Separation of gas mixtures by thermal diffusion - American Chemical

820

BORIS LEAF AND F. T. WALL

SEPARATION OF GAS MIXTURES BY THERMAL DIFFCSION BORIS LEAF A N D F. T.WALL hroyes Chemical Laboratory, Universaty of Illinois, Urbana, Illinois

Downloaded by STOCKHOLM UNIV on September 9, 2015 | http://pubs.acs.org Publication Date: August 1, 1942 | doi: 10.1021/j150422a004

Received June $2, 19.42

Thermal diffusion is the term used to describe the processes which occur in a gas mixture subjected to a temperature gradient. In general, the processes result in a non-uniform composition in the vaiious parts of the mixture. The development of the kinetic theory for non-uniform gas mixtures by Enskog (8) and Chapman (1) led to the prediction of the thermal-diffusion effect. The effect was first verified experimentally by Chapman and Dootson (3)in 1917. The most convenient presentation of the kinetic theory including thermal diffusion is contained in the recent book by Chapman and Cowling (2). Thermal diffusion became important as a method of separating isotopes or mixtures of gases of equal weight when Clusius and Dickel (4) invented the thermal-diffusion column which makes the effect cumulative. They have completely separated the isotopes of chlorine (5) and of neon (6). Their apparatus consisted essentially of a vertical tube with a heating element along its axis and a cooling jacket surrounding it. If a gas mixture in such a tube is permitted to reach a steady state, the radial thermal-diffusion effect combined with vertical convection currents produces a difference in the values of the composition at the top and bottom. In the steady state, the separation due to thermal diffusion is opposed by the mixing due to ordinary diffusion. The kinetic theory for non-uniform gas mixtures gives the equations for calculating thermal-diffusion constants for gas mixtures the molecules of which correspond to certain simple models, such as rigid elastic spheres, point centers of force which repel each other as inverse powers of their separation, etc. (2). Methods have also been proposed for calculating the thermal-diffusion constant from the steady-state concentrations obtained in the thermal-diffusion column (9). It is the purpose of this work to investigate experimentally, with a thermal-diffusion column, various gas pairs, and to observe what correspondence there may be between the results obtained by application of the theory of the column to our data and the predictions of the kinetic theory of non-uniform gas mixtures. Investigations for this same purpose have been performed previously in this laboratory by Wall and Holley (1l ) , using pairs of gafies with the same molecular weight. THEORETICIL

If we define a separation factor for thermal diffusion, kT,as the quotient D T / D12,where DT is the thermal-diffusion coefficient and Dl2 is the ordinary diffusion coefficient, then the first approximation to kT for spherical molecules possessing only translational energy is given by the kinetic theory (2) as kTI1

5(C - 1)5152(SlZl - S222)

= 2-

Qi 21

+ Q* + 2:

Q12 215 2

(1)

SEPARATIOS OF G.46 MIXTURES BY THERMAL DIFFUSION

821

where z1 and zzare the mole fractions of gases 1 and 2 and where C1, SI, S2,Q1, Q2, and Q12depend on the masses and force fields of the molecules. If the molecules can be considered to be rigid elastic spheres, then

c = 56s1 = -1


Sz =

{23’2 2 (2) - Mz(15M2 - 7Jfi) 1) (z $ (zy - Mi(15Mi - 7zI?)) 312 Av:,2

where, if ml and m2 are the masses of the two types of molecules, M I

=

m1

ml and

o1 and

02

+ m2

are their diameters. UE?

and MP= ml 012

+ m2

is defined as

01

+

UP

=-

2 The further simplication for the case where MI = 312 = 1/2 gives

z l ~ z { z l [ ( ~ y- 1 1 [ k ~ h= 59 lox1 P ( 0,2 1) ;z:(.$+

+

- zP[(zy -

11)

[$(2y(z)z+ ;]z1z2

(3)

The first approximation to DIPis given by

[DIPII= 83d z {kTpbmk$y’*

(4)

where 7t is the total number of molecules per unit volume of mixture. We can, therefore, get a first approximation to DT as the product of [kT]l and [D12]1in equations 1 and 4. This value of Dp will be referred to as DT theoretical. The values of the diameters of the rigid elastic spheres can be obtained from viscosity data, since the first approximation to the viscosity coefficient is given by (5) for each of the pure components of the mixture.

From this we find that

822


Substitution of equation 6 into equation 4 gives


where %I and %Z are the molecular weights of the two substances and N is Avogadro's number. When ml = mz

where p is the mass per unit volume of the mixture. In using these equations to obtain D , theoretical, values of the viscosities and densities were taken at the mean temperature and pressure in the column, and for z1and zz were taken the original mole fractions in the uniform mixture before thermal diffusion. The equations used in the calculation of the thermaldiffusion constants from the steady-state concentrations in the column are those developed by Furry, Jones, and Onsager (9) and used by Wall and Holley (11). Let the column be characterked by a temperature T2 a t the hot tube and T Ia t the cold tube; let its length be 21 and the width of the annular gas chamber 2w. Then the theory of the column states that DT

4x1xlpgA W' 63vj(AT/T)

where, m before, s1and are the initial compositions in the uniform mixture and p and 7 are evaluated a t the average temperature in the column, T = (TI T 2 ) / 2 ;g is the acceleration of gravity; and for AT = Tz - TI, j ( A T / T )is a function, depending on the interactions between molecules, which is nearly unity and can be omitted. The quantity A is obtained from the steady-state concentrations in the column by

+

where z1and xp are the steady-state mole fractions a t the top (indicated by +Z) and a t the bottom (indicated by - I ) of the column. The value of DT obtained from equation 9 will be referred tQ aa DT experimental. Comparison of DT experimental with DTtheoretical provides a method of testing the predictions of the kinetic theory against experimental results interpreted by the theory of the column. It should be noted that, since the convention as to subscripts used by Furry,Jones, and Onsager is opposite to that used by Chapman and Cowling, DT theoretical from equations 1 and 4 will be opposite in sign to DT experimental from equation 9. Consequently, in the discussion of our results we shall arbitrarily change the sign of DT theoretical. It is interesting to examine the pressure dependencies of DT experimental and

SEPARATION OF GAS MIXTURES BY THERMAL DIFFUSION

823

DT theoretical. From equation 9, DT experimental varies with pressure only through its dependence on p and A : DTexperimental

-

Ap

(114

whereas, by equations 1 and 4, DT theoretical varies with pressure, through the dependence of [D12]on n.


DT theoretical

N

1 -

P

(Ilb)

In order that these two dependencies should be compatible, A should vary inversely as p2. Such a relation would require a large effect of a pressure change on the steady-state concentrations attained in the column, as seen by equation 10. This is a point to be checked by experiment. EXPERIMEKT.4L

The thermal-diffusion column was constructed of brass tubing. The inner, hot tube was 1.1 cm. in outer diameter, while the outer, cold tube had an inner diameter of 2.2 cm.; hence the width of the annular gas chamber was 0.55 cm. The length of this chamber was 273 cm. The inner tube contained a heating coil of 34 ft. of N 22 Chrome1 A wire mounted on alundum tubing and enclosed in a glass sheath. The outer tube was surrounded with a water jacket through which tap water was circulated. Its temperature was taken to be that of the exit water. The temperature of the inner, hot tube was measured by two thermocouples, the junctions of which were soldered about 8 in. from the ends. The inner, hot tube was suspended at the top by a flange resting on the outer, cold tube so that it hung freely. I t was centered by three centering pins near the bottom. Each of the gases used, with the exception of argon (99.6 per cent pure), was purified by condensing out less volatile impurities with a suitable freezing mixture, and then condensing the gas itself with liquid air and pumping off residual gaseous impurities. The gases were mixed thoroughly in a 3-liter flask and the initial composition recorded. Then the mixture was introduced into the column. As samples were withdrawn for analysis, more gas was admitted from the 3-liter flask to decrease the pressure drop. The samples, less than 40 cc. in volume, were removed from the top and the bottom at intervals usually not less than 1.5 hr. All analyses were performed by the use of a Zeiss portable interferometer of the Rayleigh type, which was calibrated to give refractive-index differences relative to one of the pure components of the mixture as a standard, according to the method of Edwards (7). The equation used in the analyses was

where a is the per cent of the second gas, R1 and R2 are the refractive indices of the standard reference gas and of the second gas, respectively, a t standard conditions, and A R is the observed refractive-index difference between the mix-

824


ture and the standard gas a t temperature T and pressure p . To use this equation, it is necessary that the pressure be exactly the same in both cells. To effect this, both cells were filled to nearly the same pressure and then, before taking the reading, they were connected. This equalized the pressure and, since diffusion is relatively slow, no mixing occurred sufficient to affect the reading obtained.


RESULTS AND DISCUSSION

The results of the experiments with the various gas pairs used are summarized in table 1. The temperature Tis the average between the values at the hot and cold surfaces, and the pressure is an average value over the course of a run. The error in the pressure reading is about f 2 cm. and that in the temperature value TABLE 1 Results with eeveral vairs o f aaaes

T

TOP

3ottom

9

'K.

cm.

440

95

51.4% Nz0

45.0

50.1

388 387 432

100 92 80

52.1% CsHs 52.0 52.0

51.5 51.4 50.9

52.6 52.5 52.4

Propane-carbon dioxide ..............

437 427 428 423 423

100 113 100 113 100

29.8% CIHI 48.1 59.8 68.2 82.0

28.8 46.5 58.4 66.9 80.6

31 .O 49.5 61.5 69.9 82.5

Argon-ethane ...........................

454

80

50.0% A

46.9

54.4

Carbon dioxide-cyclopropane. . . . . . . . . . . .

395 396

98

88

49.7% 49.7

50.9 50.9

48.5 48.7

Nitrous oxide-carbon dioxide.. .......... Propene-cyclopropane.,

.................

cog

is about 3 per cent. By the use of the thermocouples at each end of the hot tube' differences in temperature at the ends could be observed. It was found that the top was generally colder than the bottom by about 10' to 35"C.,according to the particular gas pair used. This would be expected, since the hot tube is suspended at the top and heat is readily conducted out to the surface at that end, whereas a t the bottom there is no metallic connection to the outside. Superimposed on this effect, which would cause the top to be colder than the bottom, is the opposing effect of convection currents which cause the bottom thermocouple to be bathed in a colder stream of gas than the top one. The temperature of the hot surface was taken as the average of the values a t the two ends. The error in the composition values is small, 0.1 per cent. In the case of the argon-ethane mixture, a steady state was never obtained even after 11 hr.; the separation continued to increase with time.

825

SEPARATION OF GAS MIXTURES BY THERMAL DIFFUSION


The nitrous oxidecarbon dioxide pair also did not reach a steady state in 7 hr., but here we observe that the nitrous oxide is disappearing throughout the column, leading us to conclude that it is decomposing in the presence of the hot brass tube and releasing nitrogen with a much lower refractive index. Any quantitative calculations are, therefore, meaningless for this pair. The comparison of DT theoretical with DT experimental for the various gas pairs is given in table 2. The values for the viscosities of the individual gases were taken from the data of Titani (10) and from the values compiled by Chapman and Cowling (2). The interpolation equation

--1-21

a

22

a1+w

TABLE 2

Comparison of DTtheoretical with DTezperimental DT EXPEII-

m u Xllr

Propene-cyclopropane ......................

52.0

52.0

Argon-ethane............................... Carbon dioxide-oyclopropane.,. , ,

50.0% A

,{I

49.7% 49.7

co*

4.3 4.7 6.7

1.0 0.95 0.91

12.8 11.9 12.3 9.3 6.8

1.3 2.4 2.4 2.8 1.6

36.1

2.5

0.99

1.1

1.6 1.3

was used to obtain the viscosity of the mixture from the viscosities of the individual components. Also, the density of the mixture was calculated using the ; densities of the pure gases at perfect gas law and the relation p = xlpl+ x ~ p 2 the standard conditions were taken mainly from the International Critical Tables. There is agreement as to order of magnitude between DT experimental and DT theoretical except for the argon-ethane pair, which had not reached a steady state. It will be noticed that, as the pressure decreases, the value of DT experimental decreases for the pairs propene-cyclopropane and carbon dioxidecyclopropane, while the value of DT theoretical increases. This, according to equation 11, indicates that the pressure dependency observed using the theory of the column does not agree with that predicted from the kinetic theory of nonuniform gases. One of these theories must, therefore, be incorrect on this point. The rigid elastic spherical model predicts that the large and heavy molecules will accumulate at the cold wall and will therefore concentrate at the bottom of the column. From table 1 we can therefore state that, according to this model,

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R. L. ANTHONY, R. H. CASTON, AND EUGENE GUTH


propene molecules are larger than cyclopropane molecules, and propane and cyclopropane molecules are larger than carbon dioxide molecules. These conclusions agree with molecular diameters calculated from viscosity data. The behavior of the pair carbon dioxide-cyclopropane is of especial interest, because in this case the lighter molecules concentrated a t the bottom and the heavier ones at the top of the column. The larger size of the cyclopropane molecules brought them to the bottom in spite of their smaller mass. This is in agreement with the behavior expected on the basis of kinetic theory and the sizes obtained from viscosity data. REFERENCES (1) CHAPMAN, S.: Phil. Trans. M17, 115 (1917). (2) CHAPMAN, S., A N D COWLING, T. G.: The Mathematical Theory of Non-uniform Gases. Cambridge University Press, London (1939). (3) CHAPMAN, S.,AND DOOTSON, F. W.: Phil. Mag. 33, 248 (1917). (4) CLUSIUS,K.,AND DICKEL,G.: Naturwissenschaften 26, 846 (1938). (5) CLUSIUS,K.,AND DICHEL,G : Naturwissenschaften 17, 487 (1939). (6) CLUSIUS, K.,AND DICKEL, G.: Naturwissenschaften 28,461 (1940). J.: J. Am. Chem. SOC.89, 2382 (1917). (7) EDWARDS, (8) ENSKOQ,D.: Doctoral Dissertation, Upsala, 1917. (9) FURRY, W.H., JONES, R. C., A N D ONSAQER, L.: Phys. Rev. 66, 1083 (1939). (10) TITANI,T.:Bull. Chem. SOC. Japan 6, 98 (1930). A N D HOLLEY, C. E., JR.:J. Chem. Phys. 8,949 (1940). (11) WALL,F.T.,

EQUATIONS OF STATE FOR NATURAL ASD SYXTHETIC RUBBERLIKE MATERIALS. I UNACCELERATED NATURAL S o n RUBBER’ R. L. ANTHONY, R. H. CASTON,

AND

EUGENE GUTH

Department of Physics, University of Notre Dame, Notre Dame, Indiana Received February 17, 19-49 I . IiXTRODCCTION

The most important and the most characteristic property of rubber is its longrange reversible elasticity. This unLsual property is the one which distinguishes rubber from wax, chewing gum, etc., and makes possible its widespread use for B variety of special purposes. This property is so characteristic that any material possessing it may be called a “rubber” regardless of its chemical constitution. In spite of many attempts directed toward an explanation of rubber-like elasticity, this problem-a problem in the chemical physics of solids-continued to be a very puzzling one for many years. ’ Presented before the Division of Rubber Chemistry a t the lOlst Meeting of the American Chemical Society, which was held at St. Louis, Missouri, April, 1941.

Separation of gas mixtures by thermal diffusion - American Chemical

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