Diffusion in Liquids
The recently developed Akgerman-Gainer equation to predict diffusion coefficients of gases in liquids employing significant liquid structure theory and absolute rate theory is modified and extended to estimate diffusivities in liquid-liquid systems and compared with the Wilke-Chang equation.
Although many equations exist to predict diffusion coefficients in liquids (Akgerman and Gainer, 1972a,b; Johnson and Babb, 1956; Othmar and Thakar, 1953; Wilke and Chang, 1955), the molecular transport process of a solute molecule through a solvent medium is still open to speculation. The purpose of this investigation was to modify the recently developed Akgerman-Gainer equation to predict diffusion coefficients of gases in liquids (Akgerman and Gainer, 1972) and apply it to liquid-liquid systems. This equation was developed from a liquid structure model (Eyring and Jhon, 1969) considering diffusion as a rate process and employing absolute rate theory (Glasstone et al., 1941) to estimate the diffusivities. The detailed derivation is presented elsewhere (Akgerman and Gainer 1972). According to the Akgerman-Gainer equation the diffusion coefficient is given by the expression
lutes in associated solvents; system D: diffusion coefficients of nonassociated solutes in nonassociated solvents. Figures 1-4 show comparisons of Akgerman-Gainer and Wilke-Chang equations according to the above classification. In these figures, for simplicity, overlapping points are shown as a single point. Individual system identifications and numerical tables are presented elsewhere (Akgerman unpublished results, 1974, Karahan, 1974).
where EA, the number of solvent molecules around the central solute molecule A, was given by the expression EA
= 6(VA/Vd1I6
SYSTEM
(2) OY
which was somewhat empirical in nature (Akgerman and Gainer 1972). However, considering molecules (or their force fields) to be spherical, it can be shown that theoretically kA
= ?r/arcsin ( V J ~ ~ ’vA1’3 ~/(
+ VB1l3))
(3)
The activation energy difference is given by the equation
A
0
1
,D,
2 8
10‘ CALCULATED
0
AKGERMAN
B
WlLKE
-
GAINER EQN.
- CHANG
EON.
Figure 1. Comparison of Akgerman-Gainer and Wilke-Chang equations for diffusion coefficients of associated solutes in associ-
ated solvents.
where the jumping energy for the pure solvent, E B ~or, for the pure solute, EAA~, can be calculated from the pure component properties by
T R In=
--To
T
where To is some base temperature, taken to be 273.15 K in this work. In the limit when T = To
The Akgerman-Gainer equation, so developed, contains no adjustable parameters or empirical factors. I t is applied to 96 different liquid-liquid systems and over 200 data points (Karahan, 1974) and compared with the WilkeChang equation (Wilke and Chang, 1955). To make the comparison more meaningful the data are divided into four different classes of systems from the point of view of polarity: system A diffusion coefficients of associated solutes in associated solvents; system B: diffusion coefficients of associated solutes in nonassociated solvents; system C: diffusion coefficients of nonassociated so78
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
, 3
2
1 ,D ,
x
4
5
1o5 CALCULATED
- GAINER EQN - CHANG EPN
0
AKGERMAN
0
WlLKE
Figure 2. Comparison of Akgerman-Gainer and Wilke-Chang equations for diffusion coefficients of associated solutes in nonassociated solvents.
Table I. Comparison of Akgerman-Gainer and Wilke-Chang Equation
A
B C D
d
/
I
17 Y
-
-
SYSTEM
- . . . . . . .
. DA.*
0
AKGERMAN -GAINER
10’
.
.
.
4
3
2
1
0
C
CALCULATED
EQN.
0
WILKE -CHANG
EQN.
Figure 3. Comparison of Akgerman-Gainer and Wilke-Chang equations for diffusion coefficients of nonassociated solutes in associated solvents.
30.97 21.27 39.66 21.70
40.37 31.58 40.90 27.16
76 51 28 82
Gainer equation, which is based on significant liquid structure theory and absolute rate theory, gives acceptable results for predictions of diffusion coefficients of all types of solutes (gas or liquid, associated or nonassociated) in all types of liquid solvents without employing any adjustable parameters. Therefore, it is believed that the process of diffusion occurs as proposed previously (Akgerman and Gainer, 1972). Namely, liquids contain “holes” or “vacancies” in their structure and a solute molecule diffusing through the solvent matrix jumps from one hole to another between successive equilibrium positions. The energy required for this jumping is a function of the activation energies for viscosity of the solvent and the solute in their pure state and the activation energy for diffusion can be estimated from these jumping energies by employing a probabilistic geometric mean dependent on the packing factor.
Nomenclature DAB.= diffusion coefficient of solute A in solvent B E M ; = jumping energy for a solute molecule E ~ g l= jumping energy for a solvent molecule Exxj = jumping energy of the pure component X EDAB = activation energy for diffusion E C =~ activation energy for viscosity of solvent B E,x = activation energy for viscosity of pure component
x
k = Boltzmann constant
M A = molecular weight of solute A M B = molecular weight of solvent B N = Avogadro’s number n = number of data points in a system
R = gasconstant T = temperature
To = base temperature
0
1
2
0,
I:
3
VA = molar volume of solute A VB = molar volume of solvent B Greek Letters
4
1os CALCULATED
- GAINER EQN. - CHANG EQN.
o
AKGERMAN
o
WlLKE
Figure 4. Comparison of Akgerman-Gainer and Wilke-Chang equations for diffusion coefficients of nonassociated solutes in nonassociated solvents. The root mean square error for both of the equations is separately calculated in each system employing the equation (7)
where ci is the error for each data point in a system and n is the number of total data points in that system. The results are presented in Table I. The comparatively acceptable overall results obtained by employing the Akgerman-Gainer equation for liquid-liquid systems, coupled with the previously published (Akgerman and Gainer, 1972a,b) good predictions of this same equation for gas-liquid systems, prove that the Akgerman-
Aerms = root mean square error E , = error for individual data points bg [A
= viscosity of B = packing factor, number of solvent molecules around
a central solute molecule on the same plane Literature Cited Akgerman, A., Gainer, J. L., Ind. Eng. Chem., Fundam., 11, 3 7 3 (1972a). Akgerman, A., Gainer, J. J. Chem. Eng. Data. 17, 372 (1972b). Eyring, H., M. S. Jhon, Significant Liquid Structures”, pp 13-48, 81-95, Wiley. New York, N.Y.. 1969. Glasstone. S., Laidler, K. J., Eyring, H., “Theory of Rate Processes”,Chapter IX, McGraw-Hill, New York, N.Y., 1941. Johnson, P. A., Babb, A. L., Chem. Rev., 56, 387 (1958). Karahan, M.,M.S.Thesis, Ege University, Izmir, Turkey, 1974. Othmar, D. F., Thakar, M. S., Id.€ng. Chem., 4 5 , 5 8 9 (1953).
k.,
Wilke, C. R., Chang, P., A./.Ch.€.J., 1, 264(1955).
Chemical Engineering Department Ege University Bornova, Izmir, Turkey
Aydin Akgerman
Received for reuiew February 14,1975 Accepted September 22,1975 Ind. En$. Chem.. Fundam., Vol. 15, No. 1. 1976
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