3262
TERESA 5. REE, TAIKYUE REE, AND HENRYEYRING
environment of the strontium ion in strontium bromide monohydrate is analogous to the environment of the barium ion in BaC12,BaBrz, and BaL4 except that two of the nearest halide ions around a barium ion are replaced by water molecules around the strontium ion. Each strontium atom is surrounded by three nearestneighbor bromine atoms in the same crystallographic mirror plane and by two bromine atoms and one water molecule on each of the equivalent mirror planes above and below. The distance from a strontium atom to a nearest-neighbor bromine atom ranges from 3.13 to 3.38 A. while each nearest-neighbor oxygen atom is a t a distance of 2.63 A. Thus, in strontium bromide monohydrate and in the barium halides each metal cation has nine nearest neighbors. Since in strontium bromide each strontium atom has only eight nearest neighbors,2 the strontium ion is probably not large enough to allow the stable coordination of nine bromide ions as nearest neighbors. However, as oxygen is snialler than bromine, a strontium ion is large enough to allow the stable coordination of seven bromide ions and two water mole-
cules. It thus seems likely that the greater stability of SrBrz.H20over SrBrz is due primarily to the increase in coordination number of the strontium ion by the water molecule to allow the same type of stabilization as found in the barium halide crystal lattices, Each bromine atom of type 1 has for nearest neighbors three oxygen atoms a t distances of 3.35 8. (2) and 3.34 H., four strontium atoms with distances ranging from 3.25 to 3.43 and seven bromine atoms with distances varying from 3.79 to 4.23 8. Each bromine atom of type 2 has for nearest neighbors one oxygen atom a t a distance of 3.31 i., three strontium atoms at distances of 3.13 8. (2) and 3.32 K.,and seven bromine atoms with distances ranging from 3.89 to 4.36
K.,
Acknowledgments. This work was supported by grants from the Sational Aeronautics and Space Administration and The Robert A. Welch Foundation of Texas. (4) R. Sass, T. Brackett, and E. Brackett, J . P h y s . Chem., 67, 2132 [ 1963).
Significant Structure Theory of Transport Phenomena
by Teresa S. Ree, Taikyue Ree, and Henry Eyring Department of Chemistry, University of Utah, Salt Lake City, Utah
(Received J u l y 9, 1964)
By applying the significant structure theory of liquids to transport phenomena, the equation for viscosity is derived for rigid-sphere systems. When the Enskog theory of viscosity is compared to the present theory, close agreement between the two theories is found. The diffusion coefficient is also derived in terms of viscosity and V,, the solid molar volunie a t the melting point. For simple liquids, viscosities and diffusion coefficients are calculated from the theory without using any adjustable parameters. The agreement between theory and experiment is satisfactory.
I. Introduction There are three major theoretical approaches to the study of transport phenomena of liquids and dense gases. One of these theoretical approaches was intraduced by Enskog.' Although the Enskog theory is The Journal of Physical Chemistry
derived for rigid-sphere systems, it has been applied to real dense gases in excellent agreernent with experi(1) (a) S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambridge University Press, Cambridge, 1939, Chapter 16; (b) ibid., Chapter io.
SIGNIFICANT STRUCTIJRE THEORY OF TRANSPORT PHENOMENA
ment. A second approach was made by Kirkwood.2 Starting from the Liouville equation, he obtained exact expressions for the flux vectors in terms of the so-called nonequilibrium radial distribution function. Certainly this approach to the study of transport phenomena is more satisfying from a theoretical viewpoint than any other. At present, however, only a very few calculations have been carried out, so that the method is still not developed for practical work. The third approach was made by Eyring, Ree, and co-w~rkers”~ by applying the theory of absolute reaction rates to various transport phenomena. Since the flow of a liquid is a rate process, insofar as it takes place with a definite velocity under a given condition, it is justifiable to apply the theory of absolute reaction rates to the problenis of transport phenomena. In this paper, we derive the equation of viscosity for rigid spheres from Eyring’s rate process approach, and we compare the results with Enskog’o theory. Further, we derive the equation for diffusion, and we apply our theory to real liquids. The theory is very successful. 11. Theory The significant structure theory of liquids of Eyring, Ree, and co-workers516is based on the concept of a quasi-lat tice theory containing highly mobile vacancies randomly distributed. Such a mobile vacancy confers gas-like properties on the molecule which jumps into it. These vacancies also give rise to a positional degeneracy for neighboring solid-like molecules. The numbers of solid-like and gas-like molecules are NV,/V and N ( V - V , ) / V , respectively, where V is the molar volume of the liquid, V , is the molar volume of the solid at the melting point, and N is Avogadro’s number. The significant structure theory of liquids has implicit in it a general theory of transport properties. Since there are solid-like and gas-like degrees of freedom in the liquid, both must be taken into account in calculating the viscosity. If a fraction, IC,, of the shear plane is covered by solid-like molecules and the remaining fraction, xg, by gas-like molecules, then the viscosity, q , which is the ratio of shear stress, f,to rate of strain, 8, is4r5 q =
f
-
S
== (IC&
+
Zgfg)/S
3263
The term qg in eq. 1 is taken as equal t,o the rigidsphere F iscosity at infinite dilution and is equal tolb
where d is the molecular diameter. The quantity q s in eq. 1 is calculated in accord with Eyring’s earlier procedure’ as4,5
In the above equation, as shown in Fig. 1, X2X3 is the area occupied by a molecule on which the shear stress, f,, is acting; X I is the distance between two successive molecular layers, one of which slips with the shear
-0000 t
A,
-AFigure I. Distances between molecules in a liquid. ), is the distance between two successive equilibrium positions ; XI is the distance between two successive molecular layers.
rate, s, with respect to the other; the distance for one jump along.the direction of j , is then X cos O t , et being the angle between the direction of stress and the direction of jumping; k , is the jumping frequency of the molecule to the ith neighboring hole when f, = 0 and is assumed to be equal to k’ for every position i. Expanding the exponential of eq. 3, we obtain
-Kirkwood, J . Chem. Phys., 14, 180 (1946). (3) H. Eyring, D. Henderson, B. Stover, and E. Eyring, “Statistical Mechanics and Dynamics,” John Wiley and Sons, I n o . , New- k’ork, N. Y . , 1964, Chapter 16. (4) T. S. Ree, T. Ree, and H. Eyring, Proc. KatZ. Acad. Sci. U . S., (2) J. G.
Here, the subscripts, is and g, indicate that the quantity belongs to the solid-like and the gas-like molecules, respectively. In the last equality of eq. 1, it is assumed that IC,equals the fraction of t5e sold-like molecules, V,/V, and consequently that IC, = (V - V,)/V.
48, 501 (1962).
(5) H. Eyring and T. Ree, ibid., 47, 526 (1961). Ree, T. Ree, and H. Eyring, J . Phus. Chern., 6 8 , 1163 R. Marchi and H. Eyring, J . Chem. Educ., 40, 526
(6) (a) T. 8. (1964); (b) (1963).
Volume 68, Number I 1
November, 1964
TERESA S. REE, TAIKYUE REE, AND HENRYEYRING
3264
Since the sites are randomly distributed over a solid angle, we may take the averages of cos 8( and cos2 81. Thus
v
Z COS' 8%= z-('/3)
- V,
V
where 2 is the number of nearest neighbors and is equal to 12 when-there is hexagonal packing; Z ( V - VJ/V is the number of holes in the immediate vicinity of a molecule. In eq. 3 we are able to correIate moIecular dimensions, XI, Xz, X3, and X to V, for hexagonal packing as follows ( c j . Fig. 2) xzx3
Figure 2. Hexagon normal to the direction of diffusion in a liquid or hexagonal packing. Molecule 7 is diffusing forward, hence f is acting upward. h ~ h sis the area occupied by a pair of molecules 7 and 1 in the direction normal to the paper. The other molecular pairs, 7-2, 7-3, 7-4, 7-5, and 7-6, also occupy the area of X Z X ~for each.
= d'ja2 -
2 where a' is a proportionality constant, and it is assumed that the activation energy is proportional to $ ( a ) and inversely proportional to the number of holes (V - Vs)/
4
X I = --a
2
X = a
vs.
Hence, substituting eq. 5 through 9 into eq. 4, written as
v,
a3
.\/z - N where a is the distance between two nearest neighbors. In eq. 4, k' is written in accord with Eyring's reaction rate theory' as kTF* k ' = K--exph F
-EO
kT
(7)
V (nmkT) 'laNNlf exp 2(V - V&)K
to = -
v,
F*
(27rmkT ) h3 =
Vf
[exp
(2nmkT ) af exp h2
[
9 1 2kT
=
t
[1.,109
___-
If in the above equation the rigid sphere condition is introduced , i.e .
J ( T ) (9)
($(a) = 0
(c:) N u 3 - 2.4090 (Fy] (10)
Here E and cr are the energy and the distance characteristic of the system and are listed in the 1iterature.ll In eq. 7 to is the activation energy and is written as
The Journal of Physical Chemistry
(12)
-
J(T) (8)
where vf and af are the free volume and the free area, respectively; J ( T ) is the partition function for the internal degrees of freedom, and for the hexagonal packing lattice $(a) is given bylo $(a)
z$(a)]
~~
I n the above equation Z is taken as 12 and I f is a free length and equal to v f / a f . Substituting eq. 2 and 12 into eq. 1 leads to
where K is the transmission coefficient, and F and F* are the partition functions for the molecule in the initial and activated complex st'ates, respectively, and are given by 8-10 F=
[V-''V~lis 21cT
tsis
If
= 2(a
- d)
K = l
(14)
then the viscosity for the rigid spheres becomes (7) 8.Glasstone, K. J. Laidler, and H. Eyring, "The Theory of.Rnte Processes," McGraw-Hill Book Co., Inc., New York, N. Y., 1941, Chapter 9. (8) D. Henderson, J . Chem. Phys., 39, 1857 (1963). (9) T.6.Ree, T. Ree, and H. Eyring, ibid., 41, 524 (1964). (10) R. H. Wentoff, R. J. Buehler, J. 0. Hirschfelder, and C. F. Curtiss, ibid.,18, 1484 (1950). (11) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids," John Wiley and Sons, Inc., New York, N. Y . , 1964, pp. 1110-1112.
3265
SIGNIFICANT STRUCTURE THEORY OF TRANSPORT PHENOMENA
Table I : Comparison of Calculated and Experimental Viscosities for Simple Liquids T,
Substancea and parameters
P*
g./cc.
OK.
Ar d = 2.944 A. V. = 24.98 (cc./ mo1e)a
NQ d = 3.142 h;. V. = 29.31 (cc./ mole)c
1.414b 1.397 1.392 1.39 1.37 1.31 1.22 1.16
84.25 86.25 86.90 87,30 90.0 99.5 111.o 120 127 133,5 138.7 143 147 149 64.8 71.8 78.07 90.65 99.43 109.3 113.4 122.5 123.8 125.08 126
1,lO 1.02 0.95 0.88 0.78
0.70 0.861d 0.832 0.798 0.746 0.69 0.63 0.60 0.50 0.47 0.43 0.38
hplli
?them3
mpoise
mpoise
2.82b 2.62 2.56 2.52 2.32 1.62 1.37 1.16 1.00 0.77 0.70 0.63 0.56 0.50 2 .22d 1.67 1.33 0.95 0.77 0.62 0.60 0.43 0.37 0.37 0.31
Substances and parametera
c1*
2.72 2.50 2.44 2.42 2.21 1.78 1.35 1.17 1.02 0.86 0.76 0.67 0.57 0.50 2.05 1.60 1.27 0.97 0.76 0.62 0.56 0.41 0.38 0.34 0.30
d
3.546A. V , = 40.455 ( c c . / mole)a =
CHI d = 3.237A. V. = 31.06 (cc./ mole)*
CsHi d = 4.506 h;. V , = 82.826 (cc./ rnole)O
See ref. 16. ref. 15.
See ref. 17.
c
See ref 18.
See ref. 19.
e
T. OK.
197 7 199 2 202 7 207 4 207 8 213 0 213 2 220 2 220 8 228 1 229 1 237 8 273 2 91 94 98 5 103 109 111 131 148 5 161 166 5 180 273 2 283 2 293 2 303 2 313 2 323 2 333 2 343 2 353 2
See ref. 20. 'See ref. 21.
Pt B./CC.
1 658' 1 653 1 646 1 635 1 634 1 623 1 622 1 606 1 604 1 587 1 585 1 ,564 1 468 0 452' 0 448 0 442 0 436 0 428 0 425 0 395 0 364 0 338 0 325 0 277 0 '300lA 0 8895 0 8790 0 8685 0 8576 0 8466 0 8357 0 8248' 0 8145'
See ref. 22.
I
1)..*11.
Vtheort
rnpoise
rnpoise
7 29f
7 10 6 80 6 6 6 6 5
49 46 16 10 69 5 66 5 30 5 24 4 940 3 85O 2 lo 1 86 1 61 1 45 1 25 1 18 0 94 0 68 0 5!) 0 56 0 51 9 12' 7 58 6 52 5 64 5 03 4 42 3 92 3 58 3 29
* See
8 43 8 02 7 53 6 86 6 81 6 27 6 22 5 58 5 51 4 98 4 92 4 40 2 96 1 32 1 26 1 17 1 10 1 01 0 99 0 76 0 61 0 52 0 48 0 37 7 70 6 22 5 23 4 52 3 96 3 53 3 18 2 90 2 68
ref. 23.
See
'See ref. 24.
hexagonal packing, it is six (cf. Fig. 2). Substituting eq. 6 into 16 we obtain
V - V , 5 mkT ~- "' V 1 6 d 2 ( 'IT (I5)
)
The diffusion coefficient also may be calculated by means of the following equation which has been obtained by Eyring and Ree6
D = - kT X2Xa
€-7
11
Where 5 is the effective number of neighboring molecules of the diffusing molecule lying in the same plane and perpendicular to the' direction of diffusion. For
111. Calculations I n order to carry out nuinerical calculations of the rigid-sphere viscosity of eq. 15, we use V , froin the result of machine calculations by Alder and Wainwright. 1 2 According to these authors, the phase transit ion froin solid to liquid is observed at 1.5Vo < V , < 1.71T0, where V ois the closest packing volume, being equal to NrP/ (12) (a) B. J. Alder and T.E.Wainwright, J . C h e h . Phvs., 27, 1208 (1957); (b) ibid., 33, 1439 (1960).
Volume 68,Number 11 November, 1064
3266
TERESAS. REE, TAIKYUE REE, AND HENRYEYRING
@, When we take the mean value V , 15 becomes 1 -= 1 1e
0.811
- 0.540
1 - 0.540
=
1.6V0, eq.
(i) (18)
where b
2T
= - d3N
3
The viscosity based on the Enskog theory is written asla r 1e
l
-
b [l -I- 0.8Y YV
+ 0.761Yz]
(20)
where Y = ( P V / N k T ) - 1. In order to calculate the preceding equation, we use P V / N k T obtained by ThieleI3for the compressibility equation of rigid spheres PVNLT
-
agreement between the two theories is quite good. I t is interesting to note that we obtain the virial expansion from both eq. 18 and 20, in which all of the virial coefficients are positive. Through the studies of the Enskog theory‘ and the work by Longuet-Higgins and P ~ p l e , it ’ ~ is well established that even though the rigid sphere condition is imposed for deriving the viscosity equation, the calculated results from the latter agree excellently with experimental viscosities of simple liquids. Therefore, in Table we compare ‘the viscosities calculated
Table I1 : Comparison of Calculated and Experimental Diffusion Coefficients for Simple Liquide Substance arid V,
Ar
84.31&0.13 90 100 110 120 130 140 150 CH4 100 V , = 31.06 110 (cc./mole)0 120 130 CeHe 287,7 V , = 82.826 288.2 ( cc./mole)8 298.2 308.2 318.2 cc14 298.2 V. = 8 7 . 1 308.2 (cc./mole)d 318.2
V,
l+X+XZ (1 - X)’
(21)
where X = b/4V. In Fig. 3 we compare the hard sphere viscosities calculated from our eq. 18 with those calculated from Enskog’s theory (eq. 20). As shown in this figure, the I
C
x T ,OK.
24.98 (cc./mole)a =
io3
poise
2.82* 2.32d 1.7 1.40 1.14 0.90 0.69 0.48 1.45’ 1.18 1.02 0.94 7. 0ag 6.96 5.99 5.30 4.60 8 .8ag 7.70 6.90
D . , , ~x ~ io6, om.2/sec.
ahso x io6. om.2/seo.
2 07 It 0 06c 1.77 2.30 2 43* 3.48 3 54 4.65 4 80 6.23 6 08 8.54 7 45 12.0 8 72 18.5 9 98 3.7 3 Od 4 2 5.0 6.6 6 0 8.5 7 8 1.61 1 6h 1 8 8 f O 01 1.64 2 1 5 f O 05 1.98 2 40 f 0 03 2.31 2 67 f 0 06 2.75 1.31 1 41’ 1.56 1 75 1.80 1 99
See ref. 16. * See ref. 17. See ref. 25. See ref. 26. e See ref. 20. See ref. 22. See ref. 23. See ref. 27. a See ref. 29. See ref. 28.
7 fib
I3 Ens& 1
I
I
0
0.4
0.8
b/v Figure 3. Comparison of Enskog’s theory with the authors’ theory.
T h e Journal of Physical Chemistry
1.2
I
1.6
(13) E. Thiele, J . Chem. P h y s . , 39, 474 (1963). (14) H. G. Longuet-Higgins and J. A. Pople, ibid., 25, 884 (1956). (15) “Handbook of Chemistry and Physics,” 41st Ed., C. D. Hodgman, Editor-in-Chief, Chemical Rubber Publishing Co., Cleveland, Ohio, 1959-1960. (16) K. Clousius, 2. physik. Chern., B31, 459 (1936). (17) N. F. Zhadanova, Soviet P h y s . J E T P , 4, 749 (1957). (18) F. Simon, M. Ruhemann, and W. Edwards, 2. physik. C h a m . , B6, 331 (1930). (le) N. F. Zhadanova, Soviet P h y s . J E T P , 4, 19 (1957). (20) S. Chang, H. Pak, W. Paik, S. Park, M. S. John, and W. S. Ahn, J . Korean Chem. Soe., 8 , 33 (1964). (21,) “International Critical Tables,” E. W. Washburn, Editor-inChief, McGraw-Hill Book Co., New York, N. Y., 1926. (22) A. F. Gerf and G. I. Galkov, Soviet Phys.-Tech. P h y s . , 10, 725 (1941).
SIGNIFICANT STRUCTURE THEORY OF TRANSPORT PHENOMENA
from eq. 15 with the measured values for several types of liquids over a range of temperatures. The experimental viscosities are given in column 4, and the computed viscosities are in column 5. The molecular diameter, d , listed in column 1 is calculated from the van der Waals constant, b, available in the literature1$ according to eq. 1'9. The agreement between experiment and theory is striking for viscosities of Ar and Nt, and for Clz, CH4,and CeHethe agreement is quite good. If V 8and d used in the calculations were to be fixed by using the experimental viscosities, the agreement would have been excellent in the latter three cases also, I n Table 11, we compare the diffusion coefficient calculated by using eq. 17 with the experimental data.25-28 For calculating D we use the experimental viscosity as
3267
listed in column 3. Agreement with theory and experiment is very good in general. Acknowledgment. The authors express appreciation to the National Science Foundation for financial support of this research under Grant No. GP-415. (23) J. Timmermans, " Physico-Chemical Constants of Pure Organic Compounds," Elsevier Publishing Co., Inc., New York, N. Y., 1950. (24) " Landolt-Bornstein Tabellen," Verlag von Julius Springer, Berlin, 1923. (25) J. W. Corbet and J. H. Wang, J . Chem. Phys., 25, 422 (1956). (26) J . Naghizadeh and S. A. Rice, ibid., 36, 2710 (1962). (27) K. Grauprer and E. R. €3. Winter, J . Chem. Soc., 1145 (1952). (28) H. Watts, B. J. Alder, and J. H. Hildebrand, J . Chem. Phys., 23, 658 (1955). (29) H. Sackmann and G. Kloos, 2. physik. Chem., 209,319 (1958).
Volume 68,Number 11 November, 1964
