OF ASYMMETRIC MOLECULES
P R O E F S C H R I F T
TER VERKRIJGING V A N DE G R A A D V A N D O C T O R I N DE T E C H N I S C H E W E T E N S C H A P A A N DE T E C H N I S C H E H O G E S C H O O L TE DELFT OP G E Z A G V A N DE RECTOR MAGi^llF!CUS IR. H. J. DE WIJS, H O O G L E R A A R I N DE A F D E L I N G DER M I J N B O U W K U N D E , VOOR EEN COMMISSIE U I T DE S E N A A T TE VERDEDIGEN OP
DONDERDAG 16 JANUARI 1964 DES NAMIDDAGS TE 4 UUR
D O O R
CASPER JOHAN G E O R G E SLIEKER
S C H E I K U N D I G I N G E N I E U R GEBOREN TE R O T T E R D A M
1Ö64
D I T PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR
PROF. DR. J. KISTEMAKER
Op deze plaats wil ik mijn grote erkentelijk-heid tot uitdrukking brengen aan Dr. J . L o s voor zijn hulp bij de voorbereiding van dit proefschrift.
Aan mijn ouders.
This work is part of the r e s e a r c h program of the "Stichting voor Fundamenteel Onderzoek der Materie" (Foundation for Fundamental Research on • Matter -F . O . M . ) and was made possible by financial support from the "Nederlandse Organisatie voor Zuiver We-tenschappelijk Onderzoek" (Netherlands Organization for pure Scientific Research - Z . W . O . ) .
I n t r o d u c t i o n . P a g e 7 C h a p t e r 1. T h e o r y of the t h e r m a l diffusion f a c t o r . 13 1. I n t r o d u c t i o n . 13 2. T h e m o d e l of W a l d m a n n . 25 3. T h e m o l e c u l a r i n t e r a c t i o n of a s y m -m e t r i c -m o l e c u l e s . 31 C h a p t e r II. C o l u m n t h e o r y . 1. I n t r o d u c t i o n . 2. G e n e r a l t h e o r y . A. T h e s t e a d y s t a t e . B . T h e e q u i l i b r i u m t i m e of c o u n t e r -c u r r e n t -c o l u m n s . 3. T h e p l a n e c a s e . 4. T h e c o n c e n t r i c c y l i n d e r c a s e . 5. T h e hot w i r e t y p e . 6. C o m p a r i s o n of t h e o r y and e x p e r i m e n t . C h a p t e r III. E x p e r i m e n t a l . 1. T h e c o l u m n . 2. T h e i o n i z a t i o n c h a m b e r s . 3. T h e g a s e s . 4. I n t r o d u c t o r y e x p e r i m e n t s . 5. S h o r t d e s c r i p t i o n of a s e r i e s of e x -p e r i m e n t s . 6. T h e r m a l diffusion e x p e r i m e n t s . C h a p t e r IV. T w o - b u l b a p p a r a t u s and s w i n g s e p a r a t o r . 1. T w o - b u l b a p p a r a t u s . 2. T h e s w i n g s e p a r a t o r . 3. E x p e r i m e n t a l . C h a p t e r V. D i s c u s s i o n of the e x p e r i m e n t a l r e s u l t s and c o m p a r i s o n with t h e o r y . 1. C o m p a r i s o n of the e x p e r i m e n t a l r e s u l t s . 2. C o m p a r i s o n of t h e o r y and e x p e r i m e n t . S u m m a r y . S a m e n v a t t i n g . 42 42 42 44 49 51 52 57 59 62 62 65 67 67 71 72 79 79 82 84 94 94 96 100 102 R e f e r e n c e s . 105
The partial separation of a given number of components, which occurs in a homogeneous phase under the influence of a t e m p e r a t u r e gradient, is called t h e r m a l diffusion. The phenomenon is now well known in gases and liquids and was recently observed in the solid state-^).
T h e r m a l diffusion in gases is characterized by the t h e r m a l diffusion factor a, which appears as a t r a n s p o r t coefficient in the kinetic gas theory in exactly the same way as the three other t r a n s p o r t coefficients: the viscosity ri; the t h e r m a l conductivity A and the diffusion coefficient D.
In this way t h e r m a l diffusion in gases was theoretically predicted by Enskog^) and Chapman"^), although the effect in liquids had already been discovered in the middle of the 19th century by Dufour^) and Soret ). In 1917, Chapman and Dootson ) proved the existence of t h e r m a l diffusion in gases with the naixture Hg - COg.
As the elementary separation in these p r o c e s s e s is r a t h e r small, the phenomenon was not extensively studied. The in-t e r e s in-t , was, however, largely increased, when Clusius and Dickel^) invented the so-called t h e r m a l diffusion column, in which the elementary effect can be mutiplied in a v e r y simple way.
Nowadays, t h e r m a l diffusion experiments can be done in t h r e e different a p p a r t u s e s .
1. Two-bulb apparatus, in which the elementary separation is m e a s u r e d . As the analyzing techniques ( m a s s s p e c t r o m e t e r s and radiation counters) a r e much more r e -fined during the last twenty y e a r s , it is now possible to get reliable r e s u l t s . The benefit of this instrument is that direct information about the t h e r m a l diffusion factor is obtained. With a two-bulb apparatus, designed by Mason ^•^), the diffusion coefficient can be simultaneous-ly obtained from the r a t e of appraoch to the steady s t a t e .
2. The swing s e p a r a t o r .
Clusius and Huber^°) have invented this apparatus in which the elementary effect is multiplied with a known factor. Great c a r e must be taken, however, with the experimental techniques', as e r r o r s in the multiplication are easily introduced.
8 -3. The t h e r m a l diffusion column.
Only comparative m e a s u r e m e n t s can be done with this apparatus, as geometry constants, which appear in the expression for the separation factor, cannot be c a l -culated, but should be m e a s u r e d . On large industrial scale t h e r m a l diffusion columns are not used, but in laboratories it is a v e r y manageable instrument. It is easy to construct and can produce enriched isotopes in a relatively short time. With special techniques Clusius-^-*^'-^^ ) has produced highly enriched' r a r e stable isotopes C^^A, 2i]sfe, " O ) .
The general expression for the t h e r m a l diffusion factor according to the Chapman Enskog theory can be a p p r o -ximately written as the product of the relative m a s s dif-ference of the molecules and a t e m p e r a t u r e dependent factor A, which is a function of the intermolecular potential field:
M^ - M^
a = A . M^ + M2
F o r the comparison of theory with experiment it is, thus, n e c e s s a r y to introduce molecular models, which describe the intermolecular potential fields. It appears that the t h e r m a l diffusion factor is v e r y sensitive to the choice of this model, so that t h e r m a l diffusion experiments are v e r y suitable for the determination of these force constants.
In many c a s e s it i s , however, hardly possible to state whether the discrepancies between theory and experiment a r e due to assumptions made in the Chapman - Enskog theory or to the imperfections in the molecular model. The models, which give the best adjustment, a r e the Lennard
- Jones (12-6) potential and the modified Buckingham poten-tial e.g.i3.i4)_
The most important assumptions, which have been made in the Chapman - Enskog theory, a r e the following:
a. The p r e s s u r e should not be too high, so that only binary collisions -occur. This means experimentally that the working p r e s s u r e must remain below one atmosphere.
Bogolubov •^^) has given the corrections in the Chap-man - Enskog formulas for the transport coefficients at higher p r e s s u r e s .
b. The t e m p e r a t u r e must not be too low.
De Broglie wavelength gets the same o r d e r of m a g -nitude as the mean molecular d i a m e t e r . Qdantum dif-fraction effects may then be expected in pure Hg from 55" K and in pure ^He from 35° K (see e.g.^^ ). Eperimentally these phenomena a r e observed amongst others by Ghozlan " ) in ^^j^r - ^He and ssp^r - ^He. The molecules a r e spheres without internal degrees of freedom and have spherical s y m m e t r i c potential fields; theory holds thus s t r i c t l y only for noble g a s e s . Deviations in the t h e r m a l diffusion factor may thus be expected from the facts, that
1. the molecules can change their rotational and v i -brational states during a collision;
2. the molecules have non-spherical potential fields. Especially, t h e r m a l diffusion in isotopic isobaric s y s -t e m s , where -the molecules only differ in momen-ts of inertia and potential fields and where the occupation of rotational states can be different, has attracted much attention during the last y e a r s . The first e x p e r i m e n t s on this subject have been done in 1950
-1952 by Becker and c o - w o r k e r s •'•^'•^^).
They find that the binary s y s t e m s ^^O^^C^^O -1 6 Q -1 3 ( . -1 6 Q ^^^ 1 6 Q 1 2 ( - . 1 6 Q _ i 7 o i 2 c i 6 o behave dif-ferent in a t h e r m a l diffusion column and attribute this to the fact that:
1. the m a s s distribution in the isotopic isobaric COg molecules in different;
2. the s y m m e t r y numbers of these molecules a r e different, 16 - 13 - 16 having the number 2 and 17 - 12 - 16 the number 1.
De Vries et al^°) have further investigated these p h e -nomena with the isotopic isobaric CO molecules
^•^C^^Oand ^ ^ C I S Q . Here, only the m a s s distribution is important. They find ^^C-^"O is 1,15 times more enriched than i2(^i8o with respect to 12^160, although the moments of inertia differ only 3.7%.
F r o m this it becomet apparent that the m a s s d i s t r i -bution does not cause unimportant side effects, but v e r y pronounced differences. This comes out even m o r e clearly in the work of Clusius and Flubacher ^^) with the t h e r m a l diffusion of ^^A against the hydro-chloric acid isotopes H^^CL, D^^ CL, H^'^ CL and D3'' CL.
The experiments indicate that the substitution of H by D gives nearly twice as much change in the thermnl diffusion factor as the substitution of ^^CL by ^''Cl.
1 0
-Thus, instead of the expected d e c r e a s e of a with the i n c r e a s e of m a s s according to 3 6 - 3 7 - 38 - 39 the sequence is 36 - 38 - 37 - 39. In these experiments the s y m m e t r y number plays no part either. Moreover, if one r e m e m b e r s that the substitution of 3^ C L by
•''' CL changes the moment of inertia with l e s s than 1%, while on the other hand the moment of inertia of DCL is 1.95 t i m e s that of HCL, it becomes obvious that the deviations a r e caused by the a s y m m e t r i e s in m a s s distributions.
Schirdewahn, Klemm and Waldmann 2^) have measured the t h e r m a l diffusion factor of the isotopic isobaric system Dg - HT and also tried to approximate these phenomena with dimension analysis.
A formula is derived for isotopic m i x t u r e s , which takes into account the moment of inertia of the m o l e -cules.
The main purpose of this thesis is to study the influence of a s y m m e t r y effects on the t h e r m a l diffusion factor of the binary s y s t e m s HT - ^He, DT - '^He, HT - H2, DT - Dg and D2 - HT.
We started our experiments in a t h e r m a l diffusion column with the mixtures HT - *He, DT - ^He and Tg - % e . As the r e s u l t s were r a t h e r strange and the agreement between theory and experiment not univocal, we have measured these
mixtures over a large temperature range with a swing s e -p a r a t o r (80° K - 250° K) and with a two-bulb a-p-paratus (300°K - 400°K.) Moreover, we investigated in these two apparatuses the binary s y s t e m s HT - H2 and DT - Dg to study the influence of the replacement of helium by h y d r o -gen.
In the first chapter a review of the calculation of t h e r m a l diffusion factors is given and the consequences of the d i a -tomic c h a r a c t e r of hydrogen molecules a r e discussed. Schirdewahn's treatment of isotopic mixtures is mentioned and criticized. A new theory is developed, which leaves the Chapman - Enskog theory unaltered, but indicates how a change in force constants can account for the measured phenomena.
In the second chapter a simple but v e r y adequate theory for concentric thermal diffusion columns is developed. This theory can be divided into two p a r t s :
1. general theory of a countercurrent separation column; 2. determination of column constants, typical for t h e r
-mal diffusion.
fected the calculations of these constants for different m o l e -cular models. It s e e m s , however, that they have left out of account that the discrepancies between theory and e x p e r i -ment a r e not only caused by the incomplete knowledge of the column constants, but also by imperfections in the general theory of the separation column.
In chapter III the experimental data for the column a r e given and discussed.
Next the swing s e p a r a t o r and the two-bulb apparatus a r e briefly described and the experimantal r e s u l t s a r e reported, (chapter IV).
In chapter V a comparison is made between the theory developed in chapter I and the experimental r e s u l t s of the chapters III and IV.
THEORY OF THE THERMAL DIFFUSION FACTOR §1. Introduction.
The transportcoefficients appear in the Chapman-Enskog theory as the solutions of infinite s e t s of l i n e a r equations. They a r e obtained as the ratios of two infinite determinants, whose elements are the coefficients of these l i n e a r equations.
Chapman and Cowling^6) replace in their calculations the infinite determinants by finite ones and so obtain expressions for the first approximation of the transportcoefficients. F o r higher approximations successively m o r e t e r m s a r e taken.
Kihara^'^) has evaluated the determinants in a m o r e physical way. The elements can be expressed in t e r m s of the s o -called collision - o r omega integrals and their t e m p e r a t u r e derivatives. Kihai'a now first calculates the transportcoef-ficients for a special kind of molecules: the Maxwellian molecules. These molecules interact according to a force, which v a r i e s as the inverse fifth power of the i n t e r m o l e c u l a r distance. The collision integrals are for this type of i n t e r -action independent of t e m p e r a t u r e , so that all their derivatives a r e z e r o . It appears now that in this case all elements in the determinants become z e r o , except on the main diagonal.
The first approximation of the transportcoefficients for non Maxwellian molecules is now obtained by neglecting all derivatives of the collision i n t e g r a l s , the second approxi-mation by neglecting all second and higher o r d e r derivatives.
Mason2*^) has compared both methods and states that K i h a r a ' s calculations a r e preferable in most c a s e s , because of their accuracy as well as their simplicity. In our m i x t u r e s the first Kihara approximation is roughly equal to the second one of Chapman and Cowling.
The first Kihara approximation for the thermal diffusion factor is given by the following expression:
Sj N - Sg (1-N)
cv = (6C* - 5) __^ (1,1) N^Oj + (1-N)2 0 „ + N(l-N) O
1 4 -M i 2-M2 -M h ^ i 1 0-11 -M j -M g A * M g t M i + Mgj l"i,'2 J l°'i2J (M^+Mg)^ 15 M2 (M2 - M^) ( M i + M^f ( 1 , 2 ) 2M2
^ ' " M2(Mi+M2)
I M I + M J"• "" ^^*^^ ^ •{M, + 3M2 +
+ f M , M 2 A * ( 1 , 3 ) 1 5 ( M ^ - M 2 ) 2 32M M , „ M ^ + M g f ^ i i ^ l Q^2 = + A* + 1 j —-: ( M ^ + M g ) ^ ( M i + Ma)"'^ (M^Mg)^ l ^ i ' 2 ^ * ( 1 . 4 ) S2 and Q2 a r e o b t a i n e d f r o m Sj^ and Q^ by an i n t e r c h a n g e of s u b s c r i p t s which r e f e r to the c o m p o n e n t s .M ;^ and M2 a r e the m a s s e s of the m o l e c u l e s and m t h i s t h e s i s the suffix 1 a l w a y s r e l a t e s to the t r i t i u m c o m p o u n d . C * and A * a r e r a t i o s of c o l l i s i o n i n t e g r a l s " 1 , 2 ^ "1.2 1,2 1,2 T h e o m e g a i n t e g r a l s , w h i c h a r e c o l l i s i o n c r o s s s e c t i o n s , a v e r a g e d o v e r the k i n e t i c e n e r g y of the r e l a t i v e m o t i o n of the p a r t i c l e s , c a n be d e n o t e d g e n e r a l l y a s i.j w h e r e i, j i n d i c a t e s the i n t e r a c t i o n b e t w e e n the m o l e c u l e s i and j . T h e y a r e o b t a i n e d by t h r e e s u c c e s s i v e i n t e g r a t i o n s . F i r s t the a n g l e of d e f l e c t i o n ^ of two c o l l i d i n g m o l e c u l e s i s c a l -c u l a t e d .
x ( g , b , T ) = TT - 2 b . „ - d r b ' 9 ( r )
r 2 |A«g2 ( 1 , 6 ) w h e r e
g, the r e l a t i v e v e l o c i t y of the two p a r t i c l e s b, the i m p a c t p a r a m e t e r
fx, the r e d u c e d m a s s of the two m o l e c u l e s
r^ , d i s t a n c e of c l o s e s t a p p r o a c h 9 ( r ) , the i n t e r m o l e c u l a r p o t e n t i a l f i e l d . S e c o n d l y the c r o s s s e c t i o n Q f o r d i f f e r e n t c o l l i s i o n s i s d e -t e r m i n e d f/n T\ - OTT n - r-r^a ^ -. Q ' ( g , T) = 27r (1 - c o s ' x) bdb ( 1 . 7 ) i = 1 "diffusion c r o s s s e c t i o n " i = 2 " t h e r m a l c o n d u c t i v i t y and v i s c o s i t y c r o s s s e c t i o n " . F r o m t h e s e c r o s s s e c t i o n s the o m e g a i n t e g r a l s a r e c a l c u l a t e d " ^ • ' (T) A I e-y'^ T2S+3 Qi(g) d r (1,8) 1/ 2ir/.i 0'' w h e r e y = -!^^
and s i s the a p p r o x i m a t i o n in the c o l l i s i o n i n t e g r a l .
T h e a s t e r i k m e a n s t h a t the o m e g a i n t e g r a l i s divided by the c o r r e s p o n d i n g i n t e g r a l f o r the r i g i d e l a s t i c s p h e r e m o d e l . In t h i s way t h e y a r e t a b u l a t e d a s function of the r e d u c e d
k T t e m p e r a t u r e T * = - — , ( w h e r e k = B o l t z m a n n c o n s t a n t ) in ^ 12 ref. 13 f o r d i f f e r e n t m o l e c u l a r m o d e l s . T h e m o s t a d e q u a t e d e s c r i p t i o n of t h e r m a l f a c t o r s i s o b -t a i n e d wi-th 1. a L e n n a r d - J o n e s 12-6 p o t e n t i a l 9 ( r ) = 4 e [(^)''' -(^)'} ( 1 , 9 ) w h e r e 9 (r) i s the p o t e n t i a l e n e r g y of i n t e r a c t i o n b e t w e e n two m o l e c u l e s a t d i s t a n c e r .
1 6
-T h e m e a n i n g of a in A and f 'k in '"'K follows f r o m fig, [, 1.
Fig. 1,1. The Lennard Jones (12 - 6) and the modified Buckingham potential. 2. a m o d i f i e d B u c k i n g h a m o r e x p o n e n t - 6 naodel 6 ^ t ? ( r ) 1 -s J 6 s(i — ) 6 i ^ ^m
(f)
) f-- -
> r „ )(r) = CO for r ^ r s e e fig. I, 1 ( 1 , 1 0 ) w h e r e s = the s t e e p n e s s p a r a m e t e r of the r e p u l s i v e p a r t of the p o t e n t i a l . F o r g a s e o u s m i x t u r e s the c o r r e s p o n d i n g f o r c e c o n s t a n t s h a v e to be c o m b i n e d in o r d e r to get the i n t e r a c t i o n of u n -l i k e m o -l e c u -l e s . A r i g o r o u s t r e a t m e n t of t h e s e c o m b i n a t i o n r u l e s h a s b e e n given by M a s o n and R i c e ^ ^ ) . F o r an e x p o n e n t - 6 m o d e l t h e y a r e r a t h e r c o m p l i c a t e d . In c a s e of a L e n n a r d - J o n e s p o t e n t i a l t h e y s i m p l y b e c o m e CT^ + ag '12 (1<11) •12= V^
( 1 . 1 2 ) In a l l o u r c a s e s , w h e r e one of the c o m p o n e n t s i s in t r a c e c o n c e n t r a t i o n s ( 1 , 1 ) r e d u c e s to a = - (6 C * - 5) S„ Q o ( 1 , 1 3 ) and with t h i s e q u a t i o n we h a v e c a l c u l a t e d the t h e r m a l dif-fusion f a c t o r s of the m i x t u r e sH T g - Hg, DT - Do. Dg - HT, HT - "^He, DT - ^He and T2 - He from 70^ - 1000 °K using the force constants given by Mason and Rice29), obtained from second virial coefficient and viscosity data, (taking the same values of the molecular p a r a m e t e r s for all isotopic hydrogen m o l e -cules).
TABLE I.I
Force constants for a Lennard-Jones potential. «2 0 3.287 s / k 37.00 He 2.869 10.22 Hg-He mixtures 3.078 19.44 1 TABLE I, II
Force constants for an exponent-6 model. " 2 1 ^m 3-33^ e / k 37.3 s 14.0 He 3.135 9.16 12.4 Hg-He mixtures 3.244 18.27 13.22 1
The r e s u i i s a r e given in the tables (I, lU), (I, IV), (I, V) and (I, VI) and the figures 1,2-1,7.
The second approximation of Kihara gives slightly better r e s u l t s . In dase of the isotopic hydrogen mixtures the t h e r -maldiffusion factor i n c r e a s e s about 2%, for the helium-hydrogen m i x t u r e s the improvement is 4%.30)
1 8
-TABLE 1,111
Theoretical tHermaldiffusion factors of isotopic hydrogen m i x -tures, calculated for a Lennard-Jones 12-6 molecular model.
T °K 74 89 104 118 133 148 167 185 222 259 296 333 870 740 1110 D2-H2 HT-H2 0.088 0.110 0.127 0.138 0.148 0.156 0.164 0.169 0.179 0.183 0.187 0.188 0.190 0.193 0.193 DT-D2 0. 029 0.036 0.042 0.046 0.049 0.052 0.054 0.056 0.059 0.060 0.062 0.062 0.063 0.064 0.065 HT-D2 Over the whole t e m p e -rature range ZERO •HT-Hg 0.064 0.080 0.092 0.100 0.108 0.114 0.119 0.123 0.130 0.133 0.136 0.137 0.138 0.140 0.141 •DT-Dg 0.025 0.032 0.037 0.040 0.043 0.045 0.047 0.049 0.052 0.053 0.054 0.054 0.055 0.055 0.056 •HT-D2 - 0. 026 - 0. 032 - 0.037 - 0. 040 - 0. 043 - 0. 046 - 0. 048 - 0. 050 - 0. 053 - 0. 054 - 0. 054 - 0. 055 - 0.056 - 0. 057 - 0. 057 •HT = effective mass of 3.27
TABLE I. IV
Theoretical thermaldiffusion factors of isotopic hydrogen mixtures, c a l -culated for an exponent-6 model.
T ° K 75 93 112 131 149 187 224 261 298 336 373 448 522 597 6 7 1 746 032 1119 1305 1492 D 2 - H 2 H T - H 2 0 . 0 8 4 0 . 1 0 8 0 . 1 2 5 0 . 1 3 7 0 . 1 4 5 0 . 1 5 7 0 . 1 6 4 0 . 1 6 8 0 . 1 7 0 0 . 1 7 1 0 . 1 7 2 0 . 1 7 2 0 . 1 7 2 0 . 1 7 1 0 . 1 7 1 0 . 1 7 1 0 . 1 6 9 0 . 1 6 8 0 . 1 6 6 0 . 1 6 6 D T - D 2 0 . 0 2 8 0 . 0 3 6 0 . 0 4 1 0 . 0 4 5 0 . 0 4 8 0 . 0 5 2 0 . 0 5 4 0 . 0 5 5 0 . 0 5 6 0 . 0 5 6 0 . 0 5 7 0 . 0 5 7 0 . 0 5 7 0 . 0 5 6 0 . 0 5 6 0 . 0 5 6 0 . 0 5 6 0 . 0 5 5 0 . 0 5 5 0 . 0 5 5 H T - D g Over t h e whole t e m p e -r a t u -r e r a n g e ZERO • H T - H g 0 . 0 6 1 0 . 0 7 9 0 . 0 9 0 0 . 1 0 0 0 . 1 0 6 0 . 1 1 4 0 . 1 1 9 0 . 1 2 2 0 . 1 2 3 0 , 1 2 5 0 . 1 2 5 0 . 3 2 5 0 . 1 2 5 0 . 1 2 5 0 . 1 2 4 0 . 1 2 4 0 . 1 2 3 0 . 1 2 2 1..121 0 . 1 2 1 • D T - D 2 0. 024 0 . 0 3 1 0 . 0 3 6 0 . 0 3 9 0 . 0 4 2 0 . 0 4 5 0 . 0 4 7 0 . 0 4 8 0 . 0 4 9 0 . 0 4 9 0 . 0 5 0 0 . 0 5 0 0 . 0 5 0 0 . 0 5 0 0 . 0 4 8 0 . 0 4 8 0 . 0 4 8 0 . 0 4 8 0 . 0 4 8 0 . 0 4 8 • H T - D 2 - 0 . 0 2 5 - 0. 032 - 0. 037 - 0. 040 - 0. 043 - 0. 046 - 0. 048 - 0. 049 - 0 . 0 5 0 - 0. 050 - 0 . 0 5 0 - 0 . 0 5 1 - 0 . 0 5 0 - 0. 050 - 0 . 0 5 0 - 0 . 0 5 0 - 0. 049 - 0. 049 - 0 . 0 4 9 - 0. 049 •HT = effective mass of 3.27 •DT = effective mass of 4.87
2 0
-TABLE 1. V
Theoretical thermaldiffusion factors of hydrogenhelium m i x -tures for a Lennard-Jones potential.
T ° K 80 100 120 140 160 180 200 220 240 260 280 300 500 1000 D 2 - % e H T - ^ e 0 . 0 2 9 0 . 0 3 0 0 . 0 3 0 0 . 0 2 9 0 . 0 2 9 0 . 0 3 0 0 . 0 3 1 0 . 0 3 2 0 . 0 3 2 0 . 0 3 2 0 . 0 3 2 0 . 0 3 2 0 . 0 3 2 0 . 0 3 2 D T - % e 0 . 0 9 2 0 . 0 9 7 0 . 0 9 9 0 . 1 0 0 0 . 1 0 1 0. ;02 0 . 1 0 5 0 . 1 0 6 0 . 1 0 6 0 . 1 0 6 0 . 1 0 6 0 , 1 0 6 0 . 1 0 6 0 . 1 0 6
1
T g - ' ^ H e ' H T - ^ e 0 . 1 4 5 0 . 1 5 4 0.157 0 . 1 5 9 0 . 1 6 1 0 . 1 6 4 0 . 1 6 7 0 . 1 6 7 0 . 1 6 8 0 . 1 6 8 0 . 1 6 8 0 . 1 6 8 0 . 1 6 8 0 . 1 6 8 - 0 . 0 2 9 - 0 . 0 3 1 - 0. 034 - 0. 035 - 0. 035 - 0. 036 - 0. 035 - 0. 035 - 0. 035 - 0. 035 - 0. 035 - 0. 035 - 0 . 0 3 5 - 0 . 0 3 5 • D T - * H e 0 . 0 8 4 0 . 0 8 9 0 . 0 9 0 0 . 0 9 1 0 . 0 9 2 0. 094 0 . 0 9 6 0 . 0 9 6 0. 096 0 . 0 9 6 0 . 0 9 6 0 . 0 9 6 0 . 0 9 6 0 . 0 9 6 *HT = effective mass of 3.27 •DT = effective mass of 4.87Theoretical thermaldiffusion factors of hydrogen-helium mixtures for an exponent-6 model.
T ° K 55 64 73 91 110 128 146 164 182 219 256 292 329 365 458 548 639 731 822 914 Dg- He H T - % e 0,022 0,023 0,024 0,025 0,026 0,027 0.028 0.028 0.028 0.029 0,029 0,029 0,029 0,029 • 0,029 0,029 0,029 0,029 0,029 0,029 DT-'^He 0,068 0.073 0,078 0,083 0,086 0,089 0.090 0.091 0.091 0.091 0.092 0.092 0.091 0.091 0,091 0,090 0.090 0.089 0.089 0.089 T2-^He 0.107 0.116 0.123 0.133 0.138 0.141 0.143 0.144 0.145 0.146 0.146 0.145 0.144 0.144 0.143 0.142 0.141 0.140 0.139 0.139 * H T - % e - 0.020 - 0. 023 - 0.025 - 0, 027 - 0. 028 - 0. 029 - 0. 029 - 0. 029 - 0.030 - 0. 029 - 0.029 - 0. 029 - 0. 028 - 0. 028 - 0. 027 - 0. 027 - 0. 027 - 0. 026 - 0. 026 - 0. 026 •DT-^He 0.062 0.067 0.071 0.076 0.079 0.081 0.082 0.083 0.083 0.084 0.084 0.084 0.083 0.083 0.083 0.082 0.081 0.081 0.081 0.081 •HT = effective mass of 3.27 •DT = effective mass of 4.87
2 2
-As was stated in the introduction^ these calculations only hold for mono-atomic g a s e s . All our experiments however, deal with mixtures of which at least one of the components is poly-atomic. In the first place we have thus to consider the consequences of the diatomic c h a r a c t e r of different i s o -topic hydrogen molecules.
1. The intermolecular potential field of two hydrogen mole-cules is dependent on their mutual orientations. This fact has been considered by de Boer"^-^) and Kihara'^^). Fortunately corrections for the non spherical behaviour are not n e c e s s a r y a s they a r e automatically made bj determination of force constants from experiments. 2. The molecular p a r a m e t e r s of hydrogen molecules a r e
dependent on their rotational s t a t e s . This effect is cal-culated by Knaap and Beenakker'^"^): the relative difference between the state 1 = 0 and 1 = 2 is for e 3 %o and for ( 7 - 0 , 4 °/oo (negative, because of the l a r g e r contribution of the increased attraction). F o r deuterium, tritium and mixed isotopes the influence is l e s s .
3. It has been long assumed that the force constants of different hydrogen isotopes a r e the s a m e . Recently S r i v a s -tava and Barua '^^) have calculated from second v i r i a l coefficient data the force constants of deuterium and hydrogen for an exponent-6 model (see Table I, VII).
TABLE I . V n " 2 •^2 s 14.0 14.0 0 E / k K 3 8 . 0 2 3 7 . 3 3 o 3 . 3 3 9 3 . 3 3 4
The r e s u l t s for H2 differ slightly from those of Mason and Rice, given in table I, II.
Knaap and Beenakker •^^) have determined from the dif-ference in polarizability of H9 and D^ the molecular p a r a m e t e r s of these molecules for a L e n n a r d - J o n e s poten-tial (table I, VIII).
TABLE I, VIII « 2 "2 £ / k ° K 3 7 . ÜÜ 3 8 . 6 7
.X
2. 028 2 . 9 2 74. The effect of inelastic collisions.
When a diatomic molecule collides with an other atom or molecule, it can change its rotational state. These inelastic collisions give r i s e to the so-called Eucken correction for the thermal conductivity of poly-atomic g a s e s .
Takayanagi ^^'^^j has calculated the rotational transitions in Hg, Dg and HD as a function of t e m p e r a t u r e for a Morse potential,
TABLE I.rx
Rotational transition probability in H,
r o t a t i o n a l t r a n s i t i o n 0 - » 2 1 - . 3 2 - 4 F = t r a n s i t i o n p r o b a b i l i t y 100 °K 1.5 1 0 " ^ 9 . 6 1 0 " ^ 9 . 6 1 0 ' ^ ° 2 0 0 ° K 33 1 0 " ^ 1 4 , 6 1 0 ' ^ 9. 0 1 0 " ' ' 3 0 0 °K 112 1 0 " ^ 9 . 6 1 0 - 5 112 1 0 - ' ' TABLE I,X
Rotational transition probability in D.
r o t a t i o n a l t r a n s i t i o n 0 — 2 1 - 3 2 - » 4 F = t r a n s i t i o n p r o b a b i l i t y 1 100 °K 5 . 1 1 0 - 4 2 . 3 1 0 " ^ 1 . 4 10-Ö 200 °K 3 . 0 1 0 - ^ 3 . 8 1 0 " * 0 . 7 1 0 - 4 3 0 0 ° K 6 . 4 1 0 " ^ 1.2 1 0 - ^ 3.0 10-4 1 TABLE I,XI Rotational transition
proba-bility in HD. r o t a t i o n a l t r a n s i t i o n 0 - 1 1 - 2 F 200 °K 0 . 1 3 0.05 1
r
2 4
-These theoretical data are in r a t h e r good agreement with experimental values for H2 and D2, obtained from ultrasonic sound dispersion experiments'^'^ ). It seems that this will be also the case for HD (unpublished experiments of K. W . T a -conis, J. J . M. Beenakker and C.G. Sluiter. F . O . M , annual r e p o r t 1961, page 85).
It appears thus that the number of inelastic collisions may be neglected with r e s p e c t to the elastic ones for sym-m e t r i c isotopic hydrogen sym-molecules, not however for the a s y m m e t r i c molecules. This is mainly due to the facts that a. the a s y m m e t r i c molecules have no ortho a n d ' p a r a
modi-fications, so that they can change their rotational states by one.
b. the non spherical part of the potential is much l a r g e r , as the center of gravity of the molecules does not coin-cide with -the e l e c t r i c center.
Summarizing we may conclude that the change in force constants of hydrogen for deuterium and tritium is small. These deviations a r e of little importance compared to the differences in thermaldiffusion factors calculated with the L e n n a r d - J o n e s or exponent-6 model, the latter still giving the best agreement between theory and experiment.
F u r t h e r m o r e inelastic collisions do not affect the t h e r m a l -diffusion factors of isotopic symmetric hydrogen molecule m i x t u r e s or m i x t u r e s of these molecules with helium.
This can be deduced from the fact that with the force constants of table I, II theoretical values of the thermaldif-fusion factors of the s y s t e m s *He - Hg, ^He - Dg and Ug - Hg are in agreement with experiments, done at 150°-450°K ^). This in contradiction to the thermal conductivity of hydrogen, which has to be c o r r e c t e d with 25% for inelastic collisions to obtain a satisfactory agreement between theory and ex-periment ^•^)
About the a s y m m e t r i c molecules there is hardly anything to say. The non spherical part of the intermolecular poten-tial field is much l a r g e r . As a consequence the number of inelastic collisions is increased. About the influence of both effects on the thermaldiffusion factor nothing is known yet. Unfortunately experimental data of the t r a n s p o r t p r o p e r t i e s of a s y m m e t r i c hydrogen molecules are also scanty
Only the thermal diffusionfactor of the system HD-Hg ^^), the viscosity of HD from 20°-80°K^^) and the viscosity of H2-HD a n d D a - H D m i x t u r e s from 1 4 ° - 2 9 3 ° K 3 ^ ) are m e a s u r e d .
§2. The model of Waldmann.
In the previous paragraph it was concluded that the ano-malous behaviour of a s y m m e t r i c molecules could be the r e s u l t of an a s y m m e t r i c potential field and of an i n c r e a s e of the number of inelastic collisions. In order to stay with the Chapman-Enskog theory, the a s y m m e t r i c potential must be averaged and replaced by a s y m m e t r i c one with new molecular p a r a m e t e r s . However, it is in this way impos-sible to explain the deviations of the thermaldiffusion factor with these new force constants for a Lennard-Jones poten-tial. The changes in e and a should be too large and not r e a l i s t i c any longer. In the HT-Ho system e . g . the dia-m e t e r of HT should be decreased with ~ 18%.
F r o m T* = 10 to higher t e m p e r a t u r e s the thermaldif-fusion factor becomes constant for a Lennard-Jones potential. Variations in e/k only shift the curves I in figs. 1,2-1,7
HT-Hj
. n
HL
1Ó0 300 500 7 0 0 . T ' K
Fig. I, 2. The thermal diffusion factor of the system HT H2 as a function of t e m -perature.
I. Theoretically for a Lennard Jones (12 - 6) potential. II. Theoretically for an exp-6 model.
III. Theoretically for a Lennard Jones potential with an effective mass for HT of 3.27.
IV. Theoretically for an exp-6 model with an effective mass for HT of 3.27.
A Theoretically with effective potential. o. Experimentally swing separator. X. Experimentally two-bulb apparatus.
V Theoretically Do - Ho with an effective potential.
0,20 0,15 0,10 / /
-r.
^ ^ a , OJLJ!—"—*—-26-906 Obi 0,0 2 0 o -^ o L » ' i » « I II III IV D I - D j ' A '
Fig. 1,3. The thermal diffusion factor of the system DT - D2 as a function of tem-perature.
Ill and IV: effective mass DT is 4.86. Legend see fig. I, 2.
Dj-HT
nr
IV
0 0 0 0 0 0
Fig. 1,4. The thermal diffusion factor of the system Dg - HT as a function of tem-perature.
0.04 0,02-Q 0,02 0,0/: j ^ ' — ' * ' D • N ^ ^ \ ^ 0 I 11 IV III HI-*He r'K
Fig. 1,5. The thermal diffusion factor of the system HT - He as a function of tem-perature.
D experilTlentaUy^ column experiments. Legend see fig. I, 2.
T°K
'ig. 1,6. The thenrial diffusion factor of the system DT "^He as a function of t e m -perature.
• 2 8
Fig. 1,7. The thermal diffusion factor of the system Tg He as a function of t e m -perature.
Legend see fig. I, 2.
along the temperature axis. Simple physical arguments neither succeed. One of the simplest considerations is to assume a l a r g e r diameter for the a s y m m e t r i c molecules, because they rotate around their gravity center. The more the distance between this point and the geometrical mid-point is l a r g e r , more the effective diameter should be in-c r e a s e d . This is however in in-contradiin-ction with experiments, which r e q u i r e a d e c r e a s e of the diameter, see e . g . ref. 18, 19 and 20.
The exponent-6 model however, gives more flexibility because of the extra p a r a m e t e r , the steepness s.
The influence of s on the thermaldiffusion factor is r a t h e r big, as is shown in fig. I, 8 for the HT-Hg system.
The first and only attack on these problems has been given by Schirdewahn, Klemm and Waldmann-").
A formula is derived with dimension analysis, which holds for the thermaldiffusion factor of isotopic s y s t e m s .
' M Ml - M2 Mj + M2 + C 0 © 1 - © 2 &, + 0 . (1.14) where M^, Mg are the m a s s e s of the molecules and 0^, 0 2 their moments or inertia.
0,15
Fig. I, 8. The thermal diffusion factor of the system D2(HT) - H2 as a function of temperature for different values of the steepness parameter s.
C^ and C0 are calculated with the values
a ( D „ , H j = 0.15 and a(D„, HT) = 0.028 C
M 0 . 2 5 CQ •= 0.20 at 360°K. On first sight the contribution of rotation is of the same o r d e r as the contribution of the translation. But' if C^i and C 0 are calculated with the values Qr(D2, H2J = 0.15 and a(DT, D2) = 0.042, the latter also being m e a s u r e d by Schir-dewahn, it is found that
C M = 0.05 CQ = 0 . 4 0 (1,14a)
F r o m these values it could be concluded that the influence of translation can be neglected in comparison with rotation. It s e e m s m o r e reliable to determine the constants Cw srd C0 with a l e a s t square method for all m e a s u r e d values, so that a better representation is obtained.
We changed the calculation of Waldmann in the following way: Giving the a s y m m e t r i c molecule the subscript 1, the moment of inertia are 'Respectively:
3 0
-©2 = Mg R ' ( 1 , 1 6 )
w h e r e the m e a n i n g of R and 6 c a n be s e e n f r o m fig. I, 9. T h e s e c o n d t e r m of ( 1 , 14) c a n be w r i t t e n
0-, - 0o M .f - M ff
0 J + 0 2 Mgff + M 2 M^.ff = effective m a s s .
( 1 , 1 7 )
FiR. I. Schematic drawing of an asymmetric diatomic molecule.
m^, the heavy atom. m2. the light atom.
5, the distance between the electrical midpoint and the center of gravity. 1 w h e r e M eff 1 M, 1 + M l Ó' 0 ,
J
( 1 , 1 8 ) E q u a t i o n ( 1 , 18) c l o s e l y r e s e m b l e s the e x p r e s s i o n which F r i e d m a n n " * " ) i n t r o d u c e s for m a s s e s of i s o t o p i c m o l e c u l e s , in o r d e r to e x p l a i n t h e i r v a p o u r p r e s s u r e d i f f e r e n c e s . T h i s e x p r e s s i o n i s M eff- V •" 3
MÖ' & ( 1 , 1 9 ) U s i n g ( 1 , 14a) the e q u a t i o n ( 1 , 14) c a n be a p p r o x i m a t e l y w r i t t e n a s M a = 0. 45 eff Mr Meff + M2 ( 1 , 2 0 ) F r o m ( 1 , 19) we get Mgff (HT) = 3 . 2 7 , M^ff (HD) = 2 . 7 7 , rf,tf (DT) = 4 . 8 6 . S u b s t i t u t i o n of t h e s e v a l u e s in ( 1 , 2 0 ) g i v e s the t h e r m a l -'liffusion f a c t o r s . In t a b l e I, XIa t h e s e v a l u e s a r e t a b u l a t e d t o g e t l i e r with the e x p e r i m e n t a l v a l u e s of S c h i r d e w a h n .TABLE I.Xla
The'inal diffusion factors, calculated with an effective mass, compared with the experimental values of Schiidewahn.
«(Dg, HT) = 0.045 aCHT, H2) = 0.108 a ( D T , D2) = 0.044 a(HD, H2) = 0.073 (0.028)' (0.12) (0.046) (0.078)* * Schirdewahn's values.
Although uptill now no r e a s o n exists to couple both pheno-mena, the agreement of our calculations with those of Schir-dewahn is r a t h e r good and it s e e m s that the a s y m m e t r i c molecules behave as if they have a m a s s , given by (1, 19).
Therefore we have calculated the thermaldiffusion factors of the s y s t e m s HT-Hg, DT-Dg, Dg-HT, HT-*He and DT-*He with (1, 13), taking for HT and DT their effective m a s s e s from (1,19). The r e s u l t s are given in tables (I, III)-(I, VI) and figures I, 2-1, 7.
One should of course not await from these simple a r g u -ments a complete agreement between theory and experiment, especially for the m i x t u r e s D2-HT and HT-^He. In these c a s e s the m a s s e s are equal, so the deviations due to the a s y m m e t r y of the molecules come out here most pronounced.
§3. The molecular interaction of asymmetric molecules.
It is possible to calculate the influence of the m a s s d i s tribution in isotopic hydrogen molecules on the i n t e r m o l e -cular potential field in exactly the same way as de Boer'^^) determines the non spherical potential between two s y m m e t r i c hydrogen molecules.
The repulsive part of this potential i s , according to de Boer, the sum of the interactions between the four atoms of the two molecules AB and CD, see fig. I, 10.
9 = B (e'^'^AC + ^:^r^D ^ g-a^Bc ^ g-ar^D
rep * ^ ' ' where a is 3.53 K'^ and B 87.7 10'*°K.
3 2
-Fig. 1,10. The interaction of two diatomic molecules.
If the distance between the two molecules is large com-p a r e d with the internuclear distance, 2R = AB (= 0. 75 A), the v a r i a b l e s r^,-, r^j-,, rg,- and rgp may be replaced by their projections on the connection line r of the two c e n t e r s of gravity*-^). So we get for a H2-H2 interaction
9 = 4Be ^^cosh (aR cos 9^) cosh (aR cos Ö2) The attractive part of the potential is given by
3 A Ë 52
4 r ' 1 - (1 - -x cos^
" 2 ^ ° ^ '
0 2 ) y( 1 , 2 2 )
;i,23) where
AE is the ionization energy 18 eV = 2 0 . 9 . lO'* °K.
a the average polarizability (cm"^).
y the anisotropy factor of the polarizability 0. 137.
T e r m s withy^ and quadrupole interaction have been neglected. The angle dependent equations (1, 22) and (1, 23) can be transformed into effective potentials by averaging them with a Boltzmann factor (see ref. 13, pag. 985).
Weff
S^
-<?/kT
S i n \ sin Ö2 d 9^ d 9^
rpe-'P/'^Tsin 9i sin 92 d 9^ d 92
( 1 , 2 4 )
As the temperature dependent t e r m s are small, e . g . they contribute for only 2% at 50°K tc the repulsive field'^'^), so that e-9/kT = 1^
the r e p u l s i v e and a t t r a c t i v e p a r t of the p o t e n t i a l c a n be a v e r a g e d s e p a r a t e l y .
['rep]
4 B e - a r a 2 R 2 s i n h aR ( 1 . 2 5 )['an]
re f 3 A E a ^ 4 r 6 ( 1 . 2 6 )T h e s e effective p o t e n t i a l s c a n now be i d e n t i f i e d with the c o r r e s p o n d i n g p a r t s of the e x p o n e n t - 6 m o d e l : 4 B e - a r a2R2 sinh^ aR = s ( l ) 1 - 1 s - e s ( 1 . 2 7 ) 3 AEo-^ 4 r 6 1 -s ( ) ( 1 . 2 8 ) The s a m e p r o c e d u r e c a n be a p p l i e d to the i n t e r a c t i o n of a s y m m e t r i c m o l e c u l e with an a s y m m e t r i c o n e . R e g a r d i n g CD a s the a s y m m e t r i c m o l e c u l e and r e p l a c i n g the d i s t a n c e s ^AC . ^AD . ^Bc ^ " d rgD a g a i n by t h e i r p r o j e c t i o n s on the c o n n e c t i n g line of the c e n t e r s of g r a v i t y , y i e l d s : 9 = 4 B e - a r c o s h (aR c o s 9, ) c o s h (aR c o s 62)6^^ ^°^ ^2 '^P ( 1 , 2 9 ) The a t t r a c t i v e p o t e n t i a l b e c o m e s - 3 A 1 Ö 2 4 ( r - Ó cos 02)
[ i - , . - i
c o s 9, - -^ cos^)y]
( 1 , 3 0 ) A v e r a g i n g o v e r the a n g l e 9^ of the s y m m e t r i c m o l e c u l e g i v e s s i n h aR ^rep = 4 B e -• a r a R - S A E Q - ^ c o s h (aR c o s 62) e^^ c ° ^ ^2 ( 1 , 3 1 ) 4 ( r - 6 c o s 62) 1 1 , 3 2 o - ö T + 2 •>' ^ ° ^ ^2 ; i , 3 2 )3 4
-With ( 1 , 27) and ( 1 , 28) the t o t a l p o t e n t i a l of the i n t e r a c t i o n of an a s y m m e t r i c m o l e c u l e with a s y m m e t r i c one i s w r i t t e n
6 fg s ( i - ^ ) a R
?tot = (T -^ - e 'm . —. cosh (aR c o s 6^) . e^^ ^°^ ^2 1 - ^ L s i n h a R
. ^m(^ - | T + | 7 C O S ^ 0 , | (^^ 33^ ( r - Ó c o s 92)^ J
With the i n t r o d u c t i o n of the r e d u c e d p a r a m e t e r s tp r 6 R
< ? ' ' = - r'^ = — 6'' = — , R"" = — ( 1 . 3 4 )
and b e a r i n g in m i n d that a =— and ^ = 0 . 1 3 7 , ( 1 , 3 3 ) b e c o m e s
9 ' = ^ f ~ e ^ ^ ' " ' ^ o s h ( s R > ' c o s 0 2 ) e « ^ ' ^ ° « ^ 2 1 - ^ l^ s i n h a R"" 0. 924 + 0. 228 c o s ^ 0^ (r^ - 6'' c o s 9 ) 6 ( 1 , 3 5 ) Tlie effective p o t e n t i a l i s •+i r + i 9"^ e x p ( - 9 V T ' ' ) dy
WIff ~-'^, ^''''^
J exp(-9VT-^) dy
wliere y = c o s 92-The i-educed t e m p e r a t u r e T i s e x a c t l y the s a m e a s the r e d u c e d t e m p e r a t u r e T*, which is u s e d in the c a l c u l a t i o n s for the c o l l i s i o n i n t e g r a l s . [ v"^ ! rr c a n be c a l c u l a t e d by G a u s s i a n integration'*^) ' eff • + 1 f ( z ) d z = E Wj f ( Z i ) jj i=l ( 1 , 3 7 )
In o u r c a s e s a s i x - p o i n t i n t e g r a t i o n (n = 6) i s s u f f i c i e n t . Z i = - z g = 0 . 9 3 2 4 6 9 5 1 4 2 Ui = we = 0 . 1 7 1 3 2 4 4 9 2 4 Z2 = - Z 5 = 0 . 6 6 1 2 0 9 3 8 6 5 Wg = 1 ^ 5 = 0 . 3 6 0 7 6 1 5 7 3 0 Z3 = - Z 4 = 0 . 2 3 8 6 1 9 1 8 6 1 W3 = u ^ = 0 . 4 6 7 9 1 3 9 3 4 6 T h e c a l c u l a t i o n s a r e d o n e w i t h a I. B . M . 7 0 9 0 c o m p u t e r f o r 6^ = 0 . 0 0 0 0 ( s y m m e t r i c m o l e c u l e s , H ^ , D , , T ^ ) . 6" = 0 . 0 2 2 5 ( D T ) . d"" = 0 . 0 3 7 5 ( H D ) . ó"- = 0 . 0 5 5 4 ( H T ) . a t T ' ' = 5, 10 a n d 2 0 . F o r t h e v a l u e s of t h e f o r c e c o n s t a n t s a r e t a k e n t h e e x -p e r i m e n t a l d a t a of t a b l e I, I I . T h e r e s u l t s a r e g i v e n i n tlie t a b l e s (I, X I I ) , (I, XIII) a n d (I, X I V ) .
TABLE I,XII
Effective potentials <p^ as a function of the reduced intermolecular distance r ' at the reduced temperature T' = 5 for the interaction of
a symmetric hydrogen molecule with an asyn metric one.
r'' 0 . 7 5 0 . 8 0 0 . 8 5 0 . 9 0 0 . 9 5 1.00 1.05 1.10 1.15 1.20 l . 4 u ).6o 1.80 2.UU
9^ 1
6-'^ = 0. 0000 1 0 . 1 3 9 4 . 1 0 1 6 1 . 0 8 0 5 - 0 . 3 4 0 7 - 0 . 8 8 0 9 - 1 . 0 0 2 2 - 0 . 9 3 3 5 - 0 . 8 0 2 9 - 0 . 6 6 4 8 - 0 . 5 4 0 6 - 0 . 2 2 9 7 - 0 . 1 0 4 1 - 0 . 0 5 1 4 - 0 . 0 2 7 3 &^ = 0 . 0 2 2 5 9 . 8 7 7 6 3 . 9 1 5 3 1 . 0 2 8 0 - 0 . 3 3 2 5 - 0 . 8 7 2 3 - 0 . 9 9 3 9 - 0 . 9 3 0 3 - 0 . 8 0 2 3 - 0 . 6 6 5 3 - 0 . 5 4 1 4 - 0 . 2 3 0 1 - 0 . 1 0 4 3 - 0 . 0 5 1 5 - 0 . 0 2 7 4 6^ = I). 0375 9. 3996 3 . 6 1 7 8 0 . 9 3 9 4 - 0 . 3 2 0 2 - 0 . 8 4 7 0 - 0 . 9 7 8 9 - 0. 9244 - 0. 8011 - 0. eiHUI - 0 . 5 4 2 7 - 0. 230!) - 0. 1045 - O.üölii - 11. 11274 6^ = 0. 0554 8 . 5 2 1 2 3 . 1547 0 . 7 9 0 3 - 11.3344 - il. 8025 - 0 . 9 5 0 8 - u . 9 1 2 8 - 0.7985 1 11.6672 - 0 . 5 4 5 1 - il. 2323 - 0. 1051 - 0 . 0 5 1 8 L - U. 112753 6
-TABLE l.XIII
Effective potentials 9 ^ as a function of the reduced intermolecular distance r" at the reduced temperature T'' = 10 for the interaction of
a symmetric hydrogen molecule with an asymmetric one.
r 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00 9' 6^ = 0.0000 11.6510 4.7463 1.2679 - 0.2975 - 0.8787 - 1.0011 - 0.9335 - 0.8029 - 0.6648 - 0.5405 - 0.2297 - 0.1041 - 0.0514 - 0. 0273 6" = 0. 0225 11.2244 4.6006 1.2672 - 0.2719 - 0.86 07 - 0.9923 - 0.9302 - 0.8022 - 0.6651 - 0.5412 - 0.2301 - 0. 1043 - 0.0515 - 0. 0274 6^ = 0.0375 10.5396 4.3572 1.2619 - 0.2282 - 0. 8288 - 0.9765 - 0.9241 - 0.8008 - 0.6657 - 0.5423 - 0.2308 - 0.1046 - 0.0516 - 0.0274 &^ = 0. 0554 9. 4641 3.9541 1.2403 - 0.1532 - 0.7699 - 0. 9464 - 0.9123 - 0.7979 - 0.6665 - 0. 5444 - 0.2321 - 0.1051 - 0.0518 - 0.0275 TABLE I,XIV
Effective potentials 9 " as a function of the reduced intermolecular distance r" at the reduced temperature T ' ' = 20 for the interaction of
a symmetric hydrogen molecule with an asymmetric one.
r'' 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00 9 " ó" = 0. 0000 13.0265 5.1668 1.3726 - 0.2748 - 0.8745 - 1.0006 - 0.9335 - 0.8029 - 0.6648 - 0.5405 - 0. 2297 - 0.1041 - 0.0514 - 0.0273 6^ = 0. 0225 12.6732 5.1374 1.4163 - 0.2385 - 0. 8547 - 0.9916 - 0.9301 - 0.8021 - 0.6650 - 0 . 5 4 1 1 - 0.2300 - 0.1043 - 0.0515 - 0.0274 ó" = 0.0376 12.868 5.0786 1.4899 - 0.1744 - 0.8190 - 0.9753 - 0.9240 - 0.8007 - 0.6655 - 0.5422 - 0.2308 - 0. 1046 - 0.0516 - 0.0274 6^ = 0. 0554 11.1231 4.9513 1.6130 - 0.0575 - 0.7519 - 0.9441 - 0.9120 - 0.7976 - 0.6662 - 0.5441 - 0.2321 - 0.1051 - 0.0518 - 0.0275
The same procedure is also applied to the interaction of a helium atom with different hydrogen molecules. The values of 6 " have changed because r,^ 12 is different for this c a s e .
5^ = 0.0000 He-Hg, D^, Tg interaction. 6x = 0.0231 He-DT interaction.
6'' = 0.0385 He-HD interaction.
0^ = 0.0570 He-HT interaction.
Calculations are performed with the experimentally d e t e r -mined force constants, listed in table I, II.
TABLE I,XV
Effective potentials 9 as a function of the reduced intermolecular distance r^ at the reduced temperature T^ = 5 for the interaction of
a helium atom with an as)Tnmetric hydrogen molecule.
X r 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.4Ü 1.60 1.80 2.00 9=^ ó'' = 0.0000 8.4823 3.4436 0.8375 - 0.4098 - 0.8979 - 1.0025 - 0.9374 - 0.8120 - 0.6773 - 0.5543 - 0.2390 - 0.1088 - 0.0538 - 0.0286 ó" = 0.0231 8.2368 3.2866 0.7983 - 0.4004 - 0.884C - 0.9943 - 0.9341 - 0.8113 - 0.6776 - 0.5550 - 0.2395 - 0.1090 - 0.0539 - 0.0286 &^ = 0. 0385 7.8076 3.0351 0.7311 - 0.3857 - 0.8598 - 0.9797 - 0.9279 - 0.8098 - 0.6782 - 0.5562 - 0.2403 - 0.1093 - 0.0540 - 0. 0287 6^ = 0.0570 7.0571 2.6421 0.6149 - 0 . 3 6 4 8 - 0 . 8 1 6 9 - 0 . 9 5 2 3 - 0 . 9 1 6 1 - 0.8068 - 0 . 6 7 9 1 - 0.5585 - 0.2418 - 0.1099 - 0.0542 - 0.0288
3 8
-TABI.C I . X \ I
Effective potentials 9 " as a functian of tlie reduced intermolecular distance r'^ at the reduced temperature T^ = lu for the interaction of
a helium atom with an asymmetric hydrogen molecule.
r 0.75 0.80 0.85 0.90 0.95 1.00 1.06 1.10 1.15 1.20 1.40 1.60 1.80 2.00 6^ = 0. 0000 9.7966 3.9817 0.9985 - 0.3702-- 0.8898 - 1.0013 - 0.9373 - 0.8120 - 0.6773 - 0.5542 - 0.2390 - 0.1088 - 0. 0538 - 0. 0286 ó-'^ = 0. 0231 9.4440 3.8765 1.0047 - 0.3457 0.8727 - 0.9927 - 0.9339 - 0.8112 - 0.6775 - 0.5548 - 0.2394 - 0.1090 - 0.0539 - 0.0286 6-^ = 0. 0385 8.8777 3.6977 1.0120 - 0.3036 - 0.8423 - 0.9772 - 0.9277 - 0.8095 - 0.6779 - 0.5559 - 0.2402 - 0.1093 - 0.0540 - 0.0287 ö-'' = 0. 0570 7.9877 3.3922 1.0142 - 0.2303 - 0.7863 - 0.9477 - 0.9154 - 0.8062 - 0.6784 - 0.5579 - 0.2416 - 0.1099 - 0.0542 - 0.0288 TABLE I,XVII
Effective potentials (p^ as a function of the reduced intermolecular distance r at the reduced temperature T = 2 0 for the interaction of
a helium atom with an asymmetric hydrogen m o l e c u l e .
X r 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.40 1.60 1.80 2.00 X 9 &^ - 0. 0000 10.8860 4.3202 0.1088 - 0.3496 - 0.8856 - 1.0006 - 0.9373 - 0.8120 - 0.6772 - 0.5542 - 0.2390 - 0.1088 - 0.0538 - 0.0286 6^ = 0. 0231 10.6426 4.3149 1.1304 - 0.3158 - 0.8668 - 0.9918 - 0.9338 - 0.8111 - 0.6774 - 0.5548 - 0.2394 - 0.1090 - 0.0539 - 0.0286 6^ = 0.0385 10.2298 4.2988 1.2036 - 0.2561 - 0.8331 - 0.9769 - 0.9275 - 0. 8094 - 0.6777 - 0.5558 - 0.2401 - 0.1093 - 0.0540 - 0.0287 h'^ = 0.0570 9.5281 4.2491 1.3286 - 0.1468 - 0.7695 - 0.9453 - 0.9152 - 0.8059 - 0.6781 - 0.6676 - 0.2416 - 0.1099 - 0.0542 - 0.0288
Effective force constants at a tunction of the reduced temperature T'' for the interaction ot a symmetric hydrogen molecule with an asyrrimetric one. ^ 2 . 2 X e r^ m s 0^ = 0.0000 5 • 0.992 0.999 13.1 10 0.987 1.003 13.1 20 0.993 1.002 13.5 &^ = 0. 0225 5 0.978 1.000 12.8 10 0.974 1.006 12.9 20 0.981 1.005 13.3 0^ = 0.0375 5 0.960 1.003 12.4 10 0.951 1.011 12.4 20 0.959 1.010 13.0 6^ = 0. 0554 5 0.927 1.008 11.6 10 0.912 1.020 11.7 20 0.922 1.021 12.4 TABLE I,XIX
Effective force couMants as a function of the reduced temperature T ' ' for the interaction of a helium atom with an asymmetric hydrogen molecule. - 1 . 2 X e ^m s 6^ = 0.0000 5 0.993 0.998 12.4 10 0.988 1.002 12.5 20 0.994 1.001 12.8 ó'^ = 0.0231 5 0.982 1.000 12.2 10 0.976 1.005 12.3 20 0.982 1.004 12.7 6== = 0.0385 5 0.964 1.003 11.8 10 0.954 1.010 11.9 20 0.961 1.009 12.4 6" = 0.057 0 5 0.931 1.008 11.2 10 0.916 1.019 11.3 20 0.926 1.020 12.0
4 0
-It can be seen from the values of <?^ for 6^ = 0 at T'^ =
5, 10 and 20 that the temperature dependence is g r e a t e r than
de Boer has stated^^ ). At lower temperature this will give r i s e to considerable deviations in his force constants for hydrogen.
The interval 1. 35 > r ' ' > 0. 75 of these effective potentials is used to calculate by an iteration method with the I . B . M .
7090 computer those values of r,^,^ , e^ and s of the exponent-6 model, which give the best fit. These force constants are listed in the tables I, XVIII and I, XIX. The deviations in the adjustments are in the order of 1%.
It is now possible to investigate in general the transport p r o p e r t i e s of a s y m m e t r i c molecules. But as this is beyond the scope of this thesis, we will only calculate the t h e r m a l -diffusion factors of the gas mixtures involved. The conse-quence of this r e s t r i c t i o n is that we get the worst comparison between theory and experiment, as it is a well known fact that of all transport coefficients the thermaldiffusion factor is the most sensitive to the choice of the molecular model. The r e s u l t s of the calculations are given in the tables I, XX and I, XXI and in the figures 1,2-1,7.
TABLE I,XX
Thermaldiffusion factors of isotopic hydrogen mixtures.
2.2 \ ^ 5 10 20 D2-H2 0.150 0.163 0.164 HT-Hg 0.127 0.138 0.143 DT-Dg 0.049 0.054 0.054 Dg-HT 0.0037 0.0034 0.0026 TABLE I,XXI
Thermaldiffusion factors of isotopic hydrogen-helium mixtures. h . 2 \ ^ ^ 5 10 20 HT -"'ne U.U23 0 . 0 2 6 0 . 0 3 0 DT-'*He U.084 0 . 0 9 ] 0. U96 4 T g - l l e U. 133 0 . 1 4 8 0 . 1 5 0
Finally, one r e m a r k should be made.
In the D2-H2 system the force constants for the (1,2) and (2, 2) in4;eractions are assumed to be the same and taken from table I, XVIII.
C H A P T E R I I
COLUMN THEORY
§ 1. Introduction.
The elementary effect in thermal diffusion p r o c e s s e s is so small, that this method was not used for separating isotopes until 1938. In that y e a r Clusius and Dickel'') described an apparatus in which the elementary effect is multiplied in a surprisingly simple way. This apparatus, called "thermal dif-fusion column", consists of two vertical surfaces, one hot and the other cold, so that the horizontal temperature gradient causes a partial separation by thermal diffusion. Due to the difference in density at the two surfaces and the gravitational field of the earth, a natural convection c u r r e n t is set up, which i n c r e a s e s the separation by transporting one component to the top and the other to the bottom of the column; the theory of this p r o c e s s had been worked out by various authors (43, 44, 45, 46).
In t e r m s of the cascade theory, a thermal diffusion column may be treated as a square cascade 4''), that is a cascade in which the feed into any stage along the entire length of the cascade is constant. The theory, developed by Cohen on the base of this equivalence, is, however, r a t h e r implicit and as the author explains, more or l e s s an e x c e r c i s e in formal logic. So its practical usefulness is quite drastically ob-scured and even limited. Although an outline of the application to thermal diffusion has been given ^^), tlie presentation r e m a i n s r a t h e r vague.
In this chapter a theory is developed of extended utility and simplicity, using Cohen's notation. The nature of the assumptions is shortly quoted and the theory as a whole is verified by comparing it with experimental r e s u l t s .
§ 2. General theory.
The differential equation for the separation in a thermal diffusion column is obtained by applying two times the equation of continuity, once for the whole mixture and once for the desired component, to the two flows in the column. These flows are
1. the t r a n s p o r t of the considered component under the in-fluence of a temperature gradient ^'^•^^•^°).
2. the transport in vertical direction, due to the convection current v.
In vector notation
T^ = -p D {grad N + o N ( l - N ) grad In T } + pN v (2, 1) where TQ is the t r a n s p o r t of the desired component (g/cm^
sec)
p, the total density (g/cm^)
N, the mole fraction of the considered component D, the diffusion coefficient ( c m ^ / s e c )
a, the thermaldiffusion factor
T, the absolute temperature (°K)
It is assumed that the t e m p e r a t u r e gradient has only a horizontal component and that the convection c u r r e n t occurs only vertically.
The solution of the differential equation may be found by the standard methods, but as the concentration gradients in all directions are small, the equation can be solved in an e a s i e r way, giving the typical equation for isotope separa -tion in square cascades and countercurrent colums ^•^•^^•^'^):
^6 f = -5 0 - è {PN + CiN(l-N)} (2,2)
The cascade p a r a m e t e r s c j , Cg and Cg have the following physical meaning:Cj, the "initial t r a n s p o r t " determines the enrichment in the column
Cg, takes into account the l o s s e s , due to backdiffusion in vertical direction and the dragging effect caused by the convection current
Cg, is the hold up, the amount of m a t e r i a l p e r unit length. The constants can be written down in cartesian or cylin-drical coordinates, according to the shape of the column
cartesian coordinates cylindrical coordinates
, j . . B r ' a ^ i i L l d x f \ v d x c, ^ - 2 . f ' ^ - ^ ^ d r /" pvrdr g/. Cj = B j pD dx cj = 2ff ƒ pDr dr g cm/f ( 2 , 3 ) s e e 5 " "^2 "^ *^3 g = B i Pdx 3 = B ( pvdx Cg - C j + Cg c^ =2. JVd. P = 2)1 ['•'pvrdr g/cm g/sec
4 4
-where x, r and z denote respectively the horizontal, radial and vertical' coordinate.
d, the distance between hot and cold wall (cm) B, the width of the column (cm)
P, the production rate (g/sec) r^,, the i'adius of the cold wall (cm) r , the radius of the hot wall (cm)
H
To C2 and C3 may be added another cascade p a r a m e t e r , designated c^, the value of which depends on the magnitude of the i r r e g u r a l i t i e s in the dimensions and temperature distribution of the column. The necessity for the introduc-tion was menintroduc-tioned by F u r r y and Jones '*®) (07 = Kp) and more recently by Dickel^'*-^^).
Equation (2, 2) has been solved by a number of au-thors'**'^^'^^'^®'^'^ ) for special c a s e s and, although the general solution has not yet been found, satisfactory approxi-mations have been given by Cohen (only r e s t r i c t i o n N < 1) and by Majumdar (P = 0).
A . THE STEADY STATE.
We first consider the stationnary state and shall discuss, when n e c e s s a r y , only the features of a flat column (cartesian coordinates).
In the steady state equation (2, 2) takes the form:
P Np = PN + c i N ( l - N ) - C5 1 ^ (2,4) where Np is the mole fraction of the product.
Integration with the boundary conditions: N = Nj, at z = 0, N = Np at z = Z gives
1 r (Np-No) A(^) ^
Z = t a n h - M } (2>5) eA(^) UNp-2Np No+No-(Np-No)(A J
where i/' = — , the normalized production rate (2,6) c 1 A((//) = V l + 2(// ( l - 2 N p ) +1//^' ( 2 , 7 ) ( 2 , 8 ) c 1 e = — 2c 5
At t o t a l r e f l u x the p r o d u c t i o n P i s z e r o so that 1 R — In — 2 e Ro N Z = — In — ( 2 , 9 ) with R 1-N • Rp
F r o m the definition q = — , w h e r e q i s the s e p a r a t i o n
R Q
f a c t o r , it follows that
In q = 2 e Z (2, 10) i . e . the s e p a r a t i o n f a c t o r for a t h e r m a l d i f f u s i o n c o l u m n with both e n d s c l o s e d .
The s e p a r a t i o n f a c t o r i s t h u s c h a r a c t e r i z e d by the two p a r a m e t e r s c^ and Cg, both c o n t a i n i n g the function
1 P V. dx
T h i s p r o d u c t d e t e r m i n e s the flow p a t t e r n and the m a g n i t u d e of the c o n v e c t i o n c u r r e n t . The influence of t h e s e two q u a n -t i -t i e s on -the e n r i c h m e n -t c a n be c o n v e n i e n -t l y s e p a r a -t e d by i n t r o d u c i n g the function: L = / ' I p v l d x , ( 2 , 1 1 ) / a | P V |
which r e p r e s e n t s the flow up and down the c o l u m n , J p vdx C o n s e q u e n t l y d o e s not depend on L . L S u b s t i t u t i o n in (2, 8) y i e l d s a^ "L 2 e = ( 2 , 1 2 ) Cg + ag L C^ Cg w h e r e a^ = — and a„ = .
4 6
-The maximum separation with r e s p e c t to L occurs at the optimal flow L ^
o
(2.13)
The corresponding value of 2e is then 2 e
2 Eo = p = ^ = , (2, 14)
2 V c 2 a 3 2 y c 2 C 3
So by rewritting the separation factor, the following ex-pression is obtained: 2m In q = 2 e „ Z (2,15) (2,16) 1 + m^ 2m
The function in (2, 15) gives the contribution of the 1 + m^
magnitude of flow to the separation factor.
As subsequently will be shown in paragraph 3, m appears to be a simple function of the working p r e s s u r e in the column
o
; m — p " .
The factor 2 e^ c h a r a c t e r i z e s the flow pattern, but un-fortunately the natural convection is completely determined by the shape of the column and the temperature distribution, so that it can not be chosen as an operating variable like the magnitude of the flow.
The maximum cf 2 CQ with r e g a r d to flow pattern has been calculated by Cohen'*^). The optimum flow pattern is in plane geometry entirely along the walls c . q .
J
J
X pv dx = constant 0 ^ x < d ^2, 17) X pv dx - 0. x - dand ( 2 e j ^ , , = « ^ l d (2.18) F o r natural convection we have (next paragraph)
2 ^ ° = - ¥ 2 : T d ^'-''^
so that the 2 e^ for natural convection is 83% of the onefor optimum flow pattern. It is worthwile to r e m a r k at this stage that the placing of baffles in the column can increase the efficiency of the flow pattern.
A very important function in counter current devices with a certain production r a t e , is the separative power 6U. This factor r e p r e s e n t s the quantity of separated m a t e r i a l multi-plied by the number of elements or the length of the column to produce it.
In t e r m s of cascade constants 6U can be written as
cf
6U = (2,20) 4 Cg
(2, 20) may be analogously treated as was done for the
separation factor above. This yields the following e x p r e s -sion
c^ m^
6U = (2, 21) 4 Cg 1 + m^
For natural convection
6 U = 0 , 7 0 SUgptj^m-n fio^^ pattern ( 2 , 2 2 ) The optimum value of the magnitude of flow at production is furthermore totally different from that at no production, because
6U = (5U)max for m-» 00 (2,23) In q = (In q),^^^ for m = 1 (2,24) The approach to (óU),-nax is however rapid, as can be seen
4 8
-6U = 0 . 9 0 (-6U),^^^ In q = 0 . 6 0 (In q ) ^ .
Fig. II, 1. The separation factor and separative power as a function of the magnitude of flow. I. f (m) = -II. f (m) = 6u (6u)^ 2m 1 + m2 1 + m2
F o r large m, the flow in the column is not longer l a m i n a r , but turbulent so that the separation factor d e c r e a s e s m o r e than is expected from (2, 15).
It is n e c e s s a r y at this stage to emphasize that the column theory holds for every separation p r o c e s s . It is only need-ful to specify the nature of the flow v and the elementary ef-fect in the cascade p a r a m e t e r c^.
This p a r a m e t e r can generally be written as
f (2a)odx r
O*' o^
pvdx (2,25)
where {2a)o c h a r a c t e r i z e s the separation effect, e . g . centrifuge, chemical exchange, m a s s diffusion, etc.
F o r thermaldiffusion (2 0-)^ is thus
( 2 « ) o 9 In T ax ( 2 , 2 6 ;
B . THE EQUILIBRIUM TIME OF COUNTERCURRENT COLUMNS.
The equilibrium time of a countercurrent column can be obtained by solving (2, 2) with boundary conditions, as de-termined by the column conditions.
There are two principal types of operation (fig. II, 2). In type I the column is operated with a given production P , drawn at the top from a hold-up HQ, while the bottom is fed with fresh m a t e r i a l , having the feed composition N^. In type II the column is connected with r e s e r v o i r s at the top and at the bottom.
• T O P
T y p e I BOTTOM Type II
Fig. 11,2. The two principal types of column operation.
Type I with P = 0 is equivalent to type II with H^ = oo. Columns with a given production rate are only important in c a s c a d e s . Mostly columns are operated with one or botli ends closed.
F o r mathematical convenience it is common use to take type I H„ = 0
type II H.
H i H. = 0
( 2 , 2 7 )
Equation (2, 2) with P = 0 and the boundary conditions (2, 27) has been solved by Majumdar^'^ ) by linearizing both
5 0
-the e q u a t i o n and -the b o u n d a r y c o n d i t i o n s with -the s u b s t i t u t i o n s
u = 2 e Z T = t (2, 28)
C 5 C 6
The r e s u l t i s o b t a i n e d in the f o r m of c o m p l i c a t e d infinite s e r i e S j but a c o n s i d e r a b l e s i m p l i f i c a t i o n a r i s e s ^ when one of the c o m p o n e n t s i s p r e s e n t in t r a c e c o n c e n t r a t i o n s .
T h e s o l u t i o n r e d u c e s then to
tyt)e I
q(t) = q(oo) - E Cn e ""''" ( 2 , 2 9 ) w h e r e q(oo) i s the s e p a r a t i o n f a c t o r in the s t a t i o n n a r y s t a t e
Cn = (2. 30) 1 - T ; l - e Z ( l - T ^ )
4 C 6 C 5 1
tn = t j > t o n = l , 2 , ( 2 , 3 1 )
c\ I -yl
Yj, a r e the s o l u t i o n s of the t r a n s c e n d e n t a l e q u a t i o n
t a n h y e Z = T {2, 32) s e e ref. 47 p a g e 180. type 11 1 + k E b n ( - l ) " e - ' ^ - ^ ' ^ n q ( t ) = q { c o ) . 2 ( 2 , 3 3 ) 1 + k E bn e-''^n n w h e r e q ( ^ ) - 1 N„ In q(oo) ( 2 , 3 4 ) 1 - ( - 1 ) " e - e Z ^ • « N „ - • ' , ( 2 , 3 5 )
4 c „ c ^ 1
n = 1,2.
An application of the approach to the steady state has been given by Dickel^^). He i n c r e a s e s the limit of age determination by radio carbon dating by enriching I'^Ci^O of a gaseous carbon monoxide sample with a thermaldiffusion column. As the separation factor cannot be calculated a c -curately Dickel determines the activity in the column N(t) at different t i m e s . As the function N(t) is linear for small t, extrapolation to -t = 0 yields the initial concentra-tion NQ and thus the separaconcentra-tion factor at time t.
To calculate the behaviour of the thermal diffusion column it is only n e c e s s a r y now to evaluate the constants c-^, c^, c„, c„ and P for different shapes of the column.
The most important c a s e s are: 1. plane p a r a l l e l plates
2. concentric cylinders 3. "hot w i r e "
Flat columns strictly only exist in theory as it is p r a c -tically impossible to construct them properly. Its theory can be used however in first approximation for the concen-tric cylinder type, provided that the annular gap is small and the radii of the cylinders l a r g e .
The hot wire column is obtained fi'om the concentric cylinder type by taking a = « 1.
§3. The plane case.
This type of column, for wliich the calculations are the e a s i e s t , consists of two v e r t i c a l parallel surfaces, a d i s tance d apart. The width of the plates is B cin. The t e m p -ei-ature of the surfaces are T^ and T,, (T, >T,,). The most difficult problem in calculating the befiavioui' ol' the column a r i s e s from the fact that the constants c j , c j and C3 contain the transport coefficients A, a, r], and D. If AT = T j - Tg is small, these coefficients may be I'egarded as constants in first approximation, taking tlieii' values at tlie ai-ithmetic mean t e m p e r a t u r e .
5 2 -by s o l v i n g the N a v i e r S t o k e s e q u a t i o n with s u i t a b l e b o u n d a r y c o n d i t i o n s . T h e p r o c e d u r e h a s b e e n e x t e n s i v e l y d e s c r i b e d e l s e w h e r e 43.45,46)_ -pj^g following e q u a t i o n d e s c r i b e s the v e r t i c a l flow a s a function of x.
^ = - 12 - ^ T t (2^ • ^ ^ ( ^ -^^ (2,37)
w h e r e g i s the a c c e l e r a t i o n of g r a v i t y . The c o l u m n c o n s t a n t s b e c o m e d^ P ^ a g 9 ! T7^ D ( 2 . 3 8 ) ( 2 , 3 9 ) ( 2 , 4 0 )r e s u l t s which c o i n c i d e with the c o l u m n c o n s t a n t s c a l c u l a t e d by F u r r y and J o n e s (ref. 46 p a g e 168) „ ( 0 ) ^ ( o ) (O) Ci = H , C2 = Kd , Cg = K ^ F u r t h e r m o r e Z AT 2m In q = a — - . ( 2 , 4 1 ) 2. 4 d T 1 + m^ ^ p g d 3 A T w h e r e m = \ / — = —^ (2, 42) 602 n B T A s s u m i n g that p ~ p and D ~ p"^ we ge t m ~ p "^. B e a r i n g in m i n d that q h a s i t s m a x i m u m v a l u e at m = 1, we c a n define an o p t i m a l p r e s s u r e Popt: . r. Do T p2 , = 602 p-^ ( 2 , 4 3 ) P p s d ' AT
and thus m = j l (2,44)
where Po is some reference pressure, where p = p^ and D = DQJ PO is usually taken 76 cm Hg.
From (2,42) the spacing at the optimal pressure, dp is
obtained °P'
(2,45)
but this is not the optimal spacing d,-,.; with regard to the separation factor.
Differentiation of (2,41) with respect to d yields the latter l^r 11 D T
d = 0.89 d = 7 . 5 1 \ (2,46) This is an important factor to remember, when designing
columns. The separative power is given by 0. 175 Z i' AT^" m^
6U = p D B < a -;;^\ (2,47) d I T j 1 + m2
and the optimum spacing djj" with regard to separative power becomes
d* = 1.3 d (2,48) This fact was already known experimentally; as a rule of
thumb the diameter of a column for production is chosen 1, 2-1, 3 times the diameter at zero production.
It can be seen from (2,47) that thermaldiffusion must be carried out in the gaseous state, as the product p D is here 102 times larger, than in the liquid state. Gaseous thermal-diffusion with Uraniumhexafluoride failed however, but liquid thermaldiffusion proved to be succesfuU ^^).
The equations in this paragraph were thus derived, as-suming, that AT was small. Furry and Jones *6) andFleisch-mann and Jensen^^) claim that the formulas are fair approxi-mations, even when AT is large.
