Ed., Interscience Publishers, Inc., New York, N. Y., 1961.
Volume 69,Number 10 October 1966
