Self-Diffusion Coefficients and the Type of Liquids Keishi Gotoh Deparfmenf of Chemical Process Engineering, Hokkaido University, Sapporo, Japan
The compressibility of liquids obtained from a cell model is applied to the modified Enskog theory for the self-diffusion coefficient of hard sphere dense fluids. The results explain the probable difference in the self-diffusion coefficients due to the type of liquids (liquid metals and nonmetallic, saturated liquids). The Arrhenius plot suggests that in liquid metals the activation energy for self-diffusion is attributed to only one of the nearest neighbors of a diffusing atom; in nonmetallic liquids it is attributed to about half of the nearest neighbors.
Many statistical theories have been proposed for transport processes of simple liquids (Hirschfelder et al., 1954;Rice et al., 1968); a short history of recent studies on self-diffusion is given in the literature (Collings et al., 1971; Protopapas et al., 1973). Some of them are based upon the velocity autocorrelation calculation (Brown and Davis, 1971; O’Reilly, 1971) and others are modifications of the Enskog theory of hard sphere fluids (Dymond, 1974; Frisch and McLaughlin, 1971; Protopapas et al., 1973). In most cases the molecular dynamics calculations of Alder et al. (1970) play an important role as the rigorous computer experiment for hard sphere fluids. Protopapas e t al. (1973) compared theoretical selfdiffusion coefficients of many investigators and of their own with experiments for various liquid metals, in which they used the Enskog theory modified by a correction factor of Alder et al. (1970). Ely and McQuarrie (1974) calculated the viscosity and thermal conductivity coefficients of argon by use of the Enskog theory with the compressibility factor obtained via the statistical mechanical perturbation theory, which covers the range from gas to what we call the gas-like liquid, but not to the liquid region. These theories deal with some particular substances and do not explain directly the probable difference in the self-diffusion coefficients due to the type of the liquids. For more practical purpose, Hildebrand (1971; Hildebrand and Lamoreaux, 1973) modified Batschinski’s equation for viscosity (Frenkel, 1955) and he therein discussed the selfdiffusion coefficient. Ertl and Dullien (1973a,b) subsequently reported that Hildebrand’s equation for the self-diffusion coefficient requires a small modification, namely
mass, CH is the correction factor, and YH can be obtained from (Ely and McQuarrie, 1974) y = z + 7 ( ” aT z) v-1
(3)
2 is the compressibility factor for which the Carnahan-Starling expression is applicable in the cn3e of the hard sphere system (Carnahan and Starliag, 1969), yielding
where +H = (r/6)u3n is the picking density of the hard sphere fluid, n is the number density, and the subscript H denotes the hard sphere system. Fcr ease of later calculations, we herein express appr0ximate.y the correction factor of Alder e t al. (1970) in the following simple form. CH = -0.268
+ 1 1 . 2 4 4 -~ 1 9 . 1 4 4 ~(0.2 ~ 5 4~ 5 0.5)
(5)
Protopapas et al. (1973) o b t i n e d the self-diffG-ion coefficients of liquid metals from eq 2 and 4, in which they used a 4~ vs. T relation deterinined semiempirically so as to fit to the experimental self-dipfusion coefficients. Since we are dealing with not only spherical nonpolar liquids but also nonspherical polar liquids, and since molecules are in contact with one another in the liquid state, we set (m/p)ll3in place of cr in eq 2 (cf. Li and Chang, 1955; Eyring et al., 1964),where p is the density. Hence eq 2 reduces to the self-diffusion coefficient of liquids, D
v-v
D = B ( Y )a where V is the volume, V Ois the minimum volume, B is a constant, and the exponent cy is about 1.5 on the average for benzene, monohalogenated benzene derivatives, n-heptane, and n-decane. However, the physical meaning of these two constants remains unknown. The object of this study is to discuss the probable difference in the self-diffusion coefficients due to the type of liquids (liquid metals and nonmetallic, saturated liquids) rather than arguing the accuracy of the prediction for individual substances. Self-Diffusion Coefficient Following Protopapas e t al. (1973), we use the modified Enskog theory for the self-diffusion coefficient of hard sphere dense fluids.
where u is the hard sphere diameter, k is the Boltzmann constant, T i s the absolute temperature, m is the molecular 180
Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
in which the subscript H is removed from C and y , X is the molecular shape parameter (A = 1for spheres), T*= kT/t, and t is the Lennard-Jones (6,12) pair potential parameter. In eq 6 , f i is inserted into the both sides so as to yield the following expressim for nonmetallic liquids. (6a) where XkT,/t = 8 4 1 9 (Gotoh, 1972), M is the molecular weiglit, No is Avogadro’s number, and T , is the critical temperature. Equation 6 with X = 1 is applied for liquid metals, while for nonmetallic liquids the expression, eq 6a, is used for D + . The following equation of state can apply for liquids (Gotoh, 1972; Renon et al., 1967). (7)
where 4 = 4 4~ = (a/6) 4u3 p/m is the molecular packing
,
100
,
,
,
I
,
,
,
I
t
,
I
,
1 1
/ A
0.05
I
Y
0.01
d
05
01
9
(-)
Figure 1. Comparison of .y for hard sphere systems (eq 4) and for -, eq 9 ( A = 1). liquids (eq 9): - - - -,eq 4 with 4~ = $/A; /
I
I
I
I
I
I
I
0.1
005
(V-V,)/V.
density of the liquid, and u is set equal to the Lennard-Jones (6,12) pair potential parameter. Equation 7 is the same in form as that obtained by Flory (1965), and Renon et al. (1967) derived it based on a modification of Prigogine's cell theory and on a three-parameter theorem of corresponding states. Substitution of eq 7 into eq 3 yields
(-1
Figure 2. Volume dependence of the self-diffusion coefficient of liquids: - - - -, data line of Ertl and Dullien (1973a,b);-, eq 6 with eq 9,10,11,and 13,or approximation by eq 14; TA, TB = eq 13a is used in place of eq 13, respectively, for X = 1and 1.695.
AT* = 11.145233uP(1- up1/3) (0.6 5 up 5 0.9) In the case of spherical molecules ( X = l),we can compare y with YH.After substituting 4~ = 4 1 4 into eq 4, we can calculate the Y H vs. 4 relation which agrees closely with that obtained from eq 9 in the range of 4 = 0.3 to 0.6 (see Figure 1). In the view that 4 should not exceed the closest packing density, eq 9 is superior to eq 4 in the high density region. Despite simple in form, eq 9 can apply not only to spherical nonpolar liquids but also to nonspherical polar liquids, and, moreover, it gives the probable bounds due to X (A = 1 to 1.695).Accordingly, we use eq 9 in later calculations. Substitution of 4~ = A@/* into eq 5 yields C = -0.268
+ 5 . 8 8 5 ~-~5 . 2 4 7 ~ ~ ~
(10)
where A4 is used in place of 4 so as to extend its use to nonspherical polar liquids (cf. Gotoh, 1972). In this way we can calculate the self-diffusion coefficient of liquids from eq 6 using a relation between u p and AT.. For nonmetallic, saturated liquids, we apply the following modified Guggenheim relation (Gotoh, 1974).
+ 43 (1 - T,) + 47 (1 -
3 . 1 =~ 1~
-
-
(11)
where T, = TIT,= 9 X T . / ( 8 4 ) . Alternatively, there is no general relation available between the temperature and the packing density of liquid metals. Although Protopapas et al. (1973) derived a relation using the effective hard sphere diameter, it was based on a viewpoint different from the present one: the packing density of liquid metals a t the melting point is 0.472, which is obtained semiempirically by adjusting theoretical self-diffusion coefficients to the experiments; while the present theory is based on the fact that the packing density of liquid metals at the melting point is 0.65, which is obtained from the first peak in the radial distribution curve. For liquid metals we herein assume eq 12a derived from eq 7 by setting Z = 0. Namely, we consider the cell theory and neglect the pressure effect for liquid metals. As we shall see later, this assumption yields reasonable results as well as the method of Protopapas et al. (1973).
(12a)
or 0.982
up = 2 [l
+ (1 - 2 4 AT*/3)'/2] (0.4 5 AT* 5 1) (12b)
where eq 12b is obtained from a relation other than eq 12a, and the numerical value 0.982 is set so as to fit eq 12b to eq 12a (Gotoh, 1972). In this way we have obtained all the expression required for our present purpose. Hereafter we consider the effects of volume and temperature on the self-diffusion coefficient for a variety of liquids.
Volume Dependence Ertl and Dullien (1973a,b) measured relations between the self-diffusion coefficient and the relative volume expansion for n-paraffins, benzene, and its derivatives (Figure 2), of which data lines cover the range from the melting point to the boiling point at atmospheric pressure. The minimum volume V Ois empirically determined by extrapolating fluidity-volume plots to the point where the fluidity becomes zero. It is herein postulated that V Ocorresponds to the closest packed state of the substance, Le., XC$ = d \ / 2 / 6 . We thus obtain
v - Vo = 1 - u p
(13) VO UP As can be seen from eq 6, 9, 10, and 11,D+ of nonmetallic, saturated liquids depends on up and A. u p relates to the liquid volume as eq 13. Hence D+ can be expressed by ( V - Vo)/Vo and X as depicted in Figure 2 comparing with data lines of Ertl and Dullien (1973a,b).Since the change in up from the melting point to the boiling point is small and (V - V0)IVoexpands its range widely, the D+ vs. ( V - Vo)/Vorelation becomes almost a straight line in Figure 2 and can be closely approximated by
(y)
0.515 V
D+ = X
V
1.56
(0.03 5 ( V - Vo)/Vo5 0.4) Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
(14) 181
Table I. Lennard-Jones (6,12)Pair Potential Parameter and Molecular Shape Parameter 0.1
Number
Liquids
elk, K
h
118 0.99 148 0.98 589 1.33 C6H6 610 1.36 C6H5F 622 1.40 C,H5C1 706 1.40 C,H,Br 745 1.40 803 1.40 ‘ 6 H5 I 9 n-Heptane 640 1.49 10 n-Decane 7 69 1.57a 11 Acetone 589 1.45 12 Ammonia 448 1.40 13 Water 762 1.49 14 Methanol 651 1.59 UPitzer’s acentric factor of Chiu et al. (1973) is used.
2
1
(xi*?
( - )
Figure 3. Arrhenius plot of the self-diffusion coefficient of liquids. Data for liquid metals are cited from Protopapas et al. (1973). The numerical numbers for data points of the nonmetallic liquids denote the liquids listed in Table I; the dBta are cited from Dymond (1974), Ertl and Dullien (1973a,b),Ertl (19741, O’Reilly (1968; 1971; 1974), O’Reilly and Peterson (19711, Pruppacher (1972), Reddy and Doraiswamy (1967),and Rice et al. (1968). Curve a = eq 6 with eq 9,10 and 12; curves b, c = eq 6 with eq 9,10, and 11, respectively for X = 1 and 1.695; curves A, B = eq 4 is used in place of eq 9, respectively, for curves a and b.
Although D+ contains the liquid density, the numerical value of p1I3does not change so widely for the liquid state from the melting point to the boiling point. Accordingly, eq 14 can be regarded as the same in form as eq 1. The exponent is 1.56 irrespective of the type of the liquids. As the probable range of the molecular shape parameter is from 1.0 to 1.695, all the data should fall within the two theoretical lines. The agreement with the experiments is not so good. This seems to be so because we have postulated that Vo corresponds to the closest packed state of the substance. If VOcorresponds to the state of the packing density A4 = 0.7. in place of 0.74 ( = ~ f i / 6 ) ,for example, eq 13 becomes
and the theoretical curves in Figure 2 shift upward as denoted by TA and TB, respectively, for X = 1.0 and 1.695. The small change in the value assumed for VIVO affects largely the theoretical value of D+. Hence the detail in the effect of X is not discussed here. Table I lists tlk and X for various liquids: elk is obtained from the critical temperature and X is obtained from Pitzer’s acentric factor (Gotoh, 1972). T e m p e r a t u r e Dependence From eq 6 to 12 we know that D+ can be determined by XT* and A; since we set X = 1, D+ of liquid metals becomes a function of T* alone. Figure 3 compares the D+ vs. (AT*)-’ relations for a variety of liquids in the Arrhenius plot, in which some of the data are displayed only by two points which should be connected with each other by a line. The data cover the range below the boiling point a t atmospheric pressure. t/k and X of the nonmetallic liquids are listed in Table I, and elk of the liquid metals are obtained from the gross mean condition, T* = 0.412, for various liquid metals a t the melting point 182
Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
1
Ar
2 3 4 5 6 7 8
CH, CCl,
(Gotoh, 1972). Figure 3 shows the explicit difference in D+ between the nonmetallic liquids and the liquid metals. If we express the data points of the liquid metals by a single line, D+ = DOexp(-ElkT), the slope of the line gives the activation energy of E sc: t , while the nonmetallic liquids give E = (3 4 ) t h which corresponds roughly to one-half of the coordination number in the liquid state (Gotoh, 1971; 1972). Accordingly, we can say in the overall view that in the liquid metals the activation energy for the self-diffusion is attributed to only one of the nearest neighbors of a diffusing atom. This energy corresponds to a complete dissociation of a pair of atoms and could be attributed to a small displacement of a number of neighbors, while in the nonmetallic liquids the activation energy is attributed to about one-half of the nearest neighbors of a diffusing molecule. From eq 6,9,10, and 11we can calculate the D+ vs. (AT*)-’ relations for nonmetallic, saturated liquids as shown in Figure 3 (curves b and c); for liquid metals we use eq 12 in place of eq 11and we obtain curve a. The theoretical curves explain fairly well the difference in the self-diffusion coefficients due to the type of liquids. If we use the Carnahan-Starling expression with 4~ = d / f i in place of eq 9, curves a and b become curves A and B, respectively (see broken curves): the CarnahanStarling expression can apply for liquid metals but not for the nonmetallic liquids. N
Conclusion The Enskog theory of hard sphere dense fluids is applied to the self-diffusion coefficients of liquid metals and nonmetallic, saturated liquids (eq 6), in which the compressibility relation obtained from a cell model (eq 9) is used. Despite a simple calculation, the result explains fairly well the difference in the self-diffusion coefficients due to the type of liquids. The experimental data in the Arrhenius plot, Figure 3, gives the following overall view that in the liquid metals the activation energy for the self-diffusion is attributed to only one of the nearest neighbors of a diffusing atom, while in the nonmetallic liquids the energy is attributed to about one-half of the nearest neighbors of a diffusing molecule. Acknowledgment The referee’s comments are gratefully acknowledged. Nomenclature C = correction factor (eq 5 and lo), dimensionless D = self-diffusion coefficient D+ = defined by eq 6 or 6a, dimensionless k = Boltzmann’s constant M = molecular weight
m = molecular mass N o = Avogadro's number n = plm, number density of molecules T = absolute temperature T , = critical temperature T , = TIT,, dimensionless T. = k TIC, dimensionless V = volume V o = minimumvolume up = X @ / ( ~ 4 / 6dimensionless ), y = defined by eq 3, dimensionless 2 = compressibility factor, dimensionless Greek Letters C , O = Lennard-Jones (6,12) intermolecular pair potential parameters X = molecular shape parameter ( X = 1.0 to 1.695; X = 1 for spheres), dimensionless p = liquiddensity @H = ( x / 6 ) $ n , bulk-mean particle volume fraction, dimensionless @ = *&-I, molecular packing density of liquid, dimensionless Subscript H = hard sphere system
Chiu. C. H., Hsi, C., Ruether, J. A,. Lu, B. C. Y.. Can. J. Chem. Eng., 51, 751 (1973). Collings, A. F., Watts, R. O., Woolf. L. A,, Mol. Phys.. 20, 1121 (1971). Dymond, J. H., J. Chem. Phys., 60,969 (1974). Ely, J. F., McQuarrie, D. A., J. Chem. Phys., 60,4105 (1974). Ertl, H.. Dullien, F. A. L., J. Phys. Chem., 77, 3007 (1973a). Ertl, H., Dullien, F. A. L., A.l.Ch.E. J., 19, 1215 (1973b). Ertl, H., Mol. Phys., 28, 1637 (1974). Eyring, H., Henderson, D., Stover, B. J., Eyring, E. M.. "Statistical Mechanics and Dynamics," p 458, Wiley. New York. N.Y., 1964. Flory, P. J., J. Am. Chem. Soc., 87, 1833 (1965). Frenkel, J., "Kinetic Theory of Liquids." p 204, Dover, New York, N.Y., 1955. Frisch, H. L., McLaughlin, E., J. Chem. Phys., 55,3706 (1971). Gotoh. K., Nature(London) Phys. Sci., 231, 108 (1971). Gotoh. K., Nature(London) Phys. Sci., 239, 154 (1972). Gotoh. K.. lnd. Eng. Chem., Fundam., 13, 287 (1974). Hildebrand. J. H., Science, 174, 490 (1971). Hildebrand, J. H., Lamoreaux. R. H., J. Phys. Chem., 77, 1471 (1973). Hirschfelder, J. O.,Curtiss. C. F., Bird, R. B., "Molecular Theory of Gases and Liquids," pp 611, 647, Wiley, N e w York, N . Y . , 1954. Li, J. C. M.. Chang, P., J. Chem. Phys., 23, 518 (1955). O'Reilly, D. E., J. Chem. Phys., 49, 5416 (1968). O'Reilly, D. E., J. Chem. Phys., 55,2876 (1971). O'Reilly, D. E., Peterson, E. M., J. Chem. Phys., 55, 2155 (1971). O'Reilly, D. E., J. Chem. Phys., 60, 1607 (1974). Protopapas. P., Andersen. H. C., Parlee, N. A. D., J. Chem. fhys., 59, 15 (1973). Pruppacher. H. R., J. Chem. fhys., 56, 101 (1972). Reddy, K . A., Doraiswamy, L. K., Ind. Eng. Chem., Fundam., 6,77 (1967). Renon, H., Eckert, C. A,. Prausnitz. J. M., lnd. fng. Chem., Fundam., 6,52 (1967). Rice, S. A.. Boon, J. P., Davis, H. T., "Comments on the Experimental and Theoretical Study of Transport Phenomena in Simple Liquids." in H. L. Frisch and Z. W. Salsburg, Ed., "Simple Dense Fluids," pp 251, 332, Academic Press, New York, N.Y.. 1968.
Literature Cited Received for review January 20,1975 Resubmitted July 31,1975 Accepted April 21,1976
Alder, B. J., Gass, D. M.. Wainwright, T. E., J. Chem. Phys., 53, 3813 (1970). Brown, R.,Davis, H. T., J. Phys. Chem., 75, 1970 (1971). Carnahan. N. F., Starling, K. E.,J. Chem. fhys., 51, 635 (1969).
Vapor-Liquid Equilibrium from a Hard-Sphere Equation of State R. De Santis," F. Gironi,'
and L. Marrelli'
lstituto di Chimica Applicata e Metallurgia, Facolti di lngegneria dell'Universita di Cagliari, Italy
An equation of state is proposed which combines the hard-sphere repulsion compressibility factor of Carnahan and Starling and an attraction term. The temperature dependence is derived for the latter from the saturated liquid densities of the pure compounds. For the case of multicomponent systems, simple mixing rules have been used for the constants relating to pure compounds and vapor-liquid equilibria have been estimated in a wide range of temperatures and pressures. Extension to mixtures requires the evaluation of an arbitrary binary constant.
Introduction The estimation and interpretation of multicomponent vapor-liquid equilibria a t low or moderate pressures are usually carried out on the basis of activity coefficients which characterize deviations from the ideality of liquid mixtures. The possibility of estimating activity coefficients depends on knowledge of the relation between the excess Gibbs free energy and the system composition, pressure, and temperature. Except for the case of solutions containing nonpolar molecules with low acentric factors, affording the use of the regular solution theory (Hildebrand et al., 19711, the excess free energy is interpreted by means of simple semiempirical theories using arbitrary binary constants. However, if a single equation of state could hold for the whole range from ideal gas to saturated liquid, it would be possible to derive fugacities I s t i t u t o di Chimica Applicata e Industriale, FacoltA di Ingegneria dell'universitk di Roma, I t a l y .
of liquids without resorting to activities. This possibility would be granted by an equation, best with few constants and sufficient accuracy and generality. The procedure has a number of advantages as compared with the other method: it avoids defining different reference states for the activity coefficients when some component is supercritical and, therefore, nonexistent as liquid at the given pressure and temperature; it eliminates the heterogeneity in treating deviations from ideal behavior of the two phases which causes frequent discrepancies around the critical point, where differences of volumetric properties between the two phases vanish; it yields the explicit dependence of the equilibrium from the state variables (pressure, temperature, liquid, and vapor composition), whereas semiempirical theories yield only the dependence on the liquid phase composition; this involves difficulties, as in correlating isobaric vapor-liquid equilibria when, lacking experimental heats of mixing, the temperature dependence of activities is not known; it allows deriving the properties of mixtures from the knowledge of the Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
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