Maurice 1. Martin
.I
and Peter Dunlop University of Adelaide Adelaide, southAupalia
II
A Student Apparatus for Measuring the Second Virial Coefficients of Vapors
A
survey of the available texts on experimental physical chemistry revealed that there were no experiments described for measuring the second virial coefficients of vapors. Since we believed that such an experiment would be useful to those students taking a course in statistical thermodynamics, we decided to construct a suitable apparatus and to have the students test and develop i t throughout the term. This paper gives the necessary theory, the details of apparatus construction, and the data obtained for n-hexane a t two temperatures. Theory
The second virial coefficient of a real gas or vapor may be defined by the relation
where P is the pressure, V the volume of n moles of the gas, R the gas constant, and B ( T ) the second virial coefficient. At sufficiently low pressures the higher virial coefficients in the expansion may be neglected. The measurements required for calculating B ( T ) in eqn. (1) are T , n, and correspouding values of P and V . However by making two sets of measurements, one with an ideal gas and the other with the vapor of uuknown virial coefficient (1, Z), it is possible to obtain an expression for B ( T ) which requires pressure and temperature measurements alone, provided that the same unknown volumes are used in each case. When the gas chamber is filled with n moles of real gas theu a t low compression
and at high compression
and after eliminating n from these two equations the following expression may be derived for the second virial coefficient
Figure 1. A photograph of the vifial coefficient apparatus. containing lhe n-hexone is attache2 ot TI.
The bulb
zero (3) in the vicinity of 50°C, nitrogen is a suitable reference gas. Another equally valid equation of state for a nonideal gas is the relat,ion ( P V / n R T ) = 1 + R f ( P )+ . . . (7) Using the same procedure as above, it is possible to show (1) that
where B'(T) = [B(T)/RT]
If the corresponding relation obtained for an ideal gas is ( V d V , ) = (P,'/P,')
=
r
(5)
then eqns. (4) and (5) yield (1)
(9)
if the truncated form of eqn. (1) is valid for the system of interest. Thus, by using eqns. (6) and (8) to compute B ( T ) and B 1 ( T ) ,respectively, eqn. ( 9 ) should he satisfied within experimental error if true virial coefficients are actually measured. Apparatus
Thus B ( T ) may be obtairicd from pressure measurements alone. Since its virial coefficient is essentially
A photograph of the apparatus is shown in Figure 1. Convenient quantities of gases or vapors were introVolume 46, Number 9, September 1969
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duced into the apparatus through TI and compressed in the right hand arm of the U-tube manometer after closing TS. Compression was carried out by admitting clean dry nitrogen through T8 into the mercury reservoir, and thus the mercury menisci on both sides of the U-tube were raised. Because of the large mass of mercury1 used, this reservoir was supported in a cradle which had been machined from polypropylene. The pressure in the compression chamber was reduced by attaching TS to a vacuum pump. The heights of the mercury columns were read with respect to a calibrated invar scale by means of a Gaertner Telescope attached to a meter cathetometer frame. This telescope had an extremely small depth of focus. The scale was situated in the same plane as the arms of the U-tube. Two of the engraved lines on the invar scale were used to indicate reproducible reference volumes, V1 and Va, in the compression chamber of the manometer which was constructed of precision bore Pyrex tubing with an internal diameter of 1.6 cm. During the pressure measurements the mercury a t the meniscus was illuminated with a lamp contained in a long glass tube. A slit, approximately in. in width, was placed between the lamp and the meniscus and adjusted so that a fine beam of light just passed over the top of the meniscus. It was by no means an easy matter to locate the maximum of a mercury meniscus. The apparatus was so designed that the manometer together with its supporting, kinematically-suspended (4) frame could be uncoupled from the rest of the apparatus, lifted out of the bath, and later replaced exactly in its original position. All taps in the system were metal Nupro bellows valves2 and thus the system was completely free of grease. Wherever possible I/dn. o.d., type 316, stainless steel tuhing was employed, and joints formed with Swagelok fittings3 using Teflon ferrules. Onequarter-inch Kovar metal-glass seals were used to join the Swagelok fittings to the Pyrex manometer. The apparatus was evacuated with a 2-in. Edwards oil diffusion pump and a backing pump. A 3-in. stainless steel (type 316) connector was arranged to come very close to T4 of the virial coefficient apparatus. The vacuum system could be shut off very quickly with a 1-in. Edwards butterfly valve and disconnected from the virial coefficient apparatus by means of a Swagelok connector. This vacuum system gave an ultimate pressure of better than 10-Vorr and was left in operation day and night. By using silicone oil, it was unnecessary to have a by-pass for the diffusion pump. Air pressures were measured by means of Pirani and ionization gauges. The temperature was controlled to better than +0.0l0C by means of an automatic proportional control unit.
through TI. Before use the nitrogen was dried and then stored a t approximately 0.5 atm pressure in a 5-1 Pyrex flask. The n-hexane was part of a sample prepared for a research project to measure binary diffusion coefficients. An analysis by means of gas phase chromatography indicated that this sample contained more than 99.gs% n-hexane. The sample was purified by degrading the branched isomers (5) and the impurities with chlorosulphonic acid. The compression ratio, r, in eqn. (6) was measured with pure, dry nitrogen. It was not always possible to set the mercury meniscus a t either of the two positions used to define the fixed volumes VI and V2 in eqn. (6). I t was usually more convenient to measure several pressures and the corresponding levels on the compression (RHS) chamber, and to obtain the required pressure by interpolation. Very small volume increments were used to ensure that the interpolation would he linear. We believe that with this apparatus the pressures obtained are accurate to zt0.01 tom. The value of r used in the results reported below was approximately 0.52. After thoroughly degassing the apparatus once more the hexane sample, which had been degassed by vacuum sublimation over sodium (6) and stored in a small flask sealed with a metal bellows valve, was introduced through TI and the pressures PI and Pzcorresponding to V, and V2,respectively, determined as before. When these pressures had been corrected to standard gravity and the density of mercury at O0C, eqn. (6) was used to calculate the second virial coefficient. The values of B(T) obtained a t 40°C and 55'C were -1750 and -1550 cm3 mole-', respectively. I n each case eqn. (9) was satisfied by the data. For comparison, the corresponding values read from a graph of the data obtained by McGlashan and Potter (2b) a t these temperatures were -1700 and -1450 cm3 mole-'. For this system careful students were able to measure B(T) values with a precision of *5%. The present data and those of McGlashan and Potter are shown in Figure 2. Once minor faults in the apparatus had been removed, i t was quite easy for the students to measure
Experimental
After thoroughly degassing the apparatus with the oil diffusion pump, samples of dry, high purity nitrogen (99.9gs%) or dry, degassed n-hexane were introduced I
Thr rnwwr). an: d i w l l r d four t u n e before (we. N o p w Co., I X R I S n w a c l h d , Clrvrlaud, O l ~ t ~ i m l A(rum -
Ohio 44110. "Obtainable
from Crawford Fittinm, Niagara Falls. Ontario.
Canada.
616
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Journal o f Chemical Education
Figure 2. Second virial coefficient. or o function of tomporotvro for n-hexone; 0 McGlarhon m d Potter, 0 results with the present opporatus.
r, PI, and PZin two working days, since they were provided with a dry, degassed sample of n-hexane. As part of the experiment the students were asked to ohtain the LennardJones potential parameters from the second virial coefficient data for xenon reported in Hirschfelder, Curtiss, and Bird (7), and to speculate as to whether this potential function cou!,j he to the virial coefficient data McGlashan and Potter for n-alkanes. Literature Cited
(1) (a) Cox, J. D., AND ANDON,R . J. L., Trans Famday Soe., 54, 1622 (1958); (b) Cox, J. D., l'rans. Fa~adaySoc.,
56, 959 (1960); (c) Cox, J. D., Trans. Faraday Soe., 57, 1674 (1961). (2) (a) HAMANN, S. D., AND PEARSIP, J. F., Tmns. Faraday SOL, 48, 101 (1952). (b) McG~.Asrrn~, M. L., A N D POWER, I). J. B., Proe. Roy. Soe. A, 267,478 (1962). (3) K r : ~ s s F. , G., "Temperature," Reinhold, New York, 1941, p. 46. (4) STRONG. J., "Procedures in Experimenhl Physics," Prentice Hall, N. J., 1938, p. 585. (5) SHEPARD, A. F., HENNE,A. L., AND MIDGELY, T.,J . Am. Chem. Soc., 53, 1948 (1931). E. L., HARRIS,K. R., PEPELA, C. N., (6) BELL,T. N., CUSSLER, AND DUNLOP, P. J., J . Phya. Chem., 72, 4693 (1968). J. O., CURTISS,C. F., AND BIRD, R. B., (7) IIIRSCHFELDER, "Molecular Thwry of Geses and Liquids," John Wiley & Sons, Inc., New Yark, 1954, p. 1967.
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