PREDICTION OF DIFFUSION COEFFICIENTS FOR LIQUID n-ALKANE MIXTURES A. L.VAN GEET
A. VV. ADAMSON
Semi-empirical relations are used to correlate dzffusion coeficientsf o r n-alkanes.
W i t h the correlations,
particular dtffusion coe$cients can be estimated to within 5% of the true value, and the authors suggest that the correlations can be extended f o r systems yet unstudied
iffusion coefficients estimated with the StokesEinstein equation are accurate only for dilute components. In the case of mixtures of normal alkanes, however, there are sufficient data now available to formulate a number of regularities in properties, including diffusional behavior. It is the purpose of this paper to show that these can be developed in a useful way. We shall consider three types of diffusion Coefficients. All are given the Fickian definition of permeation, per unit concentration gradient. Self-dzj%sion is the permeation of a labeled species in its own medium; no net chemical gradient is present. Following a suggestion of Mills ( 8 ) , the term tracer dijusion will be used for the diffusion of a labeled species which 'is not in its own medium. As with self-diffusion, there is no net chemical gradient in the labeled species. The tracer diffusion coefficient is a property not just of the species in question, but also of the nature of the medium. In both cases some form of labeling (isotopic, magnetic, or other) must be used. Analytical diffusion occurs when net concentration gradients are present; the composition of the system is a function of space and of time. Differential dzfusion coefficients are defined for a particular composition, and may be regarded as the ratio of permeation to concentration gradient of a species in the limit of infinitely smalI change in concentration along the gradient. We have found semi-empirical relationships that provide a fairly accurate connection between the three types of diffusion coefficients (70). These relationships are developed here in more detail, together with specific
D
62
INDUSTRIAL AND ENGINEERING CHEMISTRY
procedures for applying them to the prediction of diffusion coefficients. We limit ourselves to normal alkane mixtures. Within this limitation, we believe that all the different tracer and differential diffusion coefficients for any liquid mixture of (2-5 to C-32 hydrocarbons can be anticipated to an accuracy of about ten percent over the temperature range -20' C. to 250' C. Where the data are available to check the predictions, an accuracy of five percent or better is found. Self-Diffusion Coefficient
For any one hydrocarbon, the self-diffusion coefficient obeys the exponential form D = D o exp ( - E / R T ) quite well. In addition, not only do the values of D at a given temperature vary with chain length, n, in a regular way, but a simple relationship between D o and E has also been noted (2). This is log Do = -3.28 = -3.28
+ 0.82 X E / 2 . 3 R + 0.179 E (kcal.) (Doin cm.2/sec.) (1)
The implication of Equation 1 is that, if diffusion be regarded as an activated process, then the entropy and energy of activation are linearly related. Substituting DO = D exp E / R T , one obtains at 25' C. log D
=
-3.28
- 0.55 E
As a consequence of these regularities, the data on self-diffusion may be presented in very compact form. The nomograph given in Figure 1 was constructed by a standard procedure (6) and the linearity of the n-scale
Regularities in the properties of the norma alkanes permit the is a reflection of the validity of Equation 1. The nomograph gives the self-diffusion coefficients for nalkanes from C-5 to C-32 and temperatures from 50° C. to 300" C. to about 5% accuracy.
C F i D , = CNtD,Vc/V = D,V,CN,/V
-
I
Tracer Diffusion Coofficient
A particular tracer diffusion coefficient, D,, must approach its own self-diffision coefiXent in value, Dp, as the composition of the medium approaches that of pure ith species. For a two component system then, D1 and D Zapproach D f and D i as N1 and Na approach unity, respectively. An additional requirement is the phenomenological one that D1 and Da approach the differential diffusion coefficient in value as N1 and Na approach zero, respectively. The second regularity which we select is the semiempirical one DiVi = DaV* = D,Vi (2) where c o m p e n t s 1, 2, . ., i are those present in a particular mixture and the D values are for that mixture. In our case, the partial molar volume Vi is equal to the molar volume of pure component i. This equation was obtained by Bearman (4) for systems forming regular solutions, and we have found that it succeeds in the case of n-alkane mixtures (70). At 25' C., the molar volume of an n-alkane containing n carbon atoms is given to within a few tenths of a percent by Vca (cm.'/mole at 25' C.) = 16.4 (n 2) for 5 n 30. At other temperatures, the molar volume is given by
.
+
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