Effects of Shapes of Solute Molecules on Diffusion - ACS Publications

Article pubs.acs.org/JPCB

Effects of Shapes of Solute Molecules on Diffusion: A Study of Dependences on Solute Size, Solvent, and Temperature T. C. Chan,* H. T. Li, and K. Y. Li Department of Applied Biology and Chemical Technology, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong ABSTRACT: Diffusivities of basically linear, planar, and spherical solutes at infinite dilution in various solvents are studied to unravel the effects of solute shapes on diffusion. On the basis of the relationship between the reciprocal of diffusivity and the molecular volume of solute molecules with similar shape in a given solvent at constant temperature, the diffusivities of solutes of equal molecular volume but different shapes are evaluated and the effects due to different shapes of two equal-sized solute molecules on diffusion are determined. It is found that the effects are dependent on the size of the solute pairs studied. Evidence of the dependence of the solute-shape effects on solvent properties is also demonstrated and discussed. Here, some new diffusion data of aromatic compounds in methanol at different temperatures are reported. The result for methanol in this study indicates that the effects of solute shape on diffusivity are only weakly dependent on temperature.

1. INTRODUCTION Diffusion rates of molecules in solutions are generally recognized as important to many chemical, biological, and industrial processes. The role that molecular shape plays in diffusion in dense fluids, however, has long been a challenging problem that is still not very well-understood at present. It appears that there are relatively few systematic investigations reported on the effects of solute shapes on diffusion in liquid solutions. In particular, very little is known about the dependences of such effects on solvent, solute size, and temperature in the literature. Previously, Hayduk and Buckley1 in their study of a large number of literature diffusivities in n-hexane and carbon tetrachloride at 298.2 K discovered that the diffusivities of essentially linear solute molecules were generally about 30% greater than those of the essentially spherical ones having approximately the same molecular size. Their finding thus indicates that the effect of solute shape on diffusion is significant enough to receive attention. It should be pointed out that a solute capable of forming hydrogen bond or association with solvent molecules can produce a diffusive entity that is not only different in size but also in shape as the solute monomer. In such case, the effect due to change in shape of the diffusing body may not be ignored or underestimated. In a former experimental investigation,2 we measured the limiting mutual diffusivities of three structural isomers of xylene in acetone, ethanol, and n-tetradecane at 298.2 K. By comparing the precisely measured data with the reported diffusivities of spherical tetramethyltin (Me4Sn), which is of nearly the same molecular volume as a xylene, the result showed that all planar xylenes diffuse faster than the spherical Me4Sn in each of the solvents at constant temperature. It was also found that the diffusivities of the relatively more linear p© XXXX American Chemical Society

xylene were consistently greater than those of the other two planar m- and o-isomers in the same solvents. The differences in diffusivity between p-xylene and Me4Sn are approximately 5%, 23%, and 26% in acetone, ethanol, and n-tetradecane, respectively. Subsequently, we measured diffusivities of other nonassociated planar solutes of different sizes ranging from benzene to anthracene in the above solvents at the same temperature.3,4 Comparisons with the literature diffusivities of the nonassociated spherical solutes ranging from carbon tetrachloride to tetraethyltin under the same conditions again indicated that the planar solutes indeed diffused faster than the spherical solutes. Similar results of faster diffusion rates for planar than for spherical solute molecules were also observed experimentally in solvents propane-1,2-diol and 2-methylpentane 2,4-diol by Tyrrell et al.5,6 and in molecular dynamics (MD) calculations by Hoheisel.7 The result of Hoheisel suggested that the effect of shape is dependent on the solvent density. In another MD simulation study of the diffusion of ellipsoids in a sea of spheres, Bagchi et al.8 also found that the diffusivities of the oblates with aspect ratios close to one (essentially spherical in shape) were lower than those with small aspect ratios (essentially planar in shape). The authors attributed the result to the differences in translation-rotation coupling of the solutes. From the literature survey, it appears that previous studies of the effects of solute shapes on diffusion are generally more qualitative than quantitative. The major findings insofar can be summarized as that the trend of diffusivity of solutes with different shapes is linear > planar > spherical. Nonetheless, this result is inconsistent with the well-known fact that the diffusion Received: October 28, 2015

A

DOI: 10.1021/acs.jpcb.5b10550 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B Table 1. D12 (10−9 m2s−1) of Planar and Spherical Solutes in Polar Solvents at Different Temperatures planar solutes solvent

T (K)

methanol

283.15

298.15

313.15

a

ethanol

298.15

acetone

298.15

spherical solutes D12

benzene chlorobenzene toluene ethylbenzene naphthalene 1,2,4-trichlorobenzene 1,3,5-trimethylbenzene biphenyl benzene chlorobenzene toluene ethylbenzene m-xylene m-dichlorobenzene naphthalene 1,3,5-trimethylbenzene 1-methylnaphthalene biphenyl benzene toluene m-xylene m-dichlorobenzene naphthalene 1,3,5-trimethylbenzene phenanthrene benzene chlorobenzene toluene ethylbenzene m-xylene m-dichlorobenzene o-xylene o-dichlorobenzene naphthalene 1,2,4-trichlorobenzene 1,3,5-trimethylbenzene biphenyl anthracene benzene chlorobenzene toluene ethylbenzene m-xylene m-dichlorobenzene o-xylene o-dichlorobenzene naphthalene 1,2,4-trichlorobenzene 1,3,5-trimethylbenzene biphenyl anthracene

2.06 ± 1.87 ± 1.91 ± 1.76 ± 1.61 ± 1.55 ± 1.57 ± 1.44 ± 2.61 ± 2.40 ± 2.42 ± 2.23 ± 2.24 ± 2.21 ± 2.08 ± 2.02 ± 1.90 ± 1.85 ± 3.26 ± 3.05 ± 2.82 ± 2.78 ± 2.61 ± 2.51 ± 2.21 ± 1.79 ± 1.61 ± 1.62 ± 1.45 ± 1.44 ± 1.43 ± 1.40 ± 1.37 ± 1.33 ± 1.30 ± 1.32 ± 1.20 ± 1.04 ± 4.18a 3.71 ± 3.75 ± 3.45 ± 3.42 ± 3.38 ± 3.39 ± 3.35 ± 3.25 ± 3.13 ± 3.16 ± 2.89 ± 2.85 ±

0.02 0.02 0.02 0.01 0.02 0.02 0.01 0.01 0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.01 0.02 0.02 0.07 0.06 0.03 0.02 0.05 0.05 0.04 0.01 0.01 0.02 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.03 0.03 0.02 0.03 0.04 0.04 0.03 0.03 0.03 0.03 0.03 0.03

refs

T (K)

D12

this work this work this work this work this work this work this work this work 21 21 21 21 this work this work 21 21 21 21 23 23 this work this work 23 23 23 3 3 3 3 2 2 2 2 25 26 26 26 3 3, 27 3 3 3 2 2 2 2 3 3 28 3 3

283.15

CCl4 Me4Sn Et4Sn Pr4Sn Bu4Sn

1.76 1.62 1.27 1.05 0.89

± ± ± ± ±

0.02 0.02 0.02 0.01 0.01

20 20 20 20 20

298.15


2.27 2.06 1.66 1.38 1.18

± ± ± ± ±

0.03 0.02 0.04 0.01 0.01

22 20 20 20 20

313.15


2.84 2.60 2.13 1.74 1.53

± ± ± ± ±

0.03 0.02 0.02 0.02 0.02

19 24 24 24 24

298.15

CCl4 Me4Sn Et4Sn Bu4Sn

1.47 ± 0.01 1.25 0.950 0.604

22 20 20 20

298.15


3.61 3.38 2.92 2.36 2.00

± ± ± ± ±

22 20 20 20 20

refs

0.03 0.03 0.04 0.03 0.02

Average value from refs 3 and 27.

rates of nonspherical macromolecules are normally lower than those of the spherical ones.9,10 It should be noted that most of the previous studies, however, focused mainly on small systems. The above “contradiction” leads us to believe there are still a lot of uncertainties about the shape effects that could be clarified with more broadly based investigations. In this work, we

present an approach to unravel the effects of shapes in relation to different solute sizes, solvents, and temperatures by employing a wide range of experimental diffusivities measured in this laboratory and those reported in the literature. The objective is to provide a more complete picture for understanding the effects of solute shapes on diffusion. B


Article

The Journal of Physical Chemistry B Table 2. Relative Molecular Mass and VDW Volume of Solute and Solvent Molecules solute planar

spherical

linear

benzene chlorobenzene toluene m-dichlorobenzene o-dichlorobenzene ethylbenzene m-xylene o-xylene naphthalene 1,2,4-trichlorobenzene 1,2,4-trimethylbenzne 1,3,5-trimethylbenzene 1-methylnaphthalene biphenyl anthracene phenanthrene hexamethylbenzene pyrene CCl4 Me4Sn Et4Sn Pr4Sn Bu4Sn n-pentane n-hexane n-heptane n-octane

solvent M1a

V1 (10−3nm3)b

78.11 112.6 92.14 147.0 147.0 106.2 106.2 106.2 128.2 181.4 120.2 120.2 142.2 154.2 178.2 178.2 162.3 202.3 153.8 178.9 235.0 291.1 347.2 72.15 86.18 100.2 114.2

81.10 97.18 97.64 113.2 113.2 113.8 114.2 114.2 125.4 129.3 130.7 130.7 141.9 152.4 169.7 169.7 180.3 187.0 89.90 114.9 179.6 244.2 308.9 91.35 107.5 123.7 139.9

methanol ethanol acetone carbon tetrachloride cyclohexane n-hexane n-tetradecane

M2a

V2 (10−3nm3)b

32.04 46.07 58.08 153.8 84.16 86.18 198.4

34.82 50.99 62.24 89.90 97.02 107.5 236.9

a

Relative molecular masses M1 and M2 are from ref 29. bValues of V1 and V2 are calculated from group increments given in refs 11 and 19, except Sn from ref 30.

2. EXPERIMENTAL SECTION Measurements of the limiting mutual diffusivities (D12 values) of the disc-shaped aromatic compounds in methanol were performed by using the Taylor dispersion technique. In recent years, this technique has been widely adopted for measuring mutual diffusivities,11−17 as it can provide not only accurate but also fast acquisition of diffusion data. The basic principles of the technique have been reviewed by Tyrrell and Harris.18 The experimental setup and procedures in this work were the same as those described in our previous investigation.19 In the present study, four or more measurements were carried out to determine an average diffusivity value. The experimental precision is approximately ±1%. Here, the solvent methanol (Aldrich, 99.9%+) was degassed prior to use in an ultrasonic bath. It was then filtered through a stainless-steel 20 μm solvent filter before transferring into the solvent delivery system. The solute 1,3,5-trimethylbenzene (Riedel-de Haën, 98%) was purified before use by fractional distillation. Chlorobenzene and benzene were Aldrich chemicals with purity above 99.9%. These solutes as well as toluene (E. Merck, 99.5%), 1,2,4trichlorobenzene (Aldrich, 99%+), m-dichlorobenzene (E. Merck, 99%+), naphthalene (BDH, 99%), biphenyl (KochLight, 99%), m-xylene (Aldrich, 99%+), and ethylbenzene (BDH, 99%) were used as received.

limiting mutual diffusivities of the nonassociated planar and spherical solutes in polar solvents, which were obtained from different sources in the literature. For diffusion of solutes with similar shape in a given solvent at constant temperature, it has been shown3,4,11,21,26 that a linear relationship exists between the reciprocal of diffusivity and the van der Waals (VDW) volume of the solute, that is, −1 D12 /109 m−2s = aV1/10−3 nm 3 + b

(1)

D−1 12

where refers to the reciprocal of the limiting mutual diffusivity, a and b are constants, and V1 represents the van der Waals (VDW) volume of solute. The values of V1 and relative molecular mass M1 of the solutes in this study are displayed in Table 2. Although the linear dependence as expressed by eq 1 is not expected to be valid over a very wide range of solute sizes, it has nevertheless been demonstrated to be applicable for planar solutes11 with V1 values between approximately 81 and 509 × 10−3nm3 (i.e., from benzene to rubrene) in n-hexane at 313.2 and 333.2 K and for spherical solutes3,4 between 90 and 309 × 10−3nm3 (from carbon tetrachloride to tetrabutyltin) in various solvents at 298.2 K. The linear relationships for the solutes with similar shape in Table 1 are illustrated in Figure 1. Table 3 shows the literature diffusivities of the essentially linear, planar, and spherical solutes in various nonpolar solvents at 298.2 K. The linear dependence lines for these data are plotted in Figures 2 and 3. In Tables 1 and 3, only the diffusivity data of solutes that can be utilized to find the shape effects under the same solvent and temperature conditions are collected from the literature.

3. RESULTS AND DISCUSSION The measured diffusivities, together with their average errors, are presented in Table 1. Also listed in the table are other C


Article

The Journal of Physical Chemistry B

study. In these tables, the solutes are all inert compounds which cannot form strong association (e.g., hydrogen bonding) with solvent molecules in this work. There are in fact four pairs of solutes that are different in molecular mass but nearly the same in size and shape. They are toluene and chlorobenzene, mdichlorobenzene and m-xylene, 1,2,4-trimethylbenzene and 1,2,4-trichlorobenzene, and o-dichlorobenzene and o-xylene (see Table 2 for the V1 and M1 values of the solutes). The diffusivities of the solutes in each of the above pairs, however, are approximately the same within experimental errors in the same solvent at constant temperature. These data indeed further verify the insignificant effects of solute mass and weak dipole moment on diffusivity.2,3,11,22 The different solute-shape effects calculated for each solvent and temperature at V1/10−3 nm−3 = 75, 120, 165, 210, and 255 are shown in Table 5. In this table, the symbols DP, DS, and DL represent the diffusivities of planar, spherical, and linear solutes, respectively. The range of solute sizes are conservatively chosen for this study so that the diffusivity values computed by extrapolation using eq 1 and the constants in Table 4 do not deviate much from the mean deviation of ±1.3%. The average error of the effects or ratios given in Table 5 can therefore be reasonably estimated at about ±2.6%. As shown in Table 5, the effects range from 0.92 to 1.24 or approximately from −8% to +24%, with most of them greater than the uncertainty (±2.6%) of the calculated ratios. Hence, the magnitude of the effects in general, as compared to the overall uncertainty, can still allow useful analysis of the effects in this work. An interesting feature of the result in Table 5 is that the effect consistently decreases as the solute size increases. Some of the typical trends for DP/ DS versus V1 at 298.2 K are demonstrated in Figure 4. Similar plots for DL/DS and DL/DP are shown in Figure 5. For small solutes, all effects shown in Table 5 as well as in Figures 4 and 5 are positive with the trend linear > planar > spherical, which is consistent with the results reported in the literature for small or nonmacromolecular solutes (see Section 1). As the compared solutes of different shapes are larger, however, the ratios tend to decrease toward unity (i.e., zero effect) and even below. It should be noted that a ratio less than 1 signifies that the effect is reversed. This behavior of the effects for a wider range of solute sizes thus clarifies the inconsistent or contradictive results that we have pointed out in section 1 between macromolecules and small-sized solutes. One possible reason for the variation of the effects with solute size is due to the differences in the translation-rotation coupling of the solute molecules compared, which is probably dependent not only on the solute shape but also on the size of the solute. It is of interest to study the solvent dependence of the effects of solute shape on diffusion. By comparing the computed DP/ DS ratios for each of the solute sizes in Table 5, we have found that the values are quite different in various solvents. Nonetheless, they do not appear to be directly related to the density of the solvents in this study. The values of density d2, viscosity η, and other properties of the solvents in the present work are listed in Table 6. Considering only the DP/DS values for a given solute size, however, the shape effects as shown in Table 5 are also found not likely accountable by using the famous hydrodynamic Stokes−Einstein (SE) or Stokes− Einstein-Sutherland relation:

Figure 1. Plot of 1/D12 vs V1 for spherical and planar solutes in polar solvents: (a) ethanol at 298.2 K, (b) methanol at 283.2 K, (c) methanol at 298.2 K, (d) methanol at 313.2 K, and (e) acetone at 298.2 K. Dash lines represent linear regressions for spherical solutes (○), and solid lines are for planar solutes (⧫).

To ensure that the nonpolar aromatic solutes are very disclike, slightly linear compounds such as p-xylene and n-propylbenzene are not included for the present study. Also, a reported diffusivity of tetraethyltin in carbon tetrachloride is not used, primarily because the reciprocal value of it deviates from the linear line of eq 1 by more than 6.6%. The constants and statistics for the linear regressions using eq 1 are shown in Table 4. The results of the fittings are generally quite satisfactory. Overall, the mean of the average deviations for the 20 linear regressions is only ±1.3%, which is quite close to the overall precision of the experimental data. The maximum absolute deviations are also not very high in general; the mean of them is approximately 2.7%. Hence, fairly accurate diffusivity of a solute of any size can be evaluated by interpolation or extrapolation of the regression line concerned, provided that the extrapolation is not done very far outside the range of the actual solutes studied. One of the difficulties associated with experimental investigations of the effect of shape on diffusion is that in real systems there are not many solutes of the same size but with very different shapes for direct comparison. With the help of eq 1, however, such difficulty can be solved by calculating the diffusivities of any sized (virtual) solutes required for understanding more about the shape effects. In particular, this computational method can allow the effects in a wide range of solvents, solutes sizes, and temperatures to be studied. The effect of solute shape on diffusion can actually be expressed by ratio of diffusivities of two solutes of the same size but different shapes under the same conditions. This ratio expression for representing only the shape but not other effects is possible because mutual diffusivities are known to be insensitive to the mass and dipole moment of the solute molecules.2,3,11,22 Evidence can actually be observed also in Tables 1−3 of this

D12 = D

kBT Cπr1η

(2) DOI: 10.1021/acs.jpcb.5b10550 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B Table 3. D12 (10−9 m2s−1) of Planar, Spherical, and Linear Solutes in Nonpolar Solvents at 298.2 K planar solutes solvent CCl4

cyclohexane

n-hexane

n-tetradecane

a

benzene toluene naphthalene mesitylene biphenyl anthracene phenanthrene benzene chlorobenzene toluene ethylbenzene naphthalene 1,2,4-trichlorobenzene mesitylene biphenyl phenanthrene hexamethylbenzene benzene chlorobenzene toluene ethylbenzene naphthalene 1,2,4-trichlorobenzene 1,2,4-trimethylbenzne mesitylene biphenyl anthracene phenanthrene pyrene benzene chlorobenzene toluene ethylbenzene m-xylene m-dichlorobenzene o-xylene o-dichlorobenzene naphthalene 1,2,4-trichlorobenzene biphenyl

spherical solutes

D12a

refs

1.47 1.404 1.200 1.193 1.074 1.026 1.03 ± 0.01 1.89 1.71 ± 0.02 1.72 ± 0.02 1.56 ± 0.02 1.44 ± 0.01 1.39 ± 0.01 1.40 ± 0.01 1.28 ± 0.02 1.199 ± 0.002 1.09 ± 0.01 4.72 ± 0.04 4.22 ± 0.04 4.24 3.94 ± 0.04 3.76 ± 0.03 3.50 ± 0.04 3.57 ± 0.03 3.54 ± 0.03 3.29 ± 0.03 3.09 3.08 ± 0.03 2.86 1.18 ± 0.01 1.06 ± 0.01 1.05 0.953 ± 0.008 0.943 ± 0.005 0.936 ± 0.006 0.892 ± 0.007 0.888 ± 0.006 0.889 ± 0.007 0.832 ± 0.008 0.792 ± 0.009

31, 32 36 36 36 36 36 31 25, 27, 39−41 11 25, 11 11 11 11 11 11 25 11 11 11 47−49 11 11 11 11 11 11 50 31 50 3, 4 3, 4 3, 4, 48 3, 4 2 2 2 2 3, 4 3, 4 3, 4

linear solutes

D12a CCl4 Me4Sn Pr4Sn Bu4Sn

1.29 1.22 0.83 0.70

33, 34 37 37 37


1.49 1.43 1.10 0.89 0.75

± ± ± ±

0.03 0.03 0.01 0.01

34, 42 37 37 37 37


3.87 3.60 2.91 2.39 2.04

± ± ± ±

0.05 0.02 0.01 0.02

24, 31, 43 24 24 24 24


0.897 0.812 0.608 0.475 0.388

± ± ± ± ±

0.006 0.007 0.003 0.006 0.005

D12a

refs

refs

n-pentane n-hexane n-heptane n-octane

1.57 1.49 1.34 1.26

35 31, 35, 38 35, 36 35

n-pentane n-hexane n-heptane n-octane

4.59 ± 0.04 4.20 3.78 ± 0.03 3.47 ± 0.03

31 43−46 31 31

24 24 24 24 24

Average value where two or more refs are given.

where kB refers to the Boltzmann constant, T is the temperature, r1 represents the radius (i.e., size) of the diffusing solute, η is the viscosity of solvent, and C denotes a number (6 for “stick” and 4 for “slip” boundary conditions).18 For a given solute of fixed size, eq 2 can be alternatively written as D12 /T = a1η−1

molecules. On the other hand, it has been reported by numerous investigators24,37,50,53,55−60 that the solvent dependence of diffusivity is more appropriately expressed by

D12 /T = a 2η−t

(4)

where a2 denotes a solute-dependent constant, and t is a fractional number that is normally between 2/3 and 1.55,56 Eq 4 has been widely applied as solvent dependence of diffusivity in recent studies of diffusion-related processes in dense fluids.61−75 Often cited as the fractional Stokes−Einstein (FSE) relation in the literature. The implication of eq 4 to the solvent dependence of the solute-shape effects in the present study is that the DP/DS ratios for a fixed solute size are not necessarily constant in different solvents. The reason is because the value of the fractional exponent t for a planar solute may not be the same as that for a spherical solute of the same size. Hence, the viscosity term may not be canceled out in the

(3)

where a1 is a solute-dependent constant equal to kB/Cπr1. Eq 3 thus also represents the solvent dependence of diffusion for the SE relation. For solutes of same size but different shapes, eq 3 predicts that the DP/DS ratios should be constant (regardless of solvent), since the same viscosity (η) dependence of DP and DS would be canceled out in the ratio. Nonetheless, Table 5 shows that the ratios are different for different solvents. These results thus provide further evidence in support of many previous findings20,24,37,52−55 that the SE relation is invalid for diffusion of solutes either small or not very large as compared to solvent E


Article


Figure 2. Plot of 1/D12 vs V1 for spherical and planar solutes in nonpolar solvents: (a) n-tetradecane and (b) cyclohexane, both at 298.2 K. Dash lines represent linear regressions for spherical solutes (○), and solid lines are for planar (⧫) solutes.

DP/DS ratios. In fact, we have recently shown by FSE that the t values for planar solutes ranging from benzene (V1/10−3 nm−3 = 81.1) to biphenyl (V1/10−3 nm−3 = 152.4) in various solvents at 298.2 K are fairly constant and approximately equal to 0.712 ± 0.008.11 However, Evans et al.24 previously found that the t values are quite different for spherical solutes of different sizes. The value actually increases with spherical solute size from 0.449 for the smallest argon to 0.942 for the largest tetradodecyltin. Applying eq 4, the ratio DP/DS can be expressed as follows: DP /DS = Aη−(t p− ts)

Figure 3. Plot of 1/D12 vs V1 for spherical, planar, and linear solutes in nonpolar solvents at 298.2 K: (a) carbon tetrachloride and (b) nhexane. Dash lines represent linear regressions for spherical solutes (○); solid and dotted lines are for planar (⧫) and linear (×) solutes, respectively.

D12 /T = a3Φ−yη−z

where a3 represents a constant dependent on solute, Φ is a function of solvent properties except the thermodynamic viscosity η, and y = 1 and z = 2/3 for planar solutes. The function Φ in eq 7 can be written as19

(5)

where A is a constant for the pair of solutes of the same size, and tp and ts are the fractional exponents for the planar and the spherical solutes, respectively. Eq 5 can be simplified as P

S

D / D = Aη

−B

(7)

Φ = (ρ21/3 M 21/4)/(εr1/48Vf̅ /Vm̅ )

(6)

(8)

where ρ2 refers to solvent’s molar density, εr denotes solvent’s relative permittivity (dielectric constant), and V̅ m/V̅ f represents the reciprocal of solvent’s free volume fraction. This free volume fraction of solvent is given11 by the expression:

where B is equal to (tp − ts). By using the logarithm form of eq 6, we have attempted to correlate DP/DS with η for each of the solute sizes in Table 5. The results, as summarized in Table 7, are fairly satisfactory. The mean of the average deviations of the fits for the five solute sizes is only ±2.4%, whereas the mean of the maximum absolute deviations is about 5.3%. Hence, the solvent dependence of the solute shape effects can be reasonably regarded as interpretable in terms of the FSE relation. In a recent study,19 we have developed a relation that is even more accurate than the FSE relation for representing the solvent dependence of diffusion of the planar solutes. This relation is a modification of FSE, which takes not only viscosity but also density, relative permittivity, and molecular mass of solvents into account in mutual diffusion. It can be expressed in simple form as

Vf̅ /Vm̅ =

Vm̅ − NAV2 Vm̅

(9)

where V̅ m denotes solvent’s molar volume, NA refers to the Avogadro constant, and V2 represents the VDW volume of the solvent molecule. The value of V̅ f/V̅ m is actually equivalent to the fraction of the free volume available for each solvent molecule in a liquid. Note that the molar density ρ2 is equal to the number density of solvent molecules divided by the Avogadro constant, and that the relative permittivity is related to solvent molecule’s dipole moment in the Debye equation.9 Hence, Φ is essentially a molecular function of solvent. We F


Article

The Journal of Physical Chemistry B Table 4. Statistics and Parameters of Linear Regressions Using Eq 1 solute planar

spherical

linear

solvent

T (K)

no. of data

intercept (b/10−1)

slope (a/10−3)

correl. coef.

av. devi. (±%)

max. abs. devi. (%)

acetone carbon tetrachloride cyclohexane ethanol methanol methanol methanol n-hexane n-tetradecane acetone carbon tetrachloride cyclohexane ethanol methanol methanol methanol n-hexane n-tetradecane carbon tetrachloride n-hexane

298.15 298.15 298.15 298.15 283.15 298.15 313.15 298.15 298.15 298.15 298.15 298.15 298.15 283.15 298.15 313.15 298.15 298.15 298.15 298.15

13 7 10 13 8 10 7 12 11 5 4 5 4 5 5 5 5 5 4 4

1.444 3.920 2.216 2.068 2.312 1.896 1.659 1.132 3.683 1.764 4.885 3.637 2.792 3.302 2.717 2.244 1.569 4.716 3.225 0.826

1.295 3.450 3.743 4.323 3.084 2.319 1.709 1.260 6.157 1.021 3.001 3.112 4.431 2.557 1.858 1.400 1.071 6.737 3.369 1.467

0.979 0.996 0.995 0.986 0.992 0.996 0.994 0.995 0.959 0.993 0.998 0.998 0.999 0.999 0.999 0.999 0.999 0.999 0.991 0.999

1.56 1.08 1.21 1.62 1.24 0.80 1.14 1.14 2.54 2.20 1.54 1.89 1.11 0.67 0.26 0.80 0.99 1.66 1.00 0.46 1.25 ± 0.56

4.10 2.26 2.65 4.84 2.42 1.75 2.35 3.49 5.71 4.80 2.23 4.29 2.07 1.43 0.50 1.49 2.06 3.49 1.98 0.91 2.74 ± 1.41

mean

Table 5. Calculated Effects of Solute Shape on Diffusion for Different Sizes of Solute Molecules a

ratio DP/DS

DL/DP DL/DS a

solvent n-hexane acetone methanol methanol methanol CCl4 cyclohexane ethanol n-tetradecane n-hexane CCl4 n-hexane CCl4

T (K) 298.15 298.15 313.15 298.15 283.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15

75 1.14 1.05 1.11 1.13 1.13 1.10 1.19 1.15 1.18 1.08 1.13 1.23 1.24

120 1.08 1.00 1.05 1.06 1.06 1.05 1.10 1.12 1.16 1.02 1.11 1.10 1.17

V1 (10−3 nm3) 165 1.04 0.96 1.01 1.01 1.02 1.02 1.05 1.10 1.14 0.99 1.09 1.03 1.12

210 1.01 0.94 0.99 0.98 0.99 1.00 1.01 1.09 1.14 0.97 1.08 0.98 1.09

255 0.99 0.92 0.97 0.96 0.97 0.99 0.98 1.08 1.13 0.95 1.08 0.94 1.06

DP, DS, and DL are diffusivities of planar, spherical, and linear solutes, respectively.

have demonstrated in our earlier work19 that eq 7 with y = 1 and z = 2/3 can be used to correlate the diffusivities of each of the planar solutes (ranging from benzene to biphenyl) in various solvents to within an average deviation of ±2.5%. For spherical solutes, however, we have found that the values of y and z are different for different solute sizes, with y smaller and z larger as size of the solute is increased. Because solutes of different shapes can have different y and z values, it is therefore reasonable to expect that the DP/DS ratios for each solute size in various solvents can be fitted with the following equation: DP /DS = F Φ−Gη−H

dependence of the solute-shape effects. It should be noted, however, that eq 10 actually has one more parameter than eq 6 does. Nonetheless, many investigators76−79 are of the view that microscopic contributions of solvent molecules play a significant role in the diffusion of small or nonmacromolecular solutes. It is noteworthy that the function V̅ f/V̅ m in eq 9 is based on the free volume theory of Hilderbrand,80 whereas the 1/4 is modified from the theoretical result of product ρ1/3 2 M2 81 March. The dependence on εr is an empirical finding19 motivated by the experimental holographic fluorescence recovery results of Quitevis et al.62,63 Details of the development of eq 7 and eq 8 are provided in ref 19. It should be pointed out that the effects due to shapes of solvent molecules are not taken into account in the expression for Φ in this work, however, mainly because there is a lack of such information in the literature. Nonetheless, the previous success of eq 7 (with the function Φ expressed by eq 8) has demonstrated that this molecular modification of FSE is useful and accurate enough

(10)

where F, G, and H are constants. We have in fact carried out the correlations using eq 10, and the results are shown also in Table 7. For the five solute sizes studied, the mean of the average deviations is only 1.78%, and the average of the maximum absolute deviations is 3.88%. The results indicate that eq 10 is indeed more satisfactory than eq 6 for accounting the solvent G


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diffusion, an area which is still very much unknown in the literature. Table 5 shows the calculated DP/DS ratios for different solute sizes in methanol at the above three temperatures. For each size, the ratios at these temperatures of methanol are quite nearly the same within the general uncertainty of the calculated ratios, although the values seem generally lower at higher temperatures. For the range between 283.2 and 313.2 K, the effect of solute shape on diffusivity can hence be considered as fairly insensitive to temperature. Because a change in temperature can actually cause change in solvent properties, it is interesting to apply eq 10 (i.e., the solvent dependence of the shape effects) using the constants in Table 7 to predict DP/DS for each solute size at different temperatures. As shown in Table 8, the results agree very well with the experimental DP/DS ratios (computed by using eq 1 and the constants in Table 4) for methanol at the three temperatures. The implication is that the temperature dependence of the solute-shape effects can also be predicted by eq 10 or the solvent dependence of the effects of solute shape on diffusivity. The good agreement has thus encouraged us to extend the calculation using eq 10 to a wider range of temperatures from 268.2 to 328.2 K. The results are also given in Table 8. For this wider temperature range, it is more clearly observed that the effects of solute shape on diffusion in methanol are weakly dependent on temperature. The DP/DS ratios generally decrease 2 to 6% from low to high temperatures between 268.2 and 328.2 K, and it appears that the decrease is slightly greater for larger solutes.

Figure 4. Plot of DP/DS vs V1 for different solvents at 298.2 K: (a) ntetradecane, (b) ethanol, (c) CCl4, (d) methanol, and (e) acetone. V1/ 10−3nm3 = 75 (Δ), 120 (●), 165 (□), 210 (⧫), and 255 (○).

for representing the solvent dependence of diffusion. As eq 10 is developed from eq 7, it is therefore reasonable that the solvent dependence of the solute-shape effects or the DP/DS ratios can be well-represented by this equation. The diffusivities reported in the literature for spherical solutes in methanol are available at 283.2, 298.2, and 313.2 K (see Table 1). One of the reasons for our present measurements of the planar aromatic solutes in methanol is to combine both the spherical and planar solute data to provide a study on the temperature dependence of the solute-shape effects on

4. CONCLUSION Diffusivity of a molecule in liquid phase is normally affected by various factors. One of them is known to be the shape of the diffusing molecule. In this work, we present an investigation on the effects of solute shape on diffusion, and evidence of the dependences of such effects on solute size, solvent, and temperature is demonstrated. Our approach is by employing the diffusivities of essentially nonassociated planar, spherical, and linear solutes in different solvents, which were either

Figure 5. Plot of DL/DP (I) and DL/DS (II) vs V1 at 298.2 K: (a) CCl4 and (b) n-hexane. V1/10−3 nm3 = 75 (Δ), 120 (●), 165 (□), 210 (⧫), and 255 (○). H


Article

The Journal of Physical Chemistry B Table 6. Values of d2, η, εr, ρ2, and V̅ f/V̅ m of Solvents at Different Temperatures T (K)

solvent

d2 (g cm−3)a

η (mPa s)a

εr b

ρ2 (10−3 mol cm−3)

V̅ f/V̅ m

268.15 283.15 298.15

methanol methanol acetone carbon tetrachloride cyclohexane ethanol n-hexane methanol n-tetradecane methanol methanol

0.8136 0.8006 0.7856 1.583 0.7731 0.7873 0.6563 0.7872 0.7578 0.7733 0.7589

0.852 0.667 0.308 0.893 0.901 1.083c 0.296 0.539 2.128b 0.447 0.377

38.46 35.25 20.49 2.228 2.016 24.85 1.882 32.61 2.028 30.55 29.07

25.39 24.99 13.53 10.29 9.186 17.09 7.616 24.57 3.820 24.14 23.69

0.4675 0.4760 0.4930 0.4428 0.4633 0.4752 0.5070 0.4848 0.4551 0.4939 0.5033

313.15 328.15 a

Calculated from ref 51, except where noted otherwise. bCalculated from ref 29. cValues from ref 37.

Table 7. Results of Correlations Using Eq 6 and Eq 10 for Solvent Dependence of DP/DS constant −3

V1/10

eq 6

3

10

A

nm

75 120 165 210 255 75 120 165 210 255

−2

1.15 1.10 1.07 1.04 1.03

± ± ± ± ±

B/10 0.01 0.01 0.01 0.01 0.02

−3.71 −5.29 −6.52 −7.46 −8.23

± ± ± ± ±

G/10−2

F

1.20 1.16 1.14 1.12 1.11

± ± ± ± ±

0.04 0.03 0.03 0.04 0.04

5.30 6.22 7.03 7.61 8.17

V1/10

nm

max. abs. dev. (%)

2.05 1.90 2.38 2.65 3.04 1.76 1.31 1.62 1.95 2.25

5.14 4.50 4.87 6.06 5.88 4.73 3.52 3.32 3.86 3.96

268.15

283.15

298.15

313.15

328.15

75

1.13 1.08

165

1.05

210

1.03

255

1.01

1.12 (1.13) 1.06 (1.06) 1.02 (1.01) 0.99 (0.98) 0.98 (0.96)

1.11 (1.11) 1.05 (1.05) 1.01 (1.01) 0.98 (0.99) 0.96 (0.97)

1.11

120

1.12 (1.13) 1.07 (1.06) 1.03 (1.02) 1.01 (0.99) 0.99 (0.97)

± ± ± ± ±

3.63 2.62 2.83 3.35 3.84

−4.61 −6.35 −7.72 −8.75 −9.61

± ± ± ± ±

1.77 1.28 1.38 1.63 1.87

of macromolecular solutes are generally known to be opposite. Similar results of the solute-size dependence of the shape effects are also obtained for the linear-planar and linear-spherical solute pairs in this study. We believe that this size dependence of the effects is attributable to the differences in the translationrotation coupling of the solute molecules compared. The solvent dependence of the solute-shape effects has also been investigated by considering only the diffusivities of solutes of the same size but different shapes in various solvents. It is found that the shape effects between planar and spherical solutes are different from solvent to solvent by up to nearly 20% for each of the five solute sizes studied in this work. We have shown that the solvent dependence of the solute-shape effects is accountable fairly well by eq 6 in terms of the FSE relation and more accurately by eq 10, an equation based on a molecularly modified FSE relation. It is further demonstrated here that eq 10 is capable of correlating the shape effects for each size of the solute pairs compared in this study to within an average deviation of ±2.3%, whereas the mean of the average deviations for the five different sizes is only ±1.8%. Finally, the diffusivity ratios of planar to spherical solutes in methanol at different temperatures indicate that the effects of solute shape on diffusion are only weakly, if not insignificantly, dependent on temperature. The temperature dependence of the shape effects is also predictable by eq 10 or the solvent dependence of the effects, since temperature change leads to change also in solvent properties.

temperature (K) 3

av. dev. (±%)

1.78 1.54 1.70 1.93 2.15

Table 8. Values of DP/DS at Different Temperatures in Methanola −3

H/10−2

1.04 1.01 0.97 0.95

a

Values are calculated by eq 10; those in parentheses are calculated from experimental data of DP and DS.

measured in this laboratory or collected from various sources in the literature, to calculate the effects due to different shapes of solutes with equal size. By studying the computed effects of solute shape on diffusion in each individual solvent at constant temperature, we have discovered that the shape effects are solute-size dependent. For small solute molecules of the same size, planar solutes generally diffuse faster than spherical solutes. As the solutes get larger, however, the diffusion rates of planar solutes are reduced or even reversed, as compared to those of the spherical ones of the same size. This trend of the solute-size dependence of the effects of solute shape on diffusion clarifies why the diffusivities of relatively small molecules were consistently found greater for planar than for spherical solutes in the literature,2−7 while those

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Corresponding Author

*E-mail: [email protected]. I


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The Journal of Physical Chemistry B Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to J. G. Lu and S. H. Lam for their technical assistance. We thank the Research Committee of the Hong Kong Polytechnic University for financial support of this work under Grant 5-ZJF1.

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