J. A. CAHILLAND A. V. GROSSE
518
Viscosity and Self-Diffusion of Liquid Thallium from Its Melting Point to About 1300°K.
by J. A. Cahill and A. V. Grosse The Research Institute of Temple University, Philadelphia, Pennsylvania
(Received August ,$7; 1964)
~
The viscosity (7) of liquid thallium was measured by the oscillating crucible method between its melting point (577°K.) and about 1300°K. The values of r] at the melting point and at 1000°K. were 2.65 and 1.05 cp., respectively. Self-diffusion, at the same temand 0.80 X 10-4 peratures, was calculated using the Frenkel theory to equal 0.16 X cm. 2/sec., respectively. The experimental activation energy for viscosity, H,, equals 2500 cal./g.-atom; an empirical relationship between H, and the melting point gives the value 2270 cal./g.-atom.
Neither the viscosity nor the self-diffusion of liquid thallium has been measured so far as we could ascertain. In view of its low melting point, the experimental determination of the viscosity should be rather simple. It affords an opportunity to check the empirical relati~nshipl-~between the melting point and the activation energy of viscosity. It also would complete the subgroup IIIb of the periodic system, since both the viscosity and self-diffusion of gallium and indium have been measured.
Experimental The viscosity was measured by the oscillating crucible method as described first by Andrade4 and later, among others, by Yao and Kondic,6and reviewed by Bockris.6 A general treatment of this method, including a discussion of the experimental techniques required for precise measurements, has been published by Thresh.’ The apparatus used was similar to the one described by Thieles and consisted of a graphite crucible (5 cm. 0.d. X 3.5 cm. i.d. X 9 cm. deep) suspended in a bi$Zar arrangement by a 0.018-mm. tungsten wire. It was calibrated with several liquids and checked with mercury. The relationship between the logarithmic decrements, X - A d (sec.-l), and period of oscillation, T (sec.), of the system and the properties of the liquid is
The Journal of Physical Chemistry
where the constant c = 4.02 X cm.2/sec. g., Dm.*,and DT = density at the melting point and TOK., respectively, and 7 = viscosity in poises. The amount of thallium used in this determination was ~ 3 5 g. 0 or ~ 3 cc. 0 The density values determined by Schneider, Stauffer, and Heymers from the melting point to 920°K. were selected and may be expressed by the equation
D (g./cc.) = 12.16
- 15.21 X 10-4T OK.
(2)
Accordingly, the density varies from 11.29 g./cc. at the melting point (577°K.) to 10.64 g./cc. at 1000°K. These data were preferred to the data given in the “Liquid Metals Handbook,”1° which cover only a 30” temperature range. (1) A. V. Grosse, J. ITZOTQ. Nucl. Chem., 23, 333 (1961).
(2) A. V. Grosse, ibid., 25, 317 (1963). (3) A. V. Grosse, Science, 140, 788 (1963). (4) E. N. da C. Andrade and Y. S. Chiong, PTOC. Phy8. Sac. (London), 48, 247 (1936). (5) T. P. Yao and V. Kondic, J. Inst. Metals, 81, 17 (1952). (6) J. O’M. Bockris, J. L. White, and J. D. Mackenzie, “PhysicoChemical Measurements at High Temperatures,” Academic Press, New York, N. Y., 1959,p. 325. (7) H. R.Thresh, Am, Sac. Metals, TTan8. Quad., 55, 790 (1962). (8) M.Thiele, Doctorate Dissertation, Technical University of Berlin, 1956. (9) A. Schneider, A. Stader, and G. Heymer, NatUTWi88e?MChaft€m, 41, 326 (1954). (10) R. W. Lyons, Editor in Chief, “Liquid Metals Handbook,” I1 Ed., Department of the Navy, Washington, D. C.; NAVEXOS P-42 (Rev.), June 1952.
VISCOSITY AND SELF-DIFFUSION OF LIQUID THALLIUM
The purity of the thallium was 99.99%. Maximum amounts of impurities were, in p.p.m.: Pb, 40; Fe, 10; Cu, 10; Cd, 5; Zn, 2 ; total of others, 20. Since the thallium was contained in graphite a t high temperatures, any oxide present was converted to metallic thallium. The solubility of graphite in liquid thallium was reported to be negligible near its melting point by Moissan.ll We determined the solubility a t 900°K. by exposing a graphite rod to 10 cc. of liquid thallium contained in a stainless steel tube under argon for 4.0 hr. After the experiment, the thallium absorbed in the graphite rod was extracted with dilute HN03. Upon complete extraction, the graphite rod was weighed and only 1.5 mg. of carbon or less was ' lost. Thus, the solubility a t 900°K. is 0.03 atomic % carbon or less. The usual precautions were taken to avoid poisoning by thallium. A total of nine determinations was made during the course of five runs and these data are shown in Table I.
519
5 4
I
I I
I
I
I
I
I
Table I: : Experimental Viscosity Data Temperatur-
I-
lOOO/"K.
OK.
644 645 661 665 672 787 928 1278 1279 1283
&IOOO/T
Viscosity, cp.
1.552 1.550 1.513 1,503 1,488 1.270 1.077 0.7824 0.7818 0.7794
Figure 1. Plots of q os. 1000/T for gallium, indium, and thallium.
2.11 1.97 2.04 2.03 2.26 1.62 1.03 0.78
It follows from the empirical relationship'+ H, (cal./g.-atom) = 0.431(T,.p.)1.348 and for Tm.p.of thallium = 577"K., that
0.71
Discussion of Viscosity Data The bsst fit of the experimental points on a semilog plot is a straight line going through the points qm.p. = 2.65 and V ~ ~ O O O E . = 1.05 cp. The equation of the straight line, in terms of Andrade's first equation (see ref. 14),is
(poises) = 2.983 X 10-3ezm'BT
(4)
0.72
Uncertainties in corrections for thermal expansion, inductive electrical effects, and ambient gas viscosity limit the accuracy of this determination to an error probability of *0.05 cp. a t the melting point and kO.1 cp. a t the boiling point.
q
'K
(3)
where R = 1.9865 cal./g.-atom OK. and T is in OK. The experimental points and the straight line are illustrat,ed in Figure l.
H , = 2270 cal./g.-atom (5) This is in reasonable agreement with the experimental value of 2500 cal./g.-atom. Now that data on thallium are available, it is appropriate to compare them with those of the other metals of subgroup IIIb of the periodic system, namely, gallium and indium. The viscosity of gallium was measured by SpelW2 over the widest temperature range (30 to 1100") of any metal, using the density measurements of Hoather13 over the same temperature range. Indium was investigated by C ~ 1 p i n . l ~Their data were evaluated previously' and the constants of Andrade's first equation calculated ; both metals are compared with thallium in Table 11. (11) H. Moissan, Traite C h . Minerale, Paris, 2 , 259 (1905). (12) K.E.Spells, Proc. Phg8. SOC.(London), 48, 299 (1936). (13) W.H. Hoather, ibid., 4 8 , 699 (1936). (14) M.F. Culpin, ibid., ZOB, 1069 (1957).
Volume 69, Number B February 1966
J. A. CAHILLAND A. V. GROSSE
520
Table 11 from Andrde‘s relationship at the melting point
M.p., OK.
1.63 1.96 2.86
303 429.3 577
?oaicdt
Metal .
Ga
In
Tl
-Constants of 1st Andrade equationExptl. A,, a X 100 cal./g.-atom
4.359 3.020 2.983
The agreement between the experimental and the calculated H , values for gallium and indium is very good. I n line with increasing melting point in the sequence Ga 7In TI, the slope of the viscosity increases so that a t about 1000OK. indium’s viscosity equals that of gallium and a t temperatures above their normal boiling points thallium will be less viscous than indium. The behavior of these metals is illustrated in Figure 1; the three lines are based on the equations of Table I1 and the experimental values a t the respective melting points and a t 1000OK. The viscosities a t the melting point, as calculated from Andrade’s well-known formula,4 are also shown as squares in Figure 1; the agreement, as can be seen, is good except for gallium, where the calculated value (1.63 cp.) is below the experimental one (2.04 cp.). --+
955 1590 2500
Calcd. H,, cal./g.-atom, from eq. 4 Texpti. at the m.p.
954 1525 2270
)lexpti.
2.04 1.94 2.65
at 100O0H.
0.71 0.67 1.05
whereB, Do,’andy are constants, E v . d . is the activation energy for both viscosity and diffusion (cal./g.-atom), and R is the gas constant. It should be stressed 70
60 50
40
30
Discussion of Self-Diffusion Data The experimental determination of self-diffusion of metals is much more difIicult and less precise than that of viscosity; thus, the selfdiffusion of only about eight metals has been measuredl5J6as against about 30 for viscosity. Fortunately, the self-diffusion of gallium was measured by Nachtrieb” and that of indium by Lodding18and others. It has been known for a long time that viscosity and self-diffusion are closely related by the StokesEinstein relationship equaling in its Eyring version, i.e. VD
= k/6u T
IS0
160
I40
IX)
100
am
060
040
1000/TnK
where k is the Boltzmann constant and u the average distance between atoms or (Vatom/N~vog.)’/a. It was emphasized in a recent article19 that the selfdiffusion properties of liquid metals follow directly from viscosity measurements, provided that both viscosity, q, in poises, and self-diffusion, D, in cma2/ set., are expressed in a self-consistent manner, Le., that q/T X D equal the Stokes-Einstein relation. Applying one of the simplest theories of liquids, namely, Frenkel’s kinetics theory,20it follows that = BT~-T~+Ev.&/RT
The Journal of Physical Chemistry
(7)
Figure 2. Viscosity ( q / T ) and self-diffusion of liquid thallium.
(15) H.J. Saxton and 0. D. Sherby, Trans. Am. SOC. Metals, 55, 826 (1962). (16) H.J. Saxton and 0. D. Sherby, Department of Materials Science Report No. 62-9,Oct. 19, 1962,Stanford University, Stanford, Calif. (17) J. Petit and N. H. Nachtrieb, J. Chem. Phys., 24, 1027 (1956).
(18) A. Lodding, 2.Naturforsch., l l a , 200 (1956). (19) A. V. Grosse, Science, 145, 50 (1964). (20) J. Frenkel, “Kinetic Theory of Liquids,” Dover Publishing Co., Inc.,New York, N. Y., 1955,pp. 31-36.
VISCOSITY AND SELF-DIFFUSION OF LIQUIDTHALLIUM
that E v . d . is different from Andrade's H,. It follows further from Frenkel's theory that the constants B and Docan be calculated for any metal from the expressions
B =
hNAvog./bVatom
(9)
and
Do
=
(~/~"A.o,."/")~DV,~O,"/"
(10)
where h is Planck's constant and OD and V a t o m are the Debye temperature in OK. and the liquid atomic volume of the metal (at the melting point), respectively. It was found21that the two equations above correlate the experimental viscosity and diffusion data within experimental error for the eight metals for which self-diffusion data are a ~ a i l a b l e . ~ ~Thus, J ~ there is every reason to expect that they will describe thallium as well. Therefore, let us convert the general equations to the specific case of thallium. Since V a t o m of thallium = 18.10 cm.a/g.-atom a t its melting point while Debye's 0 = 100°,22it follows that the theoretical values for thallium are B t h e o r y = 2.205 X (poise) and Do= 3.357 X (cm.2/sec.). If we express our experimental results in terms of the Frenkel equation (7)' we obtain the expression ?I = 1.011
x
104~e+4400/~*
(11)
521
We can adjust our experimental B value (1.011 X to B t h e o r y above (2.205 X lo") by means of the in our case, e-" = 0.458 or y = Frenkel factor, 0.78. Thus eq. 11now assumes the final form rl =
2.205
x
1 0 4 e - 0 . 7 8 ~ e + 4 4 ~ / ~ ~ (12)
This form lends itself readily to describe self-diffusion in terms of the general equation (8) by means of the calculated Doand the Ev.d. together with the values of eq. 12, as D = 3.357 X 10-4e+0.78e-4WRT (13) Based on the above equation, the values for selfa t the melting point diffusion equal 0.16 X (577'K.), 0.80 X a t 1000°K., and 2.05 X cm.2/sec a t the normal boiling point. On the semilog plot the straight lines for q/T and D vs. 1/T have the same slope of 4400 cal./g.-atom, but an opposite sign, as shown in Figure 2. Acknowledgment. We gratefully acknowledge the financial support of the U. S. Atomic Energy Commission under Contract AT (30-1)2082, (21) Reference 19 gives the data for sodium and sinc; a comparison of the other six metals, including gallium and indium, is in preparation. (22) N. F. Mott and H. Jones, "The Theory of the Properties of Metals and Alloys,'' Dover Publishing Go., Inc., New York, N. Y., 1958,p. 14.
Volume 60,Number 2 February 1966
