An Empirical Relationship for Predicting the Variation with Concentration of Diffusion Coefficients in Binary Liquid Mixtures Victor Sanchez’ and Michael Cllfton Laboratoire de Chimie-Physique et Electrochimie, Laboratoire Associe au C.N.R.S. No. 192, Universitb Paul Sabatier, 3 1077 Toulouse, France
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An empirical correlation of the form D I 2 = (xlD20 x2Dlo)(l m m@)may be used to represent the variation with composition of mutual diffusion coefficients in binary liquid mixtures, where @ is the concentration dependent “thermodynamic factor” and m is an empirical constant. Examples of its application are given. The equation is accurate for all types of binary liquid systems, even for systems which are far from ideal. The equation requires fewer initial data than most other correlations, but one experimental value of 0 1 2 in the middle of the concentration range is necessary.
The study of diffusion in binary liquid systems is an area of research which has received a great deal of attention. Nevertheless, it has not been possible so far to derive a relationship which accurately describes the variation of the mutual diffusion coefficients with concentration. Various theoretical and empirical equations have been proposed but none has proved satisfactory for all types of binary liquid mixtures. Some typical correlations are those presented by Vignes (1966), Darken (1948), Rathbun and Babb (1966), Hartley and Crank (1949), Leffler and Cullinan (1970). A recent article by Kosanovitch and Cullinan (1976) presents an unusual approach to the problem, which is of great interest because of the novelty of its outlook. However, their correlation requires rather a lot of numerical data and this is a definite disadvantage from a practical point of view. The aim in setting up an equation for practical use is to provide a relationship which requires a minimum amount of data as a starting point and which gives reasonably accurate results for as many different sorts of system as possible. Many of the equations previously developed are applicable to ideal or nearly ideal binary mixtures. The equation presented here is also applicable to nonideal, associated, binary systems.
Origin of the Equation Bearman (1961) has presented a theory of diffusion based on considerations derived from statistical thermodynamics. His equations for mutual and self-diffusion coefficients in a binary liquid system may be developed into an equation in the following form (cf. Loflin and McLaughlin, 1969)
so that eq 1becomes 012
= (xi&
+ xzDiM
This equation was proposed by Darken (1948) to describe diffusion in metals, though it has, since then, been applied to binary organic mixtures. In the case of ideal mixtures eq 1is further reduced to
+
D12 = Dipid = ( x ~ D z xzD1)
c 12= 318
(~ll{2z)”2
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
(3)
(5)
However, a number of workers (e.g., Caldwell and Babb, 1956) have found that a simpler, entirely empirical equation can also be applied to experimental results for ideal systems, viz. DlZid = (X1D2O
+ xZD1O)
(6)
where DiO is the diffusion coefficient of component i in the system as its concentration approaches zero, i.e. Di0 = lim x,-0
012
= lim D, x,-0
Equation 6 has a great practical advantage over eq 5 as a definition of D1zid. The diffusion coefficients in very dilute mixtures are more easily measured than self-diffusion coefficients and, if necessary, may even be calculated from a number of correlations, e.g., Wilke and Chang (1955), Sitaram et al. (1963), King et al. (1965) and Tyn and Calus (1975).Also eq 6 requires only two values of the diffusion coefficient [Dl0 and DzO], whereas D1 and D2 are concentration dependent and have to be measured throughout the entire range of concentration. For systems which are close to ideality, Vignes (1966) has proposed the empirical relationship Diz =
where D12 is the mutual diffusion coefficient, x i is the mole fraction of component i, Di is the self-diffusion coefficient of component i in the mixture, the {are friction coefficients, and /3 is the thermodynamic factor p=--b l n a l --a h a 2 (2) Dlnxl blnxz For some solutions, which Bearman (1961) calls “regular solutions”, it is found that
(4)
(7)
It is also possible, for some systems, to use a simplified form of eq 1 012
= D12’*/3
(8)
However, when this equation is applied to nonideal mixtures it is generally found that the thermodynamic factor p causes too large a variation in the calculated value of 0 1 2 . This is true whether Dlzid is determined from eq 5, as in Darken’s equation, or from eq 6. Various authors have noted this “overcorrection” but it is also true that the curve showing fi as a function of x1 is generally very similar in shape to the graph of 0 1 2 as a function of X I . It was such observations that led
1 nC,Hw-QHB CH,COCH, - CHCI,
q m
e
2
A
CzH50H. CCI,
o
C,HSOH.C,H, CH,COCH,. H,O
3
0.5
0
1
8
Figure 1. Variation of the ratio D12/D12idwith the thermodynamic factor a. Rathbun and Babb (1966) to suggest the following relationship (9) where S = 0.6 for systems showing positive deviations from Raoult’s law and S = 0.3 for systems showing negative deviations. This equation works well for some systems but not for others. In almost all cases, however, it is superior to eq 4 and 7 .
Correlation Proposed The general form of eq 1 suggested to the present authors the possibility of graphing D12/D12id for each system as a function of the corresponding 6, using the D1zid of eq 6. It was found that, for the various systems studied, the variation of D12lD12’~with P could be approximated by a straight line passing through the point (l,l),the slope of which differed from one system to another [see Figure 11, i.e. (10)
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where D1zid is defined by eq 6 and m g must equal 1,since as x 1 approaches 0 or 1the left-hand side of eq 10 approaches 1 and p approaches 1. Therefore D12
=
(x1D2O
+ x 2 D l o ) ( 1 - m + mp)
(11)
The constant m , the slope of the line in Figure 1, is a peculiarity of each system and is in general not equal to 1. This shows that eq 3 is rarely true.
Predictive Calculations In practical applications of eq 11,it is necessary to have a value for 0 1 2 near the middle of the concentration range in order to be able to calculate m. In the calculations for this article the point chosen was the one at which reaches a minimum or maximum value. Thus eq 11requires the determination of the diffusion coefficient a t three points on the concentration scale: one at each end and one in the middle. From the values D1O and D2O, together with the value for D12 for the concentration at which /3 has its extreme value, it is possible to calculate a value for m which will allow the prediction of 0 1 2 throughout the rest of the concentration range. Calculations have been performed for a number of typical systems, using data available in the literature. Values of m for these systems are shown in Table I. Discussion The values of m given in Table I have been used to calculate values for 0 1 2 by means of eq 11.These values were calculated for each mole fraction increment of 0.1 (i.e., for x = 0.1,0.2, . . . ,0.9). Figures 2 and 3 show some representative calculated values of D12 graphed as a function of the corresponding experimentally determined values. It may be seen that the deviations between the calculated values and the observed coefficients are very small indeed. Equations 4,7, and 9, together with the combined eq 6 and 8, have also been used to calculate 0 1 2 for the same series of concentrations. The average errors between these calculated values and the experimental values for a number of systems are compared in Table 11, with the corresponding errors obtained using eq 11. In every case it is eq 11 which causes the least error, and this is true for every kind of system studied. A reasonable question to ask at this point is whether m varies not only from system to system but also with changes in temperature. Equation 11 would obviously be a much more useful correlation if m were constant or varied only slightly over a considerable range of temperature. It is likely that m shows a t least some temperature dependence since the thermodynamic factor /3 is known to move toward a value of 1 for all concentrations as the temperature rises. It is impossible to say a priori whether D12/D12id is more sensitive to temperature than p or less, so that without further experimental evidence it cannot even be decided whether m increases or decreases as the temperature rises. The experimental data for diffusivities and fugacities over a range of temperatures are
Table I. Values of the Constant m for a Number of Binary Systems and Sources of Numerical Data Data references System
m
P
Dl2
Di
Near ideal
n-Heptane-benzene n-Hexane-benzene CHCls-CC14
0.80 Brown and Ewald (1951) 0.75 Myers (1955) 1.28 McGlashan et al. (1954)
Harris et al. (1970) Harris et al. (1970) Kelly et al. (1971)
Harris et al. (1970) Harris et al. (1970) Kelly e t al. (1971)
Regular
Acetone-CHCls Acetone-CC14 Acetone-benzene Benzene-cyclohexane
0.41 0.79 1.00 0.48
Anderson et al. (1958) Anderson et al. (1958) Anderson et al. (1958) Rodwin et al. (1965)
McCall and Douglass (1967) Hardt et al. (1959) McCall and Douglass (1967) McCall and Douglass (1967)
Associated Ethanol-CC4 Methanol-benzene Ethanol-benzene Acetone-water Ethanol-water
Beckmann and Faust (1915) Bachman and Simons (1951) Othmer (1943) Scatchard et al. (1939)
0.87 Barker et al. (1953) 0.85 Scatchard et al. (1946) 0.81 Udovenko and Fatkulina (1952) 0.91 Beare et al. (1930) 0.90 Dobson (1925)
Stokes and Hammond Hardt et al. (1959) (1956) Johnson and Babb (1956) Johnson and Babb (1956) Johnson and Babb (1956) Johnson and Babb (1956) Anderson et al. (1958) Hammond and Stokes (1953)
McCall and Douglass (1967)
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977 319
Table 11. Average Error in Values for D12 Calculated Using Various Equations (10-5 cm2 s-1)
System
Eq 4, Darken (1948)
Eq 7, Vignes (1966)
Eq 9, Rathbun and Babb (1966)
Eq 6 and 8
Eq 11, this work
Near ideal
n-Heptane-benzene n -Hexane-benzene CHCls-CCld
0.02 0.11 0.12
0.13 0.32 0.17
0.09 0.10 0.11
0.06 0.18 0.08
0.04 0.08 0.05
Regu1ar
Acetone-CHCl3 Acetone-CC14 Acetone-benzene Benzene-cyclohexane
0.48 0.23 0.26 0.36
0.30 0.28 0.21 0.20
0.08 0.08 0.33 0.05
0.36 0.19 0.24 0.15
0.05 0.04 0.24 0.01
Associated
Ethanol-CCl4 Methanol-benzene Ethanol-benzene Acetone-water Ethanol-water
0.39 0.59
0.21 0.35 0.35 0.28 0.20
0.23 0.25 0.07 0.43 0.22
b.16 0.32 0.35 0.17 0.20
0.05 0.11
0.60
0.45 -
I
I
I
2
1 Di,
Figure 2. Comparison of predicted vs. experimental diffusion coefficients for regular binary systems. a t present too limited. An experimental program in which measurements of diffusion coefficients a t various temperatures were coupled with measurements of liquid-vapor equilibria at the same temperatures would have to be completed before this question could finally be resolved. Conclusion Equation 11 has important advantages as an equation to be used in predicting mutual diffusion coefficients in binary organic mixtures: it requires only a small amount of initial data and the work involved in determining the extra value of the diffusion coefficient is justified by the greatly improved accuracy of eq 11 in comparison with any other predictive relationship. Literature Cited Anderson, D. K., Hall, J. R., Babb, A. L., J. Phys. Chem., 62, 404 (1958). Bachman. K. C., Simons, E. L., J. Am. Chem. SOC.,73, 4968 (1951). Barker, J. A., Brown, I., Smith, F., Discuss. Faraday SOC.,15, 142 (1953). Beare, W. G., McVicar, G. A., Fergusson, J. B., -J. Phys. Chem.. 34, 1310 (1930). Bearman, R. J., J. Phys. Chem., 65, 1961 (1961). Beckmann, E., Faust, O., 2.Phys. Chem. A,, 69, 235 (1915). Brown, I., Ewald, A. H., Aust. J. Sci. Res. A , , 4, 198 (1951). Caldwell, C. S.,Babb, A. L., J. Pbys. Chem., 60, 51 (1956). Darken, L. S.,Trans. Am. Inst. Mining Met. Eng., 175, 184 (1948). Dobson, H. J. E., J. Chem. SOC.(London), 127, 2866 (1925).
320
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
do5, ern? s.'
0.05
0.10 0.19
1
3
C ~ C .
Figure 3. Comparison of predicted vs. experimental diffusion coefficients for associated binary systems. Hammond, B. R.. Stokes, R . H., Trans. Faraday SOC.,49, 890 (1953).
Hardt, A. P., Anderson, D. K., Rathbun. R.,Mar, B. W., Babb, A. L., J. Phys. Chem., 63, 2059 (1959). Harris, K. R., Pua, C. K. N., Dunlop. P. J.. J. Phys. Chem., 74, 3518 (1970). Hartley, G. S.,Crank, J., Trans. Faraday SOC.,45, 801 (1949). Johnson, P. A.. Babb, A. L., J. Phys. Chem., 60, 14 (1956). Kelly, C. M., Wirth, G. B., Anderson, D. K.. J. Phys. Chem., 75, 3293 (1971). King, C. J., Hsueh, L., Mao, K.-W., J. Chem. Eng. Data. 10, 348 (1965). Kosanovich, G. M., Cullinan, H. T., Ind. Eng. Chem., Fundam., 15, 41 (1976). Leffler, J., Cullinan, H. T., Ind. Eng. Chem., Fundam., 9, 84 (1970). Loflin. T.. McLauahlin. E.. J. Phvs. Chem.. 73. 166 (1969). k c a l l . D. W., D&glass;D. C.: J. Phys. Cheh., 71; 987 (1967). McGlashan, M. L., Prue, J. E., Sainsbury. I. E. J., Trans. Faraday SOC.,50, 1284 (1954). Myers, H. S.. Ind. Eng. Chem., 47, 2216(1955). Othmer, D. F., Ind. €ng. Chem., 35, 614 (1943). Rathbun, R. E., Babb, A. L., Ind. Eng. Chem., Process Des. Dev., 5 , 273 ( 1966). Rodwin, L., Harpst, J. A., Lyons, P. A., J. Phys. Chem., 69, 2783 (1965). Scatchard, G., Wood, S.E., Mochel, J. M., J. Phys. Chem., 43, 119 (1939). Scatchard, G., Wood, S. E., Mochel, J. M., J. Am. Chem. Soc., 66, 1957 ( 1946). Sitaram, R., Ibrahim, S. H., Kuloar, N. R., J. Chem. Eng. Data, 8, 198 (1963). Stokes, R. H., Hammond, B. R.. Trans. Faraday SOC.,52, 781 (1956). Tyn, M. T., Calus, W. F., J. Chem. Eng. Data. 20, 106 (1975). Udovenko, V. V., Fatkulina, L. G., Zh. Fiz. Khim., 26, 719 (1952). Vignes, A., Ind. Eng. Chem., Fundam., 5 , 189 (1966). Wilke. C. R.. Chang, P., A.I.Ch.€. J., 1, 264 (1955).
Received for review June 1, 1976 Accepted December 13,1976
