Review of relation between diffusivity and solvent viscosity in dilute liquid solutions

Chemical Engineering Science, 197 1, Vol. 26, pp. 635-646. Pergamon Press. Printed in Great Britain.

Review of relation between diffusivity and solvent viscosity in dilute liquid solutions

W. HAYDUK and S. C. CHENG Department of Chemical Engineering, University of Ottawa, Ottawa, Canada

(First received 22 July 1969; accepted 25 August 1970)

Abstract-Diffusivities measured at 25°C are reported for ethane in normal hexane, heptane, octane, dodecane and hexadecane, and at 25°C and 50°C for carbon dioxide in hexadecane. Measurements were made by means of the steady-state capillary cell technique. The general relation between diffusivities in dilute liquid solutions and solvent viscosities for non-complexing systems was re- viewed. It was found that in general diffusivity and solvent viscosity were not inversely related but that the diffusivity depended on the solvent viscosity raised to some power which was variable, depending on the diffusing substance. Niether the temperature, nor solvent molecular weight or molar volume was required to describe the observed relationship between the ditfusivities and solvent viscosities for eleven different substances, for a range of temperatures, solvents, and binary solvent compositions. The observed relationship would appear to provide an improved basis for correlating liquid diffusivities in dilute liquid solutions.

INTRODUCTION

IT IS FREQUENTLY necessary to predict diffusivities in dilute liquid solutions for equipment design. Empirical equations for predicting the pertinent diffusivities are available in many forms. Perhaps the most used are the Wilke- Chang[ 11 and Scheibel[2] equations. While for many binary systems the Wilke-Chang and Schiebel equations accord with available experimental data, it is disconcerting to find that for some other data, for unexplained reasons, large deviations are observed. Thus although the Scheibel equation is recommended for non- aqueous systems by Reid and Sherwood[3] on the basis that it yields an average deviation from experimental diffusivities of less than 20 per cent, it also yields deviations of up to about 50 per cent (or more) in some instances. Although it is possible to attribute some lack of consistency to the experimental data recognizing the experimental difficulties involved, gross deviations cannot be explained in this way.

We have measured the diffusivities of ethane at 25°C and atmospheric pressure in the normal paraffinic solvents, hexane, heptane, octane, dodecane and hexadecane, and also of carbon dioxide in hexadecane at 25°C and 50°C using

the steady-state capillary cell method [41. Although no unusual solute-solvent interactions in these solutions were expected, we found that the Wilke-Chang equation and also the empirical equation proposed by Lusis and Ratcliff[Sl (although not strictly applicable to long-chained molecules) could not adequately represent the data. The comparisons are shown in Table 1. Especially as the solvent viscosity increases, as for dodecane and hexadecane, the deviation of the predicted diffusivities becomes very large. Similar data (for the diffusivity of one solute in a variety of solvents) were reported for carbon dioxide by Davies, Ponter and Craine[6]. Those authors also found that the Wilke-Chang equation and likewise the Scheibel equation[2] predicted diffusivities that were much too low when the solvent viscosities were high. The Arnold equation[7] was recommended to represent the diffisivities of carbon dioxide.

It appeared necessary therefore, to review the basis of some of the empirical predictive equations.

Diffusion in liquids is complicated by molecular interactions of several types. In some instances the molecules of pure liquids occur in an “aggregated” form equivalent to dimers,

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Table 1. Comparison between experimentally determined and predicted diffisivities for ethane at 25°C

Solvent Diffusivity x IO5 cm2/sec

Mean Lusis and experimental Wilke-Chang Ratcliff

Hexane 5.79 6.44 7.66 Heptane 5.44 5.25 6.14 Octane 4.57 4.28 4.95 Dodecane 2.73 1.98 2.22 Hexadecane 1.95 1.09 1.11

trimers, etc. These same liquids, when dissolved in a solvent, may retain their state of aggregation or by the influence of the solvent may break down to yield a simple molecular structure. Likewise a dissolved substance may ionize in solution to form diffusing species different in size (and number) to that corresponding to its molecular formula. Whether as a dilute species or as a solvent, the state of molecular aggregation of a liquid could be expected to influence the rate of diffusion. Perhaps of greater consequence is the fact that on solution a considerable number of substances associate with the solvent. In this instance also, it would appear that the rate of diffusion would be more closely related to some parameter describing the associated species actually diffusing rather than that described by the chemical formula. Many more molecular interactions are possible in binary systems when both liquids are present in appreciable concentra- tions and especially in multicomponent systems when diffusion occurs in mixed solvents. Some attempt to account for solute-solvent association has been made by Wilke and Chang by using the association parameter. The extent of association and its effect on diffusion, however, largely remain an area of uncertainty except perhaps for the general observation that solute-solvent association is likely to reduce the rate of diffusion.

The effect of a changing temperature has been considered in a number of ways by different authors and in most empirical equations the effect of temperature on diffusion has been combined with its effect on viscosity. It may be indicative that Longsworth[8] found that in

W. HAYDUK and S. C. CHENG

general the temperature coefficient of diffusion was greater for substances of lower diffusivity. Possibly because diffusivity data have not generally been measured over wide temperature ranges, the relation between diffusivity and temperature has not been conclusively ascer- tained.

Other parameters such as molecular weight and molar volume of solute or solvent, as well as the collision diameter of the diffusing species have been found useful in describing diffusion in liquids. The functional dependence of diffusion on these parameters has often been different depending on the diffusivity data and authors concerned.

The Wilke-Chang and Scheibel equations are based on the Stokes-Einstein equation in which the diffusity and solvent viscosity are inversely related. In the Arnold equation [7], excluding the effects of other parameters, the diffusivity varies as the solvent viscosity of the -0.5 power. In the Othmer and Thakar correlation [9] for water as solvent, the diffusivity varies as the solvent viscosity to the - 1.1 power. In view of the considerable degree of uncertainty about the diffusion process in liquids we wished to analyse available data in a very fundamental manner in the hope of arriving at some fundamental relationship based on experimental fact, particularly for the relationship between diffusivity and solvent (or solution) viscosity.

First it appeared necessary that a model for diffusion in liquids should apply equally well for gases dissolved in liquids (even for a temperature exceeding the gas critical temperature) as for liquid-liquid solutions, or solid-liquid solutions, because once in solution, these all form true liquid state solutions. Next, for simplification, we limited the scope of our concern to solutions where the state of molecular aggregation for both the dilute diffusing species (solute) and the solvent (single or multicomponent) was unchanged by adding one to the other. Finally, we speculated that while the variables such as temperature, molar volumes, collision diameters etc., might be of importance in describing diffusion, their effects would be only secondary.

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The hypothesis we choose to test was that the diffusivity of a particular dilute species in any solvent depended only on the solvent viscosity, provided that the state of molecular aggregation of both the solute and solvent remained essentially unaltered on mixing. This hypothesis suggested that a unique relationship existed between a particular diffusing species in a range of solvents, temperatures, and solvent compositions, depending only on the solvent viscosity. Implied also was that the solvent viscosity alone was a characteristic property related to the difficulty of a substance to diffuse through it, regardless of the degree of solvent association.

Much of the data on which the empirical equations currently used are based, are for different solutes in a relatively few solvents. For example there are some five dozen reported diffusivities of different substances in water and a large number in benzene, methanol and ethanol. Because we wished to determine the functional relationship between solute diffusivity and solvent viscosity we selected from available data only those which were for a single substance in a range of solvents. The properties of the diffusing species therefore would remain the same for the whole series of solvents unless there were some particular solute-solvent interactions. Eleven different substances for which diffusivity data were obtained in a variety of solvents included ethane (our own data), carbon dioxide, carbon tetrachloride, the organic acids, formic, acetic, benzoic, and phenol, acetone, benzene, toluene and bromobenzene. We excluded data for water and iodine because of the extremely strong associating tendency of both.

We used a number of tests for our hypothesis which are itemized as follows:

1. Diffusivity data for each substance were plotted as a function of the solvent viscosity to see whether a single function was obtained in each case. In all cases, to a good approximation, a single function was obtained, the relation for which could be well represented by a straight line on a log diffusivity-log viscosity graph.

2. The viscosity-diffusivity relationship

obtained with different solvents was compared with that obtained when the viscosities were altered by changing the temperature. The dependence of diffusivity on temperature would be expected to be only that resulting from the temperature effect on viscosity.

3. The self-diffusion coefficients were tested (when available) to see whether the self-diffusivity corresponded to that in a solvent having an identical viscosity to that of the solute.

4. The diffusivities of a single solute in several different solvents all having similar viscosities were compared. The diffusivities likewise would be expected to be similar.

5. The diffusivities of a number of substances, each in a pair of solvents having similar viscosities, were compared to see whether the diffusivities in both solvents were essentially the same.

6. We wished to observe whether for two- component solvent solutions the solvent viscosity corresponding to a solvent composition change, yielded a viscosity-diffusivity relation similar to that for pure solvents.

The diffusivity data for the eleven different solutes in a variety of solvents, at various temperatures, in binary solvents over the complete range of compositions, and pairs of diffusivities in methanol and benzene solvents, all offered surprisingly strong support for the original hypothesis. There were few exceptions and although deviations from a single viscosity- diffusivity relation may have exceeded probable experimental errors, few exceeded about 20 per cent.

EXPERIMENTAL

The diffusivities were measured using the steady-state capillary cell method described else- where[4] and which was considered suitable for relatively soluble gases. Minor changes were made in the design of the diffusion cells and the mode of operation. The location of the stopcocks was changed to facilitate purging as shown in Fig. 1. With one stopcock near the top of the cell plug flow through the cell was possible and complete purging was thus effected with much less solvent. It was also found that the gas purged


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GAS SATURATED

LIQUID BEAD

CAPILLARY 0.4 m (PRECISION BOR

CAPILLARY I mm

(PRECISION BORE

HIGH VACUUM CAPILLARY

STOPCOCK

DEAERATED LIQUID

Fig. 1. Details of diffusion cell.

through the tee at the end of the small capillary did not need to be saturated with vapor. In fact some difficulty was encountered with condensa- tion from the vapor-saturated gas and drainage of the liquid into the measuring capillary. It appeared that when even a small amount of liquid drained into the capillary wetting its surface, the cross-sectional area was effectively reduced. In consequence, diffusivities based on the clean dry capillary area were sometimes high. The problem was eliminated and reproducibility improved by simply using dry gas. After the liquid bead was introduced, it was assumed that the confined gas become completely saturated with vapor during

the elapsed time of at least one hour before bead position readings were actually taken.

Ethane and carbon dioxide equilibrium solubilities used for calculating the diffusivities were measured in this laboratory and reported else- where[lO]. The purities of ethane, carbon dioxide and the hydrocarbons were identical to those used for the solubility measurements. The viscosities and densities of the air-saturated solvents were measured at 25°C and also at 50°C for hexadecane using calibrated Cannon- Fenske type viscosimeters, and hydrometers. The specified accuracy of the viscosimeter tubes purchased from Fisher Scientific was &0*25 per cent. The hydrometers, also purchased from Fisher Scientific, could be read to 0.0001 divi- sions. Whereas the densities determined by hydrometer were not considered highly accurate, they were measured mostly for comparison with the ethane-saturated densities and viscosities also obtained at 25°C. The latter densities were determined by immersing the hydrometer into solvents through which ethane had been bubbled at constant temperature for at least 1 hr. The procedure for measuring the gas-saturated (total pressure of one atmosphere) viscosity was modi- fied slightly to avoid placing the solution under a partial vacuum. An excess of solvent was placed in the viscometer tube and ethane was bubbled through the solvent in the tube until saturated. The tube was inverted and the excess solution allowed to escape until the level of the remaining saturated solution reached a predetermined mark on the tube. The efflux time was then measured in the normal way. The measured properties are compared with literature data in Table 2.

Table 2. Viscosities and densities of paraffinic solvents at 25”C, and hexadecane at 50°C

Solvent Viscosity (cP) Density (g/cc)

Exp. Ref. 1121 Sat’d Exp. Ref. 1121 Sat’d

Hexane 0.2969 0.2985 0.2907 0.6560 0.65481 0.6530 Heptane 0.3929 0.3967 0.3830 0.6800 0.67951 0.6770 Octane 0.5143 0.5151 0.4977 0.6986 0.69849 0.6956 Dodecane I.3608 1.378 1.3004 0.7452 0.74516 0.7425 Hexadecane 3%M9 3.095 2.9265 0.7704 0.76996 0.7680

at 50°C 1.866 0.7524

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Review of relation between diIfusivity and solvent viscosity in dilute liquid solutions

Based on the steady-state rate of shrinkage of the gas confined in the capillaries of the diffusion cell the integral binary diffusivities of the gases in the normal paraffinic solvents were calculated. The appropriate equation derived for solutions of constant mass density [4] is as follows:

D= %L pin (l+~Ao)/(l+~d’

The mass flux, diffusion path length and solution density (taken as solvent density) were readily available or could be calculated. The mass fraction of ethane and carbon dioxide at the gas-liquid interface was calculated with due consideration for the barometric pressure and solvent partial pressure using Henry’s law to extend the solubility data to the prevailing gas partial pressure. The ethane mass concentration at the end of the capillary was always calculated to be less than 1 per cent saturated; this correc- tion was not ignored, however. The diffusivities are shown in Table 3. Table 3 also contains the ethane and carbon dioxide solubilities expressed as Ostwald coefficients and mole fractions. It is noted that the solubilities for ethane in heptane and octane are mean values for the two sources, [ 101 and Thomsen and Gjaldbaek [ 11 I, because there was a difference of about 2 per cent, and 4 per cent respectively. The mean solubilities were used in calculating the diffusion coefficients.

DISCUSSION

Data were selected from the literature for the dilfusivities of eleven particular substances in a

range of solvents, temperatures, and solvent compositions. No exhaustive attempt was made to utilize all the data available in the literature; instead, only those which appeared to illustrate particular arguments were chosen. The sources of data for the various substances are listed in Table 4. The data were plotted as the logarithm of the diffusivity versus the logarithm of the solvent viscosity as shown in Figs. 2-6. Lines were drawn through the points without recourse to any mathematical fitting techniques but were shown for the purpose of qualitative discussion.

Perhaps the most convincing reason for assuming that a simple relationship exists be-

10.0 - I I

LEGEND

6.0 O@Temp. 25% = -

00 Temp. # 25’C

4.0 - @e Self-diffusion coeff.

a^ 0 Binary solvents Y 0 :

$2.0 - 0

7

C4 0.6 1.0 VISCOSITY, cps

Fig. 2. Diffusivity of carbon tetrachloride.

Table 3. Ethane solubilities and diffusivities in paraffinic solvents at 25”C, and carbon dioxide solubilities and diffusivities in hexadecane at

25°C and 50°C

Solubility Diffisivity X 105 cm%ec Solvent Solute

1 x I II III Mean

Hexane Ethane 6.09 0.0320 5.88 5.68 5.81 5.79 Heptane 5.33 0.0325 5.50 544 5.40 5.44 Octane 4.82 0.0330 4.47 460 4.65 4.57 Dodecane 3.93 0.0351 2.67 2.73 2.79 2.73 Hexadecane 3.22 0.0379 1.97 1.97 1.92 1.95 Hexadecane CO, 1.182 0.0143 2.30 2.12 - 2.21

at 50°C 1.001 0.0113 2.75 2.80 - 2.78

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.

-I 0.25 0.5 1.0 2.0 5.0

VISCOSITY, cps

Fig. 3. Diffisivity of carbon dioxide, formic, acetic, and benzoic acids.

tween solute diffusivity and solvent viscosity can be found by plotting the data of Hammond and Stokes [ 131, as shown in Fig. 2, for the ditfusivity of carbon tetrachloride in various solvents. Those authors treated their data in a much differ-

ent manner as discussed in their paper and subsequently by Tyrell[2] and also by Reid and Sherwood [3]. They divided the solvents into the three groups, one group composed of roughly spherical molecules, another of relatively elongated (paraffinic) molecules, and the third of the alcohols, methanol and ethanol. They

VISCOSITY, cps 0.4 0.6 1.0 VISCOSITY, cps

Fig. 4. Diffusivity of acetone and phenol. Fig. 6. Diffusivities of ethane, toluene and bromobenzene.

0.4 0.2 0.4 0.6 1.0

VISCOSITY, cps

Fig. 5. Diffisivities of benzene.

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Table 4. Sources of dithrsivity dam

Solute Reference

Carbon tettachloride 113,141 Carbon dioxide This work [6, 151 Formic acid [ll Acetic acid [l, 16,211 Benzoic acid [1,211 Acetone [17,18,19] Phenol [20,21] Benzene [l, 18,22,23] Ethane This work Toluene [1,21,24,25] Bromobenzene [23,25,26,27]

showed that Dp for each group of solvents was approximately a linear function of the solvent molar volume. These same data show a very good single straight line on a log D-log or. plot except for ethanol and dioxane, and even these values are within 18 per cent of the straight line. In addition the self-diffisivities of carbon tetrachloride at three temperatures (25,35 and 45°C) also fall on the same line corresponding the viscosities of carbon tetrachloride. The consistency of the data on such a plot is striking.

The data of Davies et al. [6] for the difhrsivities of carbon dioxide in a number of solvents, of Tang and Himmelblau [ 151 in the mixed solvents, carbon tetrachloride-benzene, along with our own data for the diffisivity of carbon dioxide in hexadecane at 25 and 50°C are shown in Fig. 3. It can be seen that while the diffusivity of carbon dioxide in carbon tetrachloride deviates from the line by more than 20 per cent the diffusivity- viscosity relationship for the mixed solvent approaches the line for the pure solvents. It can also be seen that the diffusivity of carbon dioxide in hexadecane at 50°C lies on the same line. The diffusivities in hexadecane and isobutanol at 25°C are essentially the same although these solvents have appreciably different molar volumes, but similar viscosities.

In Fig. 3 are also shown diffusivity data for formic acid, acetic acid and benzoic acid obtained at temperatures from 6 to 40°C in a variety of solvents as reported by Wilke and Chang[l]. While these substances are known to exist in the partially associated state we are

aware of neither the effect of the various solvents on the degree of association, nor the extent of particular solute-solvent interactions. We simply assumed therefore, that the conditions of our hypothesis were maintained for these substances. Particularly evident in Fig. 3 is the fact that the temperature coefficient of diffusivity indeed appears to follow the empirical rule stated by Longsworth[8] that if the diffusion coefficient is small the temperature variation of the diffusion coefficient is large. Of more particular concern is the fact that in every instance the diffisivity at temperatures other than 25°C appears to coin- cide with the single diffusivity-viscosity relation for each diffusing substance. It would seem then, that all that would be required to predict the diffusivities of these organic acids in other solvents would be the solvent viscosities.

The recent data of Perkins and Geankoplis [ 161 for the diffusivity of dilute acetic acid in aqueous ethanol solutions is included in Fig. 3. The data appeared to us to have considerable significance because the viscosity of ethanol-water solutions reaches a maximum value much greater than that of either of the pure solvents. It is noted that the diffusivities reach a minimum value for a corresponding solvent composition much below that obtained in either pure solvent. Indicative also is that the difhrsivity-viscosity relationship for acetic acid in the mixed solvents is roughly the same as that for pure solvents. If any deviation is considered, the diffusivities in the aqueous ethanol solutions are slightly lower than in non- associating solvents. Considerably more diffusivity data are available in various aqueous solutions but these data usually fall significantly below the diffusivities for corresponding non- associating solvents of the same viscosity.

The early diffusivity data of Dummer [ 171 (19 19) for acetone in a number of solvents are shown in Fig. 4 along with the self-diffusion coefficient of Hardt et al. [ 191 and the diffusivities in the mixed solvents, hexane-carbon tetrachloride and benzene-cyclohexane, of Leffler and Cullinan [ 181. The self-diffusion coefficient corresponds to the low viscosity of acetone. The data of Dummer were all for temperatures below


641

25°C but, except for the diffusivity in isobutanol, fall approximately on the same line. The diffusivities in the mixed solvents likewise are well expressed by the same linear relation. It can be observed that the diffusivity of acetone in carbon tetrachloride is slightly below that in cyclohexane although both have similar viscosities.

Diffusivities of phenol in a number of solvents at 25°C are also shown in Fig. 4. These data are included because the slope of the log diffusivity- log viscosity line is the greatest of all the solutes considered.

The diffusivities of benzene in various solvents, shown in Fig. 5, were included mainly for comparison with diffusivities of toluene and bromobenzene to be discussed subsequently. The data are from a variety of sources including Johnson and Babb[22] for the self-diffusion coefficients at several temperatures as well as for benzene in chloroform, Caldwell and Babb[23] for diffusivities in chlorobenzene and carbon tetrachloride at several temperatures, and Wilke and Chang[ l] for the diffusivity in bromobenzene. The recent data of Leffler and Cullinan [ 181 for diffusivities in the mixed solvents, hexane- cyclohexane and acetone-cyclohexane, are of special interest. It can be observed that while the diffusivities in chlorobenzene, carbon tetrachloride and bromobenzene are somewhat below the line drawn, all the remaining data fall about the line. Perhaps attention can be drawn to the former three solvents because the diffusivities of toluene in these same solvents also are less than what would be expected for non-associating solvents. As observed previously the diffusivity of carbon dioxide in carbon tetrachloride was also somewhat below that expected for solvents of similar viscosity. It is simply noted that the three solvents in question have densities appreciably greater than 1.0 and hence considerably greater than that of most other solvents. The diffusivities in the mixed solvents are essentially coincident with one another and correspond to the same line as that for the pure solvents, acetone, hexane and cyclohexane.

Our data for the diffusivities of ethane in the normal paraffins are shown in Fig. 6. The data


are well represented by a single straight line even although the viscosity range is about ten- fold. Diffusivities of toluene include those in normal paraffins by Wilke and Chang [ 11, in chlorobenzene and bromobenzene by Burchard and Toor[25], and in carbon tetrachloride by Lonsworth[21]. In addition the diffusivities of Holmes et al. in the mixed solvents, hexane- cyclohexane, hexane-tetradecane and cyclohexane-decane, are included. Except for the three solvents of relatively high density (chlorobenzene, carbon tetrachloride, and bromobenzene) the data are well represented by a single line.

A dotted line representing the data for benzene is shown in Fig. 6. Diffusivities of bromobenzene, mainly including early data at 7°C of Herzog [27] but also including those of Burchard and Toor[25] for chorobenzene and toluene solvents at 30°C and for chlorobenzene at 20°C of Caldwell and Babb[23], are shown. Attention is drawn to the diffusivities in the four solvents of similar viscosity, benzene, ethyl benzene, meta xylene (all at 7.3”C) and chlorobenzene (at 20°C). While the molecular weights and molar volumes are quite different (particularly for chlorobenzene) the diffusivities are nearly the same. The implication is that the solvent parameters such as molecular weight or molar volume bear no strong relation to diffusivities, at least in those solvents.

Figure 6 is useful for an interpretation con- cerning the order of decreasing diffusivity of a range of solutes in a particular solvent. While it seems consistent that ethane should have a higher diffusivity than toluene, benzene, or bromobenzene, the order of diffisivities of the last three substances, at first glance, seems anoma- lous. That toluene, whose molecular weight and molar volume are greater than those for benzene, should have a higher diffusivity than benzene seems incongruous. It may be of significance however, that the order of increasing viscosity of these substances are toluene (lowest viscosity) benzene, and bromobenzene. This latter observation caused us to consider that perhaps the viscosity of the solute- if it existed in the liquid

642


phase- might be a significant parameter in describing the order of solute diffusivities.

From the graphs in Figs. 2-6 it can be observed that the slopes of the lines appear to become more negative as the diffusivity in a particular solvent decreases. The estimated slopes along with the solute properties, the Le Bas molar volumes, critical molar volumes and viscosities are listed in Table 5. The order of increasing dithrsivity in a particular solvent has been considered to follow a decrease in the solute (Le Bas) molar volumes. In the “random walk” con- cept of diffusion as expressed by Hildebrand and co-workers[29,30] it is considered that the diffusivity varies inversely as the square of the solute molecular diameter (Du” = constant).

number of substances, at the same temperature in the two solvents, methanol and benzene. The diffusivity of each substance in the two solvents of similar viscosities should be essentially the same if the hypothesis was to be supported. Methanol, and benzene have similar viscosities, at 15°C of 0.63 CP and 0.70 cP, respectively. Suitable data are available from Wilke[26] and Reid and Sherwood[32]. It is recognized that whereas methanol has a slightly lower viscosity it is more likely to associate to some degree with various diffusing substances than benzene. The diffusivity data at 15°C are shown in Table 6 along with the ratios of the diffusivities in benzene to the corresponding ones in methanol. The average ratio was calculated to be 1 *OO indicating

Table 5. Solute properties and log D-log p slopes

Solute V (Le Bas) Viscosity

-Slope (cc/g. mole) (cclg%ole) (cP)

G;Z;; dioxide

Formic acid Toluene Acetone Carbon tetrachloride Acetic acid Benzene Bromobenzene Benzoic acid Phenol

0.44 34 94 0.06 (b) 0.49 52 148 0.09 (b) 0.66 46 73 (a) 1.6 0.71 118 318 0.6 0.84 74 213 0.3 0.85 101 276 0.9 0.86 68 171 1.2 0.88 96 260 0.6 0.89 119 324 1.3 1.01 135 300 - 1.15 103 238 (a) 6.8

(a) Estimated. (b) At elevated pressures.

The molecular diameter can be estimated from the critical volume according to Flynn and Thodos[3 11 by a simple equation. It is evident then that the molecular diameter is closely related to the critical volume. It can be seen from Table 5 that neither the critical volumes, the Le Bas volumes, nor solute viscosities actually follows the order of slopes for the whole range of solutes. The solute viscosity however, appears to be only very roughly related to the slopes of the log D-log p lines.

One final test of the original hypothesis was made utilizing the diffusivities of a considerable

that on the average the diffusivity in benzene was essentially the same as in methanol. The percentage absolute difference and the average percentage absolute difference indicated that the diffusivities varied within a maximum of 14 per cent and an average of 6 per cent from one another. It appears possible that this kind of variation was the combined result of experimental error and some (variable) degree of association in methanol. In which case the original hypothesis was supported. A similar comparison for the solvents, water and carbon tetrachloride (of similar viscosity), showed that

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Table 6. Diffisivities and ratio ;Lhg;ivities at 15”C, in methanol and

Diffusivity X lo5 (cm2/sec)

Solute methanol benzene R D

Isoamyl alcohol Benzaldehyde Bromoaniline Bromobenzene Bromophenol Chloroacetic acid Choroaniline Chloroform Chlorophenol Dibromonaphthalene m-Dinitrobenzene Dinitronaphthalene Ethylenebromide Ethylene iodide Iodoform Phthalic acid Acetyldiphenylamine p-Dibromobenzene Dichloronaphthalene

1.34 1.66 1.41 1.75 1.34 1.52 1.37 2.07 1.32 1.33 1.56 1.32 1.95 1.56 1.33 1.30 0.98 1.55 1.52

1.48 1.73 1.41 1.86 1.34 1.48 1.56 2.11 1.42 1.25 1.54 1.23 1.97 1.40 1.38 1.37 0.90 1.37 1.40

1.10 1.04 1.00 1.06 1.00 0.97 1.14 1.02 1.08 0.94 0.99 0.93 1.01 0.90 1.04

10 4 0 6 0 3

14

8 6 1 7 1

1.05 0.92

10 4 5 8

12 8

6=6

0.88 0.92

I?= 1.00

R = diffusivity in benzene/ditfusivity in methanol. D = percentage absolute difference.

diffisivities in water were consistently lower than in carbon tetrachloride and suggested what was well known, that a considerable degree of complexing occurs with most solutes in water.

CONCLUSION

On the basis of the specific observations of experimental data it can be generally concluded that a unique diffusivity-solvent-viscosity relationship, which is independent of temperature and solvent composition, exists for each different diffusing substance. The slope of the log D-log p relationship appears to depend on the diffusivity itself, the lower the ditfusivity the higher the slope. The linear relation can be simply expressed as:

D = A@.

Separate constants, A and B, would apply to each diffusing substance.

Our observations should be useful in estimating diffusivities of a particular solute in other solvents when diffusivities in several (at least two)

solvents are known. Such extrapolations will probably be of greater accuracy particularly for dissolved gases in highly viscous solvents for which currently used empirical equations tend to be erroneous.

Even for the somewhat restricted condition of our hypothesis an analysis of other effects which influence diffusion appear in order. Particularly requiring further clarification is the reason for certain apparently low diffusivities in relatively high density solvents. Many other variations in excess of experimental errors, also require explanation. However, this extremely simplified relation between diffusivity and viscosity in liquid solutions appears to be more than of qualitative usefulness for correlation and extra- polation purposes.

Acknowledgments-The authors acknowledge with thanks the financial support of the National Research Council of Canada which was made available in the form of an operating grant. Acknowledgment is made with appreciation to Mr. Victor Wong who measured the diffusivities of carbon dioxide in hexadecane.

644

Review of relation between diffisivity and solvent viscosity in dilute liquid solutions

NOTATION x solubility, mole fraction at a partial pres-

B slope, log D/log or. sure of 760 mm mercury

D’ diffusivity, cm2/sec Greek symbols

1 Ostwald coefficient, cc. gas/cc. solvent p density g/cc. L diffusion path length, cm p viscosity, cP

~~ mass flux of solute, g/cm2 set CT molecular diameter u, critical volume, ct./g. mole oAo mass fraction solute at gas-liquid interface V molar volume (Le Bas), ct./g. mole oAL mass fraction solute at end of capillary

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Resume-On rapporte les diffusivites mesurees ii 25°C pour I’tthane dans de I’hexane normal, de I’heptane, de I’octane, du dodecane et du decahexane, et a des temperatures de 25°C et 50°C pour du gaz carbonique dans du decahexane. Les mesures ont Cte faites par la technique de cellules capillaires en ttat stable. La relation generale entre les diffusivites dans les solutions liquides dilutes et les viscosites des solvants pour des systemes non complexes a ete revue. On a trouve qu’en general la diffusivite et la viscosite du solvant n’etaient pas en relation inverse, mais oue la diffusivite deoendait de la hausse de la viscositt du solvant Clevte a quelque puissance qui dtait variable, selon la substance de diffusion. Ni la temperature, ni le poids moleculaire ou le volume molaire du solvant n’etaient neces- saires pour dtcrire la relation observee entre les diffusivitis et les viscosites du solvant pour onze substances differentes, pour une gamme de temperatures, de solvants et de compositions binaires de solvants. La relation observee semble fournir une meilleure base de correlation entre les diffusivites du liquide dans les solutions liquides diluees.

Zusammenfassung-Es werden Werte fur die spezifische Diffusion von Athan in normalem Hexan, Heptan, Oktan, Dodekan und Hexadekan gemessen bei 25”C, und von Kohlendioxyd in Hexadekan bei 25°C und 50°C angegeben. Die Messungen wurden unter Anwendung der Technik, die von einer

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Kapillarzelle im station&en Zustand Gebrauch macht, durchgefiihrt. Es wird eine Ubersicht iiber die allgemeine Beziehung zwischen Diffusivilten in verdiinnten fliissigen Losungen und Losung- smittelviskosititen fiir nicht-komplexbildende Systeme wiedergegeben. Im allgemeinen konnte festgestellt werden, dass die spezifische Diffusion und die Liisungsmittelviskosit;it nicht umgekehrt proportional sind, sondern dass die spezifische Diffusion von dem zu einer gewissen Potenz erhohten Losungsmittelviskositat abhangt, und dass diese Potenz je nach der diffundierenden Substanz variiert. Weder die Temperatur, noch das Molekulargewicht oder Molvolumen des Losungsmittels waren erforderlich zur Beschreibung der beobachteren Beziehung zwischen den Diffusivitlten und Losungs- mittelviskositaten von elf verschiedenen Stoffen fur eine Reihe von Temperaturen Losungsmitteln und binare Liisungsmittelzusammensetzungen. Die beobachtete Beziehung sollte eine bessere Grundlage fur die Korrelation von Fliissigdiffusivitlten in verdiinnten fliissigen Losungen schaifen.

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Review of relation between diffusivity and solvent viscosity in dilute liquid solutions