A. P. Malinauskas S. J. Whisenhunt, Jr.' and J. 0. Searcy2
Oak Ridge National Laboratory Oak Ridge, Tennessee 37830
Determination of the Viscosity coefficients of Gases
A
simple, inexpensive apparatus is described for use in student laboratory determinations of the viscosity coefficients of gases. Furthermore, the theoretical aspects are uncomplicated and the experimental data yield quite accurate results; in the present investigations, agreement with literature values was within about 2'%. Helium, however, demonstrated a surprisingly anomalous .behavior whose nature remains unexplained. Apparatus and Experimental Procedure
The apparatus with which the studies were made is sketched in Figure 1. A U-tube was constructed by joining a standard 100 ml gas buret to 16 mm 0.d. glass tubing. The buret was further modified by removing the length of capillary a t the top and replacing i t with a large bore three-way stopcock. A second
Figure 1.
Skekh of theviscosity oppmralur.
such stopcock was joined to the first with a short length of plastic tubing. To one branch of this stopcock was attached a 7.5 cm length of 0.2 mm i.d. glass capillary, whereas the other branch, which served as a flushing port, was connected to a medicine dropper which was immersed in a small volume of the same fluid that was being employed in the U-tube. This arrangement provided a simple but effective one-way valve for the flushing operations. ltesearch fiponsored by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation. 1968 Summer Participant. Present address: Chemistry I)epartment, Kuamillc College, Knoxville, Tonneqsee. 1967 Summer Participant. Present address: Chemistry Department, Pordue University, Lafayette, Indiana.
All of the gases, with the exception of air, were taken directly from high pressure cylinders in which the purity was stated (and confirmed by mass spectrometry) to be a t least 99.9%. Gas flow into the apparatus was throttled with a needle valve in series with the cylinder regulator. In this connection the one-way valve served a dual purpose; the flow rate of the gas during flushing operations could be easily monitored simply by noting the rate of bubble formation a t the tip of the medicine dropper. Air, on the other hand, was drawn into the apparatus from the atmosphere. The ungraduated leg of the U-tube contained a mark which corresponded to the 2 ml graduation on the buret (see Fig. 1). This mark, and the 2 ml graduation, will hereafter be referred to as the fiducial marks. Prior to each experiment, the entire gas line was thoroughly flushed. Gas was then admitted into the buret and this section was similarly purged several times. Spillage of the fluid in the U-tube during this operation was avoided by lowering the leveling bulb as the gas was slowly admitted into the buret and employing the leveling bulb to force the gas out. Once the flushing operat,ions had been completed, the gas under investigation was carefully admitted into the buret until the liquid level in this arm was about 5 ml lower than the chosen starting point (t,he meniscus in the other arm, of course, was positioned with the aid of the leveling bulb a t the fiducial mark). The two stopcocks were then rotated to the positions shown in Figure 1 and the experiment begun. Initial transport of the gas through the capillary is quite rapid, so that during this period primary attent,ion is given to maintaining the liquid level in the open arm of the Utube as close to the fiducial mark as possible. When the meniscus in the buret reaches the predetermined starting point, one of two stopwatches is started. The rate of transport of the gas through the capillary is then monitored with the use of the two stopwatches as the meniscus passes predetermined marks on the buret, until the experiment is terminated. Throughout. the course of the run, however, attention should be given to maintaining the liquid le2.el in the second arm a t the fiducial mark. I n this regard we found it convenient to mount the leveling bulb on an ordinary laboratory jack. Theoretical
This experiment is in many respects similar to one which has been employed for many years in student laboratories (1). The primary difference essentially involves the manner in which the flux of gas molecules through the capillary is determined, and each has advantages and disadvantages with respect to the other. Volume 46, Number 1 1 , November 1969
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The mathematical description of the phenomenon in both cases begins with a presentation of Poiseuille's expression for laminar flow of a gas through a straight, uniform capillary of circular cross-section (3)
in which J denotes the molecular flux through the tube due to the pressure gradient dp/dz, a is the tube radius, q the viscosity coefficientof the gas, and (p/kT), by the ideal gas law, is the molecular density. Neglect of the usual "slip correction" to eqn. (1) introduces an error of a few tenths of one percent in the present circumstances. If steady-state transport of the gas is assumed, then J becomes independent of position and eqn. (1) can be integrated along the length L of the tube to yield ~ ( = dBpAp/qkT )
(2)
where Ap = p(0) - p(L) is the pressure drop across the tube, $ = [p(O) y (L)]/2 is the average pressure, and B = rra4/8L is a geometric constant. The molecular rate of flow J(rrae) is related to volumetric and pressure changes of the gas in the buret through the expression
+
J ( r a a )= -(l/kT)Ld(pV)/dt]
(3)
thus eqn. (2) can also be written in the form -d(pV)/dt
=
BpAp/s
(4)
For Ap small compared to p(L), we can neglect changes in p and $, so that eqn. (4) simplifiesto give
Discussion
It is possible in principle to evaluate the apparatus constant K by direct measurement and thus obtain absolute values of q. However, i t is experimentally more convenient, and in practice more accurate, to determine K using a gas of known viscosity. I n this work, we had chosen argon as the "calibration gas." The use of mercury as the manometric-volumetric fluid was avoided for two reasons. From a theoretical point of view, only a small portion of the buret could be employed without invalidating the assumptions concerning the constancy of pressure. This contingency can he taken into account, but the mathematics then become complicated and the experiment best performed as outlined in Reference (I). On the practical side, the procedure would necessitate handling sizeable quantities of mercury, and this circumstance should be avoided wherever possible in student laboratories. It turns out in this case that water serves as an excellent fluid from both a theoretical standpoint as well as for reasons of safety and economy. Accordingly, water was employed in the bulk of our measurements. Since the mole fraction of water vapor in the gas is of the order of 0.03, the viscosity coefficient determinations are not affected materially. In fact, the effectis probably mitigated to some extent by employing a relative method, since water vapor probably lessens the viscosity coefficients of all of the gases chosen for study here. Typical plots of the experimental data are presented in Figure 2. The uppermost plot was obtained with
(In the present work, the maximum value of Ap is about 5% of the value for p(L), that is, atmospheric pressure, but the effects of the assumptions above on the determination of q are much less.) The volume of gas in the buret a t any time t can be represented by V = V,+Al (6) where V , denotes the volume defined by the entrance side of the capillary and the fiducial mark on the buret (the 2 ml level), A is the cross-sectional area of the buret, and 1 is the distance between the liquid meniscus in the buret and the fiducial mark. I n this sense the buret is employed as a volumeter. On the other hand, the pressure drop Ap across the capillary is also related to the distance 1 by virtue of the correspondence of the fiducial marks on the two arms of the U-tube, thus wherein the manometric constant cis the product of the fluid density, the gravitational acceleration constant, and a factor to convert the buret graduations into distances (in our case, a 1 ml difference in graduation represents a distance of about 0.7 cm). I n this sense, the buret functions as a manometer. If eqns. (6) and (7) are substituted into eqn. (5), and the resultant differential expression is integrated between the initial conditions and the conditions a t time t, one obtains wherein 10 is the distance a t time t = 0, and K = Bc/A is the "apparatus constant." From eqn. (8), a plot of In I versus t should be linear, with slope inversely proportional to the viscosity coefficientof the gas. 782
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Figure 2. Time dependence of the paromater I for typical viscosity elperimenb with woter ar the manometric-volumetric fluid. 0, argon, 0 , helium; A, helium with plastic bog covering the capillary (see text).
argon, whereas the remaining three curves, to be discussed in greater detail below, represent helium data. At least three experiments were conducted with each gas; these corresponded to three different values of lo within the range defined by the 28 ml and 78 ml markings on the buret. As anticipated, the choice of 1, had no systematic effect on the results. I n many cases, however, slight curvature of the log 1 versus t plots were noted a t later times; in all instances the systematically deviating data points were neglected in the slope determinations. All of the experimental data which were taken with water in the U-tube are summarized in Table 1 and the results compared with corresponding literature values
Table 1.
Ar COI N2 Air He
0%
NP
Viscosity Coefficients at 23'C as Determined with Water as the Manometric Fluid
Gas i
**
Ar
6.959 10.635 4.814 i 8.551 i. 7.906 =t 7.956 =k 7.661 f 5.035
Table 2. Viscosity Coefficients at 23'C as Determined with Dibutyl Phtholate as the Manometric Fluid
COz Air He
+
Literature vdues. The viscosity coefficients at 23'C were interpolated from the listings in Ref. (3). At t,his temperature, the viscosity coefficient of argon is 2235 X lo-' poise. b Data taken with a plastic bag over the capillary. See text.
(3). In compiling the data summary, we found it convenient to define the quantity
whence S is simply the negative value of the slope of the linear plot. The slope ratio for argon and the gas i is therefore identical to the inverse of the viscosity ratio, that is SdSi = dna.
(10)
Values of Si which are tabulated represent the average of a t least three determinations; the uncertainties cited are the absolute average deviations of these determinations. Comparison of the experimentally derived viscosity coefficient ratios and the literature values indicates rather good agreement. However, a most remarkable effect was observed with helium. As illustrated by the data points in Figure 2 which are designated by filled circles, this gas initially obeys the linear relationship described by eqn. (8) with the prescribed slope (cf. Table I), but a t later times, a t which the pressure drop across the capillary corresponds to about 6 torr, the water level comes to a complete stop and remains at this level over a prolonged period of time (the dashed lines in Fig. 2). This stationary condition can only be regarded as one of zero net flux through the capillary, and the most obvious way in which this can occur requires that air molecules diffuse into the buret as rapidly as the helium molecules exit. Such a phenomenon is well-known and has come to be described as the Iiramers-Kistemaker effect (4). I n the present case, however, the pressure drop a t which the zero net flux condition has been observed appears to be about a hundred times too large and moreover is in the opposite direction. Although such an unexplained reversal has been observed previously for the case involving argon and carbon dioxide (5), these same workers found no reversal for the system helium-nitrogen. This pair should not differ significantly from our own case. Nonetheless, we sought to investigate this hypothesis by covering the open end of the capillary with a plastic bag and filling this bag with helium prior to performing the experiment. The bag was purposely punctured to ensure that there be no pressure buildup in the enclosed region. Fxperiments conducted with helium in this manner yielded data such as that depicted by the triangular shaped points in Figure 2, and the viscosity re-
10'Si (sec-') 7.305 7.355 11.364 11.042 8.914 8.233
( S A ~ / ~ ; ) I(si/q~rP
0.009' (1.000) 0.993 f 0.005 f 0.028 f 0.096 0.643 0.006 f0 . 0 R 9 9 . 6 6 2 f 0.006 f 0.040 0.819 i 0.00.5 0.887 i 0.004 i 0.021 =t
+
1.000 0.661 0.818 0.877
The viscosity coefficients at 2 3 T were a Literature values. interpolated from the listings in Ref. (3). At this temperature, the viscosity coefficient of argon is 2235 X 10-'poise. *Data taken with a plastic bag over the capdlary. See text.
sult which appears in Table 1. Significantly, the condition of zero net flux was not observed. Although the precise point a t which the zero net flux condition was observed was of poor reproducibility, possibly because of a dependence on the filling conditions and the difficulty in determining exactly when stoppage occurred, we anticipated a like effect at a slightly lower value of Ap (in terms of buret graduations) if dihutyl phthalate were employed rather than water. The phenomenon was not detected in this case. Dibutyl phthalate should actually be a more suitable manometric-volumetric fluid than water, since its density is only about 5% larger whereas its vapor pressure is considerably less than that of water. We therefore performed some additional experiments with this liquid. The results are given in Table 2. Agreement with the literature values is about the same as that obtained with water. (The improvement in the COXresult upon employing the plastic bag is probably fortuitous.) Summary
The apparatus described in this work can be effectively employed in student laboratories as a demonstration of laminar flow of gases through capillaries and as an aid in understanding kinetic theory principles regarding the viscosity coefficient. Although the apparatus is simple and inexpensive, the experimental data which are obtained yield quite accurate values of the viscosity coefficients. Helium was observed to possess an anomalous characteristic, in that i t was possible to obtain a condition of zero net flux with this gas in the presence of a pressure gradient. The nature of the anomaly could not be satisfactorily explained on the basis of the present investigations, thus the effect remains as a subject suitable for further experimentation. Literature Cited (1) SHOEMAKER, D. P., AND GARLAND, C . W., "Experiments in Physical Chemistry'' (2nd ed.), McGraw-Hill Book Co., Inc., New York, 1967, p. 77. ( 2 ) PRESENT,R. D., "Kinetic Theory of Gases," McGraw-Hill Book Co., Inc., New York, 1958, pp. 46-47, 61-63. v , F., "Viscosities of G a w and Gas Mixtures," ( 3 ) G o ~ u n ~ IL. Gosudarstvennoe Isdatel'stvo Fisika-Mat,ematicheskoi Literatwy, Moscow, 1959, Chap. 3. A. P., AND EYANS,It. B. 111, ( 4 ) MASON,E . A,, MALINADSICAS, J . Chem. Phys., 46,3199 (1967). L., AND SCHMITT,K. H., Z.Nalurforsch., 16% ( 5 ) WALDMANN, 1343 (1961).
Volume 46, Number 11, November 1969
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