Some simple liquids and their viscosity

Physica 84B (1976) 371-374 © North-Holland Publishing Company

LETTER TO THE EDITOR SOME SIMPLE LIQUIDS AND THEIR VISCOSITY W. HERREMAN Interdisciplinary Research Centre, Katholieke Universiteit Leuven, Campus Kortri/k, Universitaire Campus, B-8500 Kortrijk, Belgium

Received 23 June 1976 Received in final form 16 September 1976

Viscosity measurements from different investigators for the saturated liquid state of argon, neon, krypton, xenon, oxygen, nitrogen, fluorine, carbon monoxide, carbon dioxide, methane and ethane are examined. For densities greater than about twice the critical density, the density dependence of the viscosity can be described by means of the Batschinski equation. The molar volume of these liquids, where the fluidity ~ = 0, corresponds vory well with the value of the molar volume of the corresponding solidified gas at the melting point and is also related to the Van der Waals constant b.

1. Introduction

liquids ~/could be written as: = K(V-

Molecular transport in liquids is governed as well by the temperature as by the free volume of the molecules. There are many empirical expressions which relate the viscosity with temperature. Among them the most popular is the Arrhenius equation = A e B/T ,

(3)

where K is a constant and V is the volume o f the liquid. Vo is the volume of the liquid where ~ = 0, ~b being the fluidity, the reciprocal of the viscosity. When using S.I.-units, ~2is expressed in Ns/m 2, V and Vo in ma/kg. We want to investigate in how far eq. (3) is able to describe the viscosity of some simple liquids, listed in .table I, in their saturated liquid state. Therefore we write (3) in the following modified form:

(1)

where A and B are constants. This formula is able to describe the temperature dependence of ~7for simple liquids in a broad temperature range [1 ]. Less attention has been given to the free volume dependence of the viscosity. Doolittle [2] developped the following empirical expression 77 = A ' eB'(v°/vf) ,

Vo) -1 ,

q~= KIP -1 + K 2 ,

(4)

where p, expressed in kg/m 3, equals 1/V, K 1 = I l K and K 2 = - V o / K . I f K I andK2 are known we can calculate Vo:

(2)

Vo = -K~/KI

A ' and B' being constants; vo is the limiting specific volume of the liquid at absolute zero as defined by Doolittle and Doolittle and vf is the free volume. Although this formula originally was derived for n-alkanes its validity has also been proven for simple liquids [3]. The idea that the viscosity depends on the free volume of the molecules was first introduced by Batschinski [4] who observed that for many organic

(5)

and compare it with the volume o f the solidified gas at the melting point.

2. Selection of viscosity data Not all the viscosity data available in the literature were used to investigate the Batschinski behaviour of 371

372

w. Herreman / S o m e simple liquids and their viscosity

the viscosity. In general the following criterium was used for selection of the viscosity data: their completeness in view of the temperature range. Most recent measurements satisfy this condition as they have been carried out from the tripelpoint to the critical point. Only for carbon monoxide the temperature range is restricted from 68.6 K to 80.9 K. From this selection the measurements of some authors showing irregularities (f.i. considerable scatter or disagreement with other existing data) have been omitted.

.°32,.., I o.~*N~.~LL~,~,~s +

G~EVENOONK m

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t]

~.

/

/ /1 ~

/"

I I

o

I

/ I

f

o

j

i 3. Analysis

0.8

1.0

1/(2&) 1.2

1.4

1~6

I~B

Fig. 1. The fluidity of liquid oxygen as a function of o -1 . The sources used to relate the fluidity to the density by means of the Batschinski equation are given in table I. The sources include also the necessary density information or is at least citated in it. The data for argon, oxygen, nitrogen and methane of different investigators correspond very well in a broad density range. For neon and carbon monoxide the correspondence was not so good and the data of the different investigators were handled separately. On fig. 1 the fluidity of liquid oxygen is plotted as a function of p - i : for densities greater than 833 kg/m 3, which is about twice the critical density of oxygen, the fluidity satisfies the Batschinski equation. By making similar plots for the other liquids we come to a general conclusion: for densities greater than about twice the critical density, liquid argon, neon, krypton, xenon, oxygen, nitrogen, fluorine, carbon monoxide, carbon dioxide, methane and ethane show a Batschinski behaviour in their saturated liquid state. Expressions for the fluidity are given in table II. The density is expressed in kg/m 3. With these expressions and eq. (5), values for the

Sources for viscosity data References 5,6,7,8,9 1,10 8 8 5,11,12,13 5,11,14,15

References fluorine carbon monoxide carbon dioxide methane ethane

16 17,18 3 6,19,20 21

(6)

b = RTc/8Pc.

The results are given in table Ill. The Vo value for neon is the mean value of these calculated from ref. [1] (Vo = 0.01391 l/mole) and from ref. [10] (V 0 = 0.01401 l/mole), For carbon monoxide Vo is the mean value calculated from ref. ref. [17] (V 0 = 0.03013 l/mole) and from ref. [18] (Vo = 0.030331/mole). Values for Vs were taken from ref. [22] except for fluorine, carbon monoxide and

Table II Fluidity of some liquids for p greater than about 2pc @(105 m2/Ns) argon krypton xenon neon

Table I

argon neon krypton xenon oxygen nitrogen

molar volume Vo of the liquid were calculated and compared with the molar volume Vs of the solidified gas at the melting point. V0 is also compared with tht Van der Waals constant b calculated from [26]

oxygen nitrogen fluorine carbon monoxide carbon dioxide methane ethane

352p -1 - 0.214 399p -1 - 0.140 385p -1 0.110 564t~- t - 0.392 516o - 1 - 0.355 331o -1 - 0.239 3400 -1 - 0.359 385p -1 0.213 317p -1 - 0.341 371o -1 - 0.402 254p -1 0.177 217o -1 - 0.429 174o-1 - 0.255

(ref. [10]) (ref. I 11)

(ref. [17]) (ref. [18])

W. Herreman / Some simple liquids and their viscosity

l

t

I

4. Conclusion

I

Vo(10 -3 I / mole)

The empirical Batschinski equation, originally derived for organic liquids, is also a useful expression to calculate the viscosity coefficient of liquid argon, neon, krypton, xenon, oxygen, nitrogen, fluorine, b e n monoxide, carbon dioxide, methane and ethane along the saturated vapour pressure line for densities greater than about twice the critical density. Vo, calculated from this expression corresponds surprisingly well with the molar volume of the solidified gas and is linear related to the Van der Waals constant b.

c.zHS

J Aro~NC

4(

CH4

Kr CO;[

20

i

20

l

40

373

I b(1631/m°te)

60

Fig. 2. The molar volume VO as a function of the Van der Waals constant b.

References

Table 11I

[ 1] See, for example W. Herreman and W. Grevendonk, Cryogenics 14(1974) 395. [2] A.K. Doolittle, J. Appl. Physics 22 (1951) 1471. [3] See, for example W. Herreman, W. Grevendonk and A. De Beck, J. Chem. Phys. 53 (1970) 185. [4] A.J. Batschinski, Z. Phys. Chem. 84 (1913) 643. [5] A. Van Itterbeek, J. Hellemans, H. Zink and M. Van Cauteren, Physica 32 (1966) 2171. [6] J. Hellemans, H. Zink and O. Van Paemel, Physica 46 (1970) 395. [7] A. De Beck, W. Grevendonk and W. Herreman, Physica 37 (1967) 227. [8] V.P. Sluysar, N.S. Rudenko and V.M. Tretyakov, Ukr. Fiz. Zh. (USSR) 17 (1972) 1257. [9] W.M. Haynes, Physica 67 (1973) 440. [10] V.P. Sluysax, N.S. Rudenko and V.M. Tretyakov, Ukr. Fiz. Zh. (USSR) 18 (1973) 190. [11] J. Hellemans, H. Zink and O. Van Paemel, Physica 47 (1970) 45. [12] W. Grevendonk, W. Herreman, W. De Pesseroy and A. De Beck, Physica 40 (1968) 207. [ 13 ] W.M. Haynes, unpublished data. [14] W. Grevendonk, W. Herreman and A. De Beck, Physica 46 (1970) 600. [15] V.P. Sluysar, Ukr. Fiz. Zh. (USSR) 17 (1972) 529. [161 W.M. Haynes, Physica 76 (1974) 1. [171 N.S. Rudenko and L.W. Schubnikov, J.E.T.F. (USSR) 10 (1934) 1049. [18] J.P. Boon, J.C. Legros and G. Thomaes, Physica 33 (1967) 547. [19] E.T.S. Huang, G.W. Swift and F. Kurata, A.I. Ch. E.J. 12 (1966) 932. [20] W.M. Haynes, Physica 70 (1973) 410. [21] G.W. Swift, J. Lohrenz and F. Kurata, A.I. Ch. E.J. 6 (1960) 415. [22] A. Van Itterbeek, Physics of High Pressures and the Condensed Phase (North-Holland Publ. Co., Amsterdam, 1965).

Comparison of V0 with Vs and b

A Kr Xe Ne 02 N2 F2 CO CO2 CH4 C2H6

V0 (l/mole)

Vs (1/mole)

b (l/mole)

0.02428 0.02933 0.03764 0.01396 0.02310 0.02952 0.02103 0.03023 0.03070 0.03171 0.04406

0.02441 0.02893 0.03594 0.01393 0.02350 0.02950 0.02160 0.03031 0.03063 0.03160 0.04220

0.03219 0.03978 0.05105 0.01709 0.03183 0.03913 0.02875 0.03985 0.04267 ~04278 0.06380

carbon dioxide. For carbon monoxide we found a Vs value in ref. [23]. For the other two fluids values for Vs were taken from the work of Eyring and co-workers [24,25]. In his work, after having investigated five liquids, Batschinski supposed that V0 is the mean value of the molar volumes in the liquid and in the solid state at the melting point. From the table it is seen that for these condensed gases V0 is somewhat different: it corresponds very well with Vs. This correspondence was already remarked for liquid neon [1 ]. In fig. 2 a plot is made of Vo as a function of b. There is a linear relation between these two quantities: Vo = 0.0667 b + 0.00270 1/mole.

(7)

374

W. Herreman / Some simple liquids and their viscosity

[23] K. Clusius, V. Piesbergen and E. Varde, Helv. Chim. Acta 43 (1960) 2059. i~4 ]-T.IL Thomson, H. Eyring and T. Ree, J. Phys. Chem. 67 (1963) 2701.

4

[25] M.E. Zandler, J.A. Watson and H. Eyring, J. Phys. Chem. 72 (1968) 2730. [26] Handbook of Chemistry and Physics, R.C. Weast, ed. (The Chemical Rubber Co., 1972).