High-temperature isotopic thermal diffusion of neon

Physica 57 (1972) 306316 Q North-Holland Publishing Co.

HIGH-TEMPERATURE

ISOTOPIC

M. AUREA

CUNHA

THERMAL

DIFFUSION

OF NEON

and M. F. LARANJEIRA

Laboratdrio Calouste Gulbenkian de Espectrometria de Massa e Fdsica Molecular. Comissdo de Estudos de Energia Nuclear (I.A.C.) - Institute Superior T.&nico, Lisboa, Portugal Received 2 June 1971

Synopsis Isotopic thermal-diffusion factors have been determined for neon, in the temperature range 584-803 K. It is shown that the present data are consistent with our previous results and those of Fischer at lower temperatures and also consistent with the second virial coefficient, transport phenomena and crystal data. The exponential-six model has been assumed for which the thermal-diffusion data obtained provide realistic values of the parameter pi.

1. Iwkxiuction. It has been recognised by several authorsipa) that thermal diffusion of neon isotopes offers a good case for checking realistic, nondirectional intermolecular potential models. Let us consider the exponential-six model:

b(r) =

1 -

&61”

A exp[o(l I Ly.

-

+ni)l

-

(6i/r)6

, I

for I > rmax and +(Y) = 00 for r 5 rm&x, where 4(r) is the potential at a distance Y between the centers of the molecules, Ymax the value of r for which $(I) has a “spurious” maximum, E the depth of the potential well, lrn the value of r for which d(r) is minimum and LYthe additional parameter depending on the slope of the repulsive part of the potential. With the experimental data of the isotopic thermal-diffusion factor, CUT, at higher reduced temperatures, T* = T/(&/k), one is able to provide an independent determination of the parameter 01 of the model. If lowertemperature data are also available, let us say in the reduced temperature range between about 1.5 and 6, not only 01but also E might be determined. In the framework of the exponential-six model, neon is the only rare gas which, in practice, may provide values of 01and .S which can be used with more confidence.

ISOTOPIC THERMAL DIFFUSION OF NEON

307

Indeed, for helium, reduced temperatures experimentally still reliable are so high that only 01may be obtained. On the other hand, for xenon, krypton and less critically argon, the workable temperatures are not high enough to determine 01and, as a consequence, also the parameter E remains undetermined to a large extent. Therefore, experimental thermal-diffusion data for those gases may specially be used to check the theoretical consistence of a pair of parameters 01, E previously determined by other phenomena. Isotopic thermal diffusion is independent of the third parameter of the model, rm, which therefore is usually obtained from viscosity, second virial of pressure and crystal data. To a certain extent, that feature may be an advantage, at least for neon, since it may provide a less involved check of the model. If we consider a less sophisticated intermolecular potential, namely the Lennard- Jones ( 12-6) model, b(r) = 4+/W

kJ/

%$-

(2)

the situation is much more definite. Indeed, the isotopic thermal-diffusion factor, ffT, is then only dependent on the parameter E, and we have to deal only with one theoretical curve for OIT,instead of a family of curves as in the exponential-six model.

r4: 0.6 -

_-16

_-__-_

/--

L.J.w-6)

/.-'-.-._, --.I,_(

-13

a4-

Q2-

0

-Q2-

I

I 0.4

III

1

I

I 4

III

10

I

I 40

III

100

-p

Fig. 1. Kihara’s first approximation, [ao]F for the reduced isotopic thermal-diffusion factor, vevswsreduced temperature, T*. Lennard-Jones (12-6) model; ---exponential-six model with parameter (Y= 16; -.-.exponential-six model with parameter OL = 13.

M.AUREACUNHAANDM.F.LARANJEIR_~

308

Therefore, the parameter E of the (12-6) model may always be determined by thermal diffusion, as far as the model itself is consistent with the experimental data. The general trends of the Lennard- Jones (12-6) and the exponential-six models are shown in fig. 1, where the reduced isotopic thermal-diffusion factor, 010,defined by the following scales aT =

aO4'fl2[1 +y(xl-X2)

equations,

4412 +

...I.

is given against T* in semilog

(3)

ml--2 Ml2

=

ml

>m2,

ml+m2

where ml, ms and xi, x2 are, respectively, the molecular masses and the mole fractions of the isotopes and y is a small quantity slowly varying with temperature. The concentration-dependent term is usually neglected for heavy isotopic mixturess). From fig. 1 it follows that at high reduced temperatures the LennardJones model is nearly equivalent to the exponential-six model with the parameter 01 approximately equal to 16. For lower reduced temperatures, eradecreases more rapidly with temperature for the (12-6)model and shows a negative minimum at T* m 0.62. In the low-temperature range, say for T* < 2.5, the (12-6) model is approximately equivalent to the exponentialsix model with 01= 13, by making a shift of the curves parallel to the temperature coordinate by a factor of about 1.065. The different trends just mentioned about the models, may be checked with neon isotopes if sufficient, accurate data of thermal diffusion were available. However, the majority of the published data seems to be nonreliable and indeed some results are known to be affected by systematic errors29 4). As far as we have tested, the only experiments carried out within an extensive temperature range which are more nearly consistent with data from other transport phenomena, second virial and crystal data, are those of Fischer at lower temperaturess) and ours at intermediate temperatures, given in ref. 2. In the present work we have extended our previous experiments to much higher temperatures, therefore improving the independent determination of the parameter 01 for the exponential-six model. The reported values of this parameter change from 14.56) to 16.27 7). Consequently, the other parameters E and lm, also change appreciably. From the previous considerations, it is hoped that our data from isotopic thermal diffusion may help the choice of the probable value of CX,since the derivative of the reduced isotopic thermal-diffusion factor with respect to cy, &/a,, is of the order of 0.03 in the temperature range which we have observed, and individual experimental errors are of the order of l-2 per cent. Therefore, the parameter (Y for neon might be determined a priori,

ISOTOPIC

THERMAL

DIFFUSION

OF NEON

within an error less than half a unit, which is believed what might be achieved from other phenomena.

309

to be better than

2. Exfierimental firocedure. The swing separator (trennschaukel) used for the present measurements of the thermal-diffusion factor was of the Clusius type*). It consists of eight stainless-steel tubes with an inner diameter of 15 mm and 90 mm long connected in series by stainless-steel capillary tubes with an inner diameter of 0.8 mm. The upper and lower thirds of the tubes are imbedded in copper blocks which were heated up thermostatically by furnaces. The transition part, where the temperature gradient was set up, was isolated with asbestos. In each copper block there was a control system for the heating element, similar to that described in our ref. 2. The temperatures could be maintained practically constant within 1 K. The relaxation time for the isotopic separation is of the order of 40 minutes according to the van der Waerden theory 9). Initial gas samples were drawn 24 h to 48 h after the establishment of the temperature gradient, and further 4 to 8 other samples were drawn spaced by 5 to 15 hours. The samples were analysed with an MS-ZSG (AEI) mass spectrometer, after a control of background levels, and search for eventual inleakage in the thermal-diffusion system. The neon purity was 99.8% with helium as the main impurity. No traces of argon and COs, which may interfere with soNe+ and ssNe+ peaks, were found in the mass spectrum. 3. Eq5erimenta.l results. The experimental results are presented in table I, where the following quantities are given: T* and T*I, the temperatures at the cold and hot regions, respectively, p, the mean temperature assigned to the thermal-diffusion factor, WF, as determined by Davenport’slo) formula: p = (TITII)*,

TABLE I Experimental

and theoretical

-

TII

(‘;I

WI 690 780

495 509

800 815

values

@T (exp)

of thermal a0

diffusion

(exp)

x 102

for zoNe-22Ne

IaolH (LX= 15.7; elk

Dev

= 42.3)

584

2.64 & 0.07

0.555

f

0.018

0.556

2.59 & 0.04

0.544

f

0.009

0.556

561

630 670

2.64 -+ 0.03

0.555

& 0.007

640

722

2.61 + 0.02

0.549

*

0.004

0.556 0.556

926

610

752

2.65 f

0.05

0.557

f

0.011

938

596

748

2.70 f

0.02

672

803

2.70 + 0.07

0.567 0.567

f f

0.005 0.014

+0.001 +0.012 +0.001

0.556 0.556

+0.007 -0.001 -0.011

0.556

-0.011

M.AUREA~~NH_~ANDM.F.LARANJEIRA

310

01~(exp), the experimental reduced isotopic thermal-diffusion factor, as determined by eq. (3), dropping the concentration-dependent term, [a#, Kihara’s first approximation for cueand Dev, residuals [cueIF - cue(exp). The experimental thermal-diffusion factor, OlT(exp), was obtained via the total separation

factor Q, according

to the following

equations:

In Q = %olT(exp) ln(TII/TI),

(5)

Q = (+)11/(4q)1,

(6)

where n is the number of tubes of the swing separator and x2, XJ are the mole fractions (~2 + xj = 1) of saNe, ssNe, respectively, and the superscripts I and II refer to the cold and hot regions of the separator. The OlT(exp) values given in table 1 are averages of at least five observations. The experimental errors reported, of the order of l-2%, take into account only the dispersion observed in each set of experimental values. The Davenport mean temperature, eq. (4), has been assigned to OrT(exp) for the sake of simplicity. Anyway, a check has also been carried out with Brown’s formula 11))

and the more sophisticated e-clnP(~/Tcd)

=

(d4

Laranjeira formulars) *

ln( P/P)

(P1 -

P)J

(8)

with t.

and ti = (2c)f ln(Tf/T,d), where c and d are constants of the order of 0.5 and 0.6, respectively, and T, is the critical temperature. In the high-temperature range observed, the thermal-diffusion factor is nearly constant, therefore independent of temperature. Hence P remains practically undetermined by means of eq. (8), since $1 M $11, and the mean temperatures determined by the Davenport and Brown formulas have no definite physical meaning and should be regarded only as a kind of experimental reference temperature in the range TI-TII. For lower temperatures, namely for Fischer’s data on thermal diffusion reported in this paper, all the three formulas just referred to, give rise to about the same significant values for the mean temperatures.

ISOTOPIC

THERMAL

DIFFUSION

OF NEON

311

Fig. 2. Isotopic thermal-diffusion factor, OPT,as a function of temperature, for ZONeK , for the exponential-six model with 22Ne. First Kihara approximation, [OLT]~ a = 15.7; c/k = 42.3. Experimental points: + Watson et ~1.13); A Laranjeira and Kistemaker 14); 0 Fischerb) ; o Laranjeira and Cunhaz) ; * Present work.

In fig. 2 the experimental data of old are plotted against p in semilog scales. Results from Watson et al. 13)) Laranjeira and Kistemaker Id), Fischer 5) and Laranjeira et al. 2) are also plotted. The experimental values of Stier lb), Moran and Watsonl6) and Saxena et a1.17) are not considered here, since it is believed that they are affected by systematic errorssT4). 4. Discussion. As it has been mentioned in sec. 1, the experimental isotopic thermal-diffusion factor can only determine, in the framework of the exponential-six model, the parameters 01 and E. The third parameter, rm, must be obtained from other properties. Since isotopic thermal diffusion is a second-order transport phenomenon, it would be preferable to check the consistency of experimental results with those for the coefficients of self-diffusion, D, viscosity, 7, and thermal conductivity, 1, which, according to the Chapmann-Enskogrs) theory, are given respectively by the following equations: 104D =

1077

=

26.280T”

p&i%; 266.93(MT)t 7i.l

10’1 =

1989.1 T*

iwr;

fg)(a,T*) f&1,1)*(,,T*) ’ f;m,“‘(a, T*) 9(2~2)*(~, T*) ’ f$‘(a>T*) QR(%2)*(a, T*) ’

(9) (10)

(11)

M. AUREA

312

CUNH_~ AND M. F. LARANJEIRA

In these expressions M is the molecular weight, 9 the pressure in atmospheres, Q”pl)* and Q2(2y2)’are the reduced collision integrals and fg), fim) and fi*’ are slowly varying functions of temperature, close to unity. These are tabulated for the second and third order of approximation, (m), in references 6, 19, 20 and 21. In the above formulas, D, 7 and il are expressed, respectively, in cm2 s-1, poise and cal cm-l s-1 K-i. Experimental data for self-diffusion are so scarce that it is not possible to provide independent determination of the potential parameters. On the other hand, thermal conductivity data for neon are only known at high reduced temperatures, defining, at most, an independent value of the parameter 01,with appreciable uncertainty, when we bear in mind the magnitude of experimental errors. Therefore, viscosity is the usual transport phenomenon giving self-consistent, independent values of 01, E and rm. Also, crystal data 697) and the second virial coefficient, B(T), are common ways to determine potential parameters : B(T)

= b,B*(a,

T*),

(12)

with b, = (2xNa/3) r;, where B*((Y, T*) is a dimensionless quantity tabulated in references 6, 19, 20 and 21 as a function of T* for several values of 01, and No is the Avogadro number. In table II, reported values of 01, E/k and rm are presented. They are determined by Mason et aZ.6) from crystal data, second virial coefficients and viscosity; by Laranjeira et al. 2) from thermal diffusion and viscosity data; by Utting7) from crystal data and by Hogervorstss) from the most recent viscosity data of Dawe and Smithss).

TABLE

Exponential-six a

II

parameters for neon from various sources References

elk

Properties

(K) 14.5 15.5 16.138) 16.27b) 16

38.0 33.4 39.56 40.87 43 & 3

3.147 3.144 3.090 3.077 3.03 5 0.03

1

Masons) 1954 Laranjeiras) 1966 Utting 7, 1970

crystal data, D, q

Hogervorst 22) 197 1

rl

~TI

7

crystal data

a) Exponential-six parameters derived from the pairwise additive consideration of the solid state at 0 K. b) Exponential-six parameters derived from solid state at 0 K with triplet diffusion energy included in the static lattice.

ISOTOPIC

In ref. 2, we have

THERMAL

shown

DIFFUSION

that

our previous

OF NEON

results

313

for the isotopic

thermal-diffusion factor of neon at intermediate temperatures, were consistent with those of viscosity and self-diffusion assuming a set of parameters 01,E, Ye equal to 15.5, 33.4, 3.144, respectively. Referring to the parameter CY,our present results at high temperatures, given in table I, closely follow the general trend previously observed with a minor adjustment for 01= 15.7, using Kihara’s first approximation [CX@ for the reduced isotopic [a@=; where c* = f&l,

thermal-diffusion

by

“,‘/Qncl,

factor:

5 )

1)‘.

,

(13)

A

=

Q’2,2”

I

f$l,l)’

.

However, bearing in mind that [ae]F is believed to be 2% lower than the true value for the reduced isotopic thermal-diffusion factor in the temperature range observeda), we may say that a probable value of 01will be close to 15.5. As is shown in fig. 2, Fischer’s results are in reasonable agreement with those published by uss) in the temperature range from 330 to 510 K. Therefore, we have tried to obtain the parameter E combining our results and those of Fischer at low temperatures, using the superposition technique and the least-squares method. A value of E/k = 42.3 f 2.2 was obtained. This is appreciably higher than the value 33.4 previously determined from viscosity data in ref. 2. However, the former value is in better accordance with Hogervorst’s parametersas), given in table II, which have been determined from recent data for the viscositysa). The consistency of our results has been checked now, with second virial data, assuming : (a) an a priori value of 01= 15.5, which is therefore close to the minimum probable value from our thermal-diffusion data. (b) the most probable values of 01, E/k, and lm determined from second virial data24.25726). Parameters 01and Elk have been obtained by the usual superposition technique and rm by a mean least-squares method, by computer. The consistency with thermal conductivity data has been considered, assuming a fir&i values of (a) c/k and (b) of 01 and c/k from the second virial coefficient. The least-squares method then provides the corresponding value for Y*. Table III gives the set of parameters in each circumstance, and in column5 the corresponding associated standard error (r8 of the experimental fit. We conclude that we may hardly give preference to one of the sets obtained with 01= 15.5 or a = 16. Indeed, the second virial coefficient fits slightly better with the set corresponding to 01= 16 whereas for the thermal conductivity OL= 15.5 is somewhat better.

M. AUREA

314

CUNHA

AND M. F. LARANJEIKA TABLE III

Exponential-six a!

W)

es

Elk (K) 42.3 41.0 42.3 41.0 42.3

15.7 15.5 16.0 15.5 16.0

LyT)

parameters for neon, from this work

f 2.2 f 1.1 * 1.1 f 1.1 f 1.1

zi 3.043 3.041 3.059 3.025

f & & *

0.057 0.056 0.022 0.025

0.001 0.233 cm3 mole-r 0.229 3, ,, 0.416 cal cm-i s-r K-1 0.475 ,> I, *> ,I

Figs. 3 and 4 show the fit of the experimental data corresponding to 01= 16 for the second virial coefficient and thermal conductivity, respectively. The fit corresponding to 01= 15.5 is graphically practically equivalent to 01= 16. In any case, we may conclude that our results at intermediate and high temperatures and those of Fischer at low temperatures are consistent with the most recent data of transport phenomena, secondvirial and crystal data, with minor differences of the order of the experimental errors, in the framework of the exponential-six model.

I

I

I 80

I 100

I

200

I

400

I

600

r (K) Fig. 3. Experimental and theoretical second virial coefficient for neon. Experimental points: o Crommelin et al. 24) ; l Holborn and Ottoa5) ; * Holborn and Otto 2’J). Theoretical curve for exponential-six model, with (Y= 16; E/k = 42.3 and rm = 3.041. 60

ISOTOPIC

I

I

Phys.29)

DIFFUSION

I 200

100

Fig. 4. Experimental points:

THERMAL

I

and theoretical

l Saxena and Saxenaa’) with

OL=

I 400

thermal

; * Kannuluik

; * Gandhi and Saxenass).

OF NEON

Theoretical

I

I 600

conductivity and Carmanes)

315

I 1ooo

800

for neon.

I 1soo T(K)

Experimental

; o Handbook

of them.

curve for the exponential-six

model,

16; elk = 42.3 and yri, = 3.025.

We want to thank the technical staff of our laAcknowledgements. boratory, Mr. J. Casaca, J. Lourerqo and F. Neves, for the help given to the experimental part of this work. We also thank the Nticleo de Estudo e Construcao de Aparelhagem Cientifica for the technical assistance in the apparatus and Centro de Estudos de Engenharia Me&mica for the computer facilities given.

REFERENCES

1) 4

Paul, R., Howard, A. J. and Watson, W. W., J. them. Phys. 43 (1965) 1890. Laranjeira, M. F. and Cur&a, M. hurea, Portugaliae Physica 4 (1966) 281.

3)

Laranjeira,

M. F., Cunha, M. hurea

and Silva,

M. E. F., Portugaliae

Physica

5

( 1968) 49. 4)

Watson,

W. W.,

Proceedings

of the International

Symposium

5)

ration, North-Holland Fischer, A., Inaugural

6)

Mason, E. A. and Rice, W. E., J. them.

7)

Utting, B. D. and Walkley J., J. them. Phys. 52 (1970) 5470. Clusius, K. and Huber, M., 2. Naturforsch. 10a (1955) 230.

8) 9) ‘0)

on Isotopic

Sepa-

Publishing Comp. (Amsterdam, 1958) 459. Dissertation, Universitat Zurich (1959). Phys.

22 (1954) 843.

Van der Waerden, B. L., 2. Naturforsch. 12a (1957) 583. Davenport, A. N. and Winter, E. R. S., Trans. Faraday Sot. 47 (1951) 1160.

316

ISOTOPIC

THERMAL

DIFFUSION

OF NEON

11) Brown, H., Phys. Rev. 58 (1940) 661. 12) Laranjeira, M. F., Moutinho, A. and Vasconcelos, M. H., Portugaliae Physica 4 (1965) 115. 13) Watson, W. W., Howard, A. J., Miller, M. E. and Shiffrin, R. M., 2. Naturforsch. 18a (1963) 242. 14) Laranjeira, M. F. and Kistemaker, J., Physica 26 (1960) 431. 15) Stier, L. G., Phys. Rev. 62 (1942) 548. 16) Moran, T. I. and Watson, W. W., Phys. Rev. 109 ( 1958) 1184. 17) Saxena, S. C., Kelly, J. G. and Watson, W. W., Phys. Fluids 4 (1961) 1216. 18) Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, Cambridge University Press (Cambridge, 1953). 19) Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B., Molecular Theory of Gases and Liquids, John Wiley and Sons (New York, 1954). 20) Rice, W. E. and Hirschfelder, J. O., J. them. Phys. 22 (1954) 187. 21) Mason, E. A., J. them. Phys. 22 (1954) 169. 22) Hogervorst, W., Physica 51 (1971) 77. 23) Dawe, R. A. and Smith, E. B., J. them. Phys. 52 (1970) 693. 24) Crommelin, C. A., Palacios Martinez, J. and Kamerlingh Onnes, H., Commun. Phys. Lab., Leiden No. 154a (1919). 25) Holborn, Von L. and Otto, J., 2. Phys. 33 ( 1925) 1. 26) Holborn, Von L. and Otto, J., 2. Phys. 38 (1926) 359. 27) Saxena, V. K. and Saxena, S. C., J. them. Phys. 48 (1968) 5662. 28) Kannuluik, W. G. and Carman, E. H., Proc. Phys. Sot. 65B (1952) 701. 29) Handbook of chemistry and physics, 48th Edition (1968) pp. E-2. 30) Gandhi, J. M. and Saxena, S. C., Molecular Phys. 12 (1967) 57.