THEORY OF DIFFUSIOX IN LIQUIDS
981
A THEORY OF DIFFUSIOS I S LIQUIDS ALLEN E. STEARK Department of Chemistry, I'nzversity of .Missouri, Columbia, Mtssourt AND
EDWIT bI. IRISH
AND
HESRY ETRISG
Frick Chemacal Laboratory, Princeton Cnaversaty, Prznceton, N e w Jersey Receaved J a n u a r y 88, 1940 INTRODUCTION
Although for the diffusion of large molecules through a solvent composed of smaller molecules the Stokes-Einstein law of diffusion has been found to describe satisfactorily the experimental results, it is unsatisfactory in principle in case6 where solute and solvent molecules are-nearly the same size. Eyring (3) has pointed out that the process of diffusion is the same as that for viscous flow, except that in diffusion the two molecules passing each other may have different properties, whereas for viscous flow, in a pure liquid a t least, the two molecules passing each other are alike. The relation between the diffusion coefficient and the specific velocity constant given for an activated rate process was where X is the distance between two equilibrium positions of the diffusing molecule, and k' is the number of times per second that a particular molecule is engaged in passing a neighbor in any particular direction,say the direction of diffusion. The expression developed for viscosity was (3) 7 =
XikT/X2X&k'
(2)
where X has the same significance as in equation 1, namely, the distance between equilibrium positions in the direction of flow, XI is the perpendicular distance between adjacent layers of molecules, hz is the distance between adjacent molecules in the direction of flow, X 3 is the distance between molecules in the plane of flow and normal to the direction of flow, and k' is the specific rate constant. Taylor (18), using the combination D =
kTXi/X2Xsq
of equations 1 and 2, found that it gave much more reasonable results than the Stokes-Einstein relation,
D
=
kT/6~7
when the molecules diffusing into each other were similar.
982
ALLEN E. BTEARN, EDWIN hl. IRISH AND HENRY EYRING
Since viscosity was shown to exhibit an exponential variation with temperature ( 2 ) , it follows that diffusion should exhibit a like temperature variation. For solid diffusion such a variation with temperature has long been recognized, but up until this time diffusion in liquids has frequently been considered as a linear function of temperature. The difficulty of measuring diffusion rates has restricted the advancement of investigation into this problem, and even Cohen and Bruins (l),whose data on the diffusion of tetrabromoethane in tetrachloroethane seem as yet the best available for determining the variation of diffusion rate with temperature, plot their rate values against both the first and the second powers of the temperature. From absolute reaction rate theory (4) we may write for k’ in equation 1
where 3$and F,, are the partition functions of the reacting system in the activated and the normal state, respectively, AF* AS’, and AHt are, respectively, the standard free energy, entropy, and heat increases when the activated complex is formed from the reactants, Eo is the difference in energy between the lowest level in the normal and in the activated state, k is the Boltzmann constant, and h is the Planck constant. We have taken the transmission coefficient as unity. The heat of activation is calculated from the relation d Ink’ AH’ = R T ~ dT From equations 1, 3, and 4 we obtain
If we assume that the degree of freedom corresponding to flow is a translational one, and that the partition functions for other degrees of freedom are the same for the initial and activated states, then
where q is the free volume for one molecule. Capital letters are used for the volume, V , and the free volume, V, , when a mole is indicated.
THEORY OF DIFFUSION IN LIQUIDS
983
If the unit process in diffusion (or in viscous flow) is the passing of two molecules in the sense of a partial rotation of a double molecule (lo), extra space must be provided, though this may not necessarily have to be a hole the full size of a molecule. Therefore the energy of activation for diffusion will be some fraction of the energy of vaporization; AE.,t. = AEvap./n. Considering the data of Cohen and Bruins (l), Taylor (18) concluded that in diffusion the size of the hole required is approximately the arithmetic mean of that required by the two pure substances in their respective viscous flows. Substituting equation 7 into equation 5 and writing AEVa,,./nfor Eo , we obtain
For A, in the case of a pure liquid, we take (V/N)’I3,where N is Avogadro’s number. The free volume characteristic of a pure liquid has been estimated in various ways. Eyring and Hirschfelder ( 5 ) developed the relation
where AEvap,= AH,,,,
- A ( p V ) . Kincaid and Eyring (13) find uj/3
=
211/3
5 UIiq.
where I’ is the velocity of sound.’ For more concentrated solutions the change in activity coefficients of the components with concentration will affect the free energy of activation. This effect can be calculated for the simple case of molecules of approximately equal size as follows: Let AF: be the height of the energy barrier in figure 1. Then if y1 and y~ are, respectively, the activity coefficients of constituents 1 and 2, the rate a t which constituent 1 will sur1 There seems t o be no significant difference in the values of the free volumes whether calculated from equation 11 or equation 12 except in the case of water. For five organic liquids the ratio of v;/*from equation 12 t o t h a t from equation 11 averages 0.95 a n d varies from about 0.85 t o 1.07, whereas for water this ratio is about 2.5 (cf. reference 8). I n the case of t h e values of free volume used later in this paper, we specify in all cases whether they were obtained from equation 11 or equation 12.
984
ALLEN E. STEARN, EDWIN M. IRISH AND HENRY EYRING
mount the barrier in the forward direction (left t o right) will depend not only 'on AF: but also on the differences in y1 and y z on the opposite sides of the barrier. For a symmetrical barrier the additional free energy, due
DISTANCE X
FIQ.1. Potential energy curve for diffusion to non-ideality of the solution, needed by constituent 1 to reach the top of the barrier in the forward direction will be R T d In y1 dN1 X dN1 &5
-
R T d In y2 dN2 _X dN2 dx 2 -X - R T d In yl dN1
dN1
dx
5 -I-
R T d In y2 dN1 -X dN2 dx 2
where N1 and NZ are the mole fractions of constituents 1 and 2, and dN1 = -dNz. We set
and A F ~ 0 -kT e RT = B h
We may then write for the net rate' of constituent 1in the forward direction 2 When the effect of GIonly is considered, we obtain the form of activity coefficient effect given by L. Onsager and R. M.Fuoss (J. Phys. Chem. 36, 2687 (1932)).
THEORY OF DIFFUSION IN LIQUIDS
Since
(GI - G2)
