+ Estimation of Diffusion Coefficients for Gases an Vopors

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June 1955 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Yprogram of fundamental research sponsored both b y industrialcontributions and by the U. S.Army, Corps of Engineers Researchand Development Laboratories. The authors gratefully acknowl-edge the financial and technical assistance given by the manysponsoring organizations, and t he advice and criticism offered b ymembers of th e staff of th e Soil Stabilization Laboratory.

a H =

e =K =K* =L =jy =u =u* =uom =

gn =

A p =e =

NOMENCLATUREactivity of hydrogen ion, gram-mole per literactivity of sodium ion, gram-mole per literstreaming potential, stat-voltsvoid ratio, cc. of void per cc. of d ry solidpermeability coefficient in presence of counterelectro-permeability coefficient in absence of counterelectro-thickness of porous bed, om.actual length of flow pa th in porous bed, cm.superficial linear flow velocity in presence of counter-superficial linear flow velocity in t he absence of counter-linear electro-osmosis flow velocity in pores, cm. perpressure drop across porous bed, gram/cm.( sec.)dielectric constant

osmosis, sq. cm.osmosis, sq. cm.

electro-osmosis, cm. per secondelectro-osmosis, cm. per secondsecond

X

= zeta potential, stat-volts= specific conductance of liquid an d surface of pmedium, stat-mhos per cm.= viscosity of liquid, gram/(cm.)(sec .)

LITERATURE CITED(1) Booth, F., Proc. Rou. Soc. London), 03A, 533 (1950).2) Carman, P. C., rans. Inst. Chem. Engrs. London), 5(3) Cornel1 University, Final Report, Soil Solidification Res(4) Elton, G. A. H., Proc. Roy. SOC. London), 94A, 259(5) Fuerstenau, D. W., So. D. thesis, Massachusetts Instit(6) Grace, H. P., Chem. Eng.Prop. , 49, 303, 367 (1953).(7) Henniker, J. C., Reus. Mod. Phys., 21, 322 (1949).(8) Kruyt, H. R. , Colloid Science,Vol. I, Elsevier PublishinNew York, 1952.(9) Michaels, A. S., and Lin, C. S., ~ N D .ENG. CHEM., 6,(1954).(10) Muskat, M., Flow of Homogeneous Fluids through Pfifedia, McGraw-Hill, New York, 1937.(11) Prutton, C. F., and Maron, S. H., Fundamental PrincipPhysical Chemistry, p. 469, Macmillan, New York, 19(12) Ruth, B. E., ~ N D . NG.CHEM.,8, 564 (1946).

R E C ~ I V ~ Dor review June 26, 1954.

(1937).Vol. 11, 1951.(1948); 197A, 568 (1949).Technology, 1953.

ACCEPTED ecember 23

Estimation of Diffusion Coefficientsfor Gases and Vaporsc.R . WI LKE AND C . Y . L E E

Division of Chem ical Engineering, University of California, Berkeley, Calif.

S T H E theory and engineering applications of diffusionalA perations are advanced, increasing need arises for dif-fusion coefficients of gases and vapors. I n view of the scarci ty ofreliable experimental data , methods of predicting such informa-tion assume a useful role. Th e general objective of this paper isto review critically the existing methods for estimation of diffu-sivities. An effort has been made to establish therelativeaccuracyof various procedures by comparison of calculated results with abody of selected experimental data .

REVIEW OF PREVIOUS CORRELATIONSGilliland 7) developed an empirical formula based on t he h ard

sphere model of the classical kinetic theory 25). Collision di-ameters were assumed proportional to the cube roots of the molarvolumes at the normal boiling point as in the earlier method ofArnold. Examination of available dat a led to the relation:

As used in this paper, the diffusion coefficient is the proportion-ality constant in the rate equation for mass transfer in absenceof convection or other modes of transfer, defined as follows forunidirectional diffusion

where N t = ra te of diffusion of component 1of a binary mixture,D12 = diffusion coefficient, sq. cm. per secondA = cross-sectional area perpendicular to direction o fpl

gram-moles per seconddiffusion, sq. cm.

= partial pressure of component 1, atm.

RT = temperature, K.= gas constant, cc.-atm./gram-mole-o K.= distance in direction of diffusion

In using Equation 1,values of V should be obtained in t heway as were those used by Gilliland-namely, using Koppof additive volumes, with the rules and values for each elegiven by LeBas (18). Values of a tomic volumes have beenmarized in several other sources 1 , b3,sO .Arnold Method. Arnold ( 2 ) adopted the general equform proposed by Sutherland 25)containing an additionalinvolving the Sutherland constant, Clz, to account for deviafrom the hard sphere model. Upon examination of avadata to determine the best numerical constant, Arnold remended the equation:

As outlined by Arnold, the Sutherland constant, C12,mcalculated from th e expression:

The constants, Cl and CZ,are to be obtained from the absboiling temperature,

ci = 1 .47T~ ,From Equations 3 and 4 it can readily be seen th at

ci = 1.47F(TB,TB,)~



1254 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol.47, N

Oo5004

L I N E S O F CONSTANT0 03

V A L UE S O F A F O R XI.OA002

001

001 002003 005 01 02 03 0 5 I O 2 0 3 0 5 0M O L E C U L A R W E I G H T R A T I O , MIM 2Figure 1, Values of A asmole fraction of one componentapproaches zero

In absence of experimental dat a V is obtained by using Kopp'slaw of addit ive volumes, or calculated from viscosity data .

Hirschfelder, Bird, and Spotz Method. If the proper functionfor potential energy of interact ion between any two molecules isknown, the more rigorous kinetic theory as developed by Chap-man and Cowling 6 )may be employed t o give improved expres-sions for D and their temperature properties. Hirschfelder,Bird, and Spotz 8-11) have carried out such calculations fornonpolar gases which have an energy of attraction varying withthe inverse sixth power of th e distance between centers of adja-cent molecules and a repulsive energy varying with the inversetwelfth power. For pairs of nonpolar gases the first approxi-mation for the diffusion coefficient is expressed as follows

The numerical constant, B , is equal to 9.2916 Xtheory.felder, Bird, and Spotz as a function of k T / .values are listed in Table I.

according toThe collision integral, W(I)(I), as been calculated by Hirsch-

Representative

Table I. Collision Integrals for Viscosity and DiffusionkT/ v / w 2 ) 2 ;T / r ) w ) ( l ;k T / m )

0.35960.44320.54320.63010.70230.76130.85230.91720.96571.0361.0851.2221.5501.7142.095

1.3311 . 0 3 30.83350.71970.64790.69910.53730.49980.47450.44180.42110.37120.28780.25850.2085

For mixtures of two nonpolar gasee

and12 = d Z ( 8 )

For collisions between a polar and a nonpolar molecule, modi-fied forms of equations are suggested 1 4 ) . With these modifica-tions Equation 6 applies to systems containing a polar and a non-polar component. However, the energy of interaction amongpolar molecules, valence-unsaturated molecules, and cigar-shaped molecules is very different from that between nonpolarmolecules. Before Equation 6 may be considered stric tly appli-cable to systems containing such molecules, the collision integral

should be evaluated according to their respective potential futions. The labor involved is formidable and as yet this hasbeen accomplished.

The term A in Equation 6 is a correction factor, a small qutity usually less than 0.03, which varies with gas compositDetailed equations for A have been given (8, 9). A is a cplicated function of mole fraction, molecular weights, viscosithe first approximation to the diffusion coefficient, and additifunctions of kT / e I 2 . However, these equations are so lengtha t their use may not be justified for many engineering calctions, particularly when the force constants have been estimaAs a first approximation to A for a given gas system it is suggethat A values for mole fractions of each component equal to be obtained and an average value calculated, assuming A tlinear in mole fraction. Values of A a t zero mole fraction are Rented in Figure 1. This procedure is not completely safactory, as A may deviate considerably from a linear function,it should accomplish a major part of the correction. Thiscedure was adopted in subsequent applications of Equat ion 6

ESTIMATION OF FORCE CONSTANTS FOR SIMILARMOLECULESBoth ~ / knd r , may be calculated from knowledge of gas

cosity at two temperatures by solving the equation for visco9).

x 107 =where ?I = viscosity, grams per cm. second, and V/W2(2)function of k T / e which has been evaluated by HirschfelBird, and Spotz. Representative values are given in TabEquation 9 has been well substantiated by experimental evidand provides the most satisfactory basis for obtaining the foconstants.

In absence of viscosity dat a Hirschfelder, Curtiss, and Birdrecommend the following equations, which supersede those ofearlier work (8).

e l k = 0.77TC= 1.15Tb = 1.92Tm

Prior t o publication of the above equations, in the course ofpresent work examination of force constant data to checkearlier suggestions of Hirschfelder, Bird, and Spotz (8) indicthe following equations to be satis factory:

/ I C = 0.75Tc / k = 1.21Tb

Because the collision integrals a re relatively insensitive to serrors in e l k , Equations 10 and 11are considered to be equivato 13 and 14 within the limits of precision of subsequent compsons (Table IV). The relationships developed between T,Tb y Meissner (2 ') ndicate tha t the value 1.21 in Equatioshould be preferable to the value 1.39 8) , assuming that Eqtion 13 is valid, Unavailable critical and boiling data mayestimated with the equations of Meissner.

The collision diameter may be calculated from a single valuviscosity in Equation 6 using / k from Equations 13 andAlternatively Hirschfelder, Bird, and Spotz recommend:

y o = 0.833V: awhere ro = A., and V, = critical volume, cc. per gram-mole

Bromley and Wilke 4 )have tested the general form of etion proposed by Licht and Stechert ( 0 ) and obtained a rewhich in terms of y o becomes



June 1955 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

Table 11. Selected Diffusion Data for Test of Methods

SystemAir-benzeneAir-ethyl alcoholAir-ethyl etherAir-diphenylAir-I2Air-nitrobenzeneAir-n-heptadecaneAir-n-hexadecaneAir-n-octadecaneAir-n-octaneAir-materCOz-ethyl alcoholCOa-ethyl etherC0z-HzC02- CH4C02-NzCOz-NZOCOz-CsHsco2-I120CO-CZHICO-HiCO-Nzco-01Freon 12-benzeneFreon 12-ethanolFreon 12-materHe-AHe-benzeneHe-ethyl alcoholHe-NeHe-nitrobenzeneHe-HzHe-HzOH2-NHaHn-AHz-benzeneHa-CClrHi-CzHaHz-ethyl alcoholHz-CZHIHg-ethyl etherHI-HC1H2-CHIH2-h-iHz-h-20Hz-02Ha-CaHEH2-SO2H r H z 0h-2-NH1Kz-CzHaNz-HsN2-I*N2-NON2-02NzO-CsHeOs NHzOz-CCh02-C2H4NHs-C1H4A-NeCsHs-CzHeCzHs-CHI

02-benzene

Exp tl. DiffusionCoefficient SqCm./Sec. (at iTemp., O K. Atm.)29829829349 1298298298298298298298273273273.2273.2288298298297.7273273288273298298298273.2298298293298293298293293.2273296273340293293294273 2293273 2273 2300273293293300288273293273298293296298293293293293293

0.09620.1350.08960.1600.08340.08680.4230.04050.03970.06020.2600.06930 , 0 5 4 10 , 5 5 00.1530.1580.1170.08630.1640.1510.6220 . 2 1 10.1850.03850 , 0 4 7 50.1050.6410 . 3 8 40.4941 . 2 30.3721 . 6 40.9080.8490.7700.3170 . 3 4 50.4390.5780.4830 . 3 5 40.7950.6261 . 2 0 50 . 5 3 50.6970 . 4 5 00.4830 . 8 5 00 2410.1870 , 7 4 30 0700.2320 1810.0860.2530.09390.07310.1820.1770.3290 . 0 8 50.163

Examination of da ta as in the case of e l k suggested an alter-native method of estimation of ro based on molal volume a t thenormal boiling point, which may be expressed as:

ro = 1.18V1ia (17)Equation 17 appears to be somewhat more satisfactory than

Equation 15 or 16. The /3 power on V e has been retained inEquation 15, as it has a theoretical basis in the Lennard-Jonesand Devonshire theory of gases and liquids 8 ) , lthough the da taseem to correlate better us. V.O.sg. In application of Equation17 molal volumes may be estimated from Kopp's law and therules of LeBas as in the methods of Gilliland and Arnold.

COMPARISON OF EQUATIONS WITH EXPERIMENTALDIFFUSION DATATo serve as a basis for comparison in determining the relative

accuracy of t he foregoing methods, experimental diffusion da tawere selected from reliable sources for 64 systems at atmosphericpressure and near room temperature. (All methods of this paperpertain to moderate pressure systems where ideal gas behaviormay be assumed-Le., where th e product DP is independent of

Table 111. Force Cons tants and Collision Diam eteComponents of Selected SystemsTO A . from Vias qp&f;

AirAmmoniaArgonBenzeneCOZcoCClrCzHaC2H30HDiphenylEthyl etherCzHaFreon 12Heliumn-HeptadecaneH2HC1IodineCHINeonNitrobenzeneNONNzOn-Octadecanen-Octane0 2PropaneSO2Water

9 7 . 0315124.0440190110.3327230391600350b205288 b8003 3 . 3360550136.53 5 . 7539 b1199 1 . 52208200320113.2254252363356d

6 . 0 3

3.6172.6243.4185.2703 , 9 9 63.5905.8814.4184.4556.223=5,424'4.2325 l l O d2.707.92352.9683.3054.9823.8822 . 8 04.9313 . 4 73.6813.8797.963a7.4513.4335.0614 , 2 9 02.6552.649e

All values obtained f rom viscosity are ta ken from Hirschfelder, BSpotz ( a ) , unless otherwise noted. A mole complete table is preseHirschfelder, Curtiss, and Bird (IS).Values estimated from diffusion data.b Values estimated from T c .

C Values estimated from boiling temperature.d 6). e 94) .

3 02 0

w 1.0Im 05

02

i o 1

3

l-a-I D E V I A T I O N 200

005003003005 01 0 2 03 05 10 20 30

Figure 2. Comparison of experimental datawith results calculated by Gilliland equationO B S E R V E D D , S Q C M / S E C

pressure.) These data are summarized in Table 11. In vthe uncertainty and lack of precision in many investigatioported in th e literature, use of this limited number of da tselected sources covering a wide range of dif fusion coefand related properties was considered preferable to use of abody of da ta selected a t random. Also, for a large majori tysystems force constants based on viscosity (see Table 111known for the indiv idual components.

Diffusivities calculated by Equations 1 and 2 and by Eq6 with force constants from viscosity are presented in Figurand 4. Comparison between calculated and observed from Table I1 are illustrated for these three methodsmethods of Arnold and of Hirschfelder, Bird , and Spotznearly comparable results, with the latter somewhat bettermethod of Gilliland gives poor agreement for these systems.ious equations for estimating force constants were emploconnection with Equation 6 to obtain additional compari



1256 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 47, N. .v2.0

vn 1.0i7 0.5

.3Q 0.25g 0.1

W

rn

W

a 0.050.03003 005 01 0.2 0.3 0.5 I O 20 3.0O B S E R V E D D , SQ GM/SEC .

Figure 3. Comparison of experimental datawith results calculated by Arnold equation

2 0c H . B . S . M E T H O D

DEVIATION 7.0

0.03 .05 0.1 0.2 0.3 0.5 1.0 2.0 3.0O B S E R V E D 0 SQ.CM. / S E C .Figure 4. Comparison of experimental datawith results calculated by Hirschfelder,Bird, and Spotz equation

e/k and o from viscosity

Table IV. Deviations Between Experimental Data andCalculated Values for 64 SystemsAy. % Max. ,Deviation Deviationtem Method

1 Gilliland2 Arnold3 Hirschfelder, Bird and Spotz,4 T O from T c / p ? , (Eq. le , e l k5 TO from V ( E a . 17 , / k from

e / k and r o from viscosityfrom viscosity

6 r o7 TO8 M9 r o10 T O11 ro12 T O13 TO1 4 TO15 TO

viscosityfrom V (Eq. 151, a/k fromviscosityfrom V (Eq. 17 , e / k = 1.21Tb (Eq. 14)[odified Hirschfelder Bird ,Spotz, ,Eq. ,18, e l k 'and rofrom viscosityfrom T c / p c (Eq. 16), c / kfrom viscosityfrom V (Eq. 17 , /k fromviscosityfrom V e (Eq. 15), r / k fromviscosityfrom viscosity, c / k = 0.75Tc (Eq. 13from viscosity, r / k = 1.39Tb 9)from viscosity, a / k = 1.21from V ( E q . 17 , / k =1.21Tb Eq. 14Ta (Eq. 14

Equat ion12

181818181.8181818

20.08.47.010.110.311.812.03.99.47.01o.a5.47.25.17.7

46.820.521.431.324.932.731.016.032.721.937.626.024.920.524.5

0.1 0 2 0.3 0.4 0 5 0 6 0 7 0.8

Figure 5. Variation of constant in Hirschfelder, Birand Spotz equationcalculations with observed dat a. Average and maximum devtions between calculation and experiment are summarizedTable I V for all methods.

It is of interest to compare the three methods with respectthe effect of temperature on the diffusion coefficient. Th e eqtion of Gilliland is theoretically at variance with the Arnold athe Hirschfelder, Bird, and Spotz equations. The two latmethods consider possible molecular interaction upon collisand therefore in principle should give more satisfactory resulThis conclusion is borne out by the relatively meager experimtal evidence available. Table V shows the temperature behavof the carbon dioxide-air system over a range of 293 O to 1500 Kas observed by Kilbanova, Pomerantsev, and Frank-Kamenets16). Results calculated by each method, assuming the value293 K. to be correct, are also given. The Arnold and t

Hirschfelder, Bird, and Spotz methods agree closely, probabwithin the limits of experimental error, while the Gilliland eqtion gives low values a t higher temperatures.

POSSIBLE MODIFICATION OF HIRSCHFELDER,BIRD, AND SPOTZ EQUATIONAs a means of studying deviations of Equation 6 from expe

menta l results, values of the constant B were computed for easystem given in Table I11 which would make the equation agwith experiment. Force constants were based on viscosiVarious attempts were made t o relate these values of B to proerties of the systems. As shown in Figure 5 , there appears tosome correlation of the constant with the molecular weight gro[ M h k y ]. Assuming a linear relationship, the data Figure 5 may be approximated by the equation:

B = 10.7 2.46 [ L;Mfz] l IResults obtained employing Equation 18 in connection w

Equation 6 for the systems are shown in Figure 5 and Table IThe average and maximum deviations between calculated aexperimental values are significantlylese than those obtained wEquation 6, using the theoretical value of 9.2916 X lod4 or

Table V. Variation of D with Temperature for CarbDiffusion Coefficient, 9s. Cm./Sec.

Dioxide-Air SystemExptl. dat a Arnold H.B.S. GillilandT K. (smoothed) method method method

293400600800100013001500

0.1510.2730.5550.9151.321.972.45

0 :2670.5430.9051.281.972.49

0 :2660.5230.8831.972.521.28

0:iiz0.4480.6900.9701.421.77



June 1955 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y3.O2.0 MODIF IED H.B.S. ME T H O D

O B S E R V E D D , SQ. GM. / SECFigure 6 . Comparison of experimental datawith results calculated by modified Hirsch-felder, Bird, and Spotz equation

3.02.0

: .0ic; 0.5Vvo- .3

.2W-I5iY .1aV0.050.030.03 0.05 0 1 0.20.3 0 5 1.0 20 3.0

O B S E R V E D D, SQ.CM. /SECFigure 7 . Results of modified Hirschfelder,Bird, and Spotz equation using estimatedforce constants

Further results using estimated force constants are summarizedin Table IV, items 9 to 15. Figure 7 shows results obtained withEquation 18 using force constants estimated from boiling pointand molar volume. This procedure seems comparable in ac-curacy to t he Arnold method for these systems.

W ) I) = collision integral for diffusion, function of kT/eA = small correction term seldom exceeding 0.03E = maximum energy of attraction7 = viscosity, gram/cm.-sec.

Subscripts 1 and 2 denote components 1 and 2, respectiv

DISCUSSIONOne of t he reasons for using Equat ion 18was to obtain a bestpossible correlation of t he da ta using force constants based onviscosity, and then to use this relation as a basis for comparingthe various procedures for estimating the force constants.Methods of obtaining r oare compared in Tab le IV, items 8 to 11.Estimation of r o from molar volume (Equation 17) appears mostsatis factory. Use of T J P , yields slightly better results than useof critical volume. However, experimental uncertain ty in meas-uremen t of critical properties m ay con tribute to th e apparen t in-accuracy. Methods of estimating ~ / kre compared in Table

IV, items 12 to 14. Equation 13 / I C = 0.755 T,) and Equation1 4 e l k = 1.21 T i ) show essentially equivalent behavior. Asnoted reviously, Equat ions 10 and 11 should be essentiallyequivagnt to 13 and 14.It is recognized th at Equation 18 is highly empirical and th at i tsuse can be justified only on th e basis of th e improved corre-lation obtained for the present systems. It is believed that thedeviations of B from the theoretical value as shown in Figure 5represent some failure in t he theory, which should ultimately beimproved by a more fundamental approach. T o seek modifiedrelations for obtaining t he force constants for dissimilar moleculesfrom th e values for the pure components would seem moreromising ultimately th an t he more simple expedient of modi-rying the numerical constant in t he diffusion equation.NOMENCLATURE

B = numerical constant equal to 9.2916 X accordingC = Sutherlands constant for diffusion, O K.D = diffusion coefficient, sq. cm. per secondk = gas constant/molecule, Boltzmanns constant

= molecular weightP = total pressure, atm.r = molecular diameterr 2or ro = collision diameter, A .7 = absolute temperature, O K.Ta = normal boiling point, O K.T , = critical temperature, O K.T , = melting point, O K.

= function of A T / V = molar volume a t normal boiling point, cc. per gram-Vc = critical volume, cc. per gram-mole

to theory

W2 2)mole (total molar volume of each component)

REFERENCES(1) Arnold, J. H., IND.NQ.CHEM.,2, 091 (1930).(2) Boardman, L. E., and Wild, N.E., Proc . Roy. SOC Lo(3) Bradley, R. S., and Shellard, A.D., Ib id . , A198,239 (19(4) Bromley, L. A, , and Wilke, C. R., IND. NQ.CHEM.,(5) Buddenberg,J. W., and Wilke, C. R., I b i d . , 41,1345 (1946) Chapman, Sydney, and Cowling, T. G., Mathematical of Nonuniform Gases, Cambridge University Press

A162, 511 (1937).

(1951).

dington, 1939.(7) Gilliland, E. R., IND.NQ.CHEM.,6, 81 (1934).8)Hirschfelder, J. O., Bird, R. B., and Spota, E. L., Chem. R205 (1949).(9) Hirschfelder; J. O., Bird, R. E., and Spotz, E. L., J . Chem16. 968 (1948).- - -(10) Hirsohfelder, J . O., Bird, R. E., and Spotz, E. L., TranSOC.Mech. Enors., 7 1 , 921 (1949).(11) Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., MoTheory of Gases and Liquids, pp. 538-40, WileyYork, 1954.(12) Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., TranSOC.Me . Engrs., 71, 45, 8.6-4 (1949).(13) I b i d . , pp. 1110-12.(14) I b i d . , ~ .000, Eq. 8.6-3,8.6-4.(15) Xennard, E. H., Kinetic Theory of Gases, p. 195, M

(16) Kilbanova, Pomerantsev, and Frank-Kamenetskir, J(17) Landolt-Bijrnstein, Physikalisoh-chemische Tabellen(18) LeBas, G., Chem. News, 99,206 (1909).(19) Lee, C. Y. nd Wilke,C . R., IND.NG.CHEM.,6, 381 (20) Licht, W., Jr., and Stechert, D. G., J . Phys. Chem.,(21) Mack, Edward, Jr., J . Am . Chem. Soc. , 47,2 468 (1925).(22) Meissner, H. P., Chem. Eno. Progr., 45, 49 (1949).(23) Perry, J. H., Chemical Engineers Handbook, McGr(24) Rowlinson, J. S., Tran s. Faraday SOC. 7,120 (1951).(25) Sutherland,PhiL Mug., 36,5 07 (1893).(26) Titani, T.,Bull. Chem. SOC Japan),5,4 1 (1930).(27) Trautr, M., and Mtiller, W., Ann. Phy s i k , 414, 33 (1935(28) Waldmann, L., Naturwissenschaften, 32, 223 (1944).(29) Wall, F. T., and Kidder, G. A., J.Phys. Chem., 50, 235(30) Wilke, C. R., Chem. Eng. Progr., 45, 19 (1949).

Hill Book Co., New York, 1938.Phys. U.S.S.R.),12, 4-30 (1942).Springer, Berlin, (1923-35).

(1944).

Book Co., New York.

RECEIVEDor review June 1 , 1954. A C C E P T E December

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