EFFECT OF MOLECULAR PROPERTIES ON BINARY LIQUID DIFFUSION COEFFICIENTS Diffusion coefficients for several liquid systems are explained using a modified absolute rate theory equation. The modification considers the intermolecular force fields present in each system and how these forces vary with the composition of the system.
OBINSON, Edmister, and Dullien (6) have recently reported Rnew data on diffusion coefficients in binary liquid mixtures, which show that differences in the diffusion coefficients owing to changes in the molecular properties of the solute diffusing through a single solvent cannot always be accounted for by existing correlations. This communication points out that a new equation recently published by Gainer and Metzner (4)is more successful in correlating the data reported by Robinson et at. The equation as developed by Gainer and Metzner (4) is a modification of the absolute rate theory equation for the diffusivity. The equation obtained using the absolute rate theory is (5):
where = diffusivity of solute A in solvent B = Boltzmann constant = absolute temperature = viscosity of solvent PB = Avogadro’s number N = geometrical parameter, equal to 8 for ethanol, E methanol, and glycol, and equal to G for most liquids = molar volume of solvent VB = activation energy for viscosity EPB EDAB = activation energy for diffusion DAB
k T
Usually, the activation energies are assumed to be of equal value (5), thus simplifying Equation 1. Gainer and Metzner propose to estimate this quantity as follows:
and it is assumed that:
The activation energies to be used in Equation 2 can be calculated using Equations 3, 4, and 5. The ratio of the heat of vaporization due to hydrogen bonding to the total heat of vaporization appearing in Equation 5 can be found in the literature for a number of compounds (7). Therefore, diffusivities can be estimated using Equation 1, and the only data needed for the calculations are the physical properties of the substances involved. To examine the data presented by Robinson, Edmister, and Dullien (6),let us simplify the equations somewhat. We will assume that [ = 6 for all of the liquids involved and that the molar volumes of the liquids all have approximately the same value. Thus, Equation 2 simplifies to: EPB ED, = -
2
+
The authors (6) have presented data for the diffusion of two different solutes into the same solvent. This quantity would be predicted to be (from Equation 1) :
(7) Let us first consider the case of two different solutes diffusing in a solvent which does not form hydrogen bonds. In that case, EpC-== 0, and Equations G and 7 simplify to :
--DAG
[
-EPC-D)~/*
( n - D
exp.
rxx
= VX1l3
rAB
=
-I+~
~ A A
2
B B
EPB-D)~/~]
12
DBC
where
---
2
2
(8)
RT
If solutes A and B have nearly the same chemical structures, it may be assumed that EPB-D = E~A-D.In that case, Equation 8 becomes :
~
Epx-H
= viscosity activation energy due to hydrogen bond-
!??E=1 DBC
E/.L~-D
ing = viscosity activation due to dispersion forces
(9)
The data presented by Robinson et al. show:
and
EPX = EPX-DLt, EPX-H Solutes
The total energy of activation for viscosity is given by
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I&EC F U N D A M E N T A L S
Methylcyclohexane ( A ) Cyclohexanone (B) n-Octane ( A ) 1-Heptanol (B)
Solvent n-Octane (C)
0.96
Methylcyclohexane (C)
2.66
The first system studied fits this analysis well, although the second does not. The authors (6) note that perhaps the data for the diffusion of 1-1ieptanol into methylcyclohexane are in error. If this is true, perhaps this anomaly can be accounted for in that way. Let us consider a system in which the solvent is capable of forming hydrogen bonds, but one of the solutes, A, is not. In this case = 0, and again assuming that E ~ - B - D= E p A - D , Equations 6 and 7 reduce to:
From Equation 10 it c,an be seen that the ratio of diffusivities of two solutes diffusing through a hydrogen-bonded solvent is greater than unity if A refers to the non-hydrogen-bonded solute and B refers tO the hydrogen-bonded solute. This observation is confirmed by the data:
-
Methylcyclohexane (A) Cyclohexanone (B)
1-Heptanol (C)
D AC DBC 1.67
n-Octane ( A ) 1-Heptanol (B)
Cyclohexanone (C)
1.17
Solutes
Solvent
The authors also consider the diffusion of a given solute, A, through two different solvents, B and C. In this case, application of Equation 1 shows:
Salute
n-Octane (A)
DAB DAC Solvents Calcd. Ex#. (6) Methylcyclohexane 1 .70 2.28
(B)
Cyclohexanone (C) Methylcyclohexane (A)
Cyclohexanone (A)
n-Octane (B) 1-Heptanol (C)
3.50
5.75
n-Octane (B) 1-Heptanol (C)
5.40
7.33
Although in absolute value the calculated ratios are not the same as the experimental ratios, they increase in approximately the same manner as one reads down the table. Therefore, it would appear that the data can be correlated with the equations as presented by Gainer and Metzner for the prediction of diffusivities in liquids. The equations of Gainer and Metzner can also be used to predict the temperature dependence (3) and concentration dependence (2) of the diffusivity. Therefore, the authors (6) may also be able to evaluate their data from those viewpoints. literature Cited
(1) Bondi, A.,Simkin, D.J., A.Z.Ch.E. J . 9,473 (1957). (2) Gainer, J. L., unpublished manuscript. (3) Gainer, J. L., Metzner, A. B., submitted to J. Phys. Chem. (4) Gainer, J. L.,Metzner, A. B., Proceedings of Symposium on Transport Phenomena,. A.1.Ch.E.-I. Chem. E . Meetinn, - London, Vol. 6; p. 74,1965. ( 5 ) Glasstone, S., Laidler, K. J., Eyring, H., "Theory of Rate Processes." ChaD. 9. McGraw-Hill. New York. 1941. (6) Robinson, R.'L.,' Jr., Edmiste;, W. C., Dullien, F. A. L., IND. END. CHEM. FUNDAMENTALS 5,74 (1966). JOHN L. GAINER
Using Equations 2, 4, 5 , and 11 along with the physical property data presented by the authors (6), the following results were obtained :
University of Virginia Charlottesville, Va. RECEIVED for review March 23, 1966 ACCEPTEDApril 13, 1966
CORRESPONDiENCE COMPUTATION 1OF EQUILIBRIUM VAPOR COMPOSITION FROM SO LUT I0N VAPOR PRESSURE DATA OF MULTICOMPONENT SYSTEMS SIR: Mixon, Gunovvski, and Carpenter (2) have recently published an article regarding computing equilibrium vapor composition for multicomponent systems. I n a review of previous work (7, 5), they stated that Tao's method (5) appears specific to binary systems and does not seem to be easily generalized to multicomponent systems. This statement is in error, as evidenced by development (4, 6). For an n-component system, n fugacity coefficients, one for each component at a given liquid composition with a known total vapor pressure cain be determined from the following equations :
Equation 1 represents a set of n - 1 equations which are integrals of Gibbs-Duhem equation by using trapezoid rule. a designates the point with unknown vapor composition and b designates n 1 known points in the neighborhood of point a. These equations reduce to those of Tao (5) for a binary system. As an example, suppose that we wish to determine the vapor composition of a liquid mixture containing 0.715 mole fraction of methyl acetate, 0.084 chloroform, and 0.201 benzene a t 60.9' C. and 760 mm. of Hg from two known data points: x = (0.711, 0.222, 0.067), y = (0.800, 0.153,0.047) at 61.2' C. and 760 mm. of Hg; x = (0.646, 0.076, 0.278), y = (0.765,
-
VOL 5
NO. 3 A U G U S T 1 9 6 6
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