June, 1962
COLLISION DIAMETERS OF GASESAS FUNCTIONS OF TEMPERATURE
to about three times the precision in measuring the dielectric increment. The Birnbaum microwave refractometer is said to be sensitive to changes in the dielectric constant as low as 4 X lo8, and should thus be capable of measuring aerosol concentrations as low as two parts per hundred million by volume. If we assume that the density in the dispersed phase is the same as in the bulk, the sensitivity amounts to about 60 micrograms per liter of dioctyl phthalate smoke, and approximately the same concentration of other common smoke agents. Experimental work is now in progress in this Laboratory to verify this conclusion. Although the change in dielectric constant of aerosols with field strength is beyond detection by present instruments, it may be detectable in emulsions where the drop size may be larger, the.interfacial tension lower, and where high field strengths are possible without sparking. Since the effect is approximately linear in the drop radius, it may afford a convenient method of studying particle size distributions in emulsions. Another possible source of information on the drop spectrum, and a phenomenon of interest in its own right, is the dielectric relaxation. If a droplet is allowed to come to equilibrium in a steady field, and the field is suddenly removed,
797
the droplets will execute damped oscillations about a spherical shape, with frequency given by the Rayleigh formula2‘ woa = 8Y/Plao3
(11)
and a damping factor22 7
=
ao2m/5v
(12)
The polarizability of the droplets will also show this variation, so that we may expect a form of dielectric absorption near the resonant frequency, which, for colloidal droplets, falls in the megacycle range. This absorption will be of an unusual type, since unlike that due to the relaxation of polar molecules, the displacement will drop immediately to zero when the field is removed, while the “dielectric constant” will show damped oscillations about its zero-field value. Development of a quantitative theory of this effect will be hampered both by the non-linearity of the dependence of the polarizability upon the field strength, and by the electrical and hydrodynamic problems involved. However, the existence of such an effect seems assured, and its dependence upon the drop radius seems highly probable. (21) H. Lamb, “Hydrodynamias,” Dover Publishing Co., New York, N. Y., 1945, p. 475. (22) H. Lamb, i b i d . , p. G.10.
COLLISION DIAMETERS OF SOME GASES AS FUNCTIONS O F TEMPERATURE BY IRVIN GLASSMAN AND B. L. HARRIS Department of Chemical Engineering, Johns Hopkins University, Baltimore, Maryland Received J a n u a r y 16, 1061
l’tic ccllisioii diameters calculated from viecosity data for oxygen, nitrogen, argon and air are given over the range of temperature from 200 to 2500°K.
Kinetic theory states that, when a gas is assumed to be composed of smooth, rigid, elastic, spherical molecules of diameter u, the viscosity is given by the relationship = 0.1792 ( k m T ) ‘ / 2 / d
where p is the viscosity; m the mass of the molecule: k, Boltzmann’s constant; and T, the absolute temperature. The accurate derivation of this relationship is StraighCforward and is given in its entirety by Chapman and Coyling.‘ With accurate viscosity daka this relationship provides a convenient means for the determination of molecular size. Diameters, so calculated, are regarded as equivalent ones and are very useful for many sorts of approximate calculations and for comparison with size determinations by other methods. Several authors have calculated diameters by this method, but only a t 0”. (Chapman and Cowling‘ give values for helium up to 800”, but these seem to be the only data for a range of temperature.) Discrepancies between values obtained by other calculators are shown below. (1) S. Chapman and T. C. Cowling, “The Mathematical Theory of Non-Uniform Cases,” Cambridge Press. 1030, pp. IGS, 220, 221. 229.
I n this work, the molecular diameters of nitrogen, oxygen, argon and air using the latest available viscosity data and the latest accepted values of the fundamental constants are given; the variation of the diameter with temperature given by the relationship used is shown; and the values obtained with previous similar determinations are compared with values determined by completely different met hods. Recently there have been published by Vasilesco2 extensive viscosity data for air, nitrogen and argon. These determinations cover a complete range of temperatures, going as high as 2900°F.,and seem to be the most accurate and complete work on viscosity done up to this date. Vasilesco’s results were statistically analyzed by the personnel of the Air Materiel Command Heat Transfer Project at The Johns Hopkins University and published in a convenient tabular from in a recent report.* Included in this report were values for oxygen analyzed in the same manner as the other gases, but (2) V. vasilesco, Ann. Phys., 20, 137, 292 (1945). (3) August 31, 1948, Report, Air Materiel Command Projoct, Dcpt. of Chem. Eng., T h e Johns Hopkins University.
IRVIN GLASSMAN AND B. L. HARRIS
798
from much more spa.rse data4 than Vasilesco had presented. These data plus the values given by Birgej as the latest accepted values of the fundamental constants were used in this work. Calculations and Results The values of the fundamental constants used are 12 = Avogadro’s number = 6.0228 X 1020 mole-’ h: = Boltzmann’s constant = 1.38047 X 10M16erg deg.-’
The molecular weights ape given in Table I, which also shows the calculation of the mass of the molecule. I n Table I1 the results for the viscosity and TABLE I CALCULATED 1\lOLECUL.4R ?VIASSES
Gas
Mol. wt.
Nz A OS Air
28.016 39,944 32.000 (28.96)
m
x
m’/%x 10’2, g.
102‘, g.
45.5166 66.321 53.131 48.084
6.820 8.144 7.289 6.934
collisioii diameter of XZ, A, 0 2 and air, respectiyely, are given. Figure 1 shows the variation of collision diameter with temperature. r 4.0
F
WI
’
v
Air 02
I
3.0 0
I
I
500
1000 1500 2000 2500 Temperature, OK. Fig. 1.-Molecular diameter of nitrogen, oxygen, argon and air as a function of temperature.
Discussion The decrease of molecular diameter with temperature is interesting. All four gases considered show a decrease of approximately 20% over the complete range of temperature, 200-2500 OK. Chapman and Cowling’ offer the explanation of this effect by considering molecules centers of repulsive forces, not hard spheres. In Table I11 a comparison of the values obtained at 0” with values taken from other sources is given. Chapman and Cowling used the viscosity data of Schmitt6 for their calculation on nitrogen and oxygen and the data of Schultze’ for argon. Kennard used the data of Trautz8 on nitrogen, oxygen and (4) (5) (6) (7)
P. J. Higdon, Phil. Mag., 26, 961 (1938). R. T. Birge, Rev. M o d e r n Phy., 13, 223 (1941).
Schmitt, Ann. Physik, 30, 398 (1909). Schultze, ibid., 6 , 165 (1901); 6, 310 (1901). (8) Trautz and Zink, ibid., 1 , 427 (1930); Trautz and Baumann, ibid., 2, 733 (1929); Trautz and Sorg, ibid., 10, 81 (1931); Trautc a n d Melster, ibid., 1 , 409 (1930); Trautz and Binkele, ibid., 6 , 561 (1930).
Vol. 50
TABLE I1 COLLISION DIAMETERS AND VISCOSITIESAS TEMPERATURE Tylp.,
K.
Nitrogen P X 106 X?O8
200 1 2 9 . 0 250 155.1 273 166.3 298 177.9 300 178.8 400 221.0 800 257.9 600 291.1 700 321.4 800 3 4 9 . 4 900 375.5 1000 400.0 1100 4 2 3 . 3 1200 445.4 1250 ,, 4 5 6 . 0 1300 4 6 6 . 5 1400 486.7 1500 506.2 1600 525.2 1700 5 4 3 . 2 1750 5 5 2 . 0 1800 560.7 1900 5 7 7 . 7 2000 594.4 2500 671.4
3.968 3.826 3.776 3.733 3.730 3.605 3 529 3.476 3.428 3.409 3.387 3.369 3.354 3.342 3.337 3.332 3.323 3.315 3.308 3.302 3.299 3.297 3.292 3.287 3.270
Argon XYOS
160.5 195.4 210.5 226.3 227.4 285.0 385.8 381.5 423.3 462.0 498.2 532.2 564.3 594.9 609.7 624.1 652.1 679.1 705.0 730.1 742.4 754.4 777.8 800.9 907.1
X 3.887 3.725 3.665 3.617 3.614 3.469 3.379 3.318 3.274 3.240 3.213 ‘3.102 3.175 3.160 3.153 3.147 3.137 3.127 3.119 3.112 3,108 3.105 3.100 3.094 3.074
A
FUNCTION OF
Oxygen
Air
x?O6
x ;OS
x
147.1 178.3 191.5 205.9 207,O 258.1 303.2 343.7 380.8 415.0 447.0 477.0 506.5 532.6 545.6 558.4 583.2 607.0 630.0 652.2 663.0 673.7 694.4 714.8 808.9
3.841 3.688 3.632 3.587 3.584 3.460 3.364 3.307 3.265 3.234 3.209 3.190 3.176 8.159 3.154 3.148 3.138 3.129 3.120 3.115 3.112 3.109 3.104 3.099 3.080
133.1 160.3 172.0 184.2 185.1 229.2 267.8 302.6 334.3 363.5 390.9 416.6 440.9 464.1 475.3 486.2 507.4 527.8 547.5
x
TO8
3.939 3.795 3.741 3.699 3.697 3.569 3.477 3.438 3.399 3.371 3.347, 3.329 3.314 3.301 3.296 3.290 3.281 3,273 3.266 5 6 6 . 5 3.260 575.7 3 . 2 5 7 584.9 3.254 602.6 3.250 6 2 0 . 1 3.245 700.7 3 . 2 2
argon and the data of I
