thermal diffusion in isotopic gaseous mixtures

Fortschritte der Physik 15. 1-111 (1967)

Thermal Diffusion in Isotopic Gaseous Mixtures

GHEORCHE VBSARU

Institute for Atomic Physics. Sect . Cluj. Rumania

Contents

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 . Preliminary Notions . Definitions and Notations . . . . . . . . . . . . . . . . 3 2.1. Gaseous Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A . Lorentzian Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 3 B . Quasilorentzian Mixtures . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2. Molecular Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A . Rigid Elastic Sphere Model . . . . . . . . . . . . . . . . . . . . . . . 4 B . Inverse Power Model or Inversion Model . . . . . . . . . . . . . . . . . 5 C . Square-Well Molecular Potential Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5

E . LENNARD-JONES Model . . ..,. . . . . . . . . . . . . . . . . . . . . . . . 5 F . B U C K I N G H A M M ~ ~ ~ ~ . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . 6 G . Modified BTJCKJNGHAM or ex$ (- 6) Model . . . . . . . . . . . . . . . . . . 7

C . IsotopicMixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

. . . . . . D . STJTHERLAND Model . . . ., . . . .-. . . .. 5 . .i .. ': *%. . . . . . . . . . . . . . 5 . .

3

. i .

2.3. Thermal Diffusion Effect . . . . . . . , . . . . . . . . 8 . . '! . . . 2 . . . . . . . . . 8 2.3.1. Collision Integrals . . . . . . . . . . . . . . . . - . . . . . . . . . . . . 10 2.3.2. 13

3 . Thermal Diffusion and Moleoulat Models . . . . . . . . . . . . . . . . . . . . . . 17 3.1. Rigid Elastic Sphere Model . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2. Inverse Power Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3. Square-Well Molecular Potential Model . . . . . . . . . . . . . . . . . . . 24 3.4. STJTHERLAND Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5. LENNARD-JONES Model . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6. Modified BTJCKINGHAM or exp (- 6) Model . . . . . . . . . . . . . . . . . . 33

4 . Theoretical and Experimental Results . Concrete Cases . . . . . . . . . . . . . . 36 4.1. Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2. Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3. Boron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4. Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

T . D . Factor . CHAPMAN-CO~ING and KIHARA Approximations . . . . . . . .

. . . .

4.5. Nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.6.Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7. Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

1 Zeitschrift . Fortschritte der Physik". Heft 1

2 G. VKSARU

4.8. Chlorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.9. Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.10. Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.11. Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

RGferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

1. Introduction

Thermal Diffusion (shortened T.D.) in isotopic mixtures is of a large interest both from a practical point of view - being used for the separation of the isotopes and from a theoretical one, due to the close dependence of this phenomenon on intermolecular forces. The essence of this transport phenomenon lies in the fact that in the presence of a temperature gradient in a binary or multicomponent gaseous mixture - there takes place a transport of matter which produces a concentration gradient: the light component of the mixture concentrates in the region of a higher temperature and the heavy component in the region of a lower temperature. The T.D. phenomenon was theoretically predicted by ENSKOG in 1911 [ I ] and in- dependently by CHAPMAN in 1916 [ Z ] ; this phenomenon was pointed out experimentally by CHAPMAN and DOOTSON in 1917 [3 ] . Though in 1919 CHAPMAN predicted the possibility of a partial separation of the isotopes by T.D. [a] due to the small effect of separation, this phenomenon did not find its practical applications but in 1938 when CLUSIUS and DICKEL succeeded in combining this effect with a convective current in a colum thus getting a re- markable multiplication of the separation [5]. An important success regarding the isotopes separation technics by T.D. was obtained by the using of the auxiliary gas in the case of multicomponent mixtures or just of binary isotopes mixtures, when the available quantity was small and we wanted to get a high concentration or to separate completely the two isotopes [6]. In the case of a binary mixture for instance a third gas with its mass being com- prised between the masses of the two components of the mixture is looked for. The gas must behave like a third isotope and it must also be easily separable by physical or chemical methods. Thus the auxiliary gas concentrates between the isotopic components enabling in this way the obtaining of very pure isotopes till 99.9 percent. The fact that a T.D. column with a given geometry and a thermal gradient produces in a certain period of time the same separation, enables the use of T.D. for purposes of analysis. Indeed, knowing for instance the separation factor of a column and measuring the concentration of a certain isotope at one of the ends, one can find out the concentration of the same isotope at the opposite end placed within the interesting gas circuit. The magnitude of the elementary T.D. effect for a certain isotopic mixture is determined by aT - the T.D. factor’s value. This factor is very sensitive with the molecular interaction type; a temperature dependence of this factor occurs in case of more complex molecular models. In recent years, special efforts to draw up numerical tables for T.D. isotopic factor for various molecular interaction types [7-141 have been made; a series of experimental measurements on T.D. phenomenon in order to get new information on intermolecular forces [ll- 201 have been performed.

Thermal Diffusion in Isotopic Gaseous Mixtures 3

To compute the T.D. factor there are no precise theoretical expressions; they have made only successive approximations, the complexity of which grows rapidly with the approximation degree [21]. Thus there appear special problems of convergence. Recently series of works have been carried out by means of which they got approximative expressions of a higher order for this factor, their convergence being calculated through detailed numerical computations for certain typical models of intermolecular forces [21-22]. Thus it was established that the first approximation Chapman-Cowling for aT is not satisfactory for large temperature fields. In exchange, the second Kihara approximation seems to have an accuracy close to the third Chapman-Cowling approximation which has a rather complicated expression. The accuracy of this expression ist of the units order a t the third decimal [23]. Considering the complexity and the importance of the problem, an analysis and a systematisation of the existing theoretical and experimental results in order t o facilitate its employment both in practical problems connected with isotope separation and theoretical ones connected with molecular interactions are necessary. That is the purpose of the present work. '

2. Preliminary Notions. Definitions and Notations

2.1. Gaseous Mixtures

According to the molecular diameters and weights and the concentrations of a gaseous mixture components respectively, the gaseous mixtures may be classified in lorentzian, quasilorentzian and isotopic mixtures [22].

A. Lorentzian Mixtures In this category are enclosed the gaseous mixtures of which M , > M , and x, > x, or o,, > o,,, IT,,, where M is the molecular weight, x-molar fraction and o-molecular diameter of the mixture components. For these mixtures only the collisions between 1,2 molecule species are of great interest, the 2,2 type collisions are negligible. A real mixture of light and heavy gases, in which the heavy components is excessive constitutes an example of lorentzian gas. This type of mixture was studied by LORENTZ and it represents one of the few cases which enables, the absolute convergence study of approximative expressions for transport coefficients.

B. Quasilorentzian Mixtures In this category are included the mixtures with MI > M , but x, Q 2,. These are actual mixtures of light and heavy gases in which the light component is excessive. Owing to some similarities in the calculation technics to that of the lorentzian gas, this mixture is called quasilorentzian.

C. Isotopic Mixtures For these mixtures weights of the molecular components are approximatelly equal ( M , N M,) and the l , l , 2,2 and 1,2 types of molecular collisions are all over identical. In the present work we shall only refer to the isotopic mixtures.

1*

4 G. VliSARU

2.2. Molecular Models

An important characteristic of the accurate theory of T.D. lies in the fact that this transport phenomenon is strongly dependent on the molecular interaction forces. This fact makes T.D. be one of the best research means to study these forces. To choose the form of intermolecular potential with view of various transport coefficients computation, we must allow both for the necessary approximation degree and the computation difficulties connected with the use of a certain form. Out of the most important molecular models - from our point of view - we men- tion :

A. Rigid Elastic Sphere Model

is the simplest molecular model; molecules are examined as rigid elastic spheres (similar to billiard balls) of CT diameter. The forces which act between identic or nonidentic molecules within the collision process are zero but for the collision moment when they become extremely great ; so that

rp(r) = 00 for r < 0

v ( r ) = 0 for r > c

where rp ( r ) is t)he intermolecular potential and r the distance between the molecules (Fig. la). The practical interest for this models is rather reduced for the theoretic numerical data related to T.D. are different enough in comparison with the experimental data.

a ) b l

d ) Fig. 1. The form of some empirical potentials with spherical symmetry. a - the rigid elastic sphere; b - the

inversion model (repulsive punctiform centre); c - the square-well molecular potential; d - the Snrmm- L ~ N D model; e - the LENNARD-JONES potential; f - the BUCKINGEM potent i


B. Inve r se Power Model o r Inve r s ion Model

In this model, the molecule is considered as a repulsive centre of force, the potential of which varies with the inverse of the separation distance between molecules to the vth power, that is

(2) d

v(r) = ry

where d is a constant which determines the repulsion at a certain distance; v is called repulsion index. For the majority of molecules v varies between 5 and 15. The interaction potential form ist plotted in fig. 1 b. The fact that within this model the attractive part of the potential is neglected, makes q ( r ) be an unreal potential. It constitutes, however, it good approximation in the case of high temperatures, where the molecular collisions are so powerful that the attractive molecular potential may be neglected.

,

C . Square-Well Molecular Potential Model

This model represents the molecule as a rigid sphere of a diameter surrounded by an attractive field of size E (Fig. 1 c); this field extends up to the Ro distance. Thus we have

qb.1 = CQ for r < a

q ( r ) = - E for o < r < R o (3) = 0 for r > Ra.

The square-well model may be employed both to compute the repulsive forces and the attractive ones being in good agreement with the experiment in the case of complex molecules.

D. Sutherland Model

It describes the molecules as rigid spheres of a diameter which attract one another with a force, the potential of which vanes with the inverse of the separation distance between the molecules, to the d t h power, that is

q ( r ) = 00 for r < a (4)

n

This model is real enough and very simply to manipulate (Fig. Id) .

E. Lennard-Jones Model

In this model the intermolecular forces are supposed to be simultaneously repulsive and attractive, so that the interaction potential between two molecules will have the form :

d e q ( r ) = - - -

r~ rv’

6 G . V ~ S A R U

where r is the distance between the centres of the two molecules and d, c and v, v' are positive constants (Fig. le). The fist term represents the potential repulsive part and the second the attractive part. One of the very often used forms of this potential is the particular form

v ( r ) = 4 E [(;)'"- (31 where E is the depth of the potential well or the maximum value of the attraction energy which is attained at the distance r = 21/6a, [7]; a represents r value for which v ( r ) = 0. The value v' = 6 has been chosen because LONDON inferred on the basis of quantum mechanics the existence between the neutral nonpolar molecules an attraction characterized by a potential which varies inversely proportional to the separation distance between molecules to the 6th power. The choice of v = 12 was partly determined by the fact that there appeared a mathematical simplification when

This potential is known in the literature under the name LENNARD-JONES (12,6) or shortened L.J. (12,6) potential. l) There are other particular forms of this potential: L.J. (8,4), L.J. (v,2), etc. The Lennard-Jones model gives a rather good agreement with the experiment in the T.D. case for instance, and enables a rather real and simple representation on the interactions between nonpolar spherical molecules. Attempts to specialize this model have not led to the improvement of the agreement between theory and experiment. From (5) it ensues that the inversion and Sutherland models represent special cases of Lennard-Jones model: for vf = 00 we have the inversion model; for v = 00 we have the Sutherland one.

v = 2v' [24].

F. Buckingham Model

It introduces a potential of this form :

c CI q ( r ) = b exp ( - a r ) - - - - r6 rg (7)

with 4 parameters a, b, c , c' (Fig. 1 f ) . This function includes the induced-dipole- induced dipole interaction, the induced-dipole-induced-quadrupole interaction, and approximates the repulsive contribution to the potential by an exponential term. The Buckinghani model is somewhat more realistic than the L.J. (12,6) one, but more complicated from the mathematical point of view. It was especially employed in computations relating to the equation of state and less for the transport coefficients computations.

1) In this work when speaking about L.J. (12,6), L.J. (8,4) and L.J. (v,2) we refer to the interaction potential energy between the molecules; when referring to intermolecular force we have L.J. (13,7), L.J. (9,5), L.J. (v ,3) , L.J. (OO,~).


G. Modified Buckingham or exp (- 6)-Model

The L.J. (12,6) model is rather used to study the transport phenomena though the obtained results with this model are not always satisfactory. The fact that this potential does not wholly correspond is pointed out by the lack of a perfect agreement between theory and experiment even in caae of rare gases. The arbi- trary choice of v, constitutes an ampirical approximation so that we cannot expect a too high accuracy for greater range of r. There are theoretical and experimental data which show that the repulsive term is more suitable if it has an exponential form [25--311 Thus a potential of this form has been chosen :

y ( r ) = ___ 6 a [ - exp { a ( 1 - - :)}- (:r] for r z rmax

1 - -

where r, is the value of r for which q( r ) is a minimum; E - is the depth of the potential well; a - is it parameter which may be considered as a measure of the steepness of the repulsive part of the curve and rmX - is the separation distance between molecules for which the y ( r ) potential has a false maximum and then approaches - DC) as r -+ 0 (Fig. 2).

Fig. 2. The Buckingham potential; the magnitude of p (rmar) against the s is much reduced in the figwe

8 G. VASARU

The choice of the conditions for this model was determined by the fact that the maximum occurs only at high energies and thus it has not a too great influence on the ordinary thermal collisions. The form of this potential has a theoretical justi- fication in the fact that the (rm/r)6 term represents the predominant term in the expression for the attractive London dispersion potential and the exponential form of the first term from (8) is indicated by the quantum mechanical computation of the repulsive potential between molecules. The E , r,, and a parameters may be determined for a particular case by the comparison of transport coefficients experimentally observed and of second order virial coefficients with the theoretical values [24]. The potential (8) is known in the literature as Buckingham modified potential or in short exp( - 6)-potential. This potential has been more and more utilized in the computation of the transport coefficients.

2.3. Thermal Diffusion Effect

One of the simplest systems to determine T.D. effect is that, made of two reservoirs, one of V , volume maintained at T, temperature and the other of V , volume maintained at T, temperature (T, > T,) which communicate by a capillary tube. The gaseous mixture to be studied is introduced into the two reservoirs; since they communicate with each other, the pressure remains constant. In the presence of the temperature gradient which is maintained in gaseous mixtures: there occurs a separation of the components, the maximum value of this separation being obtained by attaining the steady state. In this state the transport of matter between the two reservoirs is zero, the effect being equilibrated by the opposite effect of the diffusion of concentration. If we want now to increase the T.D. effect the connection between the two reservoirs must be broken off and the content of the reservoir which comprises the diluted gas in the interesting isotope must also be evacuated. Then we expand the gas from the other reservoir and it is again subjected to the separation process until the attaining of the steady state. The successive operation of separation, evacuation and expanding are repeated several times. The diffusion equation which allows for T.D. has the form

- - v1 - 21, = - - Dl, [grad x, - kT grad In TI.

Xl xz

Since in the steady state 5, - 5, = 0 the equation (9) becomes

or grad x, = kT grad In T

Vx, = kT V In T.

(9)

In the above relations v represents the convection velocity of a certain component of the mixture; x - the molar fraction of the components; D,, - the ordinary diffusion coefficient (of concentration) ; kT - thermal dieusion ratio. If - in the steady state - the molar fraction of the component 1 in V , reservoir is (x,), and in V 2 reservoir is (x,), then the concentration difference between the


two reservoirs is given by the integral of the equation (11):

(12) T2 d X , = ( x , ) ~ - (x,) , = kT In - . Tl

This difference is called separation. The equation (12) is frequently employed to determine the T.D. ratio. From the experimental point of view it is more convenient to express the equation (11) depending on the ratio of the molar fractions, x,/x,, that is

(13) X 1

x2 V In - = (aT),,V In T

where (aT),, or simply dcT represents the thermal difiusion factor or the thermal digusion constant of species 1 and 2. The connection between kT and aT is given by re relation

kT = aTXlX2. (14)

The integration of the equation (13) leads to

where ( x , / x , ) ~ , (x,/x,), respectively, represent the ratio of the molar fractions in V , reservoir maintained a t T 2 temperature, and in V , reservoir maintained at T , temperature, respectively ; q is called separation factor. The equation (15) is frequently used to determine experimentally the T.D. factor. The equation (9) may also be written in the form:

where DT is called thermal digusion coefficient. The connection between DT, kT and aT is given by the relation

Unlike the ordinary diffusion coefficient D,, or simply D which in the first approximation is independent of x1x2, molar fractions, the T.D. coefficient DT is proportional with x lx2 product. That is why aT, the T.D. factor, was introduced. This factor is independent of pressure and in case of isotopes in the fist approximation it is also independent of the molar fractions of isotopic mixture components. For simple molecular models aT is independent of temperature, but experimentally and in case of complex molecular models there appears a variation with the temperature of this factor. Explicitly the 0 1 ~ expression consists in the ratio of an infinite determinant towards one of its minor, the elements of these determinants being collision integrals within the speed space of two molecules. The expression becomes simpler in case of isotopic mixtures, due to the fact that one may assume the force fields of all molecules as being identical, as they are determined only by the electronic con- figuration not by the mass of the nucleus.

10 G. V ~ S A R U

Another introduced quantity is the so called thermal separation ratio - or the ratio of the thermal digusion constant RT, defined as the ratio value between kT (exp) or 0 1 ~ (exp) calculated from the separation value experimentally measured (or of the same quantity calculated for a certain special molecular model) and the theoretical value corresponding to the first Chapman-Cowling approximation for rigid elastic sphere model, that is

kT (exp. or theor.) 0 1 ~ (exp. or theor.) (18) - _ RT (exp. or theor.) = -

LkT (r.e.s*)ll LOIT (r.e.s.)11

index 1 from the denominator indicates the first order approximation. Hence RT is a measure of the proximity degree of a rigid elastic sphere or in other words of this interaction's ,,rigidity" degree. We have a ,,strong" interaction when RT approaches unity and a ,,weak" one when RT is small or even negative. Since [aT (r. e. s.)ll is independent of temperature it follows that RT is simply proportional with ccT (exp or theor). RT was inserted to obtain a quantity to vary somewhat with temperature and concentration. The aT quantity being however independent of concentration is more frequently used. Since the first approximation for kT in case of rigid elastic sphere calculated through the Chapman-Cowling method differs from that calculated by Kihara method, the k; quantity called reduced thermal digusion ratio which is a function only of the reduced temperature T* = ~ T / E is sometimes introduced instead of

The relation between RT and k; is given by [lo] RT-

105 118

k; = - RT.

2.3.1. Collision Integrals

To compute T.D. factor for a certain molecular model (hence for a given ~ ( r ) ) it is necessary to know the collision integrals for the respective model. These integrals which are in general dependent on temperature, appear in the expressions of transport coefficients in general and of thermal diffusion in particular. They may be regarded as collision cross-section mediated within the domain of relative velocities of interacting molecules. Their evaluation is possible only in the case the molecular interaction law being known or assumed. Out of the collision integrals computed for various molecular models for the satisfactory description of the T. D. factor's behaviour, are of great interest the integrals calculated for those molecules which exert an attraction a t a great distance and a repulsion a t a small distance (the Lennard-Jones and modified Buckingham models). For the collisions between molecules of 1,2 types these integrals are defined by the relation [7]

m


where m

& ( I ) (9) = 2 n J ( 1 - COSI x) bd 6 , 0

W

d r

y = (Lp 2 kT

In the above relation k is the constant of Boltzmann, p - the reduced mass defined through the relation

1 1 1 P mi mi --- - +--,

y - the reduced relative initial velocity of the colliding molecules; Q(l ) (9) - the collision cross-section; x - the deflection angle of the colliding molecules in the system of coordinates of the centre of gravity; b - the collision parameter (the distance between the molecule of mi mass and the motion direction of molecule of mi mass in respect to the mi mass molecule, when this is outside the interaction zone) ; g - the initial relative speed of the molecules ; r - the distance between the molecules a t a given moment; r, - the minimum separation distance between molecules; Z,s - small whole numbers. In the case we allow for collisions between molecules of the same type (i,i) the quantities above will be marked with (i,i) index. Thus we have Qh!, vi; , etc. In the computation of transport properties for isotopic mixtures there only occm collisions of (i,i) and (j,j) types, respectively. In this case these indices are no longer necessary for the quantities above, but we also must take into account the necessity of using the reduced mass. For the molecules of rigid elastic sphere type of 0 diameter the relations (20) and (21) are reduced to [7]

For this simple model, the cross-sections are irrespective of g. All the presented potential functions may be characterized by two parameters, CT and E ; hence they may be written under the implicit form

where f (r /o) may be assumed to be the same function for all substances.

12 G. VXSARU

For convenience, in the numeric computations the dimensionless collision integrals l N ~ s ) 8 are defined so that for molecules of rigid elastic spheres type they should be equal to unity. Consequently we define the reduced variables (dimensionless)

r* = rlo

b* = b/o v* = TIE T* = ~ T / E g* = (p/2 & ) ‘ I s g The division of Q(ls)(T) and Q(Z)(g) by the corresponding values of rigid elastic sphere model (25), (26) leads to

VI : - the reduced intermolecular distance ; - the reduced collision parameter; - the reduced intermolecular potential; - the reduced temperature; - the reduced relative kinetic energy.

The physical sense of these quantities consists in the fact that L e y show the deviation of a certain molecular model from the idealized model of rigid elastic spheres. Using the reduced quantities, the relations (20) - (22) become :

m

m

In the computation of the first approximation of kT there occur the collision integrals Q(Zs2) and ~’2(~$~) (s = 1, 2, 3) for gaseous mixtures and in the second order approximation for pure gases there appear integrals as follows a2J), Q(1.8) and SZ(3,3)

(s = 1, 2, 3, 4, 5 and t = 2, 3, 4)


In the expression of the T.D. factor there appear the following three combinations of ab)* :

which are very close to unity. For some molecular models both the reduced collision integrals Q(7,s)* and the coefficients A*, B*, G* are calculated [7, 21-22].

2.3.2. T.D. Factor. CHAPMAN-COWLING and KIHARA Approximations

Chapman-Enskog theory is applied for gases in which: a) the field of molecular force presents a spherical symmetry; b) the collisions of the third and higher orders have a negligible frequency; c) the collisions are elastic ; d) the intermolecular interactions are effectuated accordhg to classical mecha-

These restrictions are not important for the majority of gases at normal pressures though for hydrogen and helium quantum corrections are necessary. According to this theory the T.D. factor in the nth order approximation may be formally expressed by the ratios of some infinite determinants whose elements are the coefficients of some linear equations. Thus for a gaseous binary mixture we have

nics.

where xl, x2 are the molar fractions of the two components of the gaseous mixture with the molecular weights of M,, M , respectively (M1 = ml/(ml + m,), M , = = m2/(ml + m,) ) M = (ml - m,)/(m, + m,) = M , - M,) ) A(n) represents a determinant of (2n + l ) t h order whose general term is aii, the index i , j varying from - n to + n including zero. The aii elements are functions complicated by the molar fractions, molecular weights and collision integrals. The computation method of aii elements versus the collision integrals was elaborated by CHAPMAN and COWLING [32]. The expressions for all aij for n = 2, n = 3, respectively, were given by MASON [lo, 211. There are two approximation methods by which we can provide expressions rapidly become convergent for T.D. factor from the formal relation (36): CHAPMAN- COWLING [32] and KIHARA methods [33]. The various orders of Chapman-Cowling approximation correspond to the succesive replacement of the infinite determinants with finite determinants of an in-

14 G. V ~ S A R U

creasing order. The convergence of the resulting series can be established but the convergence velocity considerably depends on the nature of the observed gaseous mixture and can be evaluated only through detailed numerical computations

The complexity of the calculations grows with the increase of the approximation order to such an extent, that we are obliged to limit ourselves to the first or the second term. The Kihara approximation method is mathematically less justified than the Chap- man-Cowling one, but it has a better physical ground and yields simpler expressions a t least in the early approximations. On the other hand it may easier be extended to approximations of higher order. The Kihara method is in fact a perturbation method which takes as starting point a particular interaction potential (corresponding to the Maxwellian model) for which the infinite equations system can be accuratelly solved, in finite terms. For this model all the determinant elements ail are zero unless I i I = I j I ; in this way the infinite expression (36) is reduced to a simple finite expression. For the Maxwellian case, the collision integrals (20) are independent of temperature, so that the derivatives of these integrals with respect to the temperature are zero. Furthermore all aii for any molecular model can be expressed considering the irreducible system of the collision integrals Q(lJ), Q(2,2), . . . , Q(lJ ) and their derivatives with respect to the temperature, making use of the recursion formula

r211.

1211

Under these conditions it was established that the offdiagonal elements aii are proportional only with the derivatives of Qz, z ) and are hence zero for the Maxwellian case. Thus the first approximation to the transport coefficients for a Non-Maxwellian model is obtained by neglecting all the Q(zJ) derivatives. The second approximation is obtained by neglecting the second and higher derivatives, as well as the third or higher powers of the first derivative. The method is based on the assumption that the higher derivatives of Q(zJ) get smaller. This fact is not correct for all molecular models and hence the convergence must be numerically evaluated for each particular molecular model. The nth approximation of T. D. factor for a binary mixture got by Chapman and Cowling by means of Chapman-Enskog theory is

The first approximation for 0 1 ~ is reduced to the form:

(6C* - 5) (~181- ~ 2 x 2 )

4 & I + xi&, + ~ 1 ~ 2 Q 1 2 [.dl =

where (39)

Thermal DSusion in Isotopic Gaseous Mixtures 15

M 1 - M 2 , 5 6 4 M , M,A* 12 1 5 ( M 1 + M j ( ~ - 7 B t ) + ( M , + M 2 ) 2 ( 1 1 - 5 B*)+

The expressions for S, and Q, are obtained from S, and Q1 by interchanging of the corresponding indiixs. The A*, B*, C* functions are given by the relations (33)-(35) and they are tabulated versus the reduced temperature in the work [71* For calculation purposes it is convenient that the second and higher Chapman- Cowling approximation be put under the determinant form. In the case of heavy isotopes where the relative difference of the masses M is small the equation (38) may be developed in series of powers by this quantity. We retain the first term and we get [ Z l ]

where [LU~] , , is the reduced thermal digusion factor [ZO]. This factor is independent of the mixture’s components, molecular weights and the dimensions of the molecular parameter 0.

The nth approximation of this factor may be written under the form [22]

5 DF) [ao],, = - -- 2 Dp

where D; and D; are determinants of 2n order. The connection between a0, k$ and R T is given by the relation :

(44)

hence the reduced T.D. factor introduced by CORBETT and WATSON [ZO] is identical with k$, &he reduced T.D. ratio, introduced by MASON [lo]. Development the equation (36) according to M powers, CHAPMAN gets for the first approximation the relation [34] :

[%I1 = “011 - Y M ( % - x2)lM = [.ol;M (46) where

16 G. V ~ S A R W

and

(48) 2(12B* + 5) - 3 5 - A *

= 2 5 + 2A* 16A* - 12B* + 5 5 .

A*, B* and C* having the sense indicated above. The second approximation of oc, may be expressed according to the first approximation by the relation

[.,I2 = [.,I1 [I + kol (49)

where k, has a rather complex form; for rigid elastic spheres KO = 0,061 [21], for rigid elastic spheres. The third approximation is rather complicated. The ratio [ ~ x ~ ] ~ / [ o r ~ ] ~ = 1,070.

In Kihara's scheme the first approximation for the T.D. factor, [ah],, has the form of the equation (39) with the difference that Q has now the values [35]:

P I .

To distinguish the results got by the two approximation methods, the results got through the Kihara method are primed and the results got through the Chapman- Cowling method, are unprimed. Extending the Kihara approximation method, Mason gives for the second approximation the relation :

[ah12 = [.TI1 + k;l + % * (52)

In case of heavy isotopes a& may be put down under the form of the equation (43), [a611, having the simple form [21] :

The Kihara expression may be obtained just from the Chapman-Cowling expression, taking B* = 514 which represents its value for the Maxwellian model. The second Kihara approximation has a form being similar to the equation (49)

where [moll is given by the equation (47) and kh is a corrective factor, equal to 0,072 for rigid elastic spheres [21]. Using the Kihara approximation scheme, SAXENA and PARDESHI give the following relation for the first approximation of T.D. factor for heavy isotopes [35] :


This relation is simpler than the expression of [ao]; from the equation (46) and it is useful to evaluate precisely the aT for isotopic mixtures when M is not quite small (the case of H and He isotopic mixtures).

Special cases :

When x1 + 0, the equation (36) is considerably reduced. For the first Chapman- Cowling approximation we have [36]

in the case of binary gaseous mixtures: [ ~ x T ] l = - (6C* - 5) X,lQ2,

in the case of isotopic mixtures : [aT]1 = [aoll [1 + y MI M .

in the case of isotopic mixtures: [&TI1 = [aoIl [I - y M ] I K .

(56)

(57)

(59)

When x2 - t O we have in the case of gas mixtures: [.TI1 = (6C* - 5) (58)

In Kihara's approximation we have :

For x1 - + O

for binary gas mixtures: [ah]1 = - (6C* - 5) IS,/&;, for isotopic mixtures: [a&]1 = [a;]1 [I + y ' M ] M ,

and for x2 --f 0

for binary gas mixtures [ 4 1 1 = (6 C* - 5) El/&;, for isotopic mixtures [ah], = [a;], [I - y ' M ] M .

The explicit expressions for kf and ki from the relation (52) are given in Ref. 1211.

3. Thermal Diffusion and Molecular Models

In the case of binary isotopic mixtures the expressions for the T.D. constant get simpler in comparison with those for gas mixtures with different molecular components. This simplification consists in the fact that the force fields of all the molecules may be assumed as identical. If we neglect the quanto-mechanical differences between identical and nonidentical particles we may consider an isotopic mixture as one with two species of particles which are identical excepting the fact that the particles from the first species are less heavy tha,n those from the second species. This fact enables the development of the general expression of aT given by Chap- man and Cowling in series of powers of M . The obtained results for various molecular models for T.D. constant are given below.

3.1. Rigid Elast ic Sphere Model

Let us consider a binary isotopic mixture formed of such molecules which behave like rigid elastic spheres, the molecules of the first species having the molecular weight M , and those of the second species having the molecular weight M 2 with Ml > M,. For this type of mixture using the expression of the intermolecular potential, corresponding to the rigid elastic sphere model, FURRY, JONES, and

2 Zeitschrift ,,Fortschritte der Physik", Heft 1

is G. VXSARU

ONsAmR [37] get the expression of d~~ for the Chapman-Cowling first approximation :

105 MI- Ma [KT11 = - = 0,89 M.

118 M I + M, (64)

The [&TI1 values versus the relative difference of the molecular diameters Z = (a, - az)/(ol + az) are represented in fig. 3. [38]. The variation range of Z is small, the curves are limited to 0 < Z < 5. In this case the T.D. constant does not depend on temperature.

1 4

2

0 0.5 t o

I - 45 $0 I-

Fig. 3. The T. D. factor against the relative difference of the molecular diameters for various concentration of the a s t component. (5, = 0; x1 = 0,5; = 1,O). The rigid elastic sphere model')

This model is too much simplified corresponding to the reality. Though owing to the simplicity of the calculations, the model is used as a reference model. Thus we can compare RT (exp) with RT (theor) obtained for a certain particular molecular model, this fact leading to an easier interpretation of the results got under various pressure conditions, concentrations and temperatures, for various isotopic mixtures.

3.2. Inverse Power Model For this model JONES and FURRY [39] gave the expression for the first approximation of aT

where

and

15 - + 6

59 f (v) 1

f ( ~ ) - ___ - -- v - 1 ( v - 1 ) 2

C(v) = - 21

3(v - 1) I A f ) Ac,)' [ f(v) = 1 -

1) In all the figures the commas of the values of coordinates axis or of the figures explana- tions wiU be replaced by points.


The numerical values of the A$'), A t ) collision integrals as well as the numerical values of C (v) and (6/5) f (v) (useful to compute the diffusion coefficients) calculated by Chapman for various values of v are given in the table 1.

Table 1 Quantities for the inverse power model

3 0.796 4 5 0.4220 6 7 0.3855 8 9 0.3808

10 11 0.3835 12 13 14 15 0.3931 00 0.5000

1.584 1.327 1.592 1.575

0.6541 1.2918 1.5501 1.514

0.5349 1.2335 1.4802 1.454

0.4956 1.1930 1.4316 1.412

0.4778 1.1631 1.3957 1.382 1.370 1.359

0.4642 1.1428 1.3498 0.5000 1.0000 1.2000

0.807 0.809 0.8156 0.828 0.8431 0.854 0.8648 0.874 0.8823 0.890 0.896 0.901 0.9064 1 .oooo

Here v refers to its value from the expression of the intermolecular force law for this model.

We notice that for v = 00 the relation (65) is reduced to (64). If v = 5, aT = 0; for v > 5, aT > 0 and for v < 5, aT < 0. It means that aT = 0 for the Maxwellian molecules and the T.D. factor changes sign for v = 5.

4 4

5 to ti Y-

Fig. 4. The T.D. factor against Y for various values of M and various concentration of the first component (z, = 0,O; z1 = 0,5; z1 = 1,O) for the inverse power model

For most of isotopic mixtures, aT is positive. Calculated values of T.D. factor for various M are given in fig. 4. It may be seen that grows with the increase of v and that this factor also depends on the relative difference of the masses, its variation is small for small M .

2*

20 G. VASARU

The rigid elastic spheres model may be considered as a particular case of the inversion model, if we assume that v is 00 for collisions of (l,l), (2,2) ans (1,2) type. In case of gas mixtures C ( v ) varies from 0,8 to 0,9 for values of v between 5 and 15. The T. D. constant ratio given by the inverse power model relative to corresponding value for the rigid elastic spheres model is roughly determined by the (v - 5) / (v - 1) ratio. The dependence of RT (theor.) on v in the case of isotoDic mixtures for the

Y 1 = 0,s I M

I

Fig. 5. The thermal separation ratio against Y for various M and for x, = 0,5 for the inverse power model

I

inverse power modelis givenin fig. 5. These curves may be used together with the experimental values of RT to determine v for the studied mixture. In table 2 the values of the reduced T.D. factor for higher approximations are given for the inverse power model, divided by the value of the first Chapman-Cowling approximation for binary mixtures with heavy isotopes [22]. This ratio is independent of the temperature. One can see that both approximations yield accurate results up till 1 percent ; the second Kihara approximation is nearly as precise as the first Chapman-Cowling approximation but for the case of very ,,rigid“ molecules. The equation (65) enables to evaluate v from the state equation of the gas or from the variation of the gas transport coefficient with the temperature, especially of the viscosity. Thus, Lord Rayleigh has established - using dimensional arguments - that the viscosity coefficient 7 varies with T’L where n is connected with v by the

Table 2 Reduced T.D. factors in high order approximations for a gas which follows the inverse power

model

~ ~ o l m / [ ~ o l l “oln&/[~o11 v (CHAPMAN-COWLING) (KIHARA)

m = 2 m = 3 m = l m = 2

00 1.061 1.070 1.054 1.072 12 1.038 1.042 1.031 1.043 10 1.034 1.037 1.027 1.037 8 1.027 1.030 1.021 1.030 6 1.017 1.019 1.013 1.019 4 1.000 1 .ooo 1 .ooo 1.000


relation,

This fact was also confirmed by the Chapman-Enskog theory. The insertion of the relation (68) into (65) enables to obtain aT depending on n ; so that we can predict aT value from the variation of the viscosity with the temperature. Expressing RT versus aT values given for the inverse power model (relation (65)) and that for rigid elastic spheres model (relation (64)) we have :

v - 5 RT = - C(V) v - I

or allowing for (68) 2 n 3 . 3 2 n - 1

RT = 2(1 - n) C (p) N 1.7 (1 - 12).

The RT values calculated using the relation (70) for various values of n and v are presented in table 3 [40].

Table 3 The variation of RT versus n for the inverse power model

n

1.1 1.075 1.05 1.025 1.0 0.975 0.950 0.925 0.9

4.33 4.48 4.65 4.81 5 5.21 5.44 5.71 6

- 0.159 - 0.120 - 0.081 - 0.040

0 0.041 0.082 0.124 0.166

0.85 0.8 0.75 0.7 0.675 0.65 0.6 0.55 0.5

6.71 7.67 9

11 12.42 14.32 21 41 0

0.252 0.340 0.432 0.529 0.580 0.632 0.74 0.86 1.00

Using the relation (70) BROWN [all calculated R T for a certain number of gases. The obtained data are presented in table 4 [40]. From table 4 we can draw some conclusions relating to the choice of the type of the gas mixture to obtain a certain isotope. Thus to separate the nitrogen isotopes, the use of gaseous nitrogen is more effective than the use of NH,, NO or NO,. To separate oxygen isotopes, CO is more advantageous than 0, allowing for the fact that it has a bigger RT and a higher molecular weight. The neon isotopes can be easier separated; in case of xenon there arise difficulties as this gas approaches the Maxwellian gas. The data from the table 4 which refer to cases with small values of RT are not utili- zable for practical purposes as for these cases the inverse power model is not pro- per to. The relations (64), (65) present the T.D. factor as to be independent of temperature and pressure. The independence of temperature is due to the over-simpli-

Table 4 The n and RT (n) values given for the inverse power model and for Sutherland constant G

221 0.89 333 0.80 423 0.76 498 0.71 979 0.62

48 0.644 123 0.646 210 0.652 333 0.669 423 0.667

1022 0.645

244 0.668 333 0.657 423 0.644 498 0.644

1030 0.623

347 0.92 425 0.91 525 0.83

330 0.833 475 0.985 293-523 -1.0

420 0.92 482 1.04 293-523 -1.0

294-524 -1.0

333 0.775 398 0.735 448 0.731 498 0.655

333 0.666 398 0.694 448 0.667 498 0.679

333 0.737 398 0.713 448 0.687 498 0.645 291-573 -1.0

308 0.79 498 0.64

325 0.97 349-551 -0.87

0.18 0.34 0.41 0.51 0.70

0.64 0.64 0.63 0.59 0.59 0.64

0.59 0.62 0.64 0.64 0.69

0.13 0.15 0.29

0.28 0.02

N O

- 0.06

N O

-0

0.13

0.38 0.46 0.47 0.62

0.60 0.54 0.60 0.57 0.46 0.50 0.55 0.64

-0 0.36 0.66 0.05

-0.22

138 147 142 130 140

6 21 37 66 83

173

34 61 70 82

128

250 303 254

162 496

-400

301 590

-350

-400

126 122 132 91

66 96 90

108 102 107 103 85

-550 128 121

277 -255


Table 4 (continued)

23

CH4 308 348 398 448 498

C2H4 308 498

CO, 325 525

0.860 0.825 0.795 0.769 0.728

0.958 0.801 0.92 0.85

0.23 0.30 0.35 0.40 0.48 0.07 0.34 0.13 0.25

174 168 166 163 146 260 213 240 281

GO 347 0.72 0.49 96 450 0.69 0.55 103 525 0.63 0.68 98

SO2 287-472 -1.0 N O N 425

fication of the inverse model and is not conf3rmed by experiment. The independence of pressure appears as a result of the theory and may be expected to hold to the same high degree that the coefficient of viscosity is independent of the pressure. NIER [42, 431 showed that we could expect an increase with the temperature of the T.D. factor, because together with the decrease of the distance between molecules their rigidity increases too. This tendency is obvious in the viscosity data presented in table 4 where n diminishes and RT grows with the increase of the temperature. Nier has also established that within the range of temperature between 140-450°K the T.D. factor for neon varies approximately with the square root of the absolute temperature; see table 5.

Table 5 atp and RT determined experimentally for the 20Ne-22Ne system

283 617 0.0302 f 2% 0.71

90 195 0.0165 & 8% 0.39 90 294 0.0188 * 2% 0.44

A more detailed discussion related to the theoretical and experimental results, for neon, shall be done within the next chapter. Though, the inverse power model leads to a good agreement with the experiment, it is not suitable for the interpretation of the T.D. data, particulary, of the variation with temperature of the T.D. constant. The T.D. phenomenon is much more sensitive with the molecular interaction types, the sign and value of aT depending essentially on the nature of the intermolecular forces. In this case the suitable approximations for treating the free path phenomena are not sufficient. Rough information on this variation may be obtained assuming that v vanes with temperature.

24 G. VLSARU

The inverse power model may be considered valid for high temperatures where the attraction forces may be neglected. But there are cases when the attraction forces cannot be neglected; then, other molecular models which take into account all these forces are used.

3.3. Square-Well Molecular Po ten t i a l Model

Using the relation -

3.75 M ( C - 1) ( A + 1) A (2A - B + 2.75) O(T =

HOLLERAN and HULBURT have calculated the thermal separation ratio, R T (theor.), for isotopic mixtures in case of the square - well molecular potential model versus l / R and l/T* [44]. The results are given in table 6 together with A , B and C values, used in the (71) relation for this model.

Table 6 A, B, C and RT quantities for the square-well model

\ l P * 0.2 0.4 0.6 0.8 1 .o 1.2 2.0 11R \ A 0.0 0.5870 0.4 0.4110 0.6 0.4031 0.8 0.4031

B 0.0 0.7778 0.4 0.6258 0.6 0.6178 0.8 0.6122

C 0.0 0.6603 0.4 1.1388 0.6 1.1637 0.8 1.1799

R T (equation (71))

0.4 0.6817 0.6 0.8176 0.8 0.8963

0.0 - 1.230

0.6457 0.6751 0.4281 0.4480 0.4148 0.4278 0.4114 0.4192

0.8226 0.8475 0.6556 0.6832 0.6367 0.6548 0.6235 0.6295

0.6893 0.7310 1.0857 1.0552 1.1357 1.1203 1.1716 1.1702

1,037 - 0.8648 0.4084 0.2539 0.6628 0.5737 0.8414 0.8209

0.6930 0.7008 0.4651 0.4791 0.4399 0.4501 0.4247 0.4280

0.8517 0.8508 0.7011 0.7153 0.6672 0.6765 0.6303 0.6295

0.7633 0.7842 1.0371 1.0258 1.1129 1.1095 1.1714 1.1735

0.7438 - 0.6688 0.1657 0.1121 0.5255 0.4998 0.8163 0.8148

For 1/T* = 0 and for 1/R = 1, A = 0.4; B = 0.6; C = 1.2; R T = 1.0

0.7075 0.4908 0.4587 0.4297

0.8543 0.7266 0.6840 0.6282

0.8044 1.0187 1.1092 1.1736

0.6008 0.0800 0.4909 0.8166

0.7197 0.5255 0.4861 0.4298

0.8531 0.7556 0.6935 0.6230

0.8624 1.0206 1.1245 1.1748

0.4871 0.0828 0.5297 0.8205

The R T behaviour for this model versus 1/T* is plotted in fig. 6. One can observe that RT is always less than one and when the temperature approaches zero or infinity R T approaches one and thus aT approaches the value for rigid elastic spheres.


This fact may be considered valid for real molecules, as at these utmost temperatures, all the important deflexions involve collisions between the hard repulsive cores. The temperature a t which the RT value is minimum and if this minimum is positive or negative, depends on the exact form of the intermolecular potential.

Fig. 6. TheRp

n

- 1'

for isotopes against i/T* for various values of 1/R for the square-well model

Because the l / R values for T.D. are obviously smaller than those inferred for this model from viscosity data, it is very likely that for some gases RT is negative [44]. For these gases two temperatures will exist where RT = 0.

3.4. Suthe r l and Model

The first model which tries to follow the attractive forces is the Sutherland model. Initially this model was applied for studying the viscosity coefficient. For this model JONES and FURRY gave the following expression for the T. D. constant [45] :

( 1 - i l ( Y ) &) ( 1 + {5i,(y) + 2 OLT = 0,69 M

This relation expressed in first order terms in elk T becomes

where 391 i, (v) + 407 i, (v)

413 F { v ) =

(73)

(74)

26 G. V ~ S A R U

The quantities i, (v) , i, (Y) and F (v) were computed by ENSKOQ and JAMES and are given in table 7 [as].

Table 7 The quantities i, (v), iZ (v ) and P(v) for the Sutherland model

2 0.3333 0.2667 0.5748 3 0.2663 0.2337 0.4824 4 0.2274 0.2118 0.4240 5 0.2011 0.1956 0.3831 6 0.182 0.183 0.353 7 0.1667 0.1722 0.3275 9 0.14444 0.1566 0.2900

Here Y refers to its value from the expression of the intermoleculax force for this model

The dependence on temperature of T.D. factor is predicted by the Sutherland model. For v = 7 (i.e. a value predicted by the quantum theory of van der Waals forces), JONES gives the following expression for 0 1 ~ [40] :

C C 1 + 0,9771 - T T

C 1 + 0,9110-

I - 0,9679 - M (75) 014 = 0,69 M

l + j i T

C 1 - 0,98 - T

C 1 + 0,92 m

M 0,69M (76)

where C is a constant which appears in of the viscosity with the temperature,

given by the relation

1

the Sutherland equation for the variation

C I f r

(77)

(78) E c = i2 (v) x.

3.5. Lennard-Jones Model

L.J. ( ~ , 2 ) case

It was studied by JONES. For 0 1 ~ he gives the following expression:

Thermal DifFusion in Isotopic Gaseous Mixtures

where

and

f (v) is the same function which appears also in the formulae of the inverse power model (equation (68)) and c, d are constants from the equation (6). The values of B,(v), B,(v), g ( v ) and D ( v ) for various values of v are given in t,able 8 [as].

Table 8 The values Bl(v), B,(v), g ( v ) and D(v)

3 -0.812 5 -0.331 7 -0.173 9 -0.077

11 -0.016 15 0.0564 00 0.2662

- 1.825 1 .ooo - 0.7244 0.589 - 0.4137 0.299 - 0.2474 0.126 - 0.1430 0.025 - 0.0266 - 0.081

0.3506 - 0.266

D (4

1 .ooo 0.653 0.376 0.218 0.125 0.088

- 0.216

Here v refers to its value from the expression of the intermolecular force for this model.

The equation (79) is correct only for the first order in

C

d21b-1) (2 ~T)(Y-~)/(Y-U *

The equation (79) predicts aT as independent of temperature for a value of v slightly less than 15; for sma.ller values of v, aT grows with the decreasing temperature. The expression for RT in this particular case may be put down under the form

The C[v) function from this relation is given in table 1 and H(v) in table 9 respectively [46].

Table 9 The values of H (v)

V 3 6 7 9 11 15 00 ~ ~~

H ( v ) 0.000 0.833 0.557 0.293 0.120 - 0.0822 . - 0.4822

Here v refers to its value from the expression of the intermolecular force for this model

28 G. V ~ S A R U

L.J. (8,4) case

It was studied by HAss6 and COOK [47] ; for this particular models these authors computed the values of the collision cross sections only by pure numerical methods. For L. J. (8,4) model, JONES gives the following expression for the thermal separation ratio [as] :

R T = 0.43241 [1 + 0.4026 q - 0.1392 q' - 0.1616 q3 + ***I =

& = 0.43241 1 + 0.8052 - - 0.5568- - 1.2930 (&)+ ...] (83) [ (ksT)'I. k T

where

The relation (83) is valid for small values of q. For rather high q values another expression for R T is obtained, given by:

1 (I

R T = - 0.03390 r l / a - 0.37935 - + - * *

The numerical values of R T calculated versus &/kT are given in table 10 [46].

Table 10 RT values depending on temperature

1lP ElkT RT 1lP &/kT RT

co 0.0 0.43241 0.8 0.3906 0.1617 10 0.0025 0.44898 0.6 0.6944 - 0.153 5 0.01 0.46304 0.5 1 - 0.384 4 0.0156 0.4687 0.4 1.5625 - 0.573 3 0.0277 0.4753 0.3333 2.25 - 0.629 2,4 0.0434 0.4771 0.25 4 - 0.465 2 0.0625 0.4753 0.2 6.25 - 0.232 1,6 0.0977 0.4605 0.1 25 - 0.040 1,2 0.1736 0.4036 0.05 100 - 0.023 1 0.25 0.3243 0 00 0

The comparison between data obtained for R T by means of the relations (83) and (85) and the accurate result is ilustrated in fig. 7. One can notice the rapid decrease of R T with increasing (&/kT)'Ia from the left side of this figure and the high negative values of the same parameter as the temperature decreases. The behaviour is rather unexpected and not confirmed experimentally. The theoretical results may be interpreted if they are expressed in dependence on the critical temperature, T,, of the observed gas. Since the critical temperature is


the temperature at which the attractive forces overcome the repulsive forces of the thermal agitation it may be yielded approximately by the relation:

kT, = E . (86)

For L.J. (12.6) model, LENNARD-JONES and DEVONSHIRE have shown that T, can be expressed by the relation [48] :

(87) E To = 1.22 -. k

Using the relation (87) to interpret the data from figure 7 we establish that RT is positive for temperatures greater than 1.54 T, and negative for temperatures less than 1.54 T,. The practical conclusion which results from this fact is that at the atmospheric pressure one can work with gases at temperatures greater than 0.6 T,.

Fig. 7. The dependence ofRF on &/kT for the L. J. (8.4) model. The short curves represent the results obtained on the approximative formulae (83) and (85)

The variation of [ 0 1 ~ ] ~ / 1 M , with k T / e for this model and for the L.J. (12,6) model, is given in fig. 8 [38]. The ~ T / E quantity is roughly equivalent to TIT,. Hence at the same temperature, TIT,, 0 1 ~ has the same value for all isotopic mixtures being in this way independent of interaction forces constants d and c. The supplementary conditions for this fact are: firstly, d be equal to c ; and secondly, the relative difference of the M masses must be small. The analysis of the variation of the dependence of mT on temperature shows that in case when the temperature decreases starting from high values, it grows slightly as much as the temperature does not approach a value of about 10 T, after that 0 1 ~ decreases quickly and becomes negative at about a temperature of 1.5 T, and reaching a maximum negative value at -0.5 T,.

30 G. VLSARU

Any attempt to separate isotopes by means of column which is operated a t an average temperature of 1.54 T, will lead to poor results, because the value for the corresponding T.D. term in the Furry-Jones theory, H , in this case is almost zero, as a result of the change of sign of the T.D. factor. There is, however, a sufficient temperature range, between the condensation temperature and 1.54 T,.

Fig. 8. The dependence of [arll /M t e r m s log kT/e for M < 0 , l in case of L.J. (12,6), (curve B) and L.J. (8.4),

I n case there are difficulties or it is quite impossible to operate the column within temperature ranges in which O L ~ is positive, it is practically possible to operate the column in a such way that it utilizes the negative portion of the curve from the

It can be shown by a simple quantitative evaluation that such a behaviour of aT and RT is described rather satisfactorily by a repulsive- attractive molecular model for which v = 2v’ - 1. Since v’ = 4 was too small some attempts were made for v’ = 6 and v = 12. The available experimental data (which are discussed in the next chapter) cannot constitute a support for the prediction of L.J. (8,4) model as regarding the positive values of RT. For the high negative values predicted by the model no corresponding experimental data are known. It seems that the L.J. model of the form (v,2) or (8,4) is not acceptable for the explanation of the experimental data - and there are still necessary measurements of T. D. constants to be made for isotopic mixtures, in particular for temperatures in the neighbourhood of the critical temperature. JONES auggests the necessity of studying the theoretical previsions for other L.J. models of (v,6) type. The next stage begins at the same time with the introduction of the L.J. (v,6) model, particularily of the L.J. (12,6) model and of the modified Buckingham model.

L.J. (12,6) case

HIRSCHFELDER, BIRD and SPoTz [49] have examined the case in which the potential energy of the interaction between two molecules situated at the distance r is given by the relation (6).

(curve A) models

fig. 7.


In the case of heavy isotopes the T.D. ratio, kT, is given by the relation:

M X I X 2 . 15 (2A* + 5) (6C* - 5)

[kT1l = 2A* (16A* - 12B* + 55)

A quantity frequently used in the literature is k; (the ratio of the reduced T.D.). It is given in the &st approximation by the relation [7]

[ICT11 - 59 (2A* + 5) (6C* - 5) -- "%% = [kT (r.e.s.)l1 7 A*(16A* - 12B*+ 55)' (89)

These expressions are valid for the values of M _I 15, [SO]. The values of k; versus T* calculated by equation (89) was tabulated in table 11, [Y].

Table 11 [k$,l1 function for the L.J. (12.6) potential

kT (L.J. (12.6)) kT (r.e.8.) [GI1 =

T* Ck$I;l, T* [g11

0.30 0.086 0.40 - 0.002 0.50 -0.048 0.60 -0.063 0.70 -0.057 0.80 -0.039 0.90 -0.010 1 .oo 0.019 1.10 0.047 1.20 0.080 1.30 0.108 1.40 0.141 1.50 0.169 1.60 0.197

1.65 0.211 1.75 0.230 1.85 0.253 I .95 0.277 2.1 0.305 2,3 0.337 2.5 0.370 2.7 0.397 2.9 0.420 3.1 0.443 3.3 0.461 3.5 0.475 3.7 0.489 3.9 0.497

4.0 0.507 4.2 0.516 4.4 0.525 4.6 0.533 4.8 0.543 5.0 0.547 7.0 0.591 9.0 0.607

20.0 0.625 40.0 0.625 60.0 0.622 80.0 0.621

100.0 0.619 300.0 0.612 400.0 0.611

From this table one can see that for this potential we have two inversion tempera. ture for T* = 0.4 and for T* = 0.95. The data of this table may enable the calculation of the first approximation of the T. D. constant, [&T]1, for isotopic mixtures. In table 12 we find the R T value for the L. J. (12,6) model for the case the molecules having the same value of CT (for equal abundances (molar fractions) of the components, x1 = x2 = 0.5) [38]. The value of may be obtained from RT by multiplying it by 0.89M for M -+ 0, by 0.381 when M = 0.5 and by 0.536 when M = 1. In fig. 9 we find the variation of [L%T]~ with log kTI& for various values of M , for x1 = 0.5 for L.J. (12,6) model [38]. A dependence on temperature of [&T]1, was found by JONES for the case of the L.J. (8,4) model; this dependence appears now in case of L.J. (12,6) model to exist too but not to be the same. For the L.J. (12,6) model the value k T / & for

32 G. VLsmu

Table 12 R T for the L. J. (12.6) potential

kTIe M + 0 = 0.5 M = 1,0 kT/e M + 0 M = 0.5 M = 1.0

0.3 0.086 - 0.102 5.0 0.547 0.567 0.607 0.4 - 0.002 - - 0.003 6 0.578 0.600 0.643 0.5 - 0.048 - - 0.058 8 0.604 0.627 0.674 0.65 - 0.063 - 0.067 - 0.076 10 0.616 0.640 0.689 0.8 - 0.032 - 0.041 - 0.046 20 0.625 0.650 0.702 1.0 0.019 0.020 0.022 30 0.627 0.653 0.707 1.25 0.099 0.103 0.113 40 0.625 0.652 0.706 1.60 0.197 0.206 0.223 50 0.623 0.650 0.705 2.00 0.286 0.297 0.320 80 0.621 - 0.704 2.5 0.370 0.384 0.412 100 0.619 - 0.704 3.2 0.447 0.463 0.496 200 0.615 - 0.702 4.0 0.506 0.525 0.563 300 0.612 - 0.701

which [C%T]I changes sign and in which [&TI1 is negative, is smaller than for the L.J. (8,4) model but the values of [&TI1 are higher for the L.J. (12,6) model (see fig. 8). The agreement of the L.J. (12,6) model with the experiment is not perfect. Deviations of the point where RT = 0 were stated; similarly, the values from R T < 0 range do not correspond to each other [SO]. Using the method of the two

Fig. 9. The variation of [ q l l versus log k!Z'/s in ease of L. J. (12,6) model for various values of M at z1 = 0,5

reservoi&s,the trouble seems to be of an experimental order, connected with the difficulty of establishing the temperature a t which the measurement of aT is carried out. It is possible to break this deadlock, using the following method: if R T (theor.) = f (T ) for the model we want t o verify experimentally is known, for instance, for L.J. (12,6) model, we shall admit for the experimental data the same dependence on T . Since the relation R T (theor.) = f ( T ) is numerically given we shall look for an analytic relation which approximates the theoretic numerical dependence. The same relation for R T (exp.) is admitted. Then we calculate RT exp.) = f (T) by means of this relation. Thus we obtain in a forced

way, a good agreement between the theoretical and experimental values. Afterwards we draw up the theoretical and experimental curves on the same graph. At R T

(theor) = R T (exp) we have from the first curve the absciss T. In this case ~ l k : = TIT* and measuring T and T* we get the value of elk. Performing these operations for as many T temperatures as possible, we get ~ / k = P ( T ) . I f the L.J. (12,6) model is good, the analytic relation inferred for it, is correct and hence R T (exp.) is correctly plotted and corresponds exactly to the theoretical curve. Then ~ / k = constant and does not depend on temperature. If L.J. (12,6) model is not good, it results analogously that elk will depend on the temperature. From the appearance of the dependence of elk on temperature,


conclusions can be drawn about the exactness or the inexactness of the L.J. (12,6) model. Regarding this model we can say that it does not describe adequately enough the collision of the nonpolar spherical molecules. That is why MASON in 1954 introduced another molecular model - the modified Buckingham model - which approaches better the real model of the molecular interaction.

D. L.J. (oo ,~ ) case

It was studied by SAXENA and MASON in 1958 [51]. For this model the second approximation for the reduced T. D. factor, oco, with respect to T* was calculated; the results are given in table 13.

Table 13 The reduced T. D. factor, a, for the L. J. ( q 6 ) model in the second approximation

T* [a012 T* [a012 T* Earl12 T* [a012

0.000 0.312 0.400 0.355 1.33 0.509 6.67 0.865 0.250 0.337 0.444 0.360 2 0.616 10.00 0.900 0.267 0.339 0.500 0.366 2.50 0.670 12.50 0.913 0.286 0.342 0.571 0.375 2.86 0.710 16.67 0.925 0.308 0.345 0.667 0.387 3.33 0.747 25 0.937 0.333 0.348 0.800 0.409 4.00 0.785 50 0.947 0.364 0.352 1.000 0.446 5.00 0.826 00 0.954

The L.J. (00,6) model corresponds to the molecules of the attractive rigid sphere type and it is known under the familiar form of Sutherland potential. This model gives sometimes a good representation on the form of the interaction potential for polyatomic molecules. The data from the table 13 may serve to the calculations relating to the thermal diffusion effect of the nonpolar gases.

3.6. Modified Buckingham or exp (--)-Model

MASON used this potential to compute the second approximation of the reduced T.D. ratio, [k;], for a mixture of heavy isotopes versus T* for a = 12, 13, 14 and 15 [lo]. But these numerical data involve a little algebraical error [51]. In another work MASON and RICE have calculated the values of A* and C* for this potential defined by the relations (33) and (35) which intervene in the first approximation expression of the reduced T.D. ratio, [k:],, defined by the relation (53) in the Kihara approximation, as well as, the values of this ratio for a = 16 and a = 17, table 14 [ll]. The extension of the range of a values for these values A* and C* was dictated by the necessity of calculating [kTll for nonspherical molecules which requires higher vaIues than 15 for this parameter. For spherical molecules the variation of a from 12 to 15 is sufficient. The extrapolation of the tabulation from a = 12, 13, 14, 15 to a = 16, 17 was performed with a corresponding reduction of the accuracy (the error increases up to several percents). Calculations were also carried out for higher approximation of the reduced T.D. factor. Thus, MASON has calculated [oc0lnr for m = 1,2, 3 in the Chapman-Cowling

3 Zeitschrift ,,FortscMtte der Physik", Heft 1

34 G. V l s m u

Table 14 A* andC* functions as well as the reducedT.D. ratio, lc& (first approximation) for the exp (-6)-

potential, (a = 16; a = 17)

a = 16 a = 17

T* A* C* [GI, A* C* B$11

0.0 1.006 0.2 1.047 0.4 1.089 0.6 1.107 0.8 1.108 1.0 1.104 1.4 1.097 1.8 1.093 2.5 1.091 3.5 1.094 5 1.100 7 1.107 9 1.112

12 1.118 16 1.122 20 1.125 30 1.131 40 1.134 50 1.137 70 1.141

100 1.146

0.889 0.880 0.852 0.840 0.842 0.849 0.868 0.885 0.905 0.921 0.934 0.940 0.944 0.945 0.946 0.946 0.946 0.947 0.948 0.949 0.952

0.311 0.251 0.097 0.034 0.042 0.081 0.179 0.263 0.368 0.453 0.513 0.544 0.557 0.564 0.564 0.564 0.563 0.564 0.567 0.572 0.582

1.006 1.050 1.088 1.104 1.105 1.102 1.093 1.089 1.087 1.089 1.095 1.101 1.106 1.111 1.114 1.117 1.120 1.121 1.123 1.124 1.125

0.889 0.882 0.858 0.846 0.848 0.855 0.873 0.889 0.909 0.925 0.937 0.944 0.947 0.949 0.950 0.951 0.951 0.952 0.953 0.954 0.957

3.311 0.262 0.127 0.066 0.074 0.111 0.206 0.289 0.391 0.475 0.534 0.566 0.580 0.588 0.590 0.591 0.591 0.595 0.598 0.605 0.618

Table 15 Etigher approximations for the T.D. reduced factor, oc, for the exp (-6)-potential (a = 14)

[.elm ( CHAPMAN-COWLING)

T* m = l m = 2 m = 3 m = l m = 2

0 0.306 0.5 - 0.004 1 0.025 2 0.245 3 0.361 5 0.453

10 0.497 20 0.494

0.311 - 0.004

0.025 0.244 0.364 0.464 0.515 0.513

0.312

0.025 0.244 0.364 0.464 0.517 0.515

- 0.003 - 0.311 0.312

- 0.004 - 0.004 0.025 0.025 0.251 0.244 0.371 0.365 0.466 0.465 0.511 0.517 0.507 0.515

approximation and m = 1,2 in the Kihara approximation for a = 14 [22]. The results are presented in table 15. Both second approximations seem to be satisfactory; the Kihara approximation is more prefered. The first approximation are not satisfactory at all temperatures. The values of the reduced T.D. factor, [a0I2, with respect to the reduced temperature, T*, for this potential, were calculated by SAXENA and MASON [Sl]. Both the numerical data obtained for a = 12, 13, 14 and 15 and the results for the same

Thermal Diffusion in Isotopic Gaseons Mixtures 35

T.D. factor for the L.J. (12,6) potential are presented in table 16. These data were calculated getting rid of the algebrical error introduced in the calculation of the Chapman-Cowling second approximation [lo]. The data of table 16 can serve to calculate the T.D. factor for isotopic mixtures of nonpolar gases. But they are limited to the functions of the potential energy with spherical symmetry, so that, they can be applied only to molecules which are spherical or almost spherical.

Table 16 T.D. reduced factor, a,, (second approximation) for exp(-6)-potential, (a = 12; 13; 14; 15)

and for L. J. (12,6) potential

exp (-6)-potential L.J. (12,6)

T* a = 12 a== 13 a = 14 a = 15 potential

0 0.2 0.4 0.6 0.8 190 1.4 1.8 2.0 3.0 5 7

10 14 20 30 40 50 70

100

0.312 0.205

- 0.027 - 0.104 - 0.090 - 0.045

0.059 0.148 0.184 0.310 0.409 0.439 0.451 0.448 0.436 0.422 0.414 0.407 0.403 0.400

0.312 0.226 0.016

- 0.062 - 0.051 - 0.009

0.093 0.180 0.216 0.340 0.441 0.475 0.488 0.488 0.479 0.468 0.462 0.459 0.456 0.456

0.312 0.312 0.239 0.246 0.048 0.073

-0,026 0.002 -0.016 0.010 0,025 0.050 0.123 0.147 0.208 0.231 0.244 0.267 0.365 0.388 0.465 0.490 0.501 0.525 0.517 0.540 0.519 0.545 0.515 0.544 0.507 0.538 0.502 0,538 0.500 0.539 0.501 0.542 0.506 0.549

0.312

0.000 - 0.056 - 0.034

0.017 0.124 0.216 0.253 0.391 0.501 0.545 0.572

0.582 0.584 0.582 0.580 0.578 0.580

The kinetic theory of nonspherical molecules is only at the beginning of its development so that these represented tables being presently the best up-to-date approximations. We know that the second Kihara approximation leads to values which are a little different to the third Chapman-Cowling approximation and so we can consider the Kihara approximation of this order rather accurate. But for the first approximation, the errors may increase up to 4% [24]. In fig. 10 we present the variation of [aoll for the L.J. (12,6) model and of [a0I2 for exp (-6)-model [24]. The curves show that aT is strongly dependent on temperature and it attains its maximum value at T* = ~ T / E = 10. Above this temperature aT decreases slowly, and below this temperature it decreases rapidly and may become negative. The exp (-6)-potential has threeadjusting parameters (a, E and rm) whereas the L.J. (12,6) potential has only two. Thus, using the former we can obtain a better representation of the experimental data.

3*

36 G. VASARU

Like ~ l k and r,, the parameter a must depend on the nature of the isotopic mixture. Generally, a < 15 because for a 2 15, RT is positive for any temperature. A comparison of [ k T ] 2 for the exp(-6)-potential with that corresponding to the L.J. (12,6) model, shows the great similarity between the two models : kT behaves similarly regarding its dependence on temperature, as well as the differences

from a = 12 to a = 15 and the value corresponding to the L.J. (12,6) potential, are very small. The results are similar in case of non

The intermolecular exp (- 6) -potential isotopic mixtures as well.

with three parameters (a, E , r,) proved to be able to predict by a single set of potential parameters, the properties of gases composed of spherical molecules (Ne, Ar, Kr, Xe, CH,) with fair accuracy. For gases the molecules of which deviate considerably from the spherical symmetry (N, 0, CO, CO,) a single set of potential parameters is not able to reproduce all measured properties. Since the form of the exp(-6)-potential is relatively most realistic from

Fig. 10. The variation of the reduced T. D. factor, (xo, for the L. J. (12,6) potential and exp(-6)- the physical point of wiew, and quite potential. The curve for 1,. J. (12,6) potential flexibile, the fact mentioned above is deplaced in comparison with the curves which correspond to the exp(-6)-potential with seems to indicate that the hypothesis log T* = 1 of the central intermolecular forces and

elastic intermolecular collision, made when the theory was worked out, is unsuitable t,o describe the behaviour of most of the real gases [U].

4. Theoretical and Experimental Results. Concrete Cases

4.1. Hydrogen

Hydrogen consists of three isotopes. Out of these isotopes, the tritium, T, is p- radioactive and it half life is TI,, = = 12,262 years. Both deuterium, D, and tritium, T, change chemically with the hydrogen, H, giving rise to several molecular species: H,, HD, HT, D,, DT and T,. The first isotopic mixture studied from the TD point of view was the H,-D, mixture. In 1941 HEATH, IBBS and WILD studied the variation of the T.D. ratio with the concentration of the components a t the temperature T = 327°K [52]. Both the experimental and the theoretical results for the L.J. (12,6) model are given in table 17. The agreement between the experimental values and those theoretically calculated

. is rather good, due to the fact that both diffusing components, interact following the same law.


Table 1 7 T.D. experimental and theoretic ratio calculated for L.J. (12,6) potential for H2-D, system

depending on H, concentration

% H* exp. theor. L.J. (12,6)

10 20 30 40 50 60 70 80 90

1.45 2.65 3.56 4.16 4.32 4.16 3.62 2.81 1.66

1.48 2.66 3.54 4.12 4.34 4.22 3.76 2.91 1.67

For x1 = x, = 0.5 in the temperature range of 288-373"K, HEATH and coo. found aT = 0.173, corresponding to a RT = 0.61. GREW'S experiments of the same year led him to the conclusion that in the temperature range between 700°K and 90°K the T.D. constant did not depend on temperature [53]. In 1947, MURPHEY [54] determined experimentally the values of aT for the same molecular system, for some temperature ranges, the average temperature being calculated with the relation worked out by BROWN [55]

where T, > T,. The results are given in table [18]. The comparison of aT (exp.) = 0.149 determined in the inversion model :

(90)

this work by aT (calc.) for

for Y = 11, aT = 0.148

for v = 13, aT = 0.135

shows that a good agreement for v = 11 is obtained. The experiments performed on a deuterium sample with a content of 1 % HD in the temperature range 194"-292"K gave a T.D. factor, aT = 0.061. In the case of rigid elastic spheres we have aT (r.e.s.) = 0.127, the corresponding value of RT is 0.48. It may be noticed that this values of RT is in good agreement with the corresponding value of RT from table 18. The data of the table show only a small variation of aT with temperature. DE TROYER, VAN ITTERBEECK and RIETVELD [56] performed in 1951 some T.D. experiments a t very low temperatures (the temperature of the liquid oxygen and of the liquid hydrogen respectively) so that determining the separation from the viscosity variation. Thus, the temperature of the hot reservoir was maintained a t T, = 293 OK and that of the cold one, a t T, = 90.2 OK, 16.1 OK, respectively. It was noticed that in the domain 16.1 OK < T, < 90.2 OK, the separation has smaller values than accepted, attaining the maximum value at T, = 25"K, then

38 G. V ~ S A R U

Table 18 ' Y ~ and RT experimental values for the Ha-D, system depending on temperature

::i;r} 0.132 0.44

0.140 0.143 0.48

20 77 194 118 0.0194 19.3 77 194 118 0.0193 20 194 292 237 0.0095 0.146 19.3 194 295 239 0.0091 19.1 194 295 239 0.0050 0.144 20 273 359 312 0.0066 0.150 19.3 273 - 361 314 0.0064 0.148 } 0.50

1 Table 19

0 1 ~ and RT (exp.) depending on temperature compared to the theoretical values for L.J. (12,6) potential in case of H,-D, system with equal concentrations of the components

TIOKI 293- 90 80 60 40 30 20

RT (exp.) 0.69 0.61 0.43 0.22 0.08 -0.102 'YT 0.190 0.167 0.119 0.061 0.021 - 0.028

RT (L.J.(12,6)) 0.62-0.45 0.38 0.24 0.10 0.02 - 0.06

it decreases. The separation at 16.1 O K is smaller than at 20°K. Hence, there follows a decrease of the T.D. constant with the decrease of the temperature, as well as, a change of sign. Table 19 contains the experimental values of aT and R T and the R T values calculated for the L.J. (12,6) model, for the H,-D, system with equal concentrations of its components. The values of ciT from table 19 were obtained from the curves which represent the dependence of the separation factor on In T,/T, got by DE TROYER and coo. It ensues that the experimentally RT values at low temperature are much higher than those calculated for the L.J. (12,6) model. On the other hand, the temperature at which RT changes its sign coincides well enough with the theoretical value. In this case, the inversion temperature of the T.D. factor is T* = 0.95. In 1954 GREW, JOHNSON and NEAL performed T.D. measurements, extending the temperature range up to 15"K, investigating a H,-D, mixture with 59.7% D,, and up to 20°K for a H,-D, mixture with 49.3% D,. The T.D. factor values were determined from the equation of the curve of the separation factor, using the method of the least squares for processing the experimental data. The variation of RT versus log T* is plotted in fig. 11 and of aT in fig. 12. [57, 241. The results indicate a change of sign of the T.D. factor, being in agreement with the results of DE TROYER and coo., but there is also a great discrepancy among the absolute values of the T.D. factor. GREW and coo., give in the work [57] the value ciT = 0.153 for a H,-D, mixture at 3OO0K, whereas DE TROYER gives for the same mixture of the same composition, aT = 0.190. Even the values obtained by HEATH and coo. (1941) and GREW (1941) are higher that aT = 0.153 (aT = 0.168; 0.174, respectively) but the temperature range in which the isotopic mixture was studied, was ranging between 90-700°K. The experimental data are shown in table 20.


The results from group 2 agree rather well with those from group 1, a fact that collfirms the value LYT = 0.153 for a H,-D, mixture with equal proportions of the components. That agrees well both with the results obtained by MURPHEY and those of W A L D ~ from measurements performed on the diffusion thermal-effect (Fig. 11) [58].

I Fig. 11. The variation of Rr with temperature. The Fig. 12. The variation of T.D. factor, ap, with tem-

dashed curve represents the variation of Rr perature for the H,-DI (49,3% D,) BYBtem. for the L.J. (12,6)model; the solid curve- the The solid curve - experimental ValUeB; the experimental values: O-MURPEEY’S values; dashed curve - theoretical values for the O-WALDXANN’S value exp(-6)-model for a = 14; v/k = 373°K

Table 20 The T.D. factor values for the H,-D, system at 300°K

1 49.7 0.153 49.3 0.153

2 20.5 0.148 39.0 0.155 41.4 0.153 56.2 0.157 79.0 0.168

A comparison of the experimentally values for the T.D. factor with the theoretical values obtained for the L.J. model may be easily made in the case of an isotopic mixture. Theoretically, aT is determined as a function of TIE; experimentally it is found as a function of T. If the model is satisfactory we must be able to super- pose the experimental curve of aYT depending on log T on the theoretical curve aP depending on log k TIE by a displacement along the axis of temperature, the great- ness of which is given by k l ~ . The result of such superposition for a theoretical curve, calculated for L.J. (12,6), is given in fig. 11 where RT was presented instead of uT.

40 G. VKSARU

The represented numerical values in fig. 1 1 are given in table 21. Though the L.J. (12,6) model takes into consideration - in a general form - the variation of the T.D. factor with temperature, including also the change of sign, the model is not entirely satisfactory. At low temperatures, RT is more strongly negative than predicted by the model and here the L.J. (8,4) model indeed gives a better representation. But a t high temperatures there is a great discrepancy ; the magnitude of aT and RT is firstly determined by the index v and the results show that the obtained value for v = 12 is too high.

Table 21 The RT experimental values for H,-D, system

(49,7% D,); aT (r. e . 5.) = 0,276

T [OK] 18.5 29.3 46.5 73.6 116.7 185 293 465 RT -0.43 -0.018 0.23 0.37 0.48 0.53 0.55 0.55

These conclusions are in good agreement with EVETT and MARGENAU’S theoretical results, who calculated the forces between two hydrogen molecules [59]. The total interaction of the molecules can be regarded as a superposition of three main components: the first one, consists of exchange forces which appear as a result of the interpenetration of the electronic clouds; the second one, is a force given by the quadrupole moment; the third one, consists of van der Waals attractive forces which vary inversely ratio to the 6th power of the separation distance, r , of the molecules. These forces depend on the relative orien- tation of the molecules; in order to compare the theoretical results with the experiment, EVETT and MARGENAU have taken a suitable average over all possible orientations. The interaction energy was computed for different values of v always considering v’ = 6, the values of CJ and E being those given by the theoretical curves of EVETT and MARCENAU. The results show that the cases which approximate best the theoretical curve are those with v = 8 and v = 9. The fact that the T.D. measurements show that v = 12 is too great, is in good agreement with MARCENAU’S results. The comparison shows that the L.J. model for hydrogen is a satisfactory approximation in a considerably domain of the distance between molecules. The comparison of aT or RT for the Buckingham model with the experimental values for the H,-D, mixture, shows that the best agreement is obtained when 12 < a < 13. The curve which is in the best agreement with the theoretical curve for the intermolecular energy obtained by EVETT and MARCENAU is that one with a = 12, [57]. MASON and RICE [60] have calculated the first and second approximations in the Chapman-Cowling approximation scheme, and the first approximation in the Kihara scheme for being calculated with the relation (90). To compare the theoretically results with the experimental results, these authors used the experimental data of HEATH and coo. [52]. The results are given in table 22. The data are calculated for the indicated temperature using exp (- 6)-model. One may observe that the agreement between theory and experiment is good and the approximations of a high order improve this agreement ; the Kihara first approximation yields as good results as the complicated Chapman-Cowling second approximation.

= 327°K)


Table 22 The comparison of the T.D. factors’ experimental values O L ~ for the H,-D, system with those calculated for the exp (- 6)-model in the CHAPMAN-COWLING (CC) approximation and K ~ R A

(K) approximation versus H, concentration

41

10 20 30 40 50 60 70 80 90

0.1491 0.1507 0.1525 0.1544 0.1565 0.1588 0.1612 0.1638 0.1665

0.1552 0.1567 0.1584 0.1602 0.1622 0.1643 0.1666 0.1691 0.1718

0.1579 0.1590 0.1602 0.1614 0.1628 0.1643 0.1658 0.1675 0.1693

0.161 0.160 0.170 0.173 0.173 0.173 0.172 0.176 0.184

The theoretical and experimental researches continued during the following years on other isotopic mixtures of hydrogen too. Thus in 1961 SCHIRDEWAHN, KLEMM and WALDMANN performed some measurements of the T.D. factor using for this purpose a hot wire T.D. column [61]. They operated at atmospheric pressure within the temperature range of 293-503 OK.

The purpose of the work was to study the influence of the mass distribution in the molecules of the systems HT and DT in D, and HT in H,. Due to the great difference in the moments of inertia of the isotopic molecules, the hydrogen is well suited to these temperaturesin such studies. In the chosen pressure and temperature range, the dependence of T.D. factor on temperature and pressure is not essential. The measurements performed with the D,-HT isobaric system in which - according to the theory - T.D. should not exist, pointed out the existence of a strong T.D. effect. The results of these measurements on the systems above as well as on the D,-H, and DH-H, systems may be explained if one introduces a T.D. factor under the form of a linear function of the relative difference of the masses and of the moments of inertia. The supplementary term which describes the dependence of aT on the moment of inertia, has the same order of magnitude as the term which describes the dependence on the relative difference of the masses. In many cases these terms may be joined together in one unit which contains the relative difference of the masses, this fact explaining the unobservance of the depencedence of the T.D. factor on the mass distribution. If we take as a reference system the D,-H, one, we can determine the T.D. factor, Q-, from the separation factor of a column. Using for the 80% H,-20% D, mixture the value aT = 0.15 given by MURPHEY [54], from the value of the separation factor obtained experimentally, SCHIRDE- WAHN and coo. calculated aT for various systems and obtained:

IT (HT-H,) = 0.113 &2.5%

(DT-D,) = 0.0422 & 5%

aT (D2-HT) = 0.0284 f 6%.

(91)

(92)

(93)

42 G. V L s a ~ u

For the calculation of T.D. factors for isotopic mixtures of hydrogen, the WALD- MANN relation may be used [62] :

where m is the mass of the molecule and 0 the moment of inertia of the molecule. This equation which decomposes 0 1 ~ in a translation part and in a rotation one, is an enlargement of the usual equation of FURRY, JONES and ONSAGER [37]

o(T = 0.89 RTM. (95)

Neither the equation (91) nor (92) allow for the concentration effect so that both equations represent rough approximations. From (94) we have

c m Ce CXT (DT-Dz) = - + -

9 11 (944

Taking the value of aT (Dz-H,) = 0.15 and aT (D,-HT) = 0.028 we can compute the values of C,,, and Ce by means of the relations (94a) and (94b). We get

Cm = 0.25, C8 = 0.20. (96)

In table 23 the experimental and theoretical T.D. factors for the isotopic mixtures of hydrogen are given. All these values refer to aT (D,-H,) = 0.15. LEMARECRAL using a hot wire T.D. column having 1 m length with rl = 1.65 cm rp = 0.5 mm and p = 450-1150 mmHg, the measurements being performed

Table 23 Theoretical and experimental T.D. factors for various hydrogen isotopic mixtures

aT (CHAPMAN)

System Ref. Basic gas alp (exp.) alp cf. (94) alp of. (95) [34]

D,--H, [54] H, (20% DJ 0.15 0.15 0.15 0.15

HT-Ha H, 0,11 0.12 0.15 0.15 DT-D, D2 0,042 0.046 0.050 0.050

Da-HT D2 0.028 0.028 0 0

HD-H, [63] H,(2-8% HD) 0.064-0.068 0.078 0.090 0.091

Thermal DifFusion in Isotopic Gaseous Mixtures 43

every 5 mm Hg, finds [64]

Using the values of SCHIRDEWAHN and coo. [Sl],

~ c T (HT-H,) == 0.113 k0.007 LEMARECHAL finds

LYT (HD-H,) = 0.075 & 0.003

in comparison with the values given by SCHIRDEWAHN,

Cm CB OLT(HDLH~) z= - + - = 0.078 5 7

with C, = 0.25 and CB = 0.20 calculated starting from the values of aT (H,-D,) and aT (D,-HT). If Cm and CB are calculated by means of aT (HD-D,) and aT (HT-H2) we get

C,,, = 0.017 and Cs = 0.28 and thus

OIT (HD-H,) = 0.074.

REICHENBACHER and KLEMM have determined experimentally the values of the T.D. constant for the T,-H,, DT-H2 and T,-D, systems thus completing the

Table 24 Theoretical and experimental T.D. factors for various systems of hydrogen isotopic mixtures

aT (cdc.) Concen-

System tration a p (exp.) Ref. cf. (103a) cf. (103b) cf. (104a) cf. (104b)

T2--H2

DT-H2

D2-H2

T2 traces DT traces

20% D2

HT-H2

HD-H2

T2-D2

DT-D2

D2-HT

HT traces HD traces

traces DT traces HT traces

T2

~ ~~

0.210 5 0.007 0.186 & 0.007 0.149 reference value 0.113 f 0.003 0.076 f 0.003 0.074 f 0.004 0.042 & 0.002 0.028 f 0.002

_ _ _ ~ ~ ~

[65] 0.214 0.215 0.214

[65] 0.179 0.179 0.180

[54] 0.143 0.143 0.142

0.215

0.180

0.142

[61] 0.115 0.115 0.116

[64l 0.073 0.073 0.074

[S5] 0.086 0.085 0.086

[61] 0.043 0.043 0.043

[ S I ] 0.028 0.028 0.029

0.115

0,074

0.085

0.043

0.028

44 G. VXSARU

results of both SCHIRDEWAHN and coo. and MURPHEY and LEMARECHAL. The measurements were performed by means of a T.D. column analogous to that one from the work [61] in the range 293-403°K. Quantum effects are considered negligible in this temperature range. The experimental value determined by REICHENBACHER and KLEMM for the T.D. factor of the systems mentioned above, as well as the values for the D,-H, systems determined by MURPHEY, and of HT-H,, DT-D, and D,-HT determined by SCHIRDEWARN and coo. and of HD-H, determined by LEMARECHAL are given in table 24 [65]. Moreover the theoretical values calculated by different approximation formulas given by REICHENBACHER and KLEMM, are presented. Theoretically, O L ~ was calculated by means of the relations

A m3 + 0.012 __ + 0.003 m3

5 m A 0 = 0.225 __ + 0.203 -

m 0

Am A 0 A m3 m 0 m3

[&TI3 = 0.228 __ + 0.197 - + 0.017 - -

(103 a)

(103b)

(104a)

(104b)

where Am = m, - m,, m = m, -+ m,, m, = mA + m,, m2 = mB $. mD, Am, = m, - mc, Am, = mB - mD, the molecules of the 1 type being composed of the A and C atoms and the molecules of the type 2 of the B and D atoms. The expression of the moments of inertia 0, and 0, as well as those of A @ / @ = = (0, - 0,)/(0, $. 0,) are functions of the distance between the nuclei, m and the ratios Amlm Am;lm:, Amilm; and are explicitely given in the work [65]. Here are also tabulated the numerical values of A mlm, A m:lm:, A milmi, A @ / @ , A m3/m3 and (Amlm) (Am:lm; + Amilmt) which are necessary to calculate the T.D. factors for all mentioned systems. From the table it ensues that by using the third approximation (the relation 103 b and 104b) there is not obtained any essential improvement in the reproduction of the experimental values of aT in comparison with the use of the second approximation (the relations 103a and 104a). All these approximations yield values which slightly differ from the experimental values. It ensues that the deviation does not proceed from the fact that the relative mass differences with hydrogen are too high but from lack of precision of the experimental values. The fact that the relations (103a) and (104a) reproduce rather well the experimental values, enables to conclude that the influence of the mass distribution in the molecule on the behaviour of the molecule at T.D., may be approximaterly


described if instead of the mass distribution we introduce the excentricity of the centre of gravity or the moment of inertia of the molecule. SAXENA and PARDESHI [36] have performed a series of theoretical computations for the isotopic T.D. factor, using for this purpose some expressions which contain expansion series both by the relative difference of the masses, M , and by the ratio of the masses M = M,/M,, applying to this case the modSed BUCKINQHAM potential, with parameters given by MASON and RICE, [SO]. The results are given in tables 25-30. Table 25 presents the experimental data of HEATH and coo. [52], as well as the theoretic results, which were obtained using different approximation formulas for O L ~ , both in the Chapman-Cowling and in the Kihara scheme, for T = 316.4"K. For this system the formulas including terms of reduced mass, M, are prefered to those including terms of M'. It was established that the formulas of Chapman-

Table 25 The comparison of the T.D. experimental factors with the theoretical ones calculated in the

CHAPMAN-COWLINQ and KIHARA approximations, for the H,-D, system at 316,4"K

CHAPMAN-COWLINQ approximation KIHARA approximat,ion

yo D, af (exp.) [aTI1 cf. (46) [aTI1 cf.(39) [aTI2 cf. (38) [a&]2mixture cf. (52)

100 90 80 70 60 50 40 30 20 10 0

- 0.161 0.166 0.170 0.173 0.173 0.173 0.172 0.176 0.187 -

0.155 0.157 0.159 0.161 0.163 0.164 0.166 0.168 0.170 0.172 0.174

0.147 0.149 0.149 0.152 0.154 0.156 0.158 0.161 0.163 0.166 0.169

0.153

0.156 -

- - 0.162 - - - 0.169 0.174

0.157 0.158 0.159 0.160 0.161 0.162 0.164 0.166 0.167 0.169 0.171

0.155 0.157 0.157 0.160 0.161 0.163 0.165 0.167 0.169 0.172 0.174

Table 26 The comparison of the experimental T.D. factors with the theoretical ones calculated in

various approximations for H,-D, system

29.3 - 0.0050 - 0.0058 - 0.0059 0.0008 46.5 0.063 0.027 0.028 0.024 73.6 0.102 0.076 0.079 0.076

118.7 0.132 0.117 0.122 0.119 185 0.146 0.143 0.149 0.147 293 0.152 0.155 0.161 0.162 465 0.152 0.158 0.164 0.166

1) The experimental values refer to a binary mixture with 49,7% D, and 50.3% H,

46 G. VLsmu

Table 27 The comparison of the theoretical T.D. factors calculated for the H,-D, system with D, in

traces (xl --t 0) in CHAPMAN-COWLINO and KIEARA approximations

CHAPMAN -COWLING approximation KIHARA approximation

cf. (56) cf. (57) cf. (8) of. (9) cf. (60) of. (61) mixture cf. (18) T [OK1 [.TI1 [.TI1 [“TI2 ra!r12 r4?1 [ah11 [47L [.$I2

r361 [361 ~361 185 0.156 0.161 0.159 0.162 0.157 0.161 0.159 0.160 316 0.169 0.174 0.174 0.178 0.171 0.176 0.175 0.175 465 0.171 0.176 0.177 0.176 0.173 0.177 0.177 0.178 900 0.168 0.172 0.173 0.173 0.170 0.174 0.174 0.175

Table 28 The comparison of the theoretical T.D. factors calculated for the H,-D, system with H, in

traces (x, -+ 0) in CHAPMAN-COWLING and KIHAKA approximations

CHAPMAN - COWLING approximation

cf. (68) cf. (59) cf. (14) [36] cf. (62) of. (63) cf. (21) [36]

KIEARA approximation

rK1 [‘%‘I1 lad, [.TI2 [41, r4r12 [4r1,

185 0.134 0.141 0.139 0.143 0.150 0.140 316 0.147 0.155 0.153 0.157 0.163 0.155 465 0.149 0.156 0.156 0.158 0.165 0.158 900 0.147 0.155 0.144 0.156 0.162 0.156

Table 29 T.D. factors for various isotopic systems of hydrogen depending on temperature; -A

approximation

T [“K] OLT (HD-H,) o ~ T (HD-D,) C X ~ (HT-T,)

185 0.0956 0.0681 0.0957 316 0.1035 0.0747 0.1037 465 0.1050 0.0758 0.1043 900 0.1051 0.1122 0.1020

Table 30 The comparison of theoretical T.D. factors, calculated for the H,-T, system with T, in

traces in CHAPMAN-COWLING and KIHARA approximations ~ ~ ~~ ~ ~ ~~~

CHAPMAN-COWLINO approximation KIHARA approximation

cf (56) cf. (10) cf. (8) of. (11) cf. (60) cf. (19) mixture cf. (20) rK1 “%‘I1 [ a d [“TI, [.TI2 r411 [.$I1 [a$], [41,

~361 r361 r361 D6l 13.361 185 0.232 0.238 0.237 0.255 0.233 0.240 0.237 0.238 465 0.255 0.261 0.263 0.284 0.256 0.264 0.264 0.265 900 0.251 0.257 0.269 0.278 0.252 0.260 0.259 0.260


Cowling approximation have a slight convergence and that the second approximation is 4% higher. The relation (46) is rather exact and should be prefered for the approximation calculations, both from the point of view of simplicity and precision. The Kihara approximation yields better results and a satisfactory convergence. The difference between the first approximation and the second approximation is small, but this increases when one of the mixture components is in trace concentration. Even in this case it seems that the probable error which results by neglecting the higher approximations is smaller than the errors made in the experimental determinations. The relation for [a$]1 explicitly given in the Kihara approximation by (39) with Qi and Q;, given by (50) and (51), respectively, is rather good for most of the operations at this temperature and has the advantage to be simpler than the relation (52) for [a$],. The agreement between theory and experiment is only approximative. In general the theoretical values are smaller than the experimental ones. The discrepancy might be due to the errors in determining the experimental data or to the fact that the intermolecular potential exp (- 6) does not strictly correspond to the real situation. In table 26 besides the experimental data for aT obtained by GREW and coo. [57] for the H,-D, system (an equimolar mixture) depending on the temperature, are presented as well as the theoretical results obtained from various approximation formulas for the exp (- 6)-potential. In case of low temperatures the numerical values of the T.D. factor classically computed, contain some errors owing to the neglect of the quantum effects. Hence some corrections must be performed. In case of high temperatures the difference between the first approximation, [a&]l, - relation (39) in the Kihara scheme - and the second approximation - relation (52) - is small. Here the agreement between the numerical values of [&,$I2 and the experimental data is reasonable. The differences for high temperatures are far greater than the errors in the experimental determination; this fact supports the postulate that the exp (-6)-potential is not strictly adequable to represent the real potential energy between two hydrogen molecules. We can observe that the values for [&TI1 are rather different in comparison with the values for [ah]l. The calculations confirm the validity of the [ah]l formula for all approximative problems. The errors included because of the neglect of the higher approximation are between 2-3%; for computation purposes the relation for is to be prefered. SAXENA and PARDESHI have also analysed the case of one of the components of the isotopic mixtures being in trace concentration. The results for H,-D, system relating t o t,he variation of aT with the temperature are given in tables 27 and 28. The explicit expressions for the equation noted in this table with 8, 9, 14, 18, 21, respectively, are not given in the present work; they are to be found in the work [36] at these numbers. Table 27 contains the rigorous values of [aT], and [4], as well as the values obtained with simpler expressions in terms of the reduced mass. It ensues that the relation (9) and (18) from the work [36] prove to be rather adequated from the precision point of view. In this case the results of the Kihara approximation should be prefered.

48 G. VLsnxn

In case that hydrogen is in trace concentration, the data from the table 28 are preferentially valid for the results of the Kihara approximation. Table 29 contains the values of o ( ~ dependent on temperature for the H,-HD, HD-D, and HT-T, systems in which the heavy component in trace concentration, being computed using the Kihara approximation scheme. For the second approximation one uses for this purpose a relation expressed in terms of reduced mass. The results for the H2-T, system with T, in trace concentration are given in table 30. The equation 8,10,11,19,20, respectively, used in the computations of the results in table 30 are not given in the present work. They are to be found in the work [36] at the same numbers. In the H,-T, case we prefer the formula which contains the M' ratio of the molecular masses. The analysis of the data from table 30, especially, that of the rigorous values of 0 1 ~ for the first and second approximation, shows that the convergence for both approximation schemes is of the same order, but less than in the case of the H,-D, mixture. For the latter case the Kihara approximation is preferable, too. For the H,-HT system we may apply the results obtained for the H,-D, system for both systems, have identical molecular weights and intermolecular forces. The system H,-T, with H, in the trace concentration hat not been calculated, this system heaving a small practical importance. Hydrogen-water vapour system. This system is of great interest both from the point of view of its equilibrium and nonequilibrium properties due to certain aspects connected with the heat transfer, and also from the point of view of isotope separation, because of the possibility to separate hydrogen isotopes by T.D. combined with the isotopic exchange. Saxena performed the computation of the T.D. factor for various combinations of both stable isotopic hydrogen gases and isotopic water vapours in the temperature range of 307-350°K for both extremities of the concentration region, using

Table 31 Theoretical T.D. factors calculated for various combinations of hydrogen stable isotopes and

water vapours at T = 307 "K and T = 350 "K

ocT a t 300°K aT a t 350°K

System x1 = 0 2, = 0 x1 = 0 xz = 0

HZ-HZO HZ-HDO HZ-DZO '€€D-HZO HD-HDO HD-DZO DZ-HZO DZ-HDO DS-DZO

1.193 1.206 1.217 1.078 1.104 1.112 0.970 0.993 1.012

0.812 0.812 0.813 0.807 0.808 0.809 0.803 0.803 0.804

1.290 1.303 1.316 1.167 1.187 1.203 1.052 1.076 1.097

0.812 0.812 0.813 0.807 0.808 0.809 0.803 0.803 0.804

In this table xl represents the heavy component and x2 the light component


for this purpose a relation which is independent of the molecular potential [66]. This relation is valid only for molecules with spherical symmetry. The results obtained for the T.D. factor a t 300 "K and 350 "K for various systems are given in table 31. Though the data of the table contain a certain error due to the fact that in these cases the ratio of the molecular masses is far higher than 0.1 nevertheless they may be considered pretty accurate (the used relation which is valid for the ratio of the molecular masses g 0.1 yields an error of 1-2%.) These data of aT are useful to the interpretation of the data obtained by the enrichment of the hydrogen isotopes, using the exchange reaction HD + H,O + e HDO + H, in a T.D. column.

4.2. Helium

Helium consists of three isotopes. From the T.D. point of view 3He and *He are interesting. The necessity of the enrichment of 3He in view of using it both as an instrument to study the superfluidity of liquid and to determine the nuclear properties, led to a series of studies in connection with T.D. The first determination of the T.D. factor was made by MCINTEER, ALDRICH and NIER in 1947 [67]. The previous enrichment of the isotopic mixture was made in a hot wire T.D. column which was operated at a pressure of 10 atmospheres, the temperature of wire being at 973°K. The enriched product (300 times) was then used to determine experimentally aT by the two reservoirs method. For T, = 613°K and TI = 273°K was found:

OCT == 0.059 f 0.05 (105)

a value which is considerably smaller than that assumed by JONES and FURRY (aT = 0.0758) for their 3He separation installation [40]. MORAN and WATSON performed some measurements on the T.D. factor for the 3He-4He mixtures in the temperature range of 233-571 "K [68] using for this purpose a new apparatus called swing separator, composed of 20-22 tubes interconnected by capillary tubes. The oscillation period of the gas was 15 s. This new apparatus for determining the T.D. factors was introduced by CLUSIUS and HUBER [69]. In such a system the concentration gradients are additive so that the concentration variation ( A xl),, in all the n tubes of the apparatus is [YO] :

rn

The total separation factor, Q, for a series of n tubes expressed versus the separation factor q for each tube, defined as the ratio of the quantities xl/x2 estimated a t the opposite ends of the tube, is given by [68] :

The introduction of this new device to determine the T.D. factors has led to a considerable multiplication of the elementary T.D. separation process, dropping

4 Zeitaohrift ,,Fortschritte der Physik", Heft 1

50 G. V ~ A R U

in this way, the difficult two reservoirs method which requires equilibrations, evacuations and succesive expandations of the gas and which does not enable a greater enrichment than about 5 times. In addition, the fact that it needs only a little temperature difference between the cold and hot ends, enables a more accurate determination of aT. This system needs an operation time shorter than in the case of the two reservoirs method and needs no gas manipulations during an operation. The results of the measurements of the isotopic T.D. factor, aT, for a 3He-4He mixture with 16.3% 3He are given in table 32 [68].

Table 32 Experimental T.D. factors for 3He-4He system with 16,3y0 3He, depending on temperature

203 273 233 5.83 6.20 0.49 312 361 337 2.04* 5.20 0.41 314 399 357 3.51* 5.33 0.42 333 428 384 2.85* 4.31 0.34 428 519 474 2.15* 4.20 0.33 478 570 525 1.84* 3.83 0.30 530 612 57 1 1.35* 3.45 0.27

The asterisk indicates the performing of the measurements by means a swing separator with 20 tubes, The values without asterisk by a swing separator with 22 tubes

It is interesting to reveal the rather curious behaviour of RT with the temperature : as the temperature rises, the helium atoms become "softer". The authors explain this behaviour as follows: helium has two electrons and appears very "rigid" a t low temperatures, because the maximum repulsion in this case does not depend on the superposition of the electronic clouds. At temperatures a t which the electronic clouds of helium begin to extend, repulsion actually decreases because there are no inner electronic shells which could bring it about. It ensues that R T decreases wiht the rise of temperature in the studied range (203-612 OK). The comparison of the RT values obtained from viscosity measurements are of great interest. For a gas the molecules of which are subjected to the inverse power model the viscosity is proportional to T" where n is connected to the index Y by the relation (68). In this case [all]:

RT 1.7 (1 - n) . (108)

Even when the real model is not an inverse power model, the viscosky may be expressed by CTa where n is now a number dependent on T. The comparison of the experimentally obtained results for helium with those given by the equation (108) shows that R T (calc.) with this relation (table 4) decreases slightly with the temperature: between 48°K and 423°K RT decreases from 0.64 to 0.59. MORAN and WATSON suggest that a new determination of the viscosity coefficient for 3He and 4He versus temperature should improve the agreement, as the available viscosity data were obtained with ordinary helium. SAXENA and PARDESHI have performed the calculation of the T.D. factor both in Capman-Cowling approximation and in Kihara approximation [35]. For this


purpose various approximation formulas and the intermolecular potential of L. J. (12,6) type were used. The computations were performed for a binary 3He-*He mixture with 5% 3He. The results are given in table 33.

Table 33 The comparison of theoretical T.D. factors calculated for the 3He-4He system in CHAPMAN-

COWLING and KIEARA approximations, depending on temperature

CWMAN-COWLIN~ approximation

T*

5 7 9

20 40 60 80

100 200

0.0671 0.0724 0.0744 0.0767 0.0768 0.0764 0.0763 0,0761 0.0757

0.0695 0,0751 0.0788 0.0794 0.0794 0.0791 0.0789 0.0787 0.0781

0.0677 0.0732 0.0753 0.0774 0.0774 0.0771 0.0769 0.0767 0.0762

0.0690 0.0747 0.0760 0.0803 0.0801 0.0799 0.0796 0,0794 0.0784

T*

5 7 9

20 40 60 80

100 200

0.0702 0.0742 0.0779 0.0802 0.0802 0.0799 0.0797 0.0795 0.0790

0.0718 0.0759 0.0797 0.0820 0.0820 0.0816 0.0814 0.0812 0.0806

0.0707 0.0747 0,0785 0,0808 0,0808 0,0805 0,0803 0,0800 0,0795

0,0694 0,0755 0,0778 0,0809 0,0808 0,0804 0,0805 0,0800 0,0797

0,0716 0.0779 0.0803 0,0832 0.0832 0,0827 0.0826 0.0829 -

The values of 0 1 ~ from columns 2 and 3 were computed by means of the relation (39) and (43), respectively, where for n = 1, [(x,,]~ is given by the relation (47). These values differ appreciably, a fact that shows that for helium, the terms which contain only the first power of the reduced mass are not sufficient [35]. That is why the introduction of the terms with higher powers of M is required. In the column 4 the values of 0 1 ~ are given, computed by means of the relation (46) which is valid for values of M up to the second power, inclusively. One can see that these values differ much enough from [&TI1 from the column 3 (about 2.4%) but are in good agreement with from column 2 (about 0.9%) The values of [aTI2 from the column 5 have been calculated by means of the equation (38) for n = 2. These values are approximately 4% higher than the values in the column 2, thus establishing that the convergence of the series is still rapid and that the errors which occur owing to the neglect of the third or a higher order approximation are insignificant. We must note the fact that the values from the column 4 are in a better agreement with [aTI2 from the column 5 than with from the column 2. It ensues that the formula (46) should be preferred to the (38) and (39) formulas.

4*

52 G. V ~ S A R W

From the qualitative point of view, the values calculated by the Kihara approximation show the same behaviour. Here the column 6 was computed with the relation (39) in which Q1 and Q12 are given by the relation (50) and (51), respectively; the column 7 was calculated with the relation (43) where [a0I1 is given for n = 1, according to the relation (53) ; the column 8 was calculated by the relation (43), where [a& is given by the relation (55). The column 9 contains the second approximation calculated by the relation (38) for n = 2 ; the column 10 contains - for a comparison - the results obtained with a formula which considers the terms up to the second approximation, but it includes only the f i s t power terms of the reduced mass [23]. The numerical calculations performed by SAXENA and PARDESHI for aT establish that the Kihara approximation is better than the Chapman-Cowling approximation. The relation (55) may be successfully used for case when 3He is in trace concentration, treating the sHe-4He mixture as consisting only of the heavy isotope. The results obtained in this way are rather accurate. The experimental determination of the T.D. factor for which a swing separator with 9 tubes was used, was also made by VAN DER VALK and by DE VFZES [71]. They operated at T, = 370-700°K and T, = 295°K and at a pressure of 1 atmosphere and with a pulsation time of 13.5 s. The gas contained 10% 3He. For this temperature range they found the value

(109) which is in contradiction with the results found by MORAN and WATSON, who found an important decrease of aT a t a very high temperature in this region

The values obtained by VAN DER VALK and DE VRIES are in a very good agreement with the theoretical values for the exp (- 6)-potential, using for helium the parameters given by MASON and RICE [60], which indicate a slight decrease of o ( ~ from 0.0620 at 400°K to 0.0605 at 700°K. The measurements performed by SAXENA, KELLEY and WATSON [72] by help of a swing separator on a 3He--4He mixture with 5% 3He, led to results which prove very well the value of mT found by MORAN and WATSON, [68], for the temperature T = 238°K (aT = 0.062). This value corresponds to a. = 0.434 and may be compared with the value found by VAN DER VALK and DE VRIES [ 7 l ] given by the relation (109) to which a. = 0.4319 corresponds. Besides these experimental values, the theoretic numerical values for a0 = f (T) for the exp(-6)-model with ~ l k = 9.16 and a = 12 and 13, respectively, are

LYT = 0.0617 f 0.0019

[681.

94

A d 450 550 250 350

T"K1 -- 43615~ Fig. 13. The comparison of experimental and calculated values for the reduced T.D. factor, a,, for helium. -

experimental value, MORAN and WATSON [68 ] and SAXENA, KELLEY and WATSON [72]; the curve a corresponds to the exp(-6)-potential with a = 12; the curve b corresponds to the exp(-6)-potential with a = 13; the curve c - corresponds to an pure exponential potential (relation 110); the dashed line - the values of VAN DER VALK and DE VRIES 1711


given in fig. 13 (the curves a and b). For the L.J. (12,6) potential, a, is almost constant in this temperature range (a, = 0.58), [51]. These values are much higher than any of the experimentally measured values. The too rapid decrease of a, with temperature reported by MORAN and WATSON [68] seems - according to SAXENA and coo. [72] - to be due to the fact that they used very short tubes so that they could not ensure the maintenance a t both ends of a constant temperature on an enough great length of the tube, with a definite temperature gradient along their medium section. The remixing of the gas components gets worse as the temperature increases. And because SAXENA and coo. have obtained up to now only small values for a, for helium, with the bottom ends of the tubes maintained a t 78 O K . they consider that we must allow for the possible influence of quantum effects a t low temperatures, as well as of the effect of some inexactness in the theoretical expressions of aT given by the equation (43) in the present work and of the relation (12-14) in the work [72]. In the work [72] the consequences of these two effects are analysed extensively. In the same work [72] using diffusion and viscosity data, SAXENA and coo. found for a, calculated by means of the relation (55) combined with the relation (53), the value a, = 0,602. The calculations were performed for a mixture with 5% 3He and they are in good agreement with the experimental data. The value of a. corresponds to the temperature range of 14-296 OK.

For high temperatures the repulsive forces are much more important than the long distance attractive dispersion forces. In the special case of He where the depth of the potential well is small, this fact is more real. The repulsive force is well represented by the exponential function [73], [74] :

where A and Q are force constants. For the particular case of the helium, AMDUR and HARKNESS [75] have established

q ( r ) = 6.18. I0-l0exp (-4.55 r ) (111)

for r varying within the limits 1.27 and 2.30A. Using this potential and the values tabulated by MONCHICK [74] for A* and C* the curve c from fig. 13 was obtained. The computation of AMDUR and MASON [76] for a, by means of the simple relation

both for the exp (-6)-potential and for the inverse power model, led to the value a, = 0.33 with an error of 20% in the temperature range 1000-1500OoK. Thus ist was established that the exp (-6)-potential may represent succesfully the dependence of a. on temperature. The extension of the measurements of aT and a, respectively, in the domain of low temperatures was made by WATSON, HOWARD, MILLER and SHIFFRIN. A series of constructive improvements of the swing separator have been performed, in order to correct the results. The detailed description of this device and the way of operating are to be found in the work [77].

54 G. VLSARV

The experimental results obtained for aT and ao, respectively, using a swing separator with 4 tubes in various temperature regimes for a sHe-4He mixture with 50% 3He are given in table 34, aF being calculated with the relation (43). The compari-

Table 34 Experimental T. D. factors for the 3He-*He system with 50% *He determined by means of a

4 tubed swing separator

TI [OK] T, [OK] FpK] Ax% "T all

77 195 136 6.43 f 0,06 0.0696 0.488 f 0.011 77 273 175 8.65 f 0.11 0.0688 0.482 -J= 0.006

195 273 234 2.20 f 0.08 0.0657 0.460 f 0.024 273 344 309 1.46 f 0.09 0.0637 0.446 f 0.017 273 351 312 1.63 f 0.09 0.0651 0.456 f 0.017

son of the experimentally values for a. with the theoretical results is made in fig. 14. The data in table 34 are the first experimental values on the variation of the T.D. factor for helium at low temperatures. The extrapolation of the behaviour of OLT

from these measurements to values for higher temperatures shows a good agreement with the measurements of VAN DER VALK and DE VRIES [ 7 l ] . The fact that T.D. at low temperatures is a transport property very sensitive with respect to the law of molecular interaction, enables the calculation of the potential parameters. WATSON and coo. find for a from the expression of the exp (- 6)potential the value a = 12.7 and for &/I% the value &/k = 9.16 OK. The curves a, b and c from fig. 14 are not identical with those from fig. 13.

Fig. 14. The variation of the reduced T. D. factor, a,, with temperaturefor aHe--'Re system. o - experimental values; the curve a is computed for the exp(-6)-potential with a = 12; the curve

a = 13; the curve c is coniputed for a pure exponential potential

It may be seen by that the Corresponding Curve Of a = 12.7 gives a good agreement. But we can-

values of a0 - f (T) for 3He--4He b is computed for the exp (-6)-potential with not obtain an agreement between the

system and the curves based on a L.J. type potential.

For helium, quantum effects could not be pointed out at = I36 "K either. The authors of the work [75] intend t o perform measurements by a device which at the bottom end is to be maintained at or under 4°K. It is theoretically shown that using a mixture with 50% 3He-50y0 4He one may obtain in a single tube a mea- surable A x as the term In T,/TI from the equation (106) becomes favourable when T, is so small. In his thesis, VAN DER VALK [78] observed the theoretical values of SAXENA and PARDESHI for the L.J. (12,6) model being higher than the available experimental results and that even SAXENA and RAMAN in a more recent work [79] used the lowest value of aT given by NIER (relation (105)), [67]. This fact agrees with the


conclusion of MASON and RICE [ I l l that a model of the exp (-6)-type giving a better agreement than the L.J. (12,6) model, for the transport properties in the case of high temperatures. VAN DER VAIX has made experimental determination in a rather large temperature range, using for this purpose a mixture with approximately 10% 3He. Some results were given in the work [71] for the range of 400-700"K.

A----d----d '. 6 -,

I00 l<O 200 3$0 S,50 OK 3 -

I I I I 1 1 I I I r 1 I I I I , -

$4 'J 22 2,6 log T

Fig. 16. The variation of T.D. factor with temperature. A-A-A- computed by SAXENA ans PARDESHI for L.J. (12,6) model; 0-0-0- computed by VAN DER VALK for the exp(-6)-model; -0-0- NORAN and WATSON experimental values; - x - x - x - VAN DER VAZK experimental values

Fig. 16. The variation of q ( * H e - 'He) versus neon concentration; - computed for pure "We; x - experimental values obtained with natural neon

In the thesis [78] are presented the results of the experiments performed up to 12.7"K. In the present work they are given in fig. 15 besides those obtained by MORAN and WATSON [68]. In the same figure the values calculated by SAXENA and PARDESHI for the L.J. (12,6) model and the values calculated by VAN DEB VALK for exp (- 6)-model are presented. It may be noted that for high temperatures the exp (-6)-model with the parameters given by MASON and RICE leads to the best agreement with the experimental data.

56 G. V~SARU

There are still a series of works which have a more special purpose. LARANJEIRA in his thesis [80] studied the influence of “added” gases (helium or hydrogen) on the separation of neon and argon isotopes. For this purpose, the isotopic T.D. factors for neon and argon were determined in a wide concentration domain of the added gas. VAN DER VALK and DE VRIES [81] made experimental determinations of T.D. factor in a mixture consisting of two isotopic components and a nonisotopic component. These authors determined o ( ~ for the 3He--4He system with 10% sHe, finding in pure helium the value 0 1 ~ = 0.060 at 490°K and in a mixture with 100% Ne the value 0 1 ~ = 0.011 (cf. fig. 16). The results are compared to the theory elaborated by VAN DER VALK in the work [82] for ternary mixtures.

50 + Fig. 17. T.D. factors for 8He-HD-H, system. ap103 against mole fraction of *He; the curve a aT(HD-H,); the

curve b - ar(HSaHe). The relative concentrations of the hydrogenic molecules: I I1 I11 IV V

H2 1.0 0.75 0.5 0.25 0 HD 0.0 0.25 0.5 0.75 1

In a recent work [83] as well as in his thesis [78] VAN DER VALK extended the equations obtained for ternary mixtures in the work [82] to quaternary mixtures and calculated the T.D. factors for the systems H,-HD-3He-4He and H2-D,- --3He--4He depending on the relative concentrations, using for this purpose the exp (-6)-potential. It is considered that the CHAPMAN-ENSKOG theory (for spherical molecules without free internal degrees) may also be applied to hydrogenic molecules. The computations were performed for the temperature T = 487 OK using the following potential parameters : for helium, ~ / k = 9.16”K, r, = 3.135A, a = 12.4; for hydrogen elk = 37.3”K, r, = 3.337A, a = 14; for the H,-He system: ~ l k = 18.48”K, r, = 3.236 A, a = 13.2. The results both for ternary and quaternary systems are presented in figs. 17-29. I n fig. 30 a comparison is shown between the HELLUND’S theoretical results obtained for the case of rigid elastic spheres [84] and those for the exp(-6)-model for the H,-D, mixture with D, in trace concentration and with 4He added. It is observed that for the exp (-6)-model the decrease of 0 1 ~ is not as pronounced as in case of rigid elastic spheres.

57 Thermal Diffusion in Isotopic Gaseous Mixtures

! ' O 0 m

-20- Fig. 18. T.D. factors for the sHe-4He mixtures added

to HD-H, with fixed ratio 1 : 1 of relative HD and H. concentrations. ap. 10' against mole fraction of totalhelium. The curvesa-ap(HD- Hx); the curves b-ur(HD-'He); the curves c- ur(HD-'He). The relative concentrations of He iaotopes:

*He 1 0.75 0.5 0.25 0 -50 'He 0 0.25 0.5 0.75 1

- 40 - I I1 111 1v v

- x [Helium toh

I 0 425 45 $75 Ip - x (HD)

la Ila Illa IVa Va

Ib Ilb 111 b I Vb Vb

lc lk Ilk I vc vc

) 1

Fig. 19. T.D. factors for the systems: HD added to mixtures. UT. los depending on the

molar fraction of HD. The cuWe8 a - q('He- *He); the curves ~ F ~ ~ ( ~ H ~ - H D ) . The relative concentrations of helium isotopes:

SHe 1 0.75 0.5 0.25 0 'He 0 0.25 0.5 0.75 1

I I1 I11 IV v

58 G. V~SARU

-B - Vb lVb lllb

lib

Ib -40-

I I

-0u ' 1 8 I

--- x I Hydrogen total] 0 q25 qs 47s $0

Fig. 20. T.D. factors for the systems: Hz-HD mixtures added to *He-'He mixture a fix ratio 1 : 1 of the relative concentrations of *He and 'He. up. 10' against mol fraction of total hydrogen. The curves a - ap(*He-H.) the curves b - apCHe-HD); the curves c - G ~ ( ~ H ~ - ' H ~ ) . The relative concentrations of the hydrogenic molecules :

I I1 I11 IV v H, 1 0.75 0.5 0.25 0 HD 0 0.25 0.5 0.75 1

Fig. 21. T.D. factors for the systems: 'He added to D4--H,mixtures. ap. loa against molefraction of 'He. The curves a - ap (Dz-HJ; the curves b - ap(D,-'He). The relative concentrations of the hydrogenic molecules:

I I1 I11 IV v HI 1 0.75 0.5 0.25 0

20

>

DI 0 0.25 0.5 0.75 1 - X ( % w

Thermal Diffusion in Isotopic Gaseous Nixtures 59

Fig. 22. T. D. factors for the systems: 'He added to DI-H, mixtures. a ~ . lo* againstmolefraction of We. The curves a - ap(Ds-H,); the curves b - aT(DS-*Hc). The relative concentrations of the hydrogenic molecules:

I I1 I11 IV v €I, 1 0.75 0.50 0.25 0 D; 0 0.25 0.50 0.75 1

Ib

t 80

0 425 45 QZ5 t0 - x I Helium total)

Fig. 23. T. D. factors for the systems: 8He-4He mixturesadded toDr-Ha mixturesof flxed 1: 1 ratio of relative H, and D; concentrations. U T . 10' against mole fraction of total helium.Thecurves a - ap(DS-H;); the curves b - aT(D,-'He); the curves c - ar(Dr4He); the relative concentrations of helium isotopes:

I I1 I11 IV v 'He 1 0.75 0.5 0.25 0 'He 0 0.25 0.5 0.75 1

60 G. V~SARU

56

Fig. 24. ap(We-'He). los. Upper curve - in pure He against mole fraction of 4He as indicated on the upper horizontal axis; the lower curve - in mixtures with hydrogen for the limit that the total hydrogen concentration is near 100%; lower horizontal axis indicates mole fraction of H, added in H,-D, mixtures

Fig.

I 75: 70

65 - 60 -

40

35

30

25

20

26. ap(He-H,) and ap(HD-WHe) versus H, fraction of amount of hydrogen, showing effect of *He addition at HD-H. mixtures. The curves a - ap(%e-H,); thecnrves b - ap(HD-'He). Curve I I1 111 IV v 'He 0 0.25 0.5 0.75 1

Fig. 25. n~(~He-'He) in mixture with HrHD. The curve I - in pure He; the upper horizontal axis indicated the relative concentration of 'He. The curve I1 - with 100% H,, the lower horizontal axis indicates the relative concentration of H,

0 03 $0 ---- x (H2 total) Fig. 27. ~ P ( H D - ~ H ~ ) versus total hydrogen, showing

effect of varying 'He-"He ratios for total hydrogen - total helium ratio of 1: 1. Curves IIIb I I Ic I I I d I I Ie IIIf %(*He) 0.5 0.375 0.25 0.125 0 %('He) 0 0.125 0.25 0.375 0

105 a, 10'

1" 95

90

85

811


I I I

45 t

llb 'llc

I1 d

/le

/ I f

x (H2 total)' Fig. 28. ap(DraHe) versus H. fraction of total amount

of hydrogenshowing effect of varying %e-*He ratios for total hydrogen - total helium ratio of 1:1. Curve I I I b IIIc I I Id IIIe IIIf

61

I b I l b 111 b IVb V b

V a

IVa

111 a

I1 a la

-C x (H2 total) Fig. 29. aT(8He-H2) and aT(D,--'He) versus H. frac-

tion of total hydrogen, showing effect of addition of We to D,-H, mixtures. The curves a - ap('He-HI); the curves b - ap (DraHe). Curve I I1 I11 IV V

%(*He) 0.5 0.375 0.25 0.125 0; zPHe) 0 $('He) 0 0.125 0.25 0.375 0.6

t '75 150

125

100

75 I I I

0 425 02 qn I, - X ( 4 t k )

0.25 0.5 0.75 1.

I

Fig. 30. ap(D,-H.). lo8 for mixtures to which is added a bit of D. against mole fraction of 4He. The curve I - for rigid elastic sphere; the curve I1 - for exp(-6)-potential

62 G. V&mu

The calculations have shown that the whole separation of D, and H, by T.D. is possible, using helium as an auxiliary gas. The addition of 4He does not change too much the T.D. factor; 4He concentrates between H, and D,. For the separation we may use a column or a battery of columns‘of CLUSIUS type. The superposition zone of D, and H, (if hydrogen is used) will be replaced (in case we use helium as an auxiliary gas) by two zones in which the D,-He and He-H, systems are superposed. This fact enables D, and H, to be completely separated. It is obvious that these considerations refere to H, + D, e 2HD equilibrium. In any separation instalation meant to be operated at the average temperature,

= 487 OK, for which the computations were made, the hot part (the Wire or the heater element from the column) must have a rather high temperature, at which the equilibrium above could easily be established. In a separation installation this should enable the obtaining of the three hydrogenic molecules - H,, HD and D, - the H, and HD molecules moving towards the ‘,light” end of the installation. But there is also another possibility of HD dissociation, the created D, changing its place towards the “heavy” end of the cascade and finally, we get pure H, and D, at the upper and the lower end, of the separation installation, respectively, provided that we add enough helium. As a rule 3He will act in the same way like 4He as it is a semipermeable membrane for the hydrogenic molecules. Using 3He an additional advantage is included that causes an increase of ocT for D,-H, system, especially in mixtures rich of D,, but it is rather expensive (although it is not consumed itself by the separation process). Since HD will be concentrated between 3He and 4He we should try to use of 3He and 4He in order to confine a region of the separation installation into which HD be fed. On the other hand it may serve to separate SHe and 4He. Because of the chemical equilibrium it will be necessary a continous supply with HD, mixed, for instance, with helium from the supply circuit. For the calculation of the T.D. factors which are presented graphically in figs. 17-29 only the first approximation was used. The binary systems 4He-HT, 4He-DT and 4He-T, have been studied by SLIEKER and DE VRIES and a T.D. column has been used for this purpose, which was operated at three average temperatures: = 325”K, 355°K and 390°K [85]. The results are given in table 35 besides the theoretic values computed with a L.J. (12.6) potential in the first approximation of the Kihara scheme. Finally, the last mentioned work on helium is that of GHOZLAN and Los who have studied the influence of the diffraction effects on the T.D. factor at low temperatures

Table 35 The comparison of the experimental T.D. factors determined by means of a T.D. column with the theoretical values calculated for L.J. (12,6) potential, the first KIHARA approxima-

tion, for the HT-4He; DT-4He and T,-4He systems ~ ~~

aT (exp.) at ___ aT (theor.) at System 325 OK 355°K 390°K 355°K

HT-4He 0.0061 0.0024 0.0010 0.030 DT-4He 0.091 0.076 0.071 0.114 T,--’He 0.092 0.083 0.070 0.175

I) Computed in the first Kihara approximation.


[86]. They used the two reservoirs method, the upper reservoir being maintained at a constant temperature (the room temperature) and the lower one at 12°K. The used gas was helium with traces of tritium compounds - T,, DT and HT - as well as hydrogen with traces of HT. The measurements show a strong influence of the diffraction effects on the separation, especially at those temperatures where the de Broglie wavelength of the molecules becomes of the same order of magnitude as the diameter. There is a very pronounced difference between the T.D. factor for T,-He system (where both molecules T, and He may be supposed to have a spherical symmetry) and the T.D. factors of other mixtures, where the spherical symmetry is disturbed by the mass distribution in the molecule. The choice of He-DT, He-HT and He-T, systems for this study was dictated by the fact that they have the same value of elk and Q, (elk = 19,4"K and Q = =2.742 A) but different quantum parameters. In this case the important quantum parameter is the de Broglie reduced wavelength given by

where h - the Planck's constant, E - the depth of the interaction potential well, m - twice the reduced mass of the two molecules. For the chosen systems we have :

He-T, A* = 1.65

He-DT A* = 1.72 (114) He-HT A* = 1.88.

The comparison of the experimentally obtained results with the theory was impossible to be done, as the collision integrals for these quantum parameters have not been yet calculated.

4.3. Boron

It has 4 isotopes but from the therma.1 diffusion wiewpoint only 1OB and 1lB isotopes are important. The gaseous BF, system was studied by WATSON, BUCHANAN and ELDER in 1947 [87]. Using a T.D. hot wire column with 7 meters of length, operated at normal pressure and A T = 673 OK, the separation factor was determined at the steady state for the 10BF,-llBF, system and finding the value q = 1.32h0.12. The equilibration was attained within three days. The value of the T.D. constant for this separation factor is O L ~ = 5.0 - 10-4 what corresponds to RT = 0.06. The work was performed in order to enrich the boron compounds in loB by T.D., being well-known the special importance of this isotope in the nuclear researches.

4.4. Carbon

Carbon consists of 6 isotopes. From the T.D. viewpoint only IzC, 13C and 14C isotopes are interesting.

64 G . VhSARU

The T.D. factor for the 12C160-13C160 system was determined by DAVENPORT and WINTER in 1951 [88]. The experimental results as well as the theoretical values for the L.J. (12,6) potential are given in table 36. The method of the two reservoirs was used.

Table 36 Experimental T.D. factors for the 12C160-13C160 system determined by the two reservoirs method, depending on temperature, compared to those calculated for the L.J. (12,6) potential

TlrK] T,PK] F-["K] dz. lo4 OrT * 103 RT (exp*) RT (L.J.(12.6))

294 668 432 4.1 f0.2 11.3f0.6 295 528 392 1.7 f 0.3 10.3 f 1.8 195 423 284 1.2f0.3 5.3f1.3

296 663 432 56 f5 7.0 & 0.6 195 427 286 45 1 2 7.150.3 195 429 286 45 &2 7.1 & 0.3

293 683 435 62 f8 8.9 f 1.1

~ ~~ ~

0.72 f 0.03 0.50 0.66 & 0.12 0.47 0.34 & 0,08 0.37 0.57 -f 0.07 0.50 0.45 f 0.04 0.50 0.45 f 0.02 0.37 0.45 f 0.02 0.37

It is to be noticed that the experimental results obtained for RT(exp.) are higher than those for RT(theor.). Since the methane molecule has not a field of force with a spherical symmetry it can be expected that the CHAPMAN-ENSKOO theory is not valuable, and the fact that n (relation 68) varies between 1.028 and 0.728 in the temperature range of 91.5-522.8 "K causes the inverse power model t o be inadequate for methane. From the table 4 it ensues that for the temperature of 448°K and 498°K the values of n are 0.769, 0.728, respectively, and from the table 3 we find for RT the values 0.40, 0.49, respectively. It may be seen that these values are very much differing in comparison with the experimental values (table 37, NIER'S values).

Table 37 Experimental T.D. factors for WH,- WH, system determined by the two reservoirs method,

depending on temperature; RT (exp.) is compared to RT (oalc.) for L. J. (12,6) model

295 708 448 3.5f0.1 11 & I 0.41 5 0.01 0.45 295 708 448 3.6f0.2 11 fl 0.41 f 0.02 0.45 195 433 289 2.1f0.2 7.2f0.7 0.27 f 0.03 0.30 195 435 290 2 50 .1 6.7 f0.3 0.25 f 0.01 0.30 295 573 407 - 7.4 f 0.7 0.27 f 0.03 0.42 [42] 295 728 447 - 8.0 f 0.8 0.30 f 0.03 0.45 [42]

The equation (76) is more convenient for C = 160°K. By this value for the temperature 430°K and 512°K for RT the values 0.48 and 0.54, respectively, are obtained. In this case the agreement with the experimental data is poor, too. If we use L.J. (8,4) model, taking ~ / k given by the relation (87) and the critical temperature T, = 156°K it may be graphically established that the theoretical values of RT which correspond to that range of temperatures in which NIER performed his measurements, are 0.212 and 0.168, respectively. If we multiply these values by the correction factor 1.48, we obtain RT = 0.31 and 0.25, respectively

Another way to establish elk which enables the verification of the predictions of the L.J. (8,4) model with the experiment is that in which no use of the corrective

r 461.


factor is made. It can be shown that the value Elk = 120 OK led to a better verification, but the small value of this quantity cannot be justified. DAVENPORT and WINTER have also determined the T.D. factor for the 12CH4-13CH4 system. Their results are given in table 37 beside those determined experimentally by NIER [42]. The average temperature given in tables 36 and 37 was calculated by means of the relation

with A = 0.506 - k/& [MI. This equation is more adequate in the case of using L.J. (12.6) potential, than BROWN'S one [55], but only in the range TIE = 0.5-1.5. Above it, the equation (90) given by BROWN is preferable.

+ 401

- 406

I # A -258 O K

Fig. 31. The variation of T.D. factor, a?. against density and temperature for the 1aCOs-14C0, system

The 12C02-14C02 system was studied by CASKEY and DRICKAMER within the temperature range of 258-323°K and within the pressure range of 15-160 atm. [89]. In fig. 31 aT depending on density is presented, together with the different isotherms. It is to be noticed that aT has a threefold change of sign along an isotherm. These measurements were performed in the critical region (the critical point is Tr = 304.6"K, p = 73.6 atm.) by means of a T.D. column. The same authors have measured an isotherm a t 258°K for the CH,T-CH, system. In the fig. 32 is given the aspect of the isotherm. From fig. 32 it is obvious that aT in this case does not change sign but decreases to a value near zero for a density near the critical density (e = 0.163 g/cms). This isotherm is about 340 OK above the critical temperature. Another system studied by DE VRIES, HARING and SLOTZ [go], was 14C160-12C180. These molecules have the same symmetry but Merent inertia moments.

5 Zeitschrift .Fortachritte der Physik". Heft 1

66 G. V ~ S A R U

The measurements were performed using a T.D. column of 5 m length. The separation in 14C16O,lZCl*O and W 1 6 0 isobaric mixture was measured, the first molecule being added in tracer concentration and 1zCPO is present in its natural concentration (0.2%). In this case the ratios of the T.D. factor were determined. The results are given in table 38 [go], [91].

Fig. 32. The variation of T.D. factor, ap, against density for the CH,-CK.T aystem

Table 38 The ratio of T.D. factors for 14C160-12C160 and 12C180-12C160. systems

Samples taken after ,&(laO - 12c160) . . . days arf(12C'80 - 1 W 1 6 0 )

3 3 4 4 7 13

1.12 1.17 1.17 1.16 1.15 1.18

The authors state that aT is inversely proportional to the characteristic length of the column (the column length along which the separation factor equal to e is achieved) in steady state conditions. It ensues from these results that the separation between 14C160 and lZCl60 is not the same as between l2Cla0 and 12C160 but the differences are far smaller than those for molecules of various masses [91].


VAN DER VALK and DE VRIES who have determined aT for 14CO-12C0 system have found the value LXT = 0.020 at the temperature of 410 OK in pure CO ; aT = 0.014 in a mixture with 100% C2H4; aT = 0.014 in a mixture with 100% C2H,D,; 0 1 ~ = 0.011 in a mixture with 1OOyo C$€, [all. The variation of the T.D. factor with the concentration of the added gas (C,H4, C,H,D,, C,H,) is graphically presented in figs. 33-35. We notice that the decrease of aT for the 14CO-12C0 system is a little greater in cases we add C,H4 (fig. 33) than we add C,H2D2 (fig. 34) and the agreement between the data obtained by means of the column and those obtained by means of the swing separator, is satisfactory.

- X(cZH4)

Fig. 33. The ethylene as the third component added to carbon monoxide. a#CO-'sCO) against C,H, concentration. __ cdcu- lakd; x - experimental values obtained by the swing separator; A - experimental values obtained by the T.DTcolumn

Fig. 34. 1.2 dideutero-ethylene as the third component added to the carbon monoxide. ar(14CO-1*CO) against C&D. concentration. - calculated; x - experimental values obtained by the swing separator; v - experimental values obtained by T.D. column

t I 1

Fig. 35. Ethane as the third component added to the carbon monoxide. aT(WO-**CO) against C*H. concentration. - calculated; x - experimental values obtained by the awing separator; a - experimental values obtained by the T.D. column

68 G. VXSARV

I n figs. 36-38 we find the variation of the T.D. factors theoretically computed by VAN DER VALK for 1WH,-12CH,, 1sC0,-12C0, systems, respectively, for the pure isotopic mixture and for a mixture with 100% Ne as well as for the 13C0,- -l2C0, system for the pure isotopic mixture, and for a mixture with 100% N,. In

203 400 600 1C"KI

Fig. 36. The variation of T.D. factor against temperature for the laCCH,-laCH, system. -for pure isotopic mixture; - - - - - for a mixture with 100% Ne

- 200 400 600 lpK,

. . Fig. 37. The variation of T.D. factor against tem-

perature for the 18COr12C0, system. - for pure isotopic mixture - - - - - for a mixture with 1 0 0 ~ o Ne

the calculation of these curves the theory worked out by VAN DER VALK for the ternary mixtures was used, [78]. Recently QUINTANILLA has studied in his thesis a series of systems including carbon isotopes [92] to determine the T.D. factor. He has used a T.D. column,

6 t""'

200 400 600 T L O K ] Fig. 38. The variation of T.D. factor

against temperature for W O r 'pCO, system. ___ for the pure isotopic mixture. - - - - - for a mixture with 100% Nt

using for this purpose the relation

where F is a characteristic factor depending on the geometry and the thermal regime of the column and Q,,, being the maximum value of the separation factor. The value of F was determined for = 354°K utilizing the T.D. factor for the W e --22Ne system equal to 0.0270 given by FISCHER [93] as well as the value of In Qmx for the same isotopic system equal to 0.611, worked out by QUINTANILLA. From the relation (116) for the values above, we find

F = 0.0442. (117)

The results for the Ne-CH, system are given in tables 39-46.


Table 39 Experimental T. D. factors for 20Ne-12CH3D determined with the T. D. column

Used gas: 20Ne (99.948% ZONe; 0,.045y0 'We; 0.007% ?Ne) CH,D (1.097% 13CH3D; 98.903%l2CH3D)

69

0 /o 20 Ne Tl["K] T,["K] F["K] Qmax o(T (20N+12CH3D)

10.90 283.2 452 354.5 0.294 0.0130 35.17 284.1 452 355.1 0.342 0.0153 55.13 283.5 452 354.6 0.395 0.0176 73.77 283.1 452 354.5 0.453 0.0201 92.04 283.2 452 354.5 0.520 0.0231

Table 40 Experimental T. D. factors for 20Ne-13CH4 system determined with the T. D. column

Used gas: 20Ne (99.948% 2ONe; 0.045% 21Ne; 0.007~0 22Ne) 13CH4 (99.22% 13CH4; 0.78y0 WH,)

0 /o 20 Ne Tl m1 T2 ["KI ["K] In Qmax ~x~(*~Pu'e-l~CH,)

11.89 281.4 452 353.2 0.277 0.0121 42.93 281.2 452 353.0 0.336 0.0147 74.02 281.3 452 353.1 0.409 0.0180 92.14 281.3 452 353.1 0.487 0.0213

Table 4 1 Experimental T.D. factors for 20Ne-12CH4 system determined with the T.D. column

Used gas: 20Ne (99.942% 2oNe; 0.043% 2lNe; 0.015y0 z2Ne) lZCH4 (1.094% 13CH4; 98.906% 12CH4)

Qmax aT ( 2oEe-12CH4) 0 /o 20 Ne T,[OKI T,[OK] d r K ]

14.85 285 452 355.8 0.539 0.0243 55.25 285,3 452 355.8 0.666 0.0302 92.00 285 452 355.8 0.866 0.0391

Table 42 Experimental T.D. factors for 20Ne-12CD4 system determined with the T.D. column

Used gas: 2ONe (99.948% 22Ne; 0.045% ZlNe; 0.007~0 22Ne) 12CD4 (1.060% 13CD4; 96.287% 12CD4; 2.625% 13CHD3; 0.029% WHD,)

10.49 283,O 452 354.4 0.377 - 0.0168 30.00 283,4 452 354.7 0.411 - 0.0183 49.93 283.8 452 354.6 0.440 - 0.0197 70.16 284 452 355.1 0.500 - 0.0223 92.01 284 452 355.1 0.566 - 0.0253

70 G. VLSAFXJ

Table 43 Experimental T. D. factors for 22Ne-12CH3D system determined with T.D. column

Used gas: 22Ne (98.669% 22Ne; 0.537% 2lNe; 0.794% 2oNe) WH3D (10.97% 13CH3D; 98.903%12CH3D)

0 /o 22 Ne T,[OK] T2[OK] F["K] Qmax LXT ("Nd2CH,D) ~~

11.21 280.3 452 352.5 0.699 0.0302 40.14 280.3 452 352.5 0.817 0.0354 69.91 280.3 452 352.5 1.005 0.0437 92.02 280.3 452 352.5 1.166 0.0506

Table 44 Experiment,al T. D. factors for 22Ne-13CH4 system determined with the T. D. column

Used gas: 2zNe (98.669% 22Ne; 0.537% 21Ne; 0.794% mNe) 13CH4 (99.22% 13CH4; 0.78%12CH4)

_ _ _ _ _ _ _ ~ ~ ~ ~

0 /o 22 Ne Tl[OK] T2[OK] F[OK] In Qmax OLT ("Nd'CH,)

10.17 280.8 452 352.9 0.650 0.0283 39.79 280.8 452 352.9 0.791 0.0345 69.96 280.8 452 352.9 0.968 0.0422 92.00 280.8 452 352.9 1,137 0.0496

Table 45 Experimental T.D. factors for 22Ne--'2CH4 system determined with T.D. column

Used gas: 22Ne (99.941% 22Ne; 0.059% 21Ne; 0% 20Ne) 12CH4 (1.094% 13CH4; 98.9O6%l2CH4)

In Qmx 0 /o 22 Ne T,PK] T2[OK] F[OK] ~~ ~ ~

4.92 282.7 452 354.3 0.884 0.0393 20.32 283 452 354.3 0.964 0.0428 50.59 283 452 354.3 1.124 0.0500 79.53 282.8 452 354.1 1.390 0.0615 94.99 282.9 452 354.3 1.541 0.0682

Table 46 Experimental T.D. factors for zzNe--'2CD4, system determined with the T.D. column

Used gas: W e (99.941y0 22Ne; 0.059% 2lNe; 0% 2oNe) WD4 (96.287% WD4; 1.060% 13CDI; 2.625% 12CH,D; 0.029% 13CH3D)

In QmX 0 /o 22 Ne T,["K] T2[OK] F[OK]

10.66 282 452 353.8 0.025 0.0011 40.30 282 452 353.8 0.039 0.0017 74.19 282.5 452 354 0.059 0.0026 91.79 282.8 452 354 0.079 0.0035


If we account only the dependence of the T.D. factor on the mass, we observe that both in the results obtained with 2oNe and those obtained with 22Ne, the molecule 13CH4 behaves as to be heavier than the WH,D molecule. This fact cannot be explained by the available theories; to infer the relations which enable t,he computation of the transport coefficients, we assume that the molecules interact only by central forces, without the possibility of changing the internal energy. The only one quantity which can differ in the case of isobaric- isotopic molecules is the moment of inertia (5.3. 10-40g - cm2 for 13CH4 and 6.65. 10-40g - om2 for 12CH3D [gal). Within the limits of the experimental errors we notice that in the systems neon- isobaric-isotopic molecules, the difference between the T.D. factor is independent of concentration as well as of the mass of the gaseous neon-which has been used. Thus, it ensues that in a T.D. column the 13CH4 component will be concentrated at the bottom and 12CH3D at the top of the column, respectively. The results may be qualitatively controlated by the relations of CHAPMAN and KJHARA. For the 2oNe-12CH4, 2ONe -13CH4, 22Ne-12CH4 and 22Ne-13CH4 systems, QUINTANILLA calculates OIT by means of a relation of the type (39) using the Kihara approximation and the L.J. (12.6) potential. For this purpose he used the values elk = 59 OK and a = 2.95 A for the neon-methane system; elk = = 135.6"K and u = 3.11 A for methane and elk = 35.7"K and a = 2.789 A for neon. Both the results of the theoretical calculations obtained for these systems, for the limit concentrations, Ne = 1OOyo (that is xl = 1) and Ne = 0% (that is xl = 0), and the measured values, are given in table 47. For the rest of the systems mentioned, this comparison is not possible.

Table 47 The comparison of theoretical and experimental T.D. factors for various isotopic syetems

Ne - CH4

O(T = '%' (zl = O)

System calc. exp. calc. exp.

aoNe-12CH4 0.0425 0.0414 0.0218 0.0225 z0Ne-13CH4 0.0248 0.0224 0.0083 0.0106 2*Ne-12CH, 0.0699 0.0708 0.0430 0.0385 zzNe-1SCH4 0.0526 0.0525 0.0295 0.0262

In case of z2Ne--'2CD4 system, the T.D. effect has a minimum. The excess of two mass units of heavy neon is almost exactly compensated by the cross-section and by the moment of inertia of the deuterated methane. The author shows that the results obtained by the swing separator for the zONe- -WD, system are in perfect agreement, what confirms the hypothesis that the factor F (relation (116)) is independent of the studied gas. Investigating the 2ONe --'2CD4 system it was observed that the molecules 12CD4 concentrate at the top part of the column, while themolecules l3CD4concentrate at the bottom. It ensues from the tables that the 22Ne-13CD4 system has B positive T.D. factor within the shown operation conditions, while the 22Ne-13CD4 has a aT

72 G. VLSARU

negative and thus in a ternary mixture of 13CD4-22Ne-12CD,, 22Ne will behave as a "filter". It may be seen that 22Ne may be used to increase of the separation efficiency in a column by which we want to separate the carbon isotopes, starting from deuterated methane. QUINTANILLA has also studied in the same work the T.D. in binary mixtures of various isotopic molecules of methane. The results are given below: The 13CH4-12CH3D system. Used gases: 10.07% WH, (98.685% 13CH4; 1.315% WH,) ; 89,93y0 12CH3D (98.882% 12CH3D; 1.117y0 13CH3D); For the temperature T, = 284°K it was found:

The 13CH4--12CH, system. The results are given in table 48. From this table it ensues that for this system aT is independent of concentration. The average value :

OIT = 0.00756. (119)

Table 4 8 Experimental T.D. factors for 13CH4-12CH4 system determined with the T.D. column

1.094 285 452 356 0.168 0.00757 42.808 285 452 356 0.168 0.00757 86.292 285 452 356 0.167 0.00753

The 12CD4-12CH, system. Used gases: 12CD4 (1.060% 13CD4; 96.287% 12CD4; 0.029% 13CHD3; 2.625% 12CHD3); 12CH4 (1.094% 13CH4; 98.906% 12CH4). The results are given in table 49. From this table it ensues that aT is independent of concentration. The average value :

o ~ T = 0.0236. (120)

The 13CH3D--12CH3D system. Used gases : 1.097y0 13CH3D; 98.903% 12CH3D. The average value for the temperature T, = 280"K, for aT is

CXT = 0.00729. (121)

The 13CD4-12CD4-12CHD3 system. Used cases: 1.010% 13CD4; 91.290% 12CD,; 7.700% l2CHD,. The average values for the temperature T, = 280°K, for aT is - for the 13CD,-12CD, system:

OCT = 0.00610, (122)

- for the l2CD4--l2CHD3 system:


Table 49 Experimental T.D. factors for 12CD4-12CH, system determined with

73

the T.D. column

0 12 /o CD, T, [OK] In Qmax C X ~ ( ~ ~ C D ~ - - - ~ ~ C H , )

22.69 284.9 0.529 0.0239 45.57 284 0.525 0.0237 69.66 284.9 0.521 0.0235 90.45 284 0.519 0.0234

- for the 13CD4-12CHD3 system :

LYT = 0.0125.

These results are compared with the first Kihara approximation,

m1- m2 15(6C* - 5) m, - m, = c,

m1+ m2 "1 3. m2 [LYhll = 16A*

and with the relation (94) which also contains the moments of inertia Both relation (125) and (94) show that the T.D. factor for isotopic mixtures is independent of the concentration. The results for the 13CH4-12CH4 and 12CD4-12CH, systems indicate this fact. Using the relation (94) as well as the values given for the moments of inertia for methane by CLUSIUS [95], we get :

(126) 1

OCT (l3CH4--l2CH4) 1 Cm - 33

aT (l3CD,--l2CD4) = Cm (127)

(128)

(129)

(130)

(131)

(132)

41

1 aT(13CH3D-12CH3D) = Cm 35

4 1 LYT (l2CD4-l2CH4) = C;, - + Ce -

36 3

OLT (13CH4-12CH,D) = C, * 0 - C, * 0.106

I 39

2 40

L Y T ( ~ ~ C D ~ - ~ ~ C H D ~ ) = C, - + Ce * 0.07

OIT (l3CD,-l2CHD3) = Cm - + Ce 0.07

By means of the experimental values, the relations (126-128) enable to obtain Cm and the relations (129), (130) enable to obtain Ce, respectively. Thus we have

C, = 0,25 Ce = - 0.018. (133)

By these values the T.D. factors for the observed system were calculated and the results are given in table 50, beside the experimental values.

74 G. VXSARU

Table 50 The comparison of the experimental T.D. factors with those theoretically computed for

various methane isotopic systems

System a!r (exp.) ap (of. (94)) ap (cf. (125))

13CH,-12CH4 0.00756 0.00758 0.00758 13CD4--12CD4 0.00603 0.00610 0.00610 13CH,D-12CH,D 0.00729 0.00714 0.00714 12CD,-12CH4 0.0236 0.0217 0.0278

l2CD4-'2CHD3 0.0067 0.0051 0.0064 13CD4--12CHD, 0.0125 0.0112 0.0125

13CH4--12CH,D 0.003 -J= 0.001 0.002 0

Table 51 The comparison of the experimental and theoretical values for C, for 13CH4-12CH4 system

296-728 449 0.0080 0.264 0,424 ~421 296-573 405 0.0074 0.244 0,397 ~421 295-708 448 0.011 0.363 0,423 i?81 195-433 289 0.0072 0.238 . 0,284 L881 285-452 353 0.00756 0.250 0,352 L931

In case we allow for a L.J. (12.6) potential to be used, the C, value may be calculated by the relation

15 (6C" - 5) 16A* '

c, = (134)

Both the values obtained for the system 13CH4-12CH4 for various temperature ranges and the experimental results obtained by other authors are given in table 51. For all tables which refer to the QUINTANILLA'S work was computed by the relation (90).

4.5. Nit rogen

The nitrogen consists of 6 isotopes: From the T.D. view point only 14N and 15N are interesting. The T.D. factor for nitrogen was experimentally determined by the method of the two reservoirs by MANN in 1948 [96]. The computation of this factor was made by the relation

rn

where xi is the concentration of the heavy isotope in the cold reservoir; xl, x, - the concentration of the light isotope and the heavy one, respectively, in the initial mixture; T,, T, - the temperatures of the two reservoirs (T, > Tl);


/I = T, VA/TA V , - a corrective factor of the gas which regards the connecting capillary tube between the two reservoirs. For T, = 623"K, and TI = 195"K, and = 329°K (calculated by the relation (go)], A x for the 14N15N-14N, system is 1.41 50.14. That leads to

OIT = 0.0051 & 10% (136)

what corresponds to RT = 0.33 The value of R T for nitrogen for = 333°K was computed by JONES and FURRY [a@. Using for this purpose viscosity data he obtained RT = 0.46. Owing to the simplified molecular model used by JONES and FURRY, the discrepancy is rather great. In 1951 DAVENPORT and WINTER performed a study on the variation of mT with temperature [88]. In this work they also used the method of the two reservoirs. The experimental results beside the theoretical ones for the L.J. (12,6) model are given in table 52. The average temperature was computed by the relation (115).

10%.

Table 52 Experimental T.D. factors for 14N2--14N15N system determined by the two reservoirs method.

The comparison of RT (exp.) with RT calculated for the L.J. (12,6) potential

T , P K ] T,[OK] TPK] A s - lo4 ape lo3 RT (expel R,(L.J.(12.6))

294 678 434 2 o j 1 9.1 & 0.4 0.58 f 0.03 0.54 294 683 435 20&2 8.9 -& 0.9 0.57 & 0.06 0.54 1 95 427 286 14&1 7.1 4 0.5 0.47 -& 0.03 0.44 195 428 285 19&3 7.8 -& 1.2 0.50 0.08 0.44 195 623 329 - 5.1 f 0.5 0.33 -& 0.03 0.48

Obviously, the values obtained by these authors are greater than those of MANN, though the same method was used. The accuracy of the experimental measurements does not enable a critical analysis of the L.J. (12,6) model, though for practical purposes the agreement between the experimental values and the theoretical ones is satisfactory. MASON and RICE in the work [ l l ] show that the L.J. (12,6) model does not fully satisfy the reproduction of the measured properties of the gases composed of nonspherical molecules, (N, CO, 0, CO,). This fact is due either to t,he unsuitable form of the potential or to the inapplicability of the theory owing to the hypothesis of the intermolecular potentials with spherical symmetry and to the intermolecular elastic collisions. But we may use the exp (- 6)potential which is more realistic and it contains more adjustable parameters than the L.J. (12,6) potential. The parameters of the exp(- 6)-potential for nitrogen obtained from the data of second virial coefficients and crystal properties are a = 16.2; r, = 4.040 b; Elk = 113.5"K and those obtained from viscosity data are a = 17; r, = 4.011 A and ~ / k = 101.2"K [ll]. These parameters reproduce rather well the reduced T.D. ratio, k$, of nitrogen. The comparison between the experimental data and theory for this double set of parameters is given in fig. 39. In this figure we can see that the parameters obtained from viscosity data are in better agreement with the experiment than those obtained from the data of virial coefficients and crystal properties.

46 a0

q4 f

42-

Fig. 39. The variation of the reduced. T.D. ratio for nitrogen against log computed for the exp (-6)-potential. -computed using the parameters obtained from crystal properties and virial coefficients, --- - calculated utilizing parameters obtained from viscosity data, - experimental data [ 8 8 ] , [07]

Fig. 40. The variation of the reduced T.D. factor versus log T*

-

/ d

-

/ /

/

/ / /

The 12C160-14N, isobaric system was studied ~ ~ M U L L E R , using for t'his purpose a hot wire column of 2.6 m length and 9 mm in diameter [97]. The kanthal wire with the diameter of 0.3mm was heated at 1058°K. The operating pressure was 760 mm Hg. He worked with a 2500 cm3 reservoir a t the top of the column. The steady state was reached within 6 days. The experimentalresults are compared to the theoretical ones of the first Chapman-Cowling approximation for L.J. (12,6) potential, table 53.

Table 53 Theoretical and experimental T.D. factors for the W l 6 O -14N, isobaric system. The experi-

mental determination are performed with the T.D. column

% co aT. los (exp.) o r T . lo3 (L.J.(12,6))

9 1.42 0.54 28.4 1.68 0.58 44.5 1.71 0.64 68.1 1.34 0.70 89.7 1.79 0.75

The fact that the theoretical values are different in comparison with the experimental data is explained both through the insufficiency of the central forces hypothesis and through the fact that the inner repartition of the masses in the molecule was not taken in consideration.


The 14NH3--15NH3 system was studied in 1943 by WATSON and WOERNLEY [98]. The results are presented in table 54 [38].

Table 54 CXT and RT for 14NH3-15NH, system

239 - 0.0100 - 0.39 268 - 0.0039 - 0.15 366 0.0105 0.41

Since the ammonia molecule is polar, the interaction is more complicated than in the case of nonpolar molecules; so that it is difficult to expect a too good agreement between the theoretical data which were furnished by the L.J. (12,6) model and the experimental ones. Nevertheless the change of sign of RT takes place at a temperature near by that one predicted by the L.J. (12,6) model.

4.6. Oxygen

Oxygen consists of 6 isotopes. From the T.D. point of view l60, 170 and 180 are interesting. The study of the ~~0,-1601s0 system was performed for the first time by WHALLEY and WINTER [99] and by WHALLEY, WINTER and BRISCOE [loo] in 1949. They used the method of the two reservoirs. The results obtained in the work [loo] are given in table 55 and fig. 41 beside the theoretic results for L.J. (12,6) potential. The values of the reduced T.D. ratio, k;, obtained in the work [loo] are given in table 56 together with the theoretical values obtained from table 11 for various values of F* = kF/e,

In the tables 55 and 56, F was calculated by the relation (90). ~71.

Table 55 Experimental T. D. factors for 160,-160180 system determined by the two reservoirs method.

The comparison of RT (exp.) with RT (calc.) for the L. J. (12,6) model ~~ ~

T,["K] T,["K] F["K] aT 103 RT RT (L.J. (12,6))

295 373 264 9.9 f 0.8 0.37 0.35 295 528 389 12.8 5 0.6 0.48 0.50 196 703 443 14.5 f 0.6 0.54 0.50

Table 56 The theoretical and experimental reduced T.D. ratio for 160,-160180 system

% I 6 0 ["K] exp. theor.

97.5 284 0.367 0.371 386 0.475 0.468 443 0.538 0.499

78 G. VLSARU

In a more recent work DE VRIES and HARING [IOI] have measured the T.D. factor for lZClSO -12C160 and 14C160 -12C160 isotopic systems at the average temperature ranging from = 260 "K to T = 465 OK. They used a column specially construe-

Fig. 41. The dependence of thermal Separation ratio, Rp, on temperature. - calculated for L.J. (12.6) model; o - experimental values.

&d for this purpose. The average temperature was calculated with the relation

!F = T, + 0.25(T2 - T,). (137)

The separation factor attained with the column is given by the relation

4 = exp (aT f ) (I381 where aT is the T.D. factor andf a parameter which depends on the gas properties and the column geometry. In the work [IOI] separation of the

systems in ternary mixtures were measured using a column. Because of the low concentration of two of the three components, the mixture behaves as a quasi-double-binary mixture. Thus the separation factors q' for the l*C160-12C160 system and q for the 12C180-12C160 system could be regarded as independent from each other: q' = exp (ah f ) and

14C160-12ClSO and 12C180-12C160

q = exp(BTf) *

The T.D. factor for 12C180 -12C160 system was assumed as to be equal with 0.015 at 440 "K. That is almost the double of the value established by DAVENPORT and WINTER (table 36) for the 13C160-12C160 system but it is in a good agreement with ~ l k = 100"K, calculated from viscosity data and second virial coefficients By this value of ~ / k the T.D. factor for other temperatures was calculated. The dependence on temperature of aT for the L.J. (12,6), exp(- 6)-model, respectively, differs little enough for various parameters (with some %).

4.7. Neon Neon consists of 6 isotopes. From the T.D. point of view, 20Ne, 'We and The first measurements of the T.D. factor for 20Ne-22Ne system have been under- taken by NIER in 1940, using the method of the two reservoirs. The experimental values for mT and RT are given in table 5 [43].

are interesting.


Since mT is rapidly varying with the temperature, the values from the table 5 may be considered as average values for the given temperature interval. The high values of L X ~ for neon make the isotopes of this element to be separated in best conditions by T.D. The comparison of NIER'S experimental data to JONES' theoretical results from table 10 for the L.J. (8.4) models, shows that the values calculated for R T are much less than the experimental ones. That is due to the fact that the neon atom behaves as more rigid than it is predicted for the case v = 8. The experimental value R T = 0.71 is in good agreement with the viscosity data which predict for high temperature the value R T = 0.69 for T = 1030°K on the basis of the inverse power model. Because of the small magnitude of the attractive forces, the inverse power model may be adequated also for neon in the case of high temperatures. In order to take account of the greater hardness of neon atoms, we take the bold step of increasing the scale of the results from table 10, so that the maximum value of R T be 0.74 instead of 0.48. We then find by the graphical method that for ~ l k = 42.5"K the theoretical value of R T corresponding to the three temperature domains are R T = 0.71 ; 0.45 and 0.36. These values are in good agreement with the experimental data within the limit of the experimental errors. On the other hand, ~ / k = 42.5"K is not in agreement with the relation (87) as the critical temperature for neon is 44°K. The decrease of R T which occurs at higher temperatures than the critical temperature, may be explained by the fact that a value of ~ l k is used, which is in good agreement with the values indicated by other approximation methods. The Sutherland model under the form (73) may be suitable for neon but the necessary value for ~ l k is of about 350 OK, that is to say 10 times greater than the critical temperature. The study of the variation of T.D. factor with the temperature was performed in 1942 by STIER [102]. In this case the method of the two reservoirs was also used. The results are given in table 57.

Table 57 The T.D. factors, k~ and R T for 20Ne--22Ne system depending on temperature

90 195 129 0.0162 0.382 0.480 0.39 - 0.44 90 296 154 0.0187 -

195 296 238 0.0233 0.550 0.587 0.55 - 0.60 195 490 298 0.0254 -

302 645 432 0.0302 0.713 0.618 0.71 - 0.75 460 638 545 0.0318 -

621 819 712 0.0346 0.816 0.625 0.82 kT (calc.) was computed by help of the data from table 11.

Since the ratio of the T.D. constant varies with the temperature, there arises the question - in this case also - about the definition of the accurate average temperature for a measured value of R T . According to the kinetic theory,

80 G. V~SARU

Using the relation (90) given by BROWN, STIER shows that the R T (exp.) value follows an empirical law of the form:

T Rll=a1n- b

T 26.6'

which for neon is written:

R T = 0.25 In - (143)

These results are in good agreement with the results obtained by NIER (table 5). I n table 58 a comparison is shown between the experimental results obtained for neon with those calculated from viscosity measurements or using the Sutherland model with C = 60°K and L.J. (8,4) model, respectively.

Table 58 The comparison of R T values for neon for various molecular models

R T (n) R T (Suth.) R T (L.J. (8,4)) TPK] R T (exp.) visc. C = 60°K elk = 42.5"K

~ ~~

129 0.39 154 0.44 238 0.55 244 0.56 298 0.60 333 0.63 423 0.69 432 0.71 498 0.73 545 0.75 712 0.82

~~ ~ ~

0.38 0.36 0.46 0.45 0.61 0.62

0.76 0.66 0.59

0.62 0.64

0.64 0.81 0.71

0.85 0.73

But SRIVASTAVA and MADAN have shown that the equation (142) becomes in- consistent with the dependence on the temperature which was assumed in the de- duction of the equation (90). This fact led HOLLERAN to the introduction of this simple expression

T = (TIT2)'/* (144)

for the calculation of the average temperature [lo31 The experimental results a t neon may be reproduced rather well if we utilize the square-well model with the parameters l / R = 0.55 and 8/12 = 120°K. Thus we obtain the values given in table 59 [44].

Table 59 The comparison of the theoretical and experimental values of R T for neon in case we use the

square-well model

[OK] 129 200 300 400 500 600 712

R T (exp.) 0.39 0.50 0.60 0.68 0.73 0.78 0.82 R T (calc.) 0.39 0.49 0.60 0.68 0.74 0.78 0.82


For neon, HA& and COOK [a?'] have determined the viscosity for 9 temperatures within the range 91.66-717.6"K. The values of n (relation 68) obtained for the temperature ranges of 91.66-194.7"K; 91.66-288.1 OK and 288.1-575°K are 0.745; 0.716; 0.676 respectively. The corresponding values of RT for these values of n (according to table 3) are: 0.44; 0.50; 0.58 respectively. These values may be compared with RT values for the same temperature range given in table 5 . These data can be improved for the whole temperature range using relation (75) with C = 60°K (Sutherland model). We obtain thus RT = 0.42; 0.54; 0.77 respectively for the temperatures of 142 OK, 192 OK, 450 "K respectively. These values show a better agreement with the experimental ones than those obtained by the inverse power model. Nevertheless, the fact that the value ~ l k = 350"K, which according to relation (78) corresponds to C = 60"K, is larger than that indicated by other more exact measurements (latent heats, second virial coefficients), this fact makes us consider the Sutherland model as unsuitable. The corresponding values for ~ l k are between 30-40°K. In another work, HOLLERAN studies the possibility of calculating RT from di f - fusion data, this parameter being known to be depend on this coefficient, too [16]. The results for various temperatures, together with the experimental values are given in table 60.

Table 60 The comparison of the RT experimental values with those obtained from diffusion data

T ["K] Rp (exp.) RT (calc.)

150 0.42 0.43 200 0.49 0.49 250 0.55 0.54 300 0.60 0.58 350 0.64 0.61

The RT (exp.) values from this table were obtained by using relation [142), a and b being determined by the least squares method, from STIER'S data. [ lo l l . For RT(ca1c.) the constants were also determined by the least squares method using WINN'S diffusion data [ l04 ] . The comparison of the RT experimental data with the theoretical ones computed by L.J. (12,6) model is made in table 61. The 2°Ne-22Ne mixture a t which these data refer to, contained 90.1% 20Ne. For the computation, ~ l k was assumed to be 35.7"K and u = 2.8"A. [38].

Table 61 The comparison of the RT experimental values with those calculated for the L.J. (12,6) model

["K] 150 200 250 300 400 500 600 10 RT (exp.) 4.4 5 5.6 6 6.8 7,4 7.8 10 RT (L.J. 5.2 5.65 5.90 6.02 6.14 6.20 6.22 (12,6))

We must notice that here the agreement between theory and practice is not a very good one.

6 Zeitschrift ,,Fortschritte der Physik", Heft 1

82 G. VXSARW

In 1958 MORAN and WATSON performed measurements of T.D. factors using the T.D. column [105]. The values found for aT are 2.15 - 10-2 for T, = 298°K and 2.25. for T, = 358°K. In the same year the author performed measurements of T.D. factors using the swing separator [68]. As initial gas the 2°Ne-21Ne-222Ne system with 9.75% 22Ne and W e in negligible quantities was used. The results are presented in table 62. The values with asterisk were determined by a swing separator with N = 20; the rest with N = 22, N was the number of the swing separator tubes. The cal- culat'ion of aT was done with the relation (106).

Table 62 Experimental T.D. factors determined by help of the swing separator for 20Ne--21Ne-22Ne

system depending on temperature

203 273 238 1.02* 2.00 338 392 365 0.63 2.20 329 412 370 0.83* 2.15 348 403 378 0.64 2.30 370 453 415 0.88 2.30 403 50 1 451 0.99 2.38 467 550 505 0.81 2.45 516 608 558 0.79 2.40 567 658 609 0.72 2.50

*) Determined by a swing separator with 20 tubes.

0.48 0.52 0.51 0.54 0.54 0.56 0.57 0.56 0.59

Table 63 Experimental and calculated values for the reduced T.D. factor, (xo, for neon.

a. calculated with

["K] a, (exp.) exp(-6)l) exp(-6)2) L.J. (12,6)3)

170 0.290 0.320 0.461 190 0.350 0.352 0.478 210 0.380 0.378 0.490 230 0.400 0.400 0.500 250 0.416 0.416 0.508 270 0.430 0.432 0.514 290 0.444 0.444 0.519 310 0.454 0.455 0.523 350 0.472 0.472 0.527 400 0.490 0.490 0.530 450 0.500 0.500 0.532 500 0.511 0.506 0.532 550 0.514 0.51 1 0.532 600 0.515 0.514 0.531

l) Calculated with a = 14; Elk = 66.6'K; r, = 2.96 A [72] 2, Calculated with a = 14.5; ~ / k = 38°K; r, = 3.147 A [U] 3, Calculated with elk = 35.7"K; T, = 2.80 A [7]

0.492 0.514 0.528 0.538 0.545 0.550 0.554 0.559 0.570 0.574 0.576 0.577 0.579 0.581


The comparison of R, values experimentally established and presented in table 62 with R T calculated from viscosity data by means of the relation (log), shows a difference between them of about 20%. The agreement is considered better than it was expected and it shows that if we have no experimental values of 0 1 ~ we may calculate them by the relation (108) till the first approximation. The agreement between the experimental data with the theoretical ones is better in case we use exp(- 6)-potential and worse for the L.J. (12,6) potential. SAXENA, KELLEY and WATSON [72] have extended the measurements of T.D. factors a t lower temperature maintaining one of the ends of the swing separator a t 78 and 195 OK, respectively. The neon used in these experiments had contained about 9.6% 22Ne and a negligible quantity of zlNe. The results obtained by these authors are presented in table 63. Their measurements were carried out with a swing separator with 19 tubes with o = 20 mm and L = 10 cm. The linking capillary tubes had ,@ = 1 mm. The approximative computations have shown that for neon at these temperatures the quantum effects were negligible, thus the interpretation of the results might be classically done. It is also noticeable in this table, that for neon the exp(-6)-potential gives such values which are nearer to the experimental ones than L.J. (12,6) potential. Recently, the system 2oNe-zzNe has been studied again, being used a mixture of 50% 20Ne-5070 ZZNe and a swing separator with 4 tubes [77] . The experiments have been carried out a t low temperatures. The results are presented in table 64.

Table 64 Experimental T.D. factors for 20Ne-We system depending on temperature (for equal con-

centrations of the components)

77 195 136 1.46 f 0.08 0.0166 0.349 f 0.020 77 273 175 2.27 5 0.05 0.0189 0.397 f 0.020 77 293 185 2.44 & 0.07 0.0193 0.415 & 0.029 77 347 212 2.81 f 0.05 0.0200 0.420 f 0.018

195 294 245 0.85 f 0.05 0.0217 0.456 5 0.030 273 351 312 0.57 f 0.04 0.0233 0.489 & 0.028

Table 65 Experimental T.D. factors for 4He -20Ne--22Ne ternary system depending on the concentra-

tion of the components

Ne Concentration TI [OK] T2 [OK] nT(20Ne-22Ne)

1.000 1.OOO 0.900 0.800 0.800 0.600 0.400 0.200 0.100

284 284 284 284 284 284 284 285 285

658 658 650 648 673 658 667 661 670

0.0277 f 0.0010 0.0267 f 0.0004 0.0242 f 0.0005 0.0230 f O.OOO8 0.0242 + 0.0011 0.0209 f 0.0009 0.0177 5 0.0010 0.0164 f 0.0009 0.0158 0.0013

6*

84 G . VLSARU

The increase of aT from 0.0166 f 0.0010 at 136°K to 0.0233 f 0.0020 at 310°K is much greater than the values given in the previous works. These T.D. factors are in good agreement with the calculated values, using an exp (- 6)-potential with ~ l k = 46.0 f 0.6 and a = 13. The measurements have been performed in order to improve the experimental data by eliminating the drawback of the swing separator which was mentioned when we discussed about the same work at helium. In his thesis LARANJEIRA [80] has studied - among other mixtures - T.D. in the ternary mixtures 2ONe -2zNe -4He and 20Ne -22Ne -H2 using the swing separator method. The results for the T.D. factors in case of the ternary system 20Ne-22Ne- 4He are given in table 65. We can notice that the isotopic T.D. factor decreases when the contents of helium decreases. The comparison of the experimental results obtained by LARANJEIRA presented in table 65 with those obtained by STIER~S made in table 66. The average temperature is calculated by means of the relation (144).

Table 66 The variation of the T.D. factor for neon isotopes with temperature

~

TI [“K] T, [“K] ?? [“K] cxT (aoNe-z2Ne) Ref:

90 90

195 195 302 460 64 1 284 284

195 296 296 490 645 638 819 658 658

132 163 240 309 441 542 752 430 430

0.0162 0.0187 0.0233 0.0254 0.0302 0.0318 0.0346 0.0277 0.0267

In fig. 42 is presented the variation of RT for the zoNe-22Ne system depending on the concentration of the added helium. The most probable linear equation which

Fig. 42. The variation of the thermal separation ratio for PnNe--2*Ne system with addition of helium. x - experimental data.


approximates the variation of 22, from this figure, inferred through the method of the least squares is

RT(exp.) = (1.07 f 0.05) XNe + (0.25 f 0.04) X4He. (145) If the molecules are considered to behave as rigid elastic spheres, the isotopic T.D. factor has the limit values

for XNe = 1 [ O I T ~ ~ , , ~ ( r .e . s . ) ]~~ = 0.0245, (146)

for X4He = 1 [0+20,22 ( r . e . s . ) ] ~ ~ = 0.0450. (147)

for X x e = 1 [aT20,22(ekp.)]~~ = 0.026, O . 0 O l 2 , (148)

for X*He = 1 [ c x ~ ~ ~ , ~ ~ ( ~ x ~ . ) ] ~ H ~ = 0.011, f 0.001,. (149)

The limit values corresponding to the equation (145) are

It is to be noticed the fact that the standard errors, corresponding to the thermal separation ratio, RT (exp.), (eqs. 145,148, 149) are greater than those experimentally observed and they are given in table 65. So it seems that in this case the linear dependence of RT (exp.) for the 2oNe-22Ne system is not correct. This fact also ensues from the distribution of the experimental points. The experimental results obtained for the ternary system 2ONe - zzNe -H, are given in table 67 and their comparison with other data from the literature is made in table 68. For this system the dependence on the temperature of aT is not satisfactorily studied. We notice that the experimental values obtained by LARAN- JEIRA are rather small.

Table 67 Experimental T.D. factors for H,--20Ne--22Ne ternary system depending on the concentration

of the components

H,Concentration T, [“K] T, [“K] ‘YT (z0Ne-22Ne)

0.000 0.000 0.250 0.400 0.500 0.500 0.600 0.700 0.800 0.900

286 284 284 284 284 284 284 284 284 284

658 658 657 668 671 668 658 661 653 653

0.0277 & 0.0010 0.0267 f 0.0004 0.0192 f 0.0009 0.0167 0.0011 0.0160 f 0.0005 0.0156 f 0.0014 0.0128 & 0.0009 0.0126 f 0.0006 0.0111 & 0.0012 0.0105 & 0.0008

Table 68 Experimental T.D. factors for the H,--20Ne--22Ne system depending on temperature

TI [“K] T, [“K] ‘YT (H,-Ne) Ref.

86 G . V~SARU

Here also, in case we take the molecules as rigid elastic spheres, the T.D. isotopic factors has the limit values

for XNe = 1 [aTzo,2z(r.e.~.)]~e = 0.0245, (150)

for XH, = 1 [aT20,22 ( r . e . s . ) ]~~ = 0.0424. (151)

Fig.

t 7 O L ' ' 1 I

200 400 600 - Tt'KI 43. The variation of the T.D. factor for *ONe-

*lNe system with the temperature. -for pure isotopic mixture. ----for a mixture with 100% CH,.

Fig. 44. The variation of aFfor aoNe--PzXe system with addition of hydrogen. ~ calculated for neon with 90% 2oNe and 10% I*Ne. 0 - experimental values with natural neon.

24

20

72 I I I I I I .I I 1

0 QZ 44 Q6 98 ~3 - (He)

Fig. 45. The variation of UT for aONe-PaNe system with the concentration of added helium. The solid curve - calculated for pure 'He added to neon with 90% %ONe - 10% PZNe. 8 - calculated for pure 'He. 0 - experimentalvalues obtainedwith natural helium and natural neon [SO]. x - experimental values obtained with natural neon and helium containing 10% We.

This fact shows that the T.D. factor for molecules of this type, increases with the increase of the hydrogen contents. On the other hand, the experimentally determined T.D. factor, aT (exp.), decreases considerably with the growth of the hydrogen concentration (table 67). In this case we didn't notice any linear dependence of RT(exp.) on the H, concentration for the system

VAN DER VALK has computed the T.D. factor variation for the 2We- 2zNe system for the pure isotopic mixture and for a mixture with lOOyo CH, depending on temperature. The results are graphically presented in fig. 43 [78]. From the work [81] we plot in figs. 44 und 45 the variation of T.D. factor

- W e .


for the ZONe-ZZNe system both versus the hydrogen concentration, and the added helium concentration. We observe that in case of the addition of H, to !We- 22Ne system (fig. 44) the agreement with the theory is quantitative. The experimental values on this graph belong to Laranjeira [80]. If He is added to the z0Ne--22Ne system (fig. 45) the agreement ist only qualitative. In fig. 45 we find for the addition of 100% He the difference between the effect of pure 3He (0) and of pure 4He. We see that pure 3He makes the T.D. factor decrease more than pure 4He. But the experiments were performed only with 10% 3He.

4.8. Chlorine

The chlorine consists of 8 isotopes. From the point of view of the T.D. only 35Cl and 37Cl are important.

Table 69 Experimental T.D. factors for H35C1-H37C1 system

194 273 229 - 0.009 f 0.007 - 0.38 600 800 685 0.010 f 0.008 0.42

SWARTZ'S measurements of aT for HC1 a t various temperatures were performed by the method of the two reservoirs. He found a single change of the aT sign, a t approximative 485 "K [108]. JONES' am determination from viscositv data 6 I showed that a room temperature aT E 0, but it increases with the increases of temperature, [as]. KRANZ und WATSON'S repeated this determination, with a greather precision. Their results are given in table 69. T was calculated with relation (90) [log]. They found that aT change sign, a t 385°K. i.e. approx. 100°K lower

4 Rr

12

0

-2

- 4

I I I I 1 1 1 1

200 300 400 500 60071R3800 than in S W A R ~ ~ results. The RT variation with temperature is given

at F = 507°K is the lower limit computed by JONES and FURRY [40] from the data obtained with the T.D. column. The agreement with KRANTZ and WATSON'S data is very good.

- 1T"KI in fig. 46. The point marked by A Fig. 46. The variation of R p with T

4.9. Argon

The argon consists of 9 isotopes. From the point of view of the T.D., 36Ar, reAr and 40Ar are interesting. The first determinations of T.D. factor for argon were performed by STIER in 1942 [lo21 by the method of the two reservoirs. He operated at a pressure of 40 cm Hg,

88 G. V&ARW

choosing an equilibration time 5 times larger than the relaxation time of the system. His results are given in table 70 beside the values calculated for RT.

Table 70 wT and RT depending on temperature, experimentally determined by the two reservoirs method,

for argon

90 195 129 0.00315 f 10% 0.07 f 10% 90 296 154 0.0709 f 5% 0.15 & 5%

195 296 238 0.0116 f 5% 0.25 & 5% 195 495 300 0.0146 5 5% 0.31 & 5% 273 623 400 0.0182 & 5% 0.39 & 5% 455 685 555 0.0218 f 5% 0.47 & 5% 638 833 720 0.0250 5 5% 0.53 5%

The empiric relation found by him showing the RT variation with temperature, for table 70, is

F RT 10.25. In- (152) 86.9.

- T being computed by the relation (90). If the viscosity of the gas is considered to vary with Tn, then [55].

RT N 1.7(1 - n) . (153)

n being given in table 4 for different temperatures. Using the n values and relation (153), STIER determined RT from viscosity data. The results are given in table 71. The same table shows - for a comparison - the theoretic results obtained by the Sutherland and L.J. (8.4) models.

Table 7 1 The comparison of RT values for various molecular models at argon

RT(n) RT(SUth.) RT (L.J.(8,4)) F-["K] RT(exp.) visc. C = 142'K elk = 124°K

129 0.07 - 0.44 154 0.15 0.04 - 0.31 238 0.25 0.18 0.27 0.01 300 0.31 0.34 0.38 0.17 400 0.39 0.41 0.51 0.33 555 0.47 0.47 0.60 0.47 720 0.53 0.53 0.68 0.53

There may be seen that the agreement between theory (SUTHERLAND model) and experiment is not satisfactory. A better agreement occurs with L.J. (8,4) model but in this case ~ l k = 124°K was used. MANN'S experimental determination, carried out by the same method with T, =

= 195 "K and T, = 623 OK; F = 329 "K (computed with relation 90) established


for mT the value 0.0164 f 10% corresponding to a RT = 0.35 & 10% [96], which is in a good agreement with STIER'S value, R T = 0.33 5%. [102). We extract from GREW and IBBS' book for 36Ar-40Ar system containing 0.31% 36Ar- the values given in table 72 [38]. Here Elk = 124°K and u = 3.42 A". In this case the aggreement is not good either.

Table 72 The comparison of RT experimental values with the theoretical ones calculated for the L.J.

(12,6) model ~ ~~ ~ ~

P K 10. RT (theor.) 10. RT (exp.)

150 0.85 200 2 250 2.88 300 3.57 400 4.52 500 5.10 600 5.43

1.2 2 2.6 3.1 3.9 4.4 4.9

STIER'S experimental data were examined by SRIVASTAVA and MADAN in 1953 [15]. They pointed out that BROWN'S empirical relation [55] for R,

(154) B RT = R, - - T

which leads to the average temperature, p, given by relation (90) and STIER'S one (142)) [ l O S ] , for the case or Ar being the form (152), are rather notsatisfactory even for the reducing the experimental data, concluding that the average temperature which STIER'S values of R, refer to, are not adequate. In relation (154) R, represents the value to which R, tends with the increase of T, B being a constant of the given gas. Starting from the relation which gives the R, value for an isotopic mixture with a small mass difference :

(155) 16.9(C* - 1) (A* + 1)

["I1 = 4(11 - 4B* + 8A*)

Table 73 Theoretical values for RT for various values of the reduced temperature

T* = kT/e

0.6 - 0.0631 0.7 - 0.0574 0.8 - 0.039 0.9 - 0.0104 1 0.0189 1.1 0.0472 1.2 0.0802 1.3 0.109 1.4 0.142 1.5 0.17

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

0.198 0.221 0.245 0.263 0.287 0.305 0.324 0.338 0.357 0.370

3 3.5 4 4.5 5 6 7 8 9

10

0.435 0.476 0.508 0.531 0.548 0.579 0.592 0.606 0.609 0.618

90 0. VXSARU

where A*, B* and C* are ratios of the collision integrals (tabulated in [as]), SRI- VASTAVA and MADAN have computed R, for different values of the reduced temperature, T*. Their results for L.J. (12,6) model are given in table 73 and graphically in fig. 47 (curve A). The shape of this curve was compared in order to discover similarities, and to find some representative and common equation types ; several empirical equations were chosen trying afterwards to apply them to the tabulations in order to see if

A

47 r 46

4' 0

0 I00 200 300 400 500 600 700 800 0 92 94 - T l ° K l Fig. 47. The variation of RT with T* Fig. 48. The variation of RT with 1/T*

they represent the data of table 73. Particularly, the equations RT = A + BIT* ; R, = A -/- BIT* + RT = A + BT* $- CT*=; RT = A f B log T* were verified, the first two by means of fig. 48. These verifications showed the unadequacy of all equations for the whole temperature range; R, = A + BIT* + C/T*?- verifies the data from T* = 4 to T* = 10 with a set of values for A, B and C and from T* = 0.8 to T* = 4 with another set of these values; for the range from T* = 0.7 to T* = 2 the equation RT = A + BT* can be used. The relation RT = A $. BIT* is never valid for all ranges. For theoretical reasons, as T* is proportional with T if E is supposed to be constant for a small temperature range we can write:

B C T T2

RT = A - - + --

and for very low temperature ranges,

This relation enables to determine the real RT values at different temperature from experimental data (further details concerning the computation technique are given in work [15]). In table 73 the experimental values for RT are given, beside the average temperature computed by various formulas.


STIER'S values in table 74 (the last three values) lead to A = 0.837; B = 265.1; C = 32450, whereas the values of lines 2, 3, 4 lead to A = 0.625; B = 114 and C = 5662. The R T values calculated for these parameters with relation (156) are given in table 75, column 2, and represented in fig. 47 (curve B).

Table 74 Effective average temperatures calculated by various formulas, corresponding to RT (exp.)

T1["K1 T2 [OK] RT (exp.) cf. (156) cf. (142) cf. (154)

90 195 0.0673& 10% 136 132 129 90 296 0.151 & 5% 173 163 154

195 296 0.248 & 5% 240 240 238 195 495 0.312 & 5% 305 311 300 273 623 0.389 & 5% 419 412 400 455 685 0.466 & 5% 558 558 555 628 833 0.534 5 5% 728 729 720

Table 75 The comparison of the RT experimental values with the theoretical values

~

I00 - 150 200 300 400 500 600 700 800

~~ ~

- 0.0132 - 0.022 0.0993 0.102 0.212 0.222 0.308 0.280 0.375 0.393 0.437 0.455 0.485 0.500 0.525 0.530 0.556 0.556

~ ~

' Elk = 117.9"K

' elk = 154.7"K

For very low temperatures (the first two values in the table), relation (157) must be used. For the range 100 5 T 5 200°K there was found A = -0.238 and B = 0.00225. In fig. 47 a good parallelism of the experimental curve B (RT versus T ) and the theoretical curve A (RT versus T*) may be noticed, fact which shows that inthis case L. J. (12,6) model is suitable. Choosing the abscisse for various RT values we find the corresponding values for T* and T, which enable to determine easily ~ l k = TIT*. The data obtained for ~ l k are given in table IV. of work [15]. From this table two average values for ~ l k may be selected, one for the high temperatures range, (300-800°K) and another for the low temperatures range (100-200°K). We obtain Elk = 154,7; 117,9"K respectively. These values were used by SRIVASTAVA and MADAN to determine RT with relation (155). The results are given in table 75, column 3, beside the RT experimental values for various temperatures. A relatively good agreement may be noticed.

92 G. VLSARU

The empirical relation which represents best the experimental data for the R T in the column 2 (table 75) is of the form of equation (156) with A = 0.737; B = 1,667 and C = 10650. Fig. 49 (curve B, representing R T versus log T) shows that a t high temperatures the experimental data are conveniently represented by the equation

(158) F

R T = 0.25 In ~

93.3

which approximately agrees with relation (152) given by STIER. Curve A in fig. 49 represents the variation of the theoretical RT, with log T*. In this figure we may notice that experimentally there is a linear R T dependence on log. T, especially, in limited ranges a t high tempera-

For comparison, in table 74 column 4, we are given the ?i' value inferred from the R T dependence on temperature by SRIVATAVA and MADAN. This was computed with equation

48

47 Rl 46 I ~5

44

43

42 ture .

Q f

' 1,9 20 2,l 42 2,3 2,4 ZS 2,6 2 7 2,8 2,9 log r -

Fig. 49. The variation of Rr with log T

1 2 F

T = -- [ -B - (B2 + 4FC)'IxI

in which (159)

for the data in lines 3-7 and with equation

1 2 In - TI

for the data in lines 1-2. Both the k$ values for the system 36Ar-40Ar with 0.307% 36Ar computed from STIER'S data [I021 and the theoretic numerical values calculated by the use of table 11, for various values of T are given in table 76 [TI.

T a b l c 76 The comparison of the experimental and theoretical T.D. ratio, k$

F["K] exp. theor. ~

129 0.0673 0.030 154 0.151 0.096 300 0.312 0.359 555 0.466 0.528 720 0.534 0.572


We saw that SRIVASTAVA and MADAN assumed RT to be dependent on temperature, as predicted by the L.J. (12,6) potential, and calculate from this the temperatures to be assigned to the experimental RT values for the case of argon. There was also found that the relation of the (142) type adapted for argon by relation (158), agrees with the experimental data in a large temperature range. It rises the interest for the direct study of the dependance on temperature, given by equation (142). Thus substituting RT from equation (142) into equation (141), HOLLERAN got the simple expression for T, given by the relation (144), [103]. The value of the constants in relation (142) was determined by the last squares method. The results obtained by means of relation

T RT = 0.2566-1n - 91.01

beside STIER'S experimental values of RT are given in table 77. Above 150°K the deviation of the two types of value is 1%.

Table 7 7 The comparison of the experimental values for RT with the theoretical ones obtained with the

relation (162)

TI rK1 T,["K] F'[oK] RT(exp.) RT(theor.)

90 195 132 0.067 0.095 90 296 163 0.151 0.150

195 296 240 0.248 0.249 195 495 311 0.312 0.315 273 623 412 0.389 0.387 455 685 558 0.466 0.465 638 833 729 0.534 0.534

F was calculated with the relation (144)

As the probable experimental errors are +5%, the values of the constants in equation (162) must be considered uncertain by several percent. The same author performed the RT computation from diffusion data, for temperature ranges between 150-350 "K for which these data existed. For comparison, the results are given in table 78 beside the RT experimental data [16]. Obviously, the agreement is quite acceptable.

Table 78 The comparison of the experimental values for RT with the theoretical ones obtained from

diffusion date

F[OK] RT(exp.) RT(theor.)

150 0.13 0.10 200 0.20 0.19 250 0.26 0.27 300 0.31 0.33 350 0.35 0.38

94 G. VLSARU

Table 79 The comparison of the reduced T. D. ratio, kT*, for various temperatures with t,he one calcula-

ted from viscosity data

- T,["K] T,[OK] T r K ] k;(exp.) k;(theor.)

90 195 132 0.060 0.078 90 296 163 0.134 0.126

195 296 240 0.221 0.214 195 495 31 1 0.278 0.274 273 623 412 0.346 0.338 455 685 558 0.415 0.408 638 833 729 0.475 0.469

was calculated with the relation (144).

In table 79 both the theoretic and the experimental values for the reduced T.D. ratio are given for various temperatures, the theoretic values being computed from viscosity data [17]. The agreement is sufficient, except for the first two low temperatures. The discrepancy occuring in this case can be explained by the fact that the relation used in calculating are not valid for this TI, for the reasons extensively exposed in work [IY]. SAXENA and SRIVAS'~AVA computed the first and second approximations for the thermal separation ratio RT for L.J. (12,6) model [IIo] . For the [.&I2 calculation they used the force constants given in table 80 and the relation

where X,, X,, X,, Y,, Y , are given in MASON'S work [lo].

Table 80 Force constants for argon

E/k"K (first approximation) Elko K (first approximation) TI ["K] T2 [OK] approximative computation accurate computation

400 600 144 440 660 151.7 400 800 156.8 500 750 166.7 average value 154.8

148.1 154.4 160 169.4 158

Table 8 1 The values of the first and second order approximation for R, for L.J. (12,6) model

400 0.375 0.3751 0.3759 1.0022 510 0.442 0.4529 0.4599 1.0155 600 0.485 0.4940 0.5028 1.0179 700 0.525 0.5273 0.4506 1.0252


The results are compared with the experimental and the theoretical ones computed for a first order approximation in table 81 [110]. Table 81 shows a small difference between [RT], and [&I2 for the L.J. (12,6) model, much smaller than the experimental errors which occur when the RT determination is made. As SAXENA and SRIVASTAVA pointed out, the difference in MADAN'S values for [RTI1 and obtained from the exp( - 6)-potential for a great number of T* values with a = 14, is approx. 3% in the temperature range refered to by the data of the table 81. Consequently, the differences between the two approximations for the two models (exp(-6)- and L.J. (12,6)) are of the same order. Concerning the study of the variation of 0 1 ~ with temperature - as we have already mentioned - SRIVASTAVA and MADAN 11151 presume as correct theoretically predicted variation, they use it in interpreting the experimental data and observe if the theoretically predicted variation is experimentally confirmed.Their method of procedure is to fit a portion of the theoretically predicted curve with a suitable polynomial in powers of TIE and then use this polynomial for the data interpretation. They use the form given by the relation (156) for determining A , B and C by means of three experimentally values given by Stier. Thus there comes out the possibility of using polynomial forms different of Brown and HOLLERAN'S ones, for limited temperature ranges [ZO]. CORBETT and WATSON [ZO] pointed out that it is possible to verify the hypothesis on the irrespectiveness on temperature of the intermolecular force constants. Thus, the right hand side of the equation which gives aT depending on temperature, I /

d In T (164) m J 1 1

can be performed by a graphical method using the actual theoretical predictions. Aparticular value of the ~ l k force constant determines the temperature scale. The integration is made between the T , and T 2 limits. The values of these integrations are graphically represented against the experimentally given values, i.e. the right hand of the equa- J~h,,I~.dhT - 5

tion (164) was represented veIXls the left hand Fig. 50. The comparison of the experimental of the same equation. This method requires

that the theory predict both the mean temperature value and of the T.D. factor and has

results obtained by STIER for so with the theoretical values (the second approximation) obtained for the exp(-6)-potential for a = 14 and

its criteria the fit to the 45" straight line on elk = 110°K

the plot mentioned before. CORBETT and WATSON applied this method to S ~ ~ ~ ~ ' ~ e x p e r i m e n t a l data for argon, for ~ / k = 1 0 0 ; 107; 110; 112; 120; 130; 140andl5O0K, respectively,forthe exp(-6) potential, with a = 14. MADAN, applying SRIVASTAVA and MADAN'S method for the calculation of these data in the case of the exp(-6)-model, established that the ~ l k force constants vary between 116.1 and 148.9"K [18]. The method of CORBETT and WATSON leads to ~ l k = 110°K as the best value of the force constants which is suitable for the whole temperature range (fig. 50).

96 G. VASARW

Experimentally, the T.D. factor was determined by MORAN and WATSON in 1958, by means of a T.D. column [105]. Normal argon with 9.7% 36Ar was used, the ratio of temperatures being T,/T, = 2. The results are given in table 82.

Table 82 Experimental T.D. factors determined with T.D. column

a T

288 1.32 *

288 1.32 * lo-' 358 1.62 * lo-'

In the same year, the same authors experimentally performed determination of T.D. factors, using a swing separator with 22 tubes. They utilized an isotopic argon mixture with 9.87% 36Ar and negligible quantities of 38Ar. Both their results the computed RT values are given in table 83 [SS].

Table 83 Experimental CXT and RT determined by means a swing separator

T,["K] T,[oK] FPK] ~OOCXT RT

203 273 233 0.85 0.18 351 416 383 1.73 0.37 377 464 417 1.92 0.41 424 522 473 1.97 0.42 496 588 542 2.22 0.48 593 693 630 2.40 0.51

Table 84 The comparison of the values of T.D. reduced factors, a,, with those calculated for various

molecular models at argon

&,(theor.)

170 190 210 230 250 270 290 310 350 400 450 500 550 600

0.065 0.062 0.098 0.096 0.125 0.127 0.154 0.158 0.182 0.186 0.208 0.213 0.232 0.237 0.257 0.258 0.300 0.296 0.344 0.338 0.378 0.368 0.405 0.394 0.424 0.416 0.438 0.432

0.119 0.155 0.190 0.221 0.249 0.272 0.294 0.316 0.350 0.385 0.41 1 0.432 0.448 0.461

0.058 0.155 0.192 0.226 0.259 0.287 0.311 0.332 0.371 0.411 0.442 0.464 0.484 0.498

calculated with a = 14; E/k = 148'K; r, = 3,68 A [72] 2) calculated with a = 14; ~ / k = 123,2"K; r, = 3,866 d [ I l l 3, calculated with Elk = 124'K; r, = 3.418 d [7]


In 1961 SAXENA, KELLEY and WATSON [72] performed experimental determinations of reduced T.D. factors for the system 36Ar-3sAr-40Ar with 36.4% 36Ar, 1.6% s8Ar and 62% 40Ar, using for this purpose the swing separator. The experimental results beside those calculated for the exp (- 6) and L.J. (12,6)-potential are given in table 84. Details on the ~ l k calculation for exp (- 6)-potential may be found in work [7ZJ It must be noticed that for this case, the exp (-6)-potential is also more adequated than the L.J. (12,6) potential. The computations show that the quantum eEects for argon, even at low temperature, are negligible. Thus it may be treated classically. PAUL, HOWARD and WATSON lately studied the T.D. isotopic factor, a0, in the temperature range 127-653"K, using for this purpose a swing separator [Yo] . In this way the temperature range in which MORAN and WATSON, SAXENA, KELLEY and WATSON performed their measurements, was extended, and the measuring precision increased by removing the before noticed drawbacks in the construction of the swing separator mentioned before. The extension of the measurements in low temperature ranges lays a particularly great interest, as a0 sensibly variates with T i n this range, thus being able to serve to the strict verification of any theoretic molecular model. For the experimental determinations, a swing separator with 4 tubes was utilized. For the measurements a t low temperatures the lower copper block of the swing separator was introduced into a mixture of acetone with carbonic ice or liquid nitrogen; the upper part of the swing separator was maintained a t various temperatures by means of a thermostatic device. The samples were extracted at 8-24 hours after the temperature gradient had been established and the equilibrium was reached. The computed equilibration time was of 3-4 hours. Table 85 presents the results obtained for the 36Ar-40Ar system with approxi- mateIy equal concentrations of the components. All the measurements were performed at a 15 cm Hg pressure.

Table 85 Experimental T.D. factors determined by means a swing separator

77 77 77 77

195 199 195 195 273 273 327 307 353 442 585

195 273 303 41 7 273 286 346 435 373 463 446 527 594 687 725

127 155 165 201 232 240 263 299 32 1 360 386 407 466 556 653

0.518 0.540 0.438 0.563 0.541 0.636 0.467 0.589 0.541 0.587 0.546 0.410 0.410 0.410 0.410

0.0033 -J= 0.0007 0.0063 f 0.0006 0.0069 -j= 0.0007 0.0116 f 0.0010 0.0032 -j= 0.0007 0.0085 & 0.0008 0.0073 f 0.0007 0.0098 f 0.0004 0.0044 & 0.0006 0.0090 f 0.0006 0.0054 f 0.0007 0.0102 + 0.0007 0.0100 f 0.0008 0.0107 f 0.0007 0.0053 f O.OOO4

0.068 f 0.014 0.095 f 0.010 0.098 -j= 0.010 0.133 f 0.012 0.182 f 0.040 0.191 f 0.020*) 0.245 & 0.024 0.240 + 0.010 0.270 0.040 0.336 f 0.022 0.335 -j= 0.040 0.371 5 0.026 0.377 f 0.030 0.476 f 0.030 0.485 f 0.035

- *) determined with a 10 tubed swing separator.

7 Zeitschrift .Fortschritte der Physik", Heft 1

T calculated with relation (144).

98 G. VISARU

The experimental results suggest that aT is better represented by a linear function on T , i.e. aT = A ( l + BT). Substituting this aT values in the relation which gives the steady state for a binary mixture :

vx1 = a T X l X Z h 27 (165)

and integrating, we obtain :

which is similar to relation (106) for n = I if T is given by relation (161).

J 4 c o

Fig. 51. The determination of the best average temperature, T (values indicated by 0 )

In order to verify this form of ?i' other measurements for small and high temperature differences were performed. In fig. 51 the experimental values are plotted both for

= (TI + T2)/2. It is clear that the points given by relation (161) line up on a smooth curve, whereas the other don't. Though the of equation (161) is quite close to the geometrical mean (relation (144)) PAUL and coo. utilised the from the equation (161) because in fig. 51 a0 versus is linear, a fact which shows the correctness of the hypothesis. I n the case of high temperatures the linear form is not preserved but this fact does not affect too much the results for two reasons: firstly, all the forms of Fgive almost the same value at high temperatures, only when the temperature gradient is very high; secondly, at high temperatures the a. variation with temperature for argon is smaller and thus any error in Tleads only to a small difference. In the case of low temperatures the situation is inverted, because of the rapid aT variation with temperature. Fig. 52 presents beside the experimental data of PAUL, HOWARD and WATSON, other previously results for the T.D. factor, depending on temperature. STIER'S

given by equation (161) and for


values have been revised according to eq. (161) and after this correction, the data give better agreement with PAUL and coo. data, than the not corrected ones. As MORAN and WATSON’S experimental data are for high temperatures, with small temperature gradients, the new temperature ascribed, in conformity with relation (161) cause a small change in their values. At high temperatures STIER’S data, as well as MORAN and WATSON’S ones, are smaller than those of PAUL and his coo. The possible source of the errors in MORAN and WATSON’S experiments has already been pointed out in this work [72]. There it was noticed that at low pressures a,, was no longer independent on pressure but it diminishes with the pressure lowering, tending to zero (in the 0.01 -0.2

Fig. 52. The variation of the isotopic T.D. factor, ao, with the temperature for the aaAr-40Ar system. - ex- perlmentallaa, HOWARD and WATSON 1701 values, A - experimental STIER value8 [I021 [!Z’ recomputed according to the relation (161), 0 - experimental Moran and Watson values [I051 --- computed for exp (--6)-potential

atm. pressure range) [112]. The experiments were performed for the He-Ar, N,- CO, and H,--C02 gaseous systems. Experimental studies upon T.D. factors were performed for complex mixtures too. Thus, in his thesis [80] LARANJEIRA studied the helium in%uence on the isotopic separation of 36Ar--40Ar. The T.D. isotopic factors for a large concentrations range of the added gas were determined. The experimental results show that for Ar the isotopic T.D. factors increased with the concentration of helium in a s6Ar-40Ar-4He system. For Ne-Ar systems, which can be regarded as quaternary mixtures, the T.D. factor for Ne isotopes was noticed to diminish considerably with the increase of the Ar concentration. This fact is in agreement with the theory as the Ar molecules are heavier and c6softer” than the Ne ones. The results for the ternary system 36Ar-40Ar-4He are given in table 86. The determinations were performed by a swing separator. Table 87 shows STIER’S 0 1 ~ values beside those of LARANJEIRA, for comparison.

7*

100 G. V~SARU

Table 86 Experimental T.D. factors for 4He-S6Ar-40Ar system, depending on the concentration of the

components

Ar Concentration TI [OK] T~ [OK] orT ( S S A r _ 4 O h )

1 .000 1 .Ooo 1 .ooo 0.900 0.700 0.500 0.300 0.100

287 286 286 286 288 289 288 288

655 666 667 667 660 663 660 660

0.0147 & 0.0010 0.0142 f 0 OOO8 0.0149 f 0.0008 0.0149 f 0.0010 0.0153 f 0.0013 0.0176 3 0.0013 0.0207 & 0.0010 0.0222 f 0.0014

Table 87 The variation of T.D. factor for argon isotopes with temperature

~~~ ~ ~~

T,OK T,'K POK mT ( s6Ar -40Ar) Ref.

90 90

195 195 273 455 638 287 286 287

195 296 296 495 623 635 835 655 666 667

132 163 240 310 412 537 729 433 437 437

0.0031 0.0071 0.0116 0.0146 0.0182 0.0218 0.0250 0.0147 0.0142 0.0149

was calculated with relation (144).

Comparing the obtained experimental results with LARANJEIRA'S elementary theory we see that RT for 3sAr-40Ar system is linearly dependent on t'he added gas (He) concentration, in first approximation. For the case of rigid elastic spheres, LARANJEIRA'S equation for the T.D. factor is of the form

xAr and ture. This equation leads to the following limit values:

representing the molar fractions of Ar and He in the studied mix-

- for X A ~ = 1 [ o I ~ ( ~ . ~ . s . ) ~ ~ , ~ ~ ] A ~ = 0.0257, (168)

- for X4He = 1 [aT (r.e.~.),,,,,]~~~ = 0.0751. (169)

There may be noticed an increase of the T.D. factor for Ar, with the increase of the He concentration, when considering the molecules as rigid elastic spheres. Let us regard now the experimental values bcT ( e ~ p . ) , ~ , ~ ~ and RT (exp.)3,,40.


From fig. 53 a linear RT dependence on the molar concentration of He may be noticed. The most probable equation determined by the least squares method is:

Fig. 53. The variation of the thermal separation ratio for a6Ar-'OAr system with the heliuni concentration. o - experimental values.

The corresponding limit values of the T.D. factor are

- for X A ~ = 1 ~(T(exp.),,,,, = 0.0145 & 0.0007 (171)

comparable to the average value 0.0146 f 0.0009 obtained from the direct ob- servations for Ar only.

- for X I H e = 1 &T(eXp.)36,40 = 0.0258 & 0.0015. (172)

The Ne-Ar system was studied as a quaternary mixture consisting of 2oNe, 2zNe, 36Ar and 40Ar. The experimental values of the T.D. factors are given in table 88. We may first notice that ~(~(exp.),~,,,, has a considerable increase with the rise of the Ne concentration, as we expected, because argon has heavier and

Table 88 Experimental T.D. factors for Ne-Ar system

Ne Concentration T, ["K] T, [OK] aT (36&-40Ar)

0.000 0.000 0.000 0.100 0.300 0.500 0.700 0.900 1.000 1.000

287 286 286 286 288 288 287 286 286 284

655 676 667 667 667 665 663 661 658 658

0.0147 f 0.0010 0.0142 f 0.0008 0.0149 0.0008 0.0148 & 0.0010 0.0185 f 0.0012

- 0.0019 & 0.0015 -0.0043 f 0.0015

0.0199 & 0.0016 0.0257 f 0.0020 0.0315 & 0.0026

0.0048 f 0.0013 0.0075 f 0.0010 0.0192 & 0.0012 0.0277 & 0.0010 0.0267 3 0.0004

102 G. VXSARIJ

softer” molecules. Inversely, aT (exp.)zo,22 decreases with the rise of the Ar concentration and even changes sign at X A ~ = 0.600. The experimental values RT (exp.)36,4,, are presented in fig. 54. The best equation representing the experimental data of this figure is

6 6

20 40 60 80 100 - %hk Fig. 54. The variation of the thermal separation ratio for 38Ar-40Ar system wit,h the addition of neon.

x - experimental values.

0 42 44 46 Q8 &-+

-calculated for PoNe. o - experimental values with natural Ne and natural Ar Fig. 55. The variation of the T.D. factor, UT, for 36Ar-40Ar system with the concentration of added Ne.

This quaternary system was also studied by VAN DER VALK and DE VRIES. The limit values found by these authors for O L T ( ~ ~ A ~ - - ~ ~ A ~ ) were 0.014 at 440°K for pure Ar and 0.034 for a mixture with lOOyo Ne [HI. Although the theory predicts a T.D. factor depending linearly on the rise of the Ne concentration in the


case of Ne - Ar system, the agreement has nevertheless a qualitative character

The variation of the T.D. factor with temperature for the Ne-Ar system is plotted in fig. 56. In fig. 57 the variation of the same factor for He-Ar system ist given [78].

(fig. 55).

200 400 600 TPKI _c

Fig. 56. The variation of the T.D. factor for a6Ar-40Ar Fig. 57. The variation qf the T.D. factor for J6Ar-40Ar system with the temperature. The solid system with temperature. The solid curve - curve - for pure isotopic mixture. The dashed for pure isotopic mixture. The dashed curve - curve - for a mixture with 100% Ne. for a mixture with 100% He

4.10. Kryp ton

The krypton consists of 25 isotopes. Out of these isotopes those which are important for the T.D. are : 82Kr, S 4 K r and seKr. In 1956 CORBETT and WATSON performed experimental determinations of reduced T.D. factors, cz,,, and of T.D. ratios, RT, for the 6 stable Kr isotopes above mentioned [ZO]. These authors used the two reservoirs method. The rsults for some temperature ranges are given in table 89. from these table was calculated with re-

Table 89 a,, and R, experimentally determined by the two reservoirs method for krypton isotopes

195 213 231 - 0.0156 - 0.0175 f 0.0038 273 550 388 0.0078 0.0088 -J= 0.051 213 715 460 0.0181 0.020 & 0.035 273 875 490 0.0376 0.042 f 0.041

was calculated with the relation (144).

104 G. VXSARU

lation (144)) Here (xo, RT, respectively, represents an average value for these isotopes. The experimental results are graphically presented in fig. 58, compared with the theoretical results obtained for the L.J. (8,4) and L.J. (12,6) models and exp (--)-model with a = 14.

I L log ( k l / E )

Fig. 58. The comparisonof the data for Kr with the predictions of L.J. (8,4) and (12,6) model for the thermal separation ratio snd of the exp (-6)-potential for the second approximation of the reduced T.D. factor. The scale of the ordinates was adjusted in order to be able to allow the comparison in an only one figure. The solid curve - for the L.J. (8,4) model, ~ / k = 169°K; - . - . for the L. J. (12,6) model, &/k = 395°K. -_- for the exp(--d)-potential, a = 14, e/k = 470'K

t - 0,15

- 4201 I I I I I L 700 200 300 400 500 600

TC'KI - Fig. 59. The variationof Rpwith temperature. The solid curve - data obtained with the relations RF = A-B/T.

- experimental CORBETT-WATSON data for F = (T~T,)''~

These comparison lead to the following values for &/k of 169, 395 and 470"K, respectively. The values for this figure are also calculated with relation (144), Obviously, the exp (-6)- and L.J. (12,6) models seem to fit the data well, but the high values of their force constants, as compared with the 190 OK value, obtained from viscosity data [7] and 171; 158"K, respectively, from virial coefficients [7], is to be noted.

Thermal Diffusion in Isotopic Gaseous Mktures 105

The comparison of CORBETT and WATSON'S experimental data for ?F given by equation (144) is performed with an empiric relation of the form RT = A - BIT in fig. 59. MADAN utilized these data for the determination of the potential parameters of the exp(-6)-model [19]. The elk value found for this potential with a = 12, is 224°K. In work [19] other values are presented which were obtained for elk by other authors from various properties (viscosity, crystal, virial coefficients). MORAN and WATSON resumed in 1958 t.he experimental determinations for the reduced T.D. factor for Kr, using for this purpose a swing separator with 22 tubes. The initially used mixture contained 0.35% 78Kr; 2.22% s°Kr; 11.50% saKr; 11.52% s3Kr; 57.2% 84Kr and 17.54% s6Kr. The experimental determined a. and R T values mediated for the variations in cencentration of 8OKr, szKr, 83Kr and s6Kr are given in table 90 [68].

Table 90 a,, and RT experimentally determined by the swing separator method for krypton isotopes

199 264 232 0.037 0.089 314 430 372 0.10 0.23 403 502 453 0.11 0.25 428 562 486 0.11 0.25 463 609 532 0.154 0.35 521 674 598 0.165 0.37

The experimental errors in these factors are approximatively 10%. Table 90 aims to replace the experimental data of table 89. In fig. 60 the experimental values given in table 90 are presented beside the theoretically computed values for the exp(-6)-model with a = 13.5 and L.J. (12,6) model [113]. FENDER pointed out that MORAN and WATSON'S experimentally determined T.D. factors are considerably smaller than the theoretical values [114].

4 I:F/ Fig. 60. The variation of the reduced T.D. factor with temperature. The solid

curve - for the exp(-6)-potential. The dashed curve - for the L.J. (12,6) potential. o - Moran-Watson experimental values [68] 0

200 400 600

Lately, SAXENA and JOSIII [115] have analized MADAN and WATSON'S experimental results, For the calculation of the reduced T.D. factor they used WHALLEY and SCHNEIDER'S potential parameters, elk = 166,7"K, r,,, = 4.130 A", (deduced from second virial coefficients), because these parameters may be correlated with the experimental data of the crystal properties, second virial coefficients, viscosity and thermal conductivity, [116]. The calculated a. values for Kr depending on temperature, are given in table 91

- Tf 'KI

[115]. The calculated aV values may be noticed to be always greater than the experimental ones, with an amount surpassing the limit of the experimental errors.

106 G. V~SAEW

Table 91 The comparison of the experimental values for a. with the theoretical values for krypton,

depending on temperature

T["K] a, (exp.) a, (theor.) Deviation yo

232 0.074 0.122 65 372 0.200 0.290 45 453 0.22 0.359 37 486 0.22 0.382 35 532 0.308 0.406 32 598 0.330 0.436 32

These deviations are owing to a wrong construction of the swing separator, by means of which MORAN and WATSON performed the experimental determinations. Thus it ensues the necessity of new T.D. factor measurements, after removing the source of the above mentioned errors.

4.11. Xenon Xenon consists of 30 isotopes. The following isotopes are interesting from the point of view of the T.D. : 124Xe, 126Xe, lZ8Xe, 129Xe, 130Xe, 131Xe, 132Xe, 134Xe and 136Xe. MORAN and WATSON determined the reduced T.D. factor, ao, for Xe isotopes, using for this purpose a swing separator with 22 tubes. The experimental oco values mediated for lZ9Xe, ls2Xe, lS4Xe and l"Xe are given in table 92, beside the RT values [68].

Table 92 The variation of T.D. factor, &o, with temperature for xenon isotopes

T,l?'Kl T,["Kl Fl?'Kl aoaverage RT

202 208 232 0.038 0.084 316 461 386 0.09 0.20 391 516 458 0.11 0.25 427 589 500 0.12 0.27 460 619 540 0.13 0.28 484 660 572 0.13 0.30

The mediation was not performed on 128Xe, 130Xe and 131Xe isotopes inasmuch as these isotopes have a little variation in concentration, so that the determined a. values are less sure than those for the other isotopes mentioned above. The errors involved in the determination of the differences in concentration in the case of the Xe isotopes were of 10% at the average temperatures, and approx. 20% at the low temperatures, respectively. Fig. 61 shows graphically the variation of a0 with temperature for Xe as well as for Ar [68]. The comparison of the experimental data is made by the theoretical data obtained for exp (- 6)-potential. The same authors also performed determinations of aT for Xe by means of a T.D. column for T2/Tl = 2 , T, = 358°K. The value found for IT in this temperature range is aT = 7.7 - [105].


FENDER pointed out in 1961 that MORAN and WATSON'S T.D. factor measurements in the case of noble gases give considerably smaller values than give the theoretic values. This fact comes out clearly from fig. 62, where the variation of the reduced T.D. factors for these gases vs. log T* is given [ l l a ] .

I I I I 0 /

Fig. 62. The variation of no for rare gases. The experimental values belong to MORAN and WATSON [681. a - neon, e/k = 35,6"K 171. - argon, e /k = = 119,8"k [7]. o - krypton, e/k = 172,7OK. 0 elk = 166,4OK. A - xenon, e/k = 221OK. The solid curve - theoretical values for L.J. (12,6) model.

These deviations may be considered as owing to the particular choice of the L.J. (12,6) potential. In 1963 SAXENA and JOSHI resume this problem in work [115]. Observing the discrepancy between theory and experiment in the case of He, Ne, Ar and based on SAXENA, KELLEY and WATSON'S work [72], who had pointed out the impossibility of correlating the experimental data of the various equilibrium and non-equilibrium properties for these gases, according to CHAPMAN-ENSKOO theory which concerns with either the exp (-6)-potential or the L.J. (12,6) one. This owing to the fact that the potential parameters inferred from the T.D. factor dependence on temperature don't enable the satisfactory reproduction of other properties. Thus SAXENA and JOSHI underline the fact that the potential parameters which enable the correlation of all the properties, except aT, lead to aT values which are always larger than the ones experimentally determined with the swing separator. Conse- quently MASON'S observation [I171 upon the possibility of a reasonable correlation of the equilibrium and non-equilibrium properties is not strictly accurate. Allowing for the mentioned discrepancies SAXENA and JOSHI also examine the data. for Xe (the case of Kr has been mentioned above) using for this purpose the L.J. (12,6) potential only.

108 G. Vtisa~u

For the computation of the reduced T.D. factor they utilize potential parameters deduced by WHaLLEY and SCHNEIDERfrOm second virial coefficients, &/k = 225,3 "K and r, = 4.568 A" [ U S ] , as these parameters may be correlated with the experimental data of crystal properties, second virial coefficients, viscosity and thermal conductivity. The calculated values for the dependence on temperature are given for Xe in table 93.

Table 9 3 The comparison of a. (exp.) with m0 (calc.) for xenon isotopes

T["K] (exp.) a0 (calc.) Deviation yo

232 0.076 0.023 61 386 0.18 0.197 11 458 0.22 0.257 17 500 0.24 0.287 20 540 0.25 0.316 24 572 0.268 0.336 27

The a,, theoretically values for Xe may be observed to be always greater than the experimental values, with a quantity surpassing the limit of experimental errors. I n the case of xenon at low temperatures the computed value for this factor is- unlike with Kr - smaller than the experimental one. This anomaly, as AMDUR and SCHATZKI pointed out [118] is due to the failure of the interaction potential a t values of low reduced temperatures for which the contribution of the dipole- quadrupole interactions is important. Thus for Xe - as in case of He, Ne Ar and Kr - the experimental a,, values are systematically smaller than the calculated ones. The source of errors probably lies - as SAXENA and JOSHI showed [119] - in the measurements carried out by the swing separator a fact which wasn't taken into account by SAXENA and coo. [re]. Thus it appears the necessity of performing new measurements, after eliminating the above mentioned source of errors regarding the construction of the swing separator.

Acknowledgments

It is a great pleasure to thank Dr. G. MULLER from Institute for Stable Isotopes from Leipzig for his careful reading of the final manuscript of this paper and the helpful dis- cussions.

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