1332 EFFL-SIOK O F GASES
-4~ I I C R O ~ I E T I I O FOR D
AT CRITICAL VELOCITIES
l \ I O L E C U L \R \VEIGIITS O F
SIDSEY W. BCKSOS
AND
GASESAND
liL4PORS’
RICHARD COSWELL
Departirieril qf C h e m i s t r y , linivcrsity of Southern California, Los Angeles 7, California Rcceiied March 10, 1948
Ever since Grahuni’s discovery of the relation between the rate of diffusion of gases through fine porous diaphragms and the density of the gases, the method of gas diffusion has been variously employed both for molecular weight measurements and for isotope separation. However, the kinetic theory shows that the physical requirements for true diffusive flow are rather strict,-so strict, in fact, that true diffusiye processes are rarely encountered in the laboratory. The mean free paths of most gases and vapor:; at atmospheric pressure and room tempersture lie in the range lo-‘ to em. To measure diffusion through membranes a t atmospheric pressures would require that the membranes bave pores with diameters less than the mean free path of the gas (i.e., 100-1000 A ) . This situation can be improved by working a t lower pressures, but in the change, additional problems are introduced such as accurate, low-pressure measurement and surface adsorption. Outside of the very careful work of Knudsen (j),who did measure diffusion in the low-pressure range, using thin platinum diaphragms having extremely small holes, there have been very few reports in the literature on diffusion through diaphragms. Indeed, there has actually been some misunderstanding and confusion caused by the application of the term “diffusion” t o n-hat are not really true diffusion processes. More common practice has been to measure the effusive flow of gases through thin orifices. There are many commercial models of effusiometers, which are supposed to measure gas density or molecular weight through dependence on flow rate. The hydrodynamic equations for effusive flow through thin orifices under adiabatic conditions are well known (6). The mass flow in grams per second, Q, under such conditions is given by:
I n this equation =I represents the a r w of a, circular hole in a thin diaphragm, the ratio of specific heats C,lC,, R the gas constant, M the molecular weight, T the absolute temperature, Po the higli p ~ ~ + u rand e P the low pressure of the l ) , the equation reeffusing gas. For very small pressure gradients ( P Po duces t o : y
Presented before the Division of Physical and Inorganic Chemistry at the 113th Meeting of the .American Chemical Society, which n-as held in Chicago, Illinois, April, 1918.
lIOLECULAR IVEIGHTS O F GASES BY EFFUSIOS
1333
In this form it is suitable for use in relating molecular weights to f l o veloc~ ities. It is, however, worthwhile pointing out that for regular laboratory practice, in addition to having AP small, the experiment should be carried out when comparing different gases a t constant AP, for otherwise the interpretation and use become very difficult. On the other hind, a t small AP the flow becomes so slow as to change from adiabatic to isothermal for any finite orifice thicknesses. Buckingham and Edu-ards (1) have made a rather extensive investigation of effusive flow under these conditions and have found that corrections of quite complicated form had to be introduced, xvhich included both viscous effects and corrections for non-adiabaticity. The conclusion to be reached from a study of their work is that the effusiometer is not a trustworthy instrument for measuring gas densities and should be applied with considerable reservation to the analysis of unknown gases. Equation 1 does, however, offer another possible avenue of approach. As the ratio P/Po is decreased from unity, the flow reaches a maximum value for the ratio:
p/po =
(r f l "-)y'y-'
Since this is a maximum, further reduction of the back pressure cannot increase the flow and certainly cannot decrease the flow as the equation paradoxically implies. The conclusion must be drawn that once the critical pressure ratio (PlPo), is reached, the flow becomes independent of back pressure (6). Under these conditions, equation 1 reduces to:
in which
f(r)is given by:
This equation shows that Qc is well adapted for measurements of molecular weights of gases if 7 is known. However, as can be seen in table 1, in whiclif(7) is shown as a function of y, for the values of y which occur commonly (from 1.2 to 1.4), the maximum change in f(7) is only 5 per cent. Only the monatomic gases fall outside of this range (7 = 1.67) and even for these the change in f ( y ) is only an additional 5 per cent. Using some intermediate values of f(r)such as 0.670, equation 3 reduces to a form in which the flow, Qc, is simply related to P , T , and M . For a given apparatus only iM will vary from gas to gas and so the relation can be employed in molecular weight determinations. Table 2 shows how the values of the critical pressure ratio (P/Po),will vary with different values of y. (P/Po),is a smooth, monotonic function of y with very small variation in the region of commonly occurring values of y . Thus for most gases, a ratio of forward to back pressure of about 2 is sufficient to ensure critical flow.
1334
SIDSET JV. 13ESSOS A S D RICHa'dID COSWELL
The follon-ing portion of this paper is a report on an experimental method for determining molecular weights of gases by measurements of their mass flow through an orifice a t critical AOTTvelocities.
I . 670 1.500 1.400 1.300 1.200
0.712 0.701 0.685 0.670 0.652
TABLE 2 Variaiion. of critical pressure ratio with y
1 670 1 500
I
I
1.400
1 250 1.100
1
0.496 0.512 0.525 0.555 0.550
I
EXPCRIMENT.1L
The effusiometer which \\-as used in the present work is shown in figure 1.
It consists of a mercury manoneter, an orifice with by-pass (B, figure l),and a vacuum-operated Toepler pump. The entire system could be evacuated t o a pressure of 2 microns by means of a Kelch Duo-Seal vacuum pump. A gas manifold, not shown in figure 1, had Pyrex bulbs in which gases could be stored and from which small samples of gas could be transferred to the manometer through the by-pass, using the Toepler pump. By keeping the volume of all parts of the apparatus small and using a Toepler pump with a volume of 300 cc., all transfers within the system could be made with two or three strokes of the pump. Three strokes were sufficient to evacuate the manometer and orifice t o a pressure of less than 0.1 mm. of mercury, starting a t atmospheric pressure. The orifice was a specially constructed platinum disc built t o order by the American Meter Company. The disc was 0.0125 cm. thick and had a very fine circular hole 0.00493 cm. in diameter. The diameter of the opening was measured with an optical micrometer. The small hole was extremely regular, despite its size. The orifice was sealed to Corning glass S o . 705AJ and then by graded seals t o the apparatus. The manometer was constructed of specially selected, 12-mm. O.D. Pyrex tubing. Tungsten wires, to the ends of which fine platinum \Tires had been
MOLE('V1,A I?. JYEIGHTS O F GASES BP EFFUSION
1335
spot-welded through nickel, were sealed into the evacuated arm of the manometer and served as electrical contacts t o indicate different gas pressures. After the manometer had been filled with mercury, the pressures corresponding to the different contacts were measured with a cathetometer to 0.01 cm. These pressures were: 27.13, 11.03, 7.02, 4.65, 2.19, and 1.00 cm., respectively. These differences were rechecked during the course of the work and found to be constant. Variations with room temperatures were checked and found to be less than 1 part in a thousand. This is consistent with the coefficients of expansion of mercury and glass.
-
FIG.1. Vacuum system
Flow rates were measured by observing the time for the pressure to fall through the values represented by the different contact points. The timing circuit is shown in figure 2, in which the bottom four contacts are shown operating. It is a further modification of a system originally described by Feskov (4) and later modified by Drake ( 2 ) . I n operation, the loTyest contact shoum in figure 1 and figure 2 is always below the mercury surface and serves as a common ground for the electric timers used. The runs were started at high pressures with all contacts under mercury. The top contact, on opening, tripped the relay and started all the timers. Then as each subsequent contact was passed, its timer would stop. I n this n-ay, using three electric timers simultaneously, three experimental points could be obtained at once. There was no difficulty with mercury sticking a t the contacts. The times were
1336
SIDNEY I\*. BEXSON A S D RICHARD COSTYELL
measured t o 0.1 bec., and time measurements were reproducible to within 1 part in a thousand for a given gas. The timers were checked against each other and found to be consistent to within 1 part in 10,000, Runs were made by first evacuating the system to 1 micron. -1sample of gas was then pumped from the gas manifold to the manometer, using the Toepler pump. The starling pressure as adjusted, again with the Toepler pump, and residual gas was displaced into the manifold. The by-pass stopcock S (figure 1) was opened, starting the run. Runs Jyere made under three different sets of conditions: first n-ith S (figure 1) turned to vacuum; then with S turned to the
FIG.2. Timing circuit
evacuated Toepler pump; and finally, in the case of condensible gases, with S turned to the Toepler pump with its well (figure I) filled with liquid air. When runs were made with the gas passing into the Toepler pump, the sampleswere conseryed and could be re-run. The entire time for a run was less than 10min. The gases used in the present n-ork were chosen to cover a wide range of molecular weights. They n-ere hydrogen, ammonia, osygen, carbon dioxide, sulfur dioxide, and dichlorodifluoromethane (Freon 12). They were researchgrade gases, taken directly from tanks and passed into the system through indicating Drierite. Both system and storage bulb u-ere flushed and evacuated
MOLECULAR X E I G H ' N OF' GASES 13Y EFFUSION
1337
twice with the gas t o be stored before being allowed to fill t o atmospheric pressure. Both hydrogen and oxygen were allowed to pass through a trap immersed in liquid nitrogen to remove condensibles. The other gases were put through two isothermal distillations t o remove non-condensibles, The system as described, although slightly more complicated, has many distinct advantages over the conventional type of effusiometer. It is a microeffusiometer capable of operating on as little as 1 cc. (S.T.P.) of a gas. By operating a t low pressures, corrections for the non-ideality of gases are avoided. Gas samples are not lost in measurement but may be recovered after runs and re-run or used for further tests. The trying difficulties due to dust particles settling on the orifice, which are described by Buckingham and Edwards (l),are completely avoided in this closed system. Finally, it is the belief of the authors that the method of gas effusion a t critical pressure ratios is a theoretically more sound procedure than effusion a t small pressure gradients. CXPERIMCXT.IL
r+c>i-I2Ys
All gases were run under as wide a variety of conditions as possible. Typical data are shown in figure 3 in which the time, t, is shown plotted against P, the pressure of the effusing gas. The data in figure 3 are for ammonia flowing into both vacuum (dotted line) and the Toepler pump (solid line) for two different starting pressures. Because, in the present system, both the volume and the pressure of the effusing gas are changing, there is no simple analytical expression relating molecular weight to the time for given pressure drops. Nor can the flow rates be obtained directly from the efflux times. The mass flow Q, in grams per second, hon-ever, can be related to the pressure and its time derivative by the following equation:
I n equation 5 M represents molecular weight, I' the absolute temperature, P the pressure, t the time, a the cross-sectional area of the manometer (0.795 cm.'), and Tiothe volume of the system behind the orifice a t zero pressure. All quantities in equation 5 are known except dP/dt, which is obtained from the experimental measurements. Instead of attempting to obtain dP/dt from the P-t plot of figure 3, it was found simpler to plot log P against t and obtain dP/dt from the slope of the resulting curve. The data of figure 3 are shown plotted in this manner in figure 4. This curve is very close t o a straight line and the slopes are easily determined. I n terms of such a plot, equation 5 transforms to:
Following the determination of Q a t various pressures, log Q was plotted against log P for all gases in order to determine the pressure dependence of Q. The results for the different gnses are s h o ~ min figures .5 to 10. As can be seen,
MOLECULAR WEIGHTS OF GASES BY EFFUSION
3
1340
SIDNEY W. RENSON AND RICHARD COSWELL
MOLECULAH WEIGHT+ OF GASI.3 BY EFFUSIOS
I
I
I
I
I
14 N
1342
SIDNEY W. B E S S O S A S D IZI('HAl~L, C'ObWELL
for the intermediate pressure region, log Q is a linear function of log P . The slope of these portions of the curves is 1.20 for all of the gases with a deviation which is less than 2 per cent and certainly within the experimental error. Thus Q in this region varies directly as P' From the intercepts of these linear portions of i ! , dependence of Q can be obtained by comparing the different gases. I n figure 11 these intercepts, log Q (at unit p r w u r e ) , are zhown plotted against
t
, lo9 Q,
00
FIQ.11. Plot of log 21.1(molecular weight) against log Q (at unit pressure)
log M . With the exception of oxygeii, the points fall on a very good straight line. The two results indicate that the relation between Q, P , and M is given by: Q =
/;p'.20J40.63
(7)
DISCUSSIOh
For purposes of discussion, the relevant physical data for the gases used are presented in table 3 (values taken from International Critical Tables). From the values listed for the mean free paths it can be seen that even at the lowest pressures used (1 em. of mercury) the orificae diameter was still about ten times the
1343
~ I O 1 2 ~ ~ ~ \VEIGHTS ' ~ I A R O F GASES BY CFFUSION
longest mean free path. Under these conditions, the flow may be considered as effusive. Inspection of figure 3 shows that back pressure has negligible effect on the flow rate? down to very low forward prasures. This may be taken as very direct, experimental evidence for the fact that in this pressure region we are indeed beyond the limit of critical flon-. Calculations of the expected back pressures in the Toepler pump system further confirm this. Except a t the lon-est points, for the higher starting pressures, the ratio P Po is ah-ays less than 0.5. Another striking experimental fact which seems to justify further the reduction of equation 3 is the conformity of the Q-P dependence for the different gases. Effusion that is not critical n-ould be expected to show a Q-P relation that would be a function of the y for the specific gas. However, whereas equation 3 for ideal effusive flow predicts a linear relation between Q and P , the empirical equation 7 indicates that Q is proportional to P' 'O. Similarly, whereas equation 3 predicts that Q should vary as 31' for different gases, equation 7 shows that under our TABLE 3 Physical data j o i gases studied GAS
I
VISCOSITY
(20°C.) X 106
'
b f E A S FREE PATH,
1
cp[c~
(20 C )
poises
.I
Hydrogen, . . . . . . . . . . . . . . . Ammonia. . . . . . . . . . . . . . . . . . Oxygen.. . . . . . . . . . . . . . . . . . . . Carbon dioxide.. . . . . . . . . . Sulfur dioxide.. . . . . . . . . . . . Freon 1 2 . . . . . . . . . . . . . . . . . . . I
.I ~
2.02 17.0 32.0 44.0 64.0 121.0
~
1 ~
1
i
89 100 201 148 125 100 (est.)
I183 7 0 1100
1
2 6 (est.)
' ~
1.410 1 310 1 401 1 304 1 28 1 19 ( e s t . )
experimental conditions Q varies as Ai!0.63, with a large deviation s h o r n by oxygen (figure 11). This deviation of oxygen cannot be accounted for by differences in y, since f(y) for oxygen is not significantly different from the values of f(y) for the other gases. Areasonable explanation for the discrepancy betTveen equations 3 and 7 may be found from a consideration of the orifice dimensions. The orifice used was not an infinitely thin wall, but actually a short tube with a radius-to-length ratio of 1/5, The effect of this finite length is to introduce viscosity as one of the flow factors. For isothermal, viscous flow of an ideal gas through a long tube, the mass flow Q is given by:
For small back pressures, Q would be expected to vary as L1fP2q-1. For larger back pressures the apparent P dependence would be somewhat less than 5 power of 2. If we n-ere to add a correction term of the form L11P2q-1 to equation 3, we might expect a net result to be similar in form to the odd powers found
1344
SIDNEY W. BENSON AND RICHARD COSWELL
in equation 7. I n the case of oxygen, with a viscosity that is relatively high, the effect would be exactly in the direction observed. It was found that a small empirical correction of the form mentioned was partially successful in bringing all of the data into reasonable concordance. It is liliely that a much thinner orifice, which can easily be manufactured, would reduce the viscosity effect to negligible proportions. Further experiments should be made to verify this. Despite the limitations imposed by high viscosities, the present system can be used to measure molecular weights for most gases, since most gases mill have comparable viscosities. Eyring (3) ha' xported using a flow system for measuring molecular weights. A study of the dimensions of his system under the pressures used shows that the f l o was ~ effusive and in the region of critical velocities. The good agreement v-hich he obtained using a diffusion-type equation is due in part to a fortuitous choice of gases and mainly t o the remarkable coincidence that the equation for diffusion through an orifice into a vacuum is identical with equation 3, except for :L numei.ica1 vonstant. The ratio of the constants for the two different equations is 1.6, the cffusirr flow being faster. Thr present system should be very uzefiil in measuring the composition of isotopic mixtures such as K,-D, or SH3-NDs. I n such mixtures, the viscosity and y effects will entirely cancel and the accuracy will depend only on the acciuury of timing, which is about 0.2 per cent >IX\l 1I i l
1. It is shown that when the i~itioof forward to back pressure for a gas effusing through an orifice exceeds 2 , R region of critical flow velocity is reached. I n this region the flow (3 is directly proportional t o M'Pf(-y),f(y) varying very little for most g a w . 2 . A rnicroeffusiomcter is h c r i b e d having a small platinum orifice and equipped with automatic electric timers. It iy qhown that thi.; instrument can be iised in most cases t o nieasuw molecular \wight> to within 4 per cent. It is expected that the use of ii thinnw orifice vouid improve the absolute accuracy and extend the range through thr. Pliininution of viycosity effects. 3. The apparatus ha? the advantages of using small gas samples (1-2 cc. a t 3.T.P. minimum) and preserving the samplt. for further work. It should also be capable in its present form of measuring the relati1 P densities of isotopic gases to within 0.2 per cent,
