arious formulae for the isotopic Thermal diffusion factor have been reviewed and a new formula has been derived. Numerical calculations have been performed for the particular case of He3-He4 in a region where quantum effects are negligible. These calculations establish the relative adequacy of a comparatively simpler formula advanced by us and will be useful to interpret the recent experimental results of Saxena, Kelley and Watson on the thermal diffusion factor as a function of temperature

Pardeshi, P A

Saxena, S C

English

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VARIOUS APPROXIMATIONS FOR THE ISOTOPIC THERMAL DIFFUSION FACTOR. I. APPLICATION TO HELIUM ISOTOPESS. C. SAXENA AND P. A*. I^RDESHlI'Chemisthy D ivision, Atomic Enkkoy Establisha^ ^ t Tkombay, Bombay, India {Received^  September 8,ABSTRACT. Various formulae for the isotopic genual diffusion factor have been reviewed and a new formula has been derived. Numoric|il calculations liave been performed for the particular case of HeS-He  ^in a region where quotum effects are negligible. These calculations establish the relative adequacy of a comparatively simpler formula advanced by us and will be useful to interpret the recent experimAital rtjsults of Saxena, Kelley and Watson on the thermal diffusion factor as a function of jjemperaturo.1. I N T R O D U C T I O NKnowledge of the thennal diffusion factor for holiiiin isotopes is important, for thermal diffusion has f)oen used by Meintcer, Aldrich and Nier (1948) and Schuette, Zucker and Watson (1950) to enrich despite its extremely low abundance (1.3 x l 0~'^%) in natural helium and still lower abundance for well helium. Measurements of the isotopic thermal diffusion factor, a^ r, as a function o f temperature are also important to investigate the intermolecular forces. Moran and Watson (1958) measured the thermal diffusion factor for in the tem­perature range 233‘^ K to 57 T'K by using an elegant apparatus “ Trennschauker^ introduced by Clusius and Huber (1955). Kecently Saxena, Kelley and Watson (1960) have extended thes(i measurements to still lower temperatures. These measurements show a mucli steeper temperature dependence for a^ t than given by the existing theoretical expressions. Approximate quantum mechanical calculations o f by Saxena (1960) reveal tliat the quantum corrections are negli­gible even at the lowest temperature (233^K). Thennal diffusion factor like othw transport properties, emerges from the theory as the ratio of infinite determinants, Chapman-Cowling (1953). There are two alternative procedures developed separately by Chapman and Cowling (1953) and by Kihara (1949) to expand these infinite determinants into infinite convergent series. The use of varying number o f terms of these series lead to the various approximations on the two schemes. As Moran and Watson (1958) compared their experimental results only with an approximate formula, valid more rigorously for. a mixture of heavy isotopes, it would be interesting to explore the possibility o f this anomaly6666 /S. G. Saxena and P. A . Pardeshiin the use of an inadequate theoretical expression. The purpose of the present paper is to evaluate ccji for He®—He  ^ according to the expressions derived on the two approximation schemes. This will establish the relative usefulness and limitations of the various formulae. A new formula has been derived on the Kihara approximation scheme, taking into consideration terms containing upto the second power of the reduced mass.2. G E N E R A L  F O R M U L A E  F O R  T H E  I S O T O P I C  T H E R M A LD I F F U S I O N  F A C T O RThe Chapman-Enskog kinetic theory of gases expresses tlie m-th Chap­man-Cowling approximation for a^ r as follows:5'.{ " ’ i i , " - ' - ) ' ] -  <■>Here X j and are the mole fractions of the two coxTipoiieiii-s of jiiolecuJar weights Ml and respectively. The quantity represents a detenninaut of (2m4-1) order, the general term of which is aij where i and j  range from -rn toH-m including zero. The minor of obtained by deleting the row and column containing aijy is denoted by the symbol To the first a])proximation, Eq.(l) can bewritten into the following convenient form:(2)whereS' -  f V a * ~  - i / j )  ]« ! = [andQr [ i r > f ^ r : ^ 4  /  r> L \ 2 B*]-\)12 B* )+ 85 (JU,M,)iThe expresaioiis for 6’,  and are obtained from those of and Q^  by an inter­change of the subscripts for the molecular masses. Here the functions are the reduced Chapman-Cowling collision integrals and the functions A*, B* and C* arc ratios of Q**."'* and have been tabulated by Hirschfelder, Curtiss and Bird (1954) as a function of the reduced temperature. Second and higherapproximations are conveniently left in the determinant form of Eq. (1) for com- putational i^urposos.For the case of heavy isotopes where the reduced mass, is small,Eq. (1) can be expanded in the powers of the reducefl mass. Retaining terms only up to the first power of the reduced mass, oiip obtains;Yariom  Approximations for the IsoU^ic, etc. 67. . .  (3)where is diineusionk%ss thermal diffusion fact^  and has been given by Mason (1954, 1957a) for m — 1, 2 and 3. For m — 1, ao|is given by1, _  1 5 (6 0 » -5 ) (2 ^ *  +  r>)| • ...Expressions for m equal to 2 and 3 are rather k^ngtliy and will not be repeated here.Eq. (2) tra}isforms into the following form for the case of heavy isotopes when one retains terms up to the seeojid powenr of the reduced mass, M, (Chapman, 1941);-  K Jifi -  yM(X, -  x^)i (5)whore^  3(5 -  A*) _  2(12Z?* +  5)^ 2{S^+ ~M*) (16^1* -  1 2 « * -! 55)*It is interesting to note i hat [oco]', is now dependent on the relative proportions of the two isotopes unlike Eq. (4).Kihara (1949) developed an alternative scheme to expand these infinite determinants into convergent infinite series. This procedure is mathematically less straightforward than that of Chapman-Cowling but it has a more physical basis and gives simpler expressions at least in the earlier approximations. Recently, Mason {1957a)has elaborated and extended this procedure and has given the expression up to the second approximation for the thermal diffusion factor. The general formula for the wi-th approximation to the thermal diffusion factor on Kihara scheme remains the same as given by Eq. (1) except that now have different meaning. The first approximation [a^ i is again given by Eq. (2) except that the Q's are now defined as follows:hQ i'=  [Iand58 S. G. Saxena and P . A . Pardeahi[ H w ; + m V  +  ^  r, JUsing the extended Kihara scdieine. Mason (1957a) gave the following formula for the second approximation to[aVl2whereandA “i“ 3^ ~^5 » (8)(«)In Eq. (7), fa^ rJi is given by Eq. (2) and the various hj^  arc similar to those given by Mason (1957a) exce])t that the subscripts characterising the molecular species are all the same.For the case heavy isotopes, a' ,^ can again be put in the form of Eq. (3), where [oto'Jx is now given by the following simpler form:r^n ^  l5(66’* - 5 ) ... (10)The second approximation (!an be written in the following form:[ao'l  ^ -  M , ( l  +  AV). ... (11)Here is given !>y the Eq. (4) and Kq' is a correction factor given by Mason (1957a).Using Kihara's scheme for approximating transport coefficients, we have worked out the following first approximation to the thermal diffusion factor for a mixture of heavy isotopes in which terms containing upto the second power o f the reduced mass have been retained:K 'V  -  M .  {  i - ... (12)This formula is slightly siujipler than the one given by Eq. (5) and as shown in the next section, is very useful for the accurate evaluation o f aj, o f isotopic mixturesVa/riou8 Approximations for the Isotopic, etc. 59where M  is not quite small. Such isotopic mixtures are the mixtures of helium and hydrogen isotopes.3. C A L C U L A T I O N  OF T H E  T H E B M A L  D I F F U S I O N  ^ F A C T O R  F O R  H E L I U MBetailod calculations for the thermal diffusion factor of helium have been performed on both the approximation schemes, aceor^ng to the various formulae discussed in the previous section. The results of t l^  calculation are tabulated in Table I and refer to the following L—1/ ( 12—6) law!|for intermolecular force;(13)Here E(r) is the potential energy of the two moleci4es at a separation distance r and cr is the molecular separation for which the ir|»raction energy is zero, e is the value of the maximum negative potential en#gy. The various collision integrals required for these calculations are tabulated by Hirschfelder, Curtiss and Bird (1954).* In those calculations of Table I where the system has been treated as a binary mixture of He* and He*, we have chosen arbitrarily the con­centration of He® as 5%, a value close to the one used in the work of Moran and Watson (1958). Further, these calculations were performed on a desk calculator and have been subjected to only spot-checking.TABLE ICalculated values of the isotopic thermal diifusion factor for HeliumChapman-Cowling approximation scheme Kihara approximation schemeT * [ cctY i ^ o [<XTh^^6 0.0671 0.0690 0.0677 0.0690 0.0702 0.0718 0.0707 0.0694 0.07167 0.0724 0.0761 0.0732 0.0747 0.0742 0.0769 0.0747 0.0765 0.07790 0.0744 0.0788 0.0753 0.0760 0.0779 0.0797 0.0785 0.0778 0.080320 0.0767 0.0794 0.0774 0.0803 0.0802 0.0820 0.0808 0.0809 0.083240 0.0768 0.0794 0.0774 0.0801 0.0802 0.0820 0.0808 0.0808 0.083260 0.0764 0.0791 0.0771 0.0799 0.0799 0.0816 0.0805 0.0804 0.082780 0.0763 0.0789 0.0769 0.0796 0.0797 0.0814 0.0803 0.0805 0.0826100 0.0761 0.0767 0.0767 0.0794 0.0795 0.0812 0.0800 0.0800 0.0829200 0.757 0.0781 0.0762 0.0784 0.0790 0.0806 0.07S6 0,0797—• It m ay b e poin ted  out that there is a  m isprint in  the value o f  the co llis io n  in teg  al q (1 D * fo r  T* 100. I t  should read as 0,5170 instead 0.6130.60 S. G. Saxena and P . A . PardesUIn columns 2, 3, 4 and 5 of Table I are recorded the values of* the different approximations to as a function of the reduced temperature (T* =  kTfe) according to Chapman-Cowling scheme; while the (jolumns 6, 7, 8, 9 and 10 report the results on Kihara scheme. Values of a^ r listed in columns 2 and 3 of Table I have been obtained acjcording to the Eqs. (2) and (4) respectively. The two sets of values differ appreciably and reveal the fact that for Helium, terms involving only first power of the reduced mass are not enough ; a result previously pointed out by Winter (1950). It is, therefore, interesting to consider terms involving still higher powers of M. Chapman (1941) derived an expression, Eq. (6), which takes into acjcoiuit the terms upto the second power of M. Calculated values according to this formula are given in column 4 of Table I. It will be seen that these values are appreciably different («=s 2.4%) from but are in goodagreement with the values (~0 .9% ). Values of \cXfp\2, computed accordingto Eq. (1) with m -- 2 are shown in column »5. These values are approximately 4% higher than the values, establishing thereby that the convergenceof the series is fast enough and the error involve^d, because of the neglect of the third and higher approximations, is small. This inference is very welcoming in view of the fact that the higher approximations will involve evaluation of seven and higher order determinants and can be safely avoided till we considerably improve the precision and accuracy of the measurements. It is also very inter­esting to note that values are in better agreement with thevalues than the values. The authors therefore feel that the simpleformula for as given by Eq. (5) is preferable to th(^  complicated form of Eq. (2) and still more cornpli- rated fonn as given by Eq. (1).Results obtained using similar approximations, but on Kihara scheme are listed in coluiuTis 6, 7, S and 9, and are seen to follow the same qualitative trend. In column 10 are tabulated for comparison the results obtained from a formula which consideTs terms upto the second approximation but retains only the first power of reduced mass, Saxena and Mason (1958). These values of aresystematically higher than the values. As the convergence for the Kiharaapproximation scheme is still faster than that of Chapman-Cowling, we are of the opinion that the neglect of the second power of the reduced mass is much more serious. A critical examination of all the values obtained on Kihara approxi­mation schenie again leads to the same conclusion that the values obtained by using the simpler formula, Eq. (12), which considers terms upto the second power o f reduced mass, is preferable to the rest of all, both for accuracy and simplicity.A critical examination of the various approximations for on the two approxi­mation schemes for realistic intermolecular potentials and for a few mathematically simple systems, was done by Mason (1957a and 1957b). Mason (1957b) considered three t3rpes of mixtures viz. (1) Lorentzian, (2) Quasi-Lorentzian and (3) Heavy isotopic mixtures* Unfortunately our system is not identical to any o f theseVarious Approieinutiions for the Isotopic, etc. 61but resembles to (3) in as much as all the interactions are identical but the two masses differ considerably. Our numerical calculations of Table I reveal that Kihara approximation scheme is preferable to Chapman-Cowling and is in con­formity with the conclusions of Mason (1957b) for mixtures of heavy isotopes.4. C O N C L U S I O N S  ‘Our numerical calculations for of helium is^p es establish that Kiliara approximation procedure is better than that <»f Ch^man and Cowling. With He® present in trace, a simple formula derived, treating the mixture of He® and He* as a heavy isotopic mixture but retaining terms i|^ to the second power of the reduced mass, will yield results within the range of experimental error. This formula is also preferable in view of comparative sii^plicity and accuracy,a o k n o w l e d o m e i I tThe authors arc thankful to Dr. J. Shankar foi’ H^ s kind interest and making available to us all the focilities.K E F E R E N C E SCliapman, S., 1941, Proc, Boy. Soo., A177, 38.Chapman, S. and Cowling, T. O., 196.1, The Mathematical Thooiy of Non-Uniform gases, Cambridge University Press, England.Clusius, K. and Huber, M., 1955, Z. Naturforsch., lOA, 130.H irschfelder,0., Curtiss, C. F., and Bird, K. B., 1954, The Molecular Theory of Gases and Liquids, John Willey & Sons, Inc., New York.Kihara, T., 1949, Imperfect Gases, originally published in Japanese (Asakusa Book Store, Tokyo) and translated into English by the U.S. oflSoe of Air Boseareh, Wright-Patterson Air Force Base, ChB]). 6; see also 1953, Bm. Modem Phye., 25, 831.Mason, E. A., 1954, J. Clwm. Phys., 22, 169.Mason, E. A., 1967a, J. Chem. Phys., 27, 75.Mason, E. A., 1957b, J. Chem. Phys., 27, 782.Meinteer, B. B., Aldrich, L. T., and Nier, A. ()., 1948, Phys. Bev., 74, 946.Moran. T. 1., and Watson, W. W., 1958, Phys. Bev., 109, 1184.Saxena, S. C., 1960, unpublished calculations.Saxona, S. C., Kelley, T. G. and Watson, W. W., 1960, In course of pubUcation. Saxena, S. C., and Mason, E. A., 1958, J. Chem. Phys., 28, 623.Schuette, 0. F., Zuoker, A. and Watson, W. W., 1950, Bev. Sci. Instr., 21, 1016. Winter, E. It. S., 1950, Trans Faraday Soe., 46, 3l.

Various Approximations for the Isotopic Thermal Diffusion Factor. I. Application to Helium Isotopes

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Various Approximations for the Isotopic Thermal Diffusion Factor. I. Application to Helium Isotopes

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