EFFECTOF MOLECULAR FLOW UPON VAPORIZATION RATE
1327
Effect of Restrictions to Molecular Flow upon Measurements of Vaporization Rate and Vapor Pressure
by Gerd M. Rosenblatt Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania (Received September 10, 1966)
16802
A steady-state model is employed to calculate the fraction of the vaporizing molecules which ultimately escape when a sample is vaporized in an evacuated tube. The results can be used to correct measurements of vapor pressure or vaporization rate for those vaporizing molecules which recondense upon the sample after wall collisions. The calculations are also useful in the design of apparatus. Analytical results are presented for a variety of experimental arrangements in which a vaporizing platelet or a sample in an uncovered container is suspended within a cylindrical tube with a condenser at one or both ends. The results may be helpful with experimental configurations other than those explicitly treated. The corrections can be large for typical experimental arrangements. Some qualitative and quantitative predictions of the calculations are confirmed by comparison with experiment. The effect of molecular flow restrictions upon measured enthalpies and entropies of vaporization is discussed.
In many physicochemical experiments a solid or liquid is vaporized within an evacuated container. Often the gaseous molecules are condensed some distance from the vaporizing sample and it becomes necessary to relate the rate of condensation to the vapor pressure of vacuum rate of vaporization of the material. In most experimental arrangements this requires correcting for those molecules which vaporize but never reach the condenser because they recondense upon the sample after colliding with a wall of the container. The net rate of vaporization actually occurring will be equal to the rate at which gaseous molecules stick to the condenser, not to the ideal vacuum rate of vaporization of the sample. As the corrections associated with many typical experimental geometries are quite large, neglecting tube flow restrictions can lead to significant error in determinations of vaporization rate or vapor pressure. In this paper a steady-state model is used to calculate correction factors for tube limitations to molecular flow. A number of experimental geometries are treated. The results apply directly to situations where a solid in the shape of a thin plate or a sample vaporizing from an open crucible is suspended within a cylindrical
tube with a sink for the molecules at one or both ends. The results also serve as an approximation for samples of other shapes suspended in cylindrical tubes or other containers of constant cross section. The equations derived should be particularly helpful in designing and interpreting experiments to study vaporization kinetics or to measure vapor pressures by the Langmuir method, although they are not limited to these applications. For example, the results could also be helpful in estimating the pressure near a vaporizing sample from that measured at a gauge some point downstream in a vacuum system. The calculations presented here apply in the lowpressure molecular-flow regime where the mean free path of the molecules is greater than or of the same order as the dimensions of the apparatus. Burrows' has discussed the effect of mean free path upon measured vaporization rate at higher pressures where intermolecular collisions should not be neglected. The results derived below are only approximate owing to assumptions required by the steady-state model employed. However, errors due to these assumptions will normally (1) G . Burrows, Vacuum, 15, 389 (1985).
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GERDM. ROSENBLATT
1328
be smaller than errors associated with applying results derived for an idealized geometry to an actual experimental situation. Both of these errors are smaller than those occurring when restrictions to molecular flow are neglected entirely. The model used permits ready evaluation of correction factors in simple algebraic form for a variety of experimental configurations from easily accessible geometric quantities alone.
Background and Definitions The equilibrium vapor pressure, P,, may be related to the vacuum rate of vaporization, r, by the equationZ P,
=
(r/aA’)(2aMRT)1’s
(1)
where r is in moles per unit time, a is the vaporization coefficient, A’ is the effective vaporizing area,a M is the molecular weight of the vapor, R is the gas constant, and T is the absolute temperature. If both the equilibrium vapor pressure and the rate of vaporization from a plane surface are known, this equation defines the vaporization coefficient. The vaporization coefficients of most metals are equal or close to unity but materials which form molecular gaseous species may have very small vaporization coefficients. The vaporization coefficient is the ratio of the number of molecules actually evaporating from a plane surface in unit time t o the number of molecules calculated to strike that surface in unit time when the surface is in equilibrium with its vapor. The vaporization coefficient is closely related t o the condensation coefficient which is in the fraction of molecules striking a plane surface which sticks to the surface. In this paper vaporization and condensation coefficients will be assumed equal; that is, it will be assumed that a is independent of the ambient partial pressure of the vaporizing species. This assumption has been discussed previ~usly.~ The effective vaporizing area, A ’ , is equal to the measured vaporizing area for a sample with a uniformly vaporizing plane surface and to the projected plane surface area of any porous or rough sample for which a = 1 . However, A’ may be larger than the projected area of porous samples with a < 1 or smaller than the measured area of a plane sample which does not vaporize uniformly owing to contamination of portions of the surface. In the latter case, particularly, distinctions between vaporization coefficient effects and surface area effects may become rather arbitrary. For a porous sample, A’ is defined as that area of plane surface which would vaporize at the same rate, r, as the sample. Equation 1 is the basis of the Langmuirz method extensively used to measure vapor pressures of solids, The J O U T Mof~ Physical Chemistry
particularly the low vapor pressures associated with high-temperature materials. In this method the sample, which may be powder in a crucible or a small piece of metal, is suspended in an evacuated tube and heated. The rate of mass loss per unit area per unit time is determined and the vapor pressure calculated using eq 1. The vaporization coefficient must be known or assumed, as must the relation between the measured sample area and the effective vaporization area if a < 1 and the surface is rough. The same kind of experiment is often carried out in studying the kinetics of sublimation. In this case comparison, using eq 1 , of the vaporization rate from a plane surface with the equilibrium vapor pressure yields the vaporization coefficient. However, if the experiment is carried out on a sample with a rough surface normally only the ratio a A ’ / A will be determined. To obtain either equilibrium vapor pressures or vaporization coefficients from measurements of vaporization rate, the other of these two quantities must be known independently. When the experimental arrangement is such that some molecules which vaporize are not measured because they recondense upon the sample, the measured net rate of vaporization, rapp,will be less than the true vacuum rate of vaporization, r. If eq 1 is used to calculate a vaporization coefficient from the measured vaporization rate, an apparent vaporization coefficient, aapp,smaller than the true value, will be obtained. Similarly, if the purpose of the experiment is to measure a vapor pressure (with the appropriate values of a and A’ known or assumed), an apparent Langmuir vapor pressure, Pap,,less than the true vapor pressure, will be calculated. Correction factors for such situations are presented below. The results are presented in terms of the dimensionless quantity
f
= rapp/r = a s p p / a=
Papp/P,
so that they can be readily applied to rates or pressures in any units. The quantity f represents the fraction of the molecules initially vaporizing which reach the condenser, that is, the fraction which contributes to the measured net vaporization rate. The fraction f is simply related to the ambient “pressure” of vapor species upon the sample, P 1 - f = P/Pe
(3)
When f = 1, all molecules reach the condenser and the sample sees a perfect vacuum, whereas when no ( 2 ) I. Langmuir, Phys. Rev., 2 , 329 (1913). (3) G.M.Rosenblatt, J . Electrochem. Soc., 110,563 (1963).
EFFECT OF MOLECULAR FLOWUPON VAPORIZATION RATE
T
‘I
SINK
SINK
(e)
(f)
P
. .
Figure 1. Experimental configurations treated in this paper.
molecules are condensed, f = 0, the sample reaches equilibrium with its vapor. The different geometries treated in this paper are shown diagramatically in Figure 1 and discussed in detail below. I n addition to the effective vaporizing area, A’, and the projected plane vaporizing area, A , two further quantities shown in these diagrams appear in the derived results. The quantity B is the cross-sectional area of the cylindrical tube containing the sample, in the same units used for A and A’. The dimensionless symbol W represents the Clausing‘ factor, the fraction of molecules entering a tube at one end under equilibrium conditions which will exit from the other, for the tube length between the vaporizing surface and the sink for vapor molecules. Accurate values of Clausing factors for cylindrical tubes are tabulated6 as a function of the length to radius ratio of the tube, Z/a (where 3 = ad). For most purposes to which the results of this paper will be applied an approximate formula due to Dushman*
w
=
[l
+ (3E/8a)]-’
(4)
which is accurate to about lo%, will be adequate.
1329
mental configurations is similar and straightforward. The same method has been applied previously to molecular flow within a Knudsen ceL397 Consider Figure 2. The sample is a negligibly thick platelet suspended in a cylindrical tube, of crosssectional area B, closed at one end. The other end of the tube contains a sink for the vaporizing molecules. The length of tube between the sample, whose plane is perpendicular to the tube axis, and the sink has a Clausing factor W . The sample is of plane area A , with each side considered to be of area A / 2 , and has an effective vaporizing area A ‘. The dotted lines in Figure 2 represent planes perpendicular to the tube axis, Plane 1 is just below the sink, plane 2 is just above the sample, and plane 3 is between the sample and the closed end of the tube. Let ut and d , represent the molecular fluxes per unit area up and down through plane i, respectively. For example, UI represents the number of molecules which pass upward through unit area of plane 1 in unit time. It is assumed that the molecular flux is isotropic radially across any horizontal plane and only varies in an axial direction. The measured vaporization rate, T ~ will ~ be ~ equal , to the number of moles of vapor condensed in unit time, ulB, while the vacuum vaporization rate, r, is given by (YA’P42nMRT)-”’. The fraction of vaporizing molecules which reach the sink, which is the desired correction, f, is therefore
j = u~B(2aMRT)’”/aA’Pe
(5)
All molecules reaching the sink are trapped and there is no molecular flux downward through plane 1: dl = 0. Apply a mass balance to the flow of vapor through
Figure 2. Diagram representing a thin sample vaporizing from both sides and suspended in a tube closed at one end. Dotted lines represent planes perpendicular to the tube axis.
Steady-State Method This section illustrates how the steady-state approach is used to evaluate the desired correction factors, f. Only one geometry will be treated in detail, that shown as (b) in Figure 1. The procedure used in setting up the steady-state equations for other experi-
(4) P. Clausing, A m . Phyzik, 12, 961 (1932). (5) 8. Duahman and J. M. Lafferty, “Bcientific Foundations of
Vacuum Technique,” 2nd ed, John Wiley and Sons, Ino., New York, N. Y., 1962, p 94. (6) Reference 5, p 91. (7) K. Motrfeldt, J . Phya. Chem., 59, 139 (1956).
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GERDM. ROSENBLATF
1330
the tube after a steady state has been attained. The number of moles of vapor condensed in unit time is equal to the net number crossing plane 2 and equals the number vaporizing minus the number condensing.
u ~ B=
- &)B -
( ~ 2
u ~ B= aA‘Pe(2nMRT)-’’’
(6)
perimental geometry. On the other hand, the purpose of the experiment may be study of vaporization coefficients and vaporization kinetics. In this case the equilibrium vapor pressure will be known independently will be and an apparent vaporization coefficient, aapp, calculated from the kinetic measurements. In this latter case, fmay be expressed as
( d z d ’ / 2 ) - (UsaA’/2) (7) u ~ B= u ~ W B (8) Equation 8 states that the fraction of molecules crossing plane 2 which reach plane 1 is given by the Clausing factor for the length of tube between the two planes. Now, examining the plane between the sample and the closed end of the tube, at steady state the number of molecules crossing this plane in both directions must be equal as there is no source or sink for vapor below plane 3 u ~ B= deB
(9)
The final equation needed can be obtained by considering that the total molecular flow upward through plane 2 is composed of molecules flowing upward through plane 3 which miss the sample plus molecules which vaporize from the top surface of the sample plus molecules which bounce from the top surface of the sample
u ~ B= u ~ ( B- A )
+
(aA’/2)Pe(2nMRT)-’’’
+ [d2(1 - a ) A . ’ / 2 ]
(10)
A similar equation could be written for the molecular flow downward through plane 3 but the same relation can be obtained by subtracting eq 10 from 7. By a tedious series of algebraic manipulations ul, u2,u3,d2, d3, and P, can be eliminated from eq 5-10 to obtain the correction f as a function of geometric quantities and either a or cyspp. The results for this particular example are presented as case (b) of the following section. Results In this section correction factors, f, for the idealized geometries depicted in Figure 1 are presented, first, in general form. Following this, those terms in the corrections which depend upon the specific geometry are discussed on a case by case basis. If the purpose of the experiment is to determine the equilibrium vapor pressure of the sample by the Langmuir method, a must be known independently. In that case the correction factor may be expressed as l/f = 1
(R
+ ( d ’ / B ) ( & - 1)
(11) is a resistance factor which depends upon the ex-
The Journal of Physical Chemistry
f =1
- (aappA’/B)(&- 1 )
(12)
The correction becomes unimportant when f + 1, that is, when essentially all molecules which vaporize reach the condenser. This occurs independently of the specific geometry when either the vaporization coefficient or the vaporizing area becomes very small. In the first case most vapor molecules which hit the sample bounce from it without condensing and eventually reach the condenser. In the second case very few of the molecules bouncing around in the tube wilI hit the relatively small sample. However, as the product aA ’ approaches B , its maximum possible value,3 the correction becomes appreciable; that is, f becomes small. The exact magnitude of the correction to be applied will depend upon the value of (R determined by the specific experimental geometry. (a) Sample Situated at Closed End of Tube. & =
1/w
(13)
This model applies to the experimental situation in which a thin piece or film of metal, grease, or liquid is at the end of a closed tube. For the special case where d’= B , the fraction, f, is the Clausing factor for the tube, W . (b) Thin Sample Vaporizing from Both Sides Is Suspended in Tube Closed at One End. =
[B/(4B - 2A
+ d’)] + (1/W)
(14) This is the case described in detail in the preceding section. This model applies when a freely Vaporizing thin solid sample, perhaps a single-crystal platelet, is suspended in a vacuum system. In applying eq 14 and 12 to problems where a is not known, very little error will be introduced by using aappfor a in eq 14. If necessary, and it rarely will be, successive iterations using newly derived values of a a p p may be carried out to obtain a more accurate value off. If the sample is large enough to block the tube completely, A / 2 = B , eq 14 reduces to l / f = 2 (aA’/B). [ ( l / W ) - 11; no molecules vaporizing into the closed end of the tube escape and the fraction vaporizing up which escapes is the same as in case (a). To compare this limiting result with the combination of eq 11 and 13 note that, for a specific crystal platelet, A’ in case (b) is twice the effective area in case (a). (R
+
EFF&CT OF MOLECULAR FLOWUPON VAPORIZATION RATE
(c) Sample with One Vaporizing Surface I s Suspended in Tube Closed at One End. Vaporizing Surface Faces Open End of Tube.
a=
1/w
(15)
This model applies when a crucible containing the sample is suspended in an evacuated tube. It also applies to a solid sample covered so that it only vaporizes in one direction, for example, a single crystal wrapped so that it only vaporizes from one crystallographic face. A t steady state this situation is indistinguishable from that considered in case (a) as the same number of molecules leave the region between the crystal and the closed end of the tube as enter it. ( d ) Sample with One Vaporizing SurfaceIs Suspended in Tube Closed at One End. Vaporizing Surface Faces Closed End of Tube. =
[ B / ( B- A)]
+ (1/W)
(16)
Equation 16 applies, for example, when the sample (which might be contained in a crucible) vaporizes upward into a closed tube or small bell jar, with the pumping system below the sample. The first term in (R accounts for the extra obstruction t o flow caused by the requirement that all molecules which escape have to get by the sample. When the sample is small, B >> A , this term is unimportant and the tube length between the sample and the sink is the primary impediment just as in case (c). (e) Thin Sample Vaporizing f r m Both Sides I s Suspended in Tube Open at Both Ends.
+ aA’ - 0.5(W1 + W2) X (2B - 2A + d’) 6i = 0.5 0.5(Wi + W2)(4B - 2A + aA’) L WlWZ(2B - 2A + aA’) r4B - 2A
1
+I/
J
(17) The situation is similar to that treated in case (b) except that the tube is considered to have a condenser at both ends. Equation 17 will apply when the vacuum chamber extends through the heated area, aa is often the case with samples heated by induction. In the limit where W 1or W2 is zero, case (e) is the same as case (b) and eq 17 reduces to eq 14. Equation 17 also becomes particularly simple when the sample is located midway between the two condensers, Wl = Wz. In that event (R = 0.5 [(l/W) 1 ]; the correction f is the same as that obtained in case (a) with the tube area replaced by 2B. (f) Sample with One Vaporizing Surface i s Suspended in Tube Open at Both Ends.
+
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This last result is applicable, for example, when a sample contained in a crucible is heated by induction. In the limit where Wz = 0, this case is identical with case (c) and eq 18 reduces to eq 15. When W1 = 0, case (f) is the same as case (d) and eq 18 reduces to eq 16. The results contained in eq 11, 13, 15, 16, and 18 can also be used to estimate the fraction of molecules effusing from a Knudsen cell which escape from the heated portion of the vacuum system. For this application A’ is the area of the cell orifice, A is the crosssectional area of the Knudsen cell, B remains the cross-sectional area of the tube, and it may be assumed that a = 1. For most experimental arrangements the correction will be negligible because A’
