Temperature dependence of thermal diffusion in CO isotopic gas mixturesJerome D. Verlin, M. Keith Matzen, and D. K. Hoffman Citation: The Journal of Chemical Physics 62, 4151 (1975); doi: 10.1063/1.430294 View online: http://dx.doi.org/10.1063/1.430294 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/62/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal diffusion factors at low temperatures for gas phase mixtures of isotopic helium J. Chem. Phys. 96, 3775 (1992); 10.1063/1.461883 Separation of Light Isotopic Gas Mixtures in the Thermal Diffusion Column J. Chem. Phys. 53, 4319 (1970); 10.1063/1.1673940 Thermal Diffusion Factor for Isotopic CO J. Chem. Phys. 52, 1007 (1970); 10.1063/1.1672996 Composition and Temperature Dependence of the Thermal Diffusion Factor in H2–He Gas Mixtures J. Chem. Phys. 49, 5537 (1968); 10.1063/1.1670084 Temperature Dependence of Thermal Diffusion Factors in Ternary Gas Mixtures Phys. Fluids 10, 992 (1967); 10.1063/1.1762252

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Temperature dependence of thermal diffusion in CO isotopic gas mixtures

Jerome D. Verlin, M. Keith Matzen," and D. K. Hoffman

Ames Laboratory-USAEC and Department of Chemistry. Iowa State University. Ames. Iowa 50010 (Received 17 December 1974)

The temperature dependence of thermal diffusion in several different isotopic gas mixtures of CO is investigated theoretically and the results are compared with experiment. The temperature range includes the inversion temperatures for all mixtures. Thermal conductivity and diffusion coefficients are also calculated. The basis of the study is a previously presented theory which provides expressions for classical collision integrals for molecules with rotational structure in terms of spherical n* integrals.

INTRODUCTION

Quantitative, theoretical prediction of thermal diffu­sive effects in isotopic, polyatomic gas mixtures pro­vides a critical test of the molecular interaction model. To a very good approximation, a unique set of potential parameters should describe all of the interactions be­tween different isotopic species in all possible mix­tures. For instance, in our present study of CO iso­topes, binary collisions between the various species 14C1SO, 12C180, 13C160, and 12C1SO should be described by a Single, noncentral interaction potential. A spheri­cal potential is inadequate for this purpose Since, con­trary to experiment, it would predict no effect in mix­tures of species with equal masses (e.g., 14C1SO_12C180) nor would it predict different effects for two mixtures such as 14C1SO_12C160 and 12C180_12C160 which have, between the mixture components, the same mass dif­ference and the same average mass.

On the other hand, rigid ovaloid bodies, which repre­sent the simplest non central interaction model for poly­atomic molecules, are not adequate either. Although model parameters can be chosen which provide a rea­sonable estimate to the effects of rotational degrees of freedom at a single (high) temperature, the predicted thermal diffusion ratio a is temperature independent. 1 As a consequence the important phenomenon of tempera­ture inversion of thermal diffusion is completely missed. The attractive part of the interaction plays a central role here and, of course, rigid ovaloid mole­cules suffer only repulsive collisions. A related diffi­culty of the rigid ovaloid model is that the spherical part of the collision cross section is very unrealistic. As a result, if there is an important spherical contribution to the property of interest (as in the case for thermal diffusion when the two species of the mixture have dif­ferent masses) the rigid ovaloid model is unsatisfactory. Hence, a soft, nonspherical interaction model is essen­tial to describe the temperature dependence of thermal diffusion in isotopic, polyatomic gas mixtures. Of course, a realistic model should also serve as a basis for accurate calculations of other transport properties.

Evaluation of the matrix elements of the linearized colliSion operator is difficult for polyatomic molecules because the inelastic collisions which they suffer are extremely complicated events. In a previous paper2 (hereafter referred to as I) an apprOximate method for

The Journal of Chemical Physics, Vol. 62, No. 10, 15 May 1975

evaluating nonspherical collision integrals for a classi­cal gas was suggested. This scheme reduces these ma­trix elements to expressions involving the familiar 0* integrals of the ldnetic theory for central potentials. In the present study we use this treatment to investigate the temperature dependence of thermal diffusion in CO isotopes over the temperature range from 80 to 300 OK. This span includes the reported inversion temperatures for all the mixtures under investigation. 3 In addition we analyze the additive contributions to the thermal diffu­sion coefficients arising from the mass, moment of in­ertia, and load differences between the species of the mixture. We also calculate the thermal conductivity and self-diffusion coefficients over approximately the same temperature range. These latter transport properties are largely determined by the spherical contribution to the cross section. In the present work we use the ther­mal conductivity to fix the parameters which govern the spherical part of the interaction. This procedure has the advantage of widening the range of applicability of the model and at the same time Leaving only the non­sphericity parameters to be determined from thermal diffusion data,

The importance of quantum effects on the rotational motion has not been rigorously assessed. We note, however, that the characteristic rotational temperature of CO is 2.8 OK and hence that the maximum contribu­tion to equilibrium thermodynamic properties from these effects is of the order of 1% at 80 OK. Unless the effect on thermal diffusion is several times greater it will not be appreciable.

THEORY

The procedure for solving Boltzmann's equation to ob­tain numerical estimates for the thermal diffusion coef­ficient has been discussed in detail in a previous paper, 1 and as a consequence we will give only a very brief ac­count of the method here. The distribution functionj", for species a can be written in the form

j", =.t!.0> {I + qJ",) , (1)

where j:> is the local Maxwell-Boltzmann distribution and qJ", is of the form

qJ", = _@y/2A.,.. v: +n ~~~)~) (2)

Here VT is the temperature gradient, ~ is the statistical

Copyright © 1975 American I nstitute of Physics 4151

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4152 Verlin, Matzen, and Hoffman: Thermal diffusion in CO

6

5

5 4 u ~ ~ 3

....£., on o 2 x -<

100 TOK

FIG. 1. Thermal conductivity of 12C160 for the temperature range 100-300" K. EXperimental data (Ref. 8) denoted by X. Solid line denotes theoretical curve calculated using Elk =144.3"K, (0)=(1.78)27[, R=1.06, andS=0.057.

temperature, n is the total number density, and d,. is the diffusion force for species 'Y. The coefficients Aa and ~~") can be expanded in a complete set of functions of the reduced linear momentum, Wa ", (ma/3/2)1/2Va, and the reduced angular momentum na '" (f3/2la )1/2La. Here Vol. is the peculiar velocity, La is the angular momentum, mOl. is the mass, and Ia is the moment of inertia of spe­cies Ci.· We retain only the following functions in a trun­cated expansion set:

lJr~ 1) = Wa , (3)

lJr~2) = Wa (% - W~) , (4)

lJr~3) = Wa (1- n~) , (5)

and

lJr~4) '" Wa • (GaGa - t~o) , (6)

where 0 is the unit tensor. The first three functions correspond, respectively, to the flux of mass and of translational and rotational energy whereas lJr~4) is the

.26

.22

.18

'0"' :'i: .14

..... N

E ...£..10

J .06

.02

.00

InT

®x x x

FIG. 2. Diffusion coefficient for 14C160_12C160 vs InT. Experi­mental data for Ref. 9 denoted by ® and experimental data of Ref. 10 (for 12C1BO_12d60) by X. (Theoretical curves for the two mixtures are nearly identical. )

0' C

.03

.02

- .01

.00 x

x

x

-.01 L--_L...._-'---_-"-_.....L._----'-_--'_---'

o .2 .4 .6 .8 1.0 1.2 1.4

In 3031T

FIG. 3. lnq vs In(303 0 KIT) for 14d60_12C130. Symbols and parameters are as in Fig. 1.

most important term Which is anisotropic in the angular momentum. These functions are odd in W 01., which is required from parity considerations. They are also even in G",. (Terms odd in the angular momentum have previously been shown to be of no practical conse­quence. 4,5) The function lJr~4) is known to contribute at most 5% to the thermal diffusion factor. However, the importance of higher polynomials in W~ and n~ has not been assessed.

The trial functions are all odd in the components of the generalized momentum vector 1/ = (011 ~, Wl1 W2),

and hence the integrands of the matrix elements of the collision operator are even in the components of 1/. In this case, according to the theory presented in I, a sui table approximation for the generalized cross sec­tion reduces the matrix elements to expressions involv­ing the usual n* integrals for a spherical potential. The particular potential we use is of the Lennard-Jones (6-12) form for which the necessary n* integrals have been tabulated as a function of reduced temperature. 6

.02

:;.01

o

o .2 .4 .6 .8 In 303fT

x x

1.0 1.2 1.4

FIG. 4. Inq vs In(303 "KIT) for 13d60_12d60. Symbols and parameters are as in Fig. 1.

J. Chem. Phys., Vol. 62, No. 10, 15 May 1975

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Verlin, Matzen, and Hoffman: Thermal diffusion in CO 4153

.05 x

x

. 04

x x

.03 x

c-c

.02

x

.01

1.4 In 303fT

FIG. 5. lnq vs In(303 "KIT) for 12C1BO_12C160.

parameters are as in Fig. 1. Symbols and

The matrix elements necessary for the calculation of thermal diffusion coefficients1 are obtained using ap­propriate projections of the generalIzed matrix elements A..v of I. Since CO molecules have two active rotational degrees of freedom, the angular momentum projection operators are of the form given in Eq. (35) of 1.

The calculations depend on four interaction parame­ters which are assumed to be independent of isotopic mass differences between the colliding species (as is consistent with the Born-Oppenheimer approximation). These parameters arise in the generalized cross sec­tion of I and can be related (in a nonrigorous way) to a Kihara type model which assumes that the molecules are surrounded by body fixed ellipsoidal cores. The potential energy of interaction is a function of the short­est distance between cores 7

; the functionality is deter-

.03 xx

x

.02 x

c-.:

x

.2 .4 .6 .8 1.0 1.2 1.4 In 303fT

FIG. 6. lnq vs In(303" KIT) for 14C 160 _12C 160 • Symbols and parameters are as in Fig. 1.

4.0

3.0

2 .

....... ID .. .. E u 0 ...... E ~

'" -1.0

0 x '0°-2.0

-3D ."~"

'. -4D

_5.0L-----l---1.----1..----L-....L--L--.l..---::.I..----l 4.0 4.2 4.4 4.6 4.8 5.0 5.2

InT

FIG. 7. (14C160_12c1BO). Temperature dependence of additive contributions to D~ (see text). Model parameters are as in Fig. 1. ( ••• load, _. - moment of inertia, - sum).

mined from the Lennard-Jones potential. When the cores are in contact at only a point, the interaction en­ergy is zero. The potential parameters are Elk, which determines the depth of the well, {a} which is the aver­age cross sectional area of the core, R which is the ratio of the major to minor axes of the core, and S which is the displacement of the center of the ellipsoid from the bond center (measured in units of the minor axis). A positive value of S means that the ellipsoid center is nearer to the carbon atom than to the oxygen atom. The parameters (Elk) and (a) basically govern the spherical part of the interaction, whereas Rand S characterize the nonspherici ty.

10D

..--. ~ 8. .. E ~ 6.0 E ~ '" 4.0 o x ~o. 2.0

o

/

/ /

/

/

// ." _/ . -.- .. -:-:;~ . ...:.:..':':"­

~?~-=.'~~.c::.. _.

_2.0'L-..-L---:L..-.....::::It::::::::::......J1....----l----L----L---L--4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8

InT

FIG. 8. (13C 160 _12C 160 ). Temperature dependence of additive contributions to D~ (see text). Model parameters are as in Fig. 1. (--- mass, ••• load, -' - moment of inertia, - sum).

J. Chern. Phys., Vol. 62. No. 10, 15 May 1975

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4154 Verlin, Matzen, and Hoffman: Thermal diffusion in CO

22.0

IB.O ~

u ., '" E 14.0

.!(

~lo.O en 0 X 6.0

.... 0 0

2.0

0

-2.0

/ ;;

/ /

4.0 4.2 4.4 4.6 4.B 5.0 52 5.4 5.6 5.B InT

FIG. 9. (12CI80 _12CI60 ). Temperature dependence of additive contributions to I{ (see text). Model parameters and symbols are as in Fig. 8.

NUMERICAL RESULTS

The spherical, Lennard-Jones parameters (Elk) and (a) are fixed by fitting the calculated thermal conduc­tivity of pure 12C160 to the experimental dataS over the temperature range from 100 to 300 oK. Since the ther­mal conductivity is a "mean free path" type transport property, it is fairly insensitive to small variations in Rand S and can independently fix (Elk) and (a). It is found that for a reasonable range of R and S values, the thermal conductivity is fit within experimental error by the parameters (Elk) = 144. 3 OK and (a) = (1. 7S)21T A2 (see Fig. 1). The calculated self-diffusion coefficient, which is also dominated by the spherical part of the in­teraction, is compared with experiment9,10 in Fig. 2. It is seen that different Elk and (a) values (primarily those of (a») are required in this case. This discrepancy in potential parameters between different transport proper­ties is observed also for monatomic gases and hence can be at least partially ascribed to the approximate nature of the Lennard-Jones potential.

The nonsphericity parameters Rand S are determined from experimental thermal diffusion studies. The data which we use are those of Boersma-Klein and de Vries3

who measured the separation factor q for' CO mixtures using an eight tube swing separator with hot end at 303 ° K and cold end at a temperature ranging from about SO to 260 OK. The mixtures which they studied are 14Clea_ 12C180, 13C1ea_ 12c1Bo, 12C 180_ 12C 1Bo, and 14C160_

12C lea. By definition,

303°11:

1.nq(T)=sjT dlnTa(T) (7)

where a (T) is the thermal diffusion ratio, and hence

a(T) = - BdlnqldlnT • (S)

The derivative was computed from an analytical curve fit of the experimental lnq data.

We calculate a{T) and lnq(T) using the values of Elk and (a) determined above and variable Rand S parame­ters. The values which give the best fit to the experi­mentallnq points are R = 1.062 and S= O. 0566. There is

some justification in giving more weight to low tempera­ture points since, by definition, Inq (T = 303 OK) = 0 in all cases; however, for simplicity, we give all experi­mental points equal weight in our work. Other fitting procedures could be used. For instance, special weight could reasonably be given to the equimass mixture since in this case there is no dominant spherical contribution due to mass differences. As a third alternative the R and S values could be chosen so that the inversion tem­peratures are in the closest possible correspondence to the temperatures reported by Boersma-Klein and de Vries. These variations in the fitting procedure leave the best choice of parameters substantially unaffected. To a degree this indicates the uniqueness of the R and S values. The agreement between experiment and theory for each of the four mixtures is shown in Figs. 3-6.

In a previous papers it was shown that the thermal dif­fusion coefficient for isotopic mixtures can be decom­posed into a sum of additive contributions arising from the differences in mass, moment of inertia, and load between the two species of the mixture. In Figs. 7-10 these separate contributions and their sum are given for each mixture. We find in general that the load contribu­tion increases with S and is very sensitive to its value. On the other hand, the moment of inertia term increases with R. Since a small R and a large S value are em­ployed in the calculations displayed in Figs. 7-10, the load contribution dominates that of the moment of iner­tia. As the R value is increased, the best fit of Inq re­quires a decreasing S value, and hence the relative im­portance of the moment inertia contribution increases. For instance, at R = 1. 25 and S = - O. 0342, the contribu­tion of the moment of inertia dominates that of the load. However, the fit in this case is much poorer (by a factor of six in the Inq error). It is of interest in this regard that Boersma-Klein and de Vries are able to empirical­ly fit their data (using a theory due to Waldmann 11) by taking into account only the mass and moment of inertia

22

IB '"U'

Q)

~ 14 u "-

..1.10 en o X 6 '0"

2

-2

/ /

/ /'

/; /

/

'I

I /

'/

->:~ . .:::::...--:;--::..::.~. .- .-.:::;: . ..--.-..:::.:.. .................

_6_~~~-J~-L __ -L __ -L __ ~ __ ~ __ L-~

4.0 4.B 5.0 5.2 5.4 5.6 InT

FIG. 10. (14C160_12c160). Temperature dependence of additive contributions to D~ (see text). Model parameters and symbols are as in Fig. 8.

J. Chem. Phys., Vol. 62, No. 10, 15 May 1975

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Verlin. Matzen, and Hoffman: Thermal diffusion in CO 4155

Table 1. Thermal diffusion inversion temperatures.

T(experimental)° K T(calculated) ° K

14dso/12C1BO 247 247

13C160/12C1S0 185 165

12C1BO/12C1S0 110 82

14C1S0/12C1S0 173 163

differences while ignoring the load.

The trends in the relative importance of the moment of inertia and load differences are qualitatively similar to what is found for rigid ovaloids. Also the R and S values we find are in the meaningful range for rigid ova­loid calculations. 12 However, as previously discussed, the rigid ovaloid model is useful only for qualitative cal­culations and hence a unique set of Rand S values can­not be determined in this case. The shift parameter S is positive, which is in agreement with the fact that oxy­gen is more electronegative than carbon. The value of S estimated from covalent radii is positive but much smaller than the one we report.

From Figs. 3-6 it is seen that the best agreement with experiment is for the equimass mixture and the poorest is for 12C180_12CI60. The fit for the remaining mixtures appears to be slightly outside of the scatter of experimental data. The lack of agreement between the­ory and experiment for 12C180_12CI60 may be associated with the fact that this is the only mixture for which one species (i. e., 12C ~) has both the largest mass and the largest load. As a consequence the behavior of the load contribution for this mixture is different from the others as can be seen in Figs. 7-10.

In Table I the theoretically calculated inversion tem­peratures and the inversion temperatures reported by Boersma-Klein and de Vries are given. The latter were obtained in an empirical manner from the Waldmann the­ory. While they are consistent with the data it is appar­ent that a rather large latitude in the value taken for the experimental inverSion temperatures is permissible. In particular it is not clear from the data given in Fig. 5

that the 12CI80_12Cl&o mixture undergoes temperature inversion at all.

CONCLUSION

In I a new formulation of the collision dynamics of classical molecules with rotational degrees of freedom was given which, by suitable approximation, provided a framework in which the collision integrals for poly­atomic gases could be expressed in terms of spherical 0* integrals. We have used this theory to calculate the thermal diffusion coefficient as a function of tempera­ture for several CO isotopic mixtures. In addition we have calculated the thermal conductivity and self-diffu­sion coeffiCients for CO. All of these calculations were made with a single set of parameters. In general the results are in good agreement with experiment. We feel that the simplicity of the method coupled with its broad range of application demonstrates the usefulness of the treatment in I.

tpresent Address: Sandia Laboratories, Albuquerque, N M. 1M• K. Matzen, D. K. Hoffman, and J. S. Dahler, J. Chem.

Phys. 56, 1486 (1972). 2J . D. Verlin, M. K. Matzen, and D. K. Hoffman, J. Chem.

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717 (1966). 4M. K. Matzen and D. K. Hoffman, J. Chem. Phys. 62, 500

(1975) . SM. K. Matzen and D. K. Hoffman, J. Chem. Phys. 62, 509

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A 72, 359 (1968). 1A distance of analogous significance can be defined when the

cores overlap. By. S. Touloukian, P. E. Liley, and S. C. Saxena, Thermal

Conductivity-Nonmetallic Liquids and Gases, Vol 3 of Ther­mophysical Properties of Matter-The TPRC Data Series, edited by Y. S. Touloukian, (IFl/Plenum, New York, 1970).

91• Amdur and L. M. Schuler, J. Chem. Phys. 38, 188 (1963). 10H. F. Vugts, A. J. H. Boerboom, and J. Los, Physica (Utr.)

50, 593 (1970). llJ . Schirdewahn, A. Klemm, and L. Waldmann, Z. Natur­

forsch. A 16, 133 (1961). 12E • R. Cooper and D. K. Hoffman, J. Chem. Phys. 53, 1100

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J. Chern. Phys., Vol. 62, No. 10, 15 May 1975

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