Diffusion of Benzene and Alkylbenzenes in Nonpolar Solvents

Feb 7, 2018 - demonstrated its utility by obtaining the D values for solutes ranging in .... 1-CiPh total. B & Tc σ1/σ2. % diff.d no.e. % diff.d no...
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Cite This: J. Phys. Chem. B XXXX, XXX, XXX−XXX

Diffusion of Benzene and Alkylbenzenes in Nonpolar Solvents Bruce A. Kowert* Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, United States ABSTRACT: The translational diffusion constants, D, of benzene and a series of alkylbenzenes have been determined in n-pentadecane, 2,6,10,14-tetramethylpentadecane (pristane), 2,2,4,4,6,8,8-heptamethylnonane (isocetane), and 2,6,10,15,19,23-hexamethyltetracosane (squalane) using capillary flow techniques. The solutes’ D values are compared with the predictions of a cylinder diffusion model as are those for (a) benzene and alkylbenzenes in n-nonane, n-decane, ndodecane, and supercritical CO2 and (b) n-alkanes and 1alkenes in n-hexane, n-heptane, n-octane, benzene, and toluene. The D values for benzene and the alkylbenzenes also are compared with the predictions of lollipop diffusion for which the phenyl ring is the candy and the alkyl chain is the handle. Both models give an average difference of less than 4% between experimental and calculated diffusion constants in solvents whose viscosities vary by a factor of more than 600 when benzene and toluene (as solutes) are omitted; the comparisons include 150 and 85 D values for the cylinder and lollipop models, respectively. The differences increase when benzene and toluene are included and are most likely because of their shapes and the shapes assumed by the models. The agreement with the models indicates that the chains of the alkylbenzenes and 1-alkenes, like those of the n-alkanes, are relatively extended. The D values for several of the solutes also are fitted to a modification of the Stokes−Einstein relation that varies their dependence on viscosity instead of chain dimensions.

1. INTRODUCTION The shape and size of a solute are important determinants of its translational diffusion constant, D. Solutes whose dimensions can be varied in a systematic manner provide good tests of diffusion models. In ref 1, we compared the D values for benzene and a series of alkylbenzenes in several n-alkanes with the predictions of cylinder diffusion.2−4 Ratios of the diffusion constants in each n-alkane were in reasonably good agreement with the calculated values as were ratios of D values for n-alkane and 1-alkene solutes, also in n-alkane solvents. We took this to indicate similarities in the diffusion of the three types of molecules. Lollipop diffusion,5 a second length-dependent model with the phenyl ring as the candy and the alkyl chain as the handle, also gave calculated ratios of diffusion constants in general agreement with experiment. The experimental diffusion constants were determined at constant temperature in each solvent;1 the calculated ratios in a given solvent were then independent of viscosity and temperature because both models’ expressions for the D values4,5 were proportional to T/η, the ratio of absolute temperature to viscosity. This allowed the fits to focus on the shape and size of the solutes, but the experimental ratios varied by only a factor of ∼6. Comparisons involving the actual D values are more desirable; they are temperature- and viscosity-dependent4−8 and have a much wider range of values. The diffusion constants considered here vary by a factor of more than 300; they include published results as well as new D values, given in Table 1, for © XXXX American Chemical Society

Table 1. Diffusion Constants of Benzene and Alkylbenzenes in Squalane, Pristane, HPMN, and n-C15 106D(expt) (cm2 s−1) solute C6 D 6 toluene ethylbenzene 1-phenylpropane 1-phenylbutane 1-phenylpentane 1-phenylhexane 1-phenylheptane 1-phenlyloctane 1-phenlylnonane 1-phenylundecane 1-phenyldodecane 1-phenyltridecane 1-phenyltetradecane 1-phenylheptadecane

squalanea e

1.99 1.73e 1.62 1.41 1.34 1.23 1.13 1.04 0.974 0.884 0.740 0.693 0.661 0.621 0.568

pristaneb

HPMNc

n-C15d

6.17

8.04 7.51 6.77 5.94

10.6 9.91 8.74f 8.08 7.61f 7.21 6.59f 5.89f 5.41 5.01f 4.32 4.12f 3.91f 3.81 3.54

5.17

4.14

4.79

3.15

3.98

2.47

3.15

2.15 1.96

2.65 2.44

At 23.0 ± 0.5 °C. bAt 25.5 ± 0.5 °C. cAt 21.0 ± 0.25 °C. dAt 24.5 ± 0.5 °C. eFrom ref 24. fNew data; the other data are from ref 1. a

Received: October 11, 2017 Revised: January 22, 2018

A

DOI: 10.1021/acs.jpcb.7b10078 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B benzene and alkylbenzenes in n-pentadecane, 2,6,10,14tetramethylpentadecane (pristane), 2,2,4,4,6,8,8-heptamethylnonane (HPMN, isocetane), and 2,6,10,15,19,23-hexamethyltetracosane (squalane). The published diffusion constants are for (a) benzene and alkylbenzenes in n-nonane, n-decane, ndodecane, and n-pentadecane;1 (b) benzene and alkylbenzenes in supercritical CO2;9,10 and (c) n-alkanes and 1-alkenes in nhexane, n-heptane, n-octane, benzene, and toluene.11−23 When discussing a particular solute or solvent, 1-CiPh for 1phenylalkanes, n-Ci for n-alkanes, and 1-Ci for 1-alkenes will be used. The new diffusion constants were determined at a constant temperature in each solvent; the D values and temperatures are given in Table 1. Depending on the solvent, the solutes are some (if not all) of benzene, toluene, ethylbenzene, and the 1CiPh, with i = 3−9, 11−14, and 17. Also given in Table 1 are the D values in n-pentadecane from ref 1 and the D values for benzene and toluene in squalane from ref 24. The comparisons in this article show that both the cylinder and lollipop models give good agreement between the solutes’ experimental and calculated diffusion constants for the nalkanes, 1-alkenes, and alkylbenzenes ethylbenzene − 1-C17Ph; an average difference of less than 4% is found. The agreement for benzene and toluene as solutes is not nearly as good because their shapes are different from those assumed by the models. The viscosities of our new solvents (HPMN, pristane, and squalane) and supercritical CO2 are the reason that this study’s range of D values is much larger than that in ref 1. In that study, the range of viscosities was from 0.300 cP (n-C6 at 25.0 °C25) to 2.57 cP (n-C15 at 24.5 °C25). In this study, the viscosities vary from 30.3 cP (squalane26 at 23.0 °C) to 0.0476 cP (supercritical CO2 at 333 K and 15.0 MPa9), a difference of a factor of more than 600. This article’s range of temperatures also is larger than that in ref 1, but the increase is relatively small; in that study, it was 294−298 K; in this study, the lowest remains 294 K (in n-C121) but the highest increases to 333 K (in CO29,10). The microcapillary flow method used to determine our new D values was developed by Bello and co-workers;27 they demonstrated its utility by obtaining the D values for solutes ranging in size from phenylalanine to hemoglobin in H2O. Refs 28−30 describe other applications of the method involving systems such as α-tocopherol and β-carotene in supercritical CO2,28 tetraalkyltin derivatives in EtOH,29 and dioxygen in nalkanes.30 The diffusion constants for benzene and several of the alkylbenzenes also have been fitted to a modification of the Stokes−Einstein relation that varies their dependence on viscosity instead of shape and size. Our data are well-suited for this analysis because of the solvents’ appreciable range of viscosities. The fits are generally good for all of the solutes, including benzene, with R2 ≥ 0.996. The results reported here are related to work in other areas, three of which are discussed in Section 3.5. They are: (1) modeling the physical properties and combustion reactions of petroleum- and bio-based fuels,31 (2) determining the internal viscosities of micelles,32−34 and (3) evaluating the efficiency of chromatographic supports.35,36

(98%), 1-C9Ph (96%), 1-C11Ph (99%), 1-C12Ph (97%), 1C13Ph (99%), 1-C17Ph (97%), n-C15 (99%), HPMN (98%), and squalane (99%) were obtained from Aldrich; benzene (C6H6, ≥99.9%) and pristane (98%) were obtained from Sigma-Aldrich; ethylbenzene (99.8%) and 1-C3Ph (98%) were purchased from Acros Organics; 1-C5Ph (96%) and 1-C14Ph (97%) were obtained from Alfa Aesar; 1-C6Ph (≥99.9%) was obtained from Fluka; and 1-C4Ph (≥99.0%) was purchased from TCI. All of the substances were used as received. Solutions were prepared by adding two to five drops of the liquid solutes to 6.0−8.0 mL of the solvents. 2.2. Data Acquisition and Analysis. The process by which the solutes’ elution profiles were acquired and used to obtain the diffusion constants, given in Table 1, is described in refs 24 and 37; the same procedure was used for the D values in n-C15 in ref 1. The data acquisition systems, including the microcapillary (76.5 μm i.d.) and variable wavelength detector, are the same as in ref 37. A wavelength of 260 nm was used for all of the solutes, except C6H6 and C6D6 (198 nm). Diffusion constants were obtained by comparing the sigmoidal experimental profiles with those calculated using Taylor’s theory, as described in refs 1, 24, 37, and 38. All of the profiles were taken at room temperature, which varied by no more than ±0.50 °C for the 5−7 days needed to acquire the profiles in a given solvent. Multiple determinations of D were made for each solute−solvent pair; their average values are given in Table 1. The uncertainties in the D values are ±4.9% in n-C15, ±1.3% in HPMN, ±1.0% in pristane, and ±3.9% in squalane. 2.3. Solute−Solvent Systems. The n-alkane and 1-alkene solutes and their solvents are mentioned in Table 2. The nTable 2. Solutes and Solvents for n-Alkanes and 1-Alkenes in n-Alkanes, Benzene, and Toluene i for n-Ci solutes

solvent n-C6a n-C7b n-C7c n-C8d benzenee benzenef tolueneg solvent n-C6h n-C8h

32, 16, 32, 32, 32, 20, 26,

24, 14, 28, 24, 29, 18, 22,

18, 16, 12, 10, 8, 7, 6, 5 12, 10, 7 18, 14, 10 16, 14, 12, 8 28, 18, 14, 12, 10, 9, 8, 7, 6, 5 14, 10, 5 18, 14, 10, 6 i for 1-Ci solutes 14, 12, 8, 6 14, 12, 10, 8, 6

T (°C) 25.0 25.0 20.0 25.0 22.2 20.0 27.0 T (°C) 25.0 25.0

a

The D value for the self-diffusion constant of 6 is from ref 11, the rest are from ref 12. bThe D value for the self-diffusion constant of 7 is the average of the two values in refs 13 and 14, the rest are from ref 15. c The D values are from ref 16. dThe D value for 8, the self-diffusion constant, is the average of the values in refs 11, 17−19; that for 12 is from ref 17; that for 14 is from ref 15; those for 16, 24, and 32 are from ref 12. eAll D values are from ref 20. fAll D values are from ref 21. g All D values are from ref 22. hThe D values for the 1-Ci in the n-Ci were calculated using the data in ref 23.

alkanes range from n-C5 to n-C32 in n-C6, n-C7, n-C8, benzene, and toluene;11−22 the 1-alkenes range from 1-C6 to 1-C14 in nC6 and n-C8.23 The D values for benzene and toluene in squalane, in Table 1, are the averages of the values in ref 24. The viscosities of the n-alkanes, HPMN, pristane, squalane, benzene, and toluene20,25,26,39−41 are given in Table 3. The solutes in supercritical CO2 are benzene, toluene, ethylbenzene, and 1-CiPh with i = 3−6, 8, and 12;9,10 the temperatures,

2. EXPERIMENTAL METHODS 2.1. Chemicals and Sample Preparation. Benzene-d6 (99.6 atom % D), toluene (99.8%), 1-C7Ph (98%), 1-C8Ph B

DOI: 10.1021/acs.jpcb.7b10078 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B Table 3. Values of the Solvents’ Viscosities, d for the Cylinder Model, and σ2 for the Lollipop Model solutes 1-CiPh 1-CiPh 1-CiPh 1-CiPh 1-CiPh 1-CiPh 1-CiPh n-Ci, 1-Ci n-Ci n-Ci n-Ci, 1-Ci n-Ci n-Ci n-Ci

solvent n-C9 n-C10 n-C12 n-C15 HPMN pristane squalane n-C6 n-C7 n-C7 n-C8 toluene benzene benzene

T (°C) 24.0 23.5 20.5 24.5 21.0 25.5 23.0 25.0 25.0 20.0 25.0 27.0 22.2 20.0

102η (P)

Table 6. Average Percentage Differences without Benzene and Toluene for Lollipop Diffusion

d (Å)a

σ2 (Å)b

0.838 0.839 0.745 0.586 0.605 0.401 0.288 0.971 0.932 0.932 0.847 0.875 0.890 0.890

0.561 0.560 0.496 0.391 0.400 0.266 0.190

c

0.673 0.857c 1.50c 2.57c 3.44d 6.61e 30.3f 0.300c 0.385c 0.407c 0.508c 0.535g 0.632h 0.651c

1-CiPh in alkanesa

1-CiPh in CO2b

σ1/σ2

% diff.d

no.e

% diff.d

no.e

2.00 2.17 2.40 2.64

3.72 3.78 4.72 5.99

50 50 50 50

3.97 3.66 2.94 3.14

35 35 35 35

B & Tc

1-CiPh total % diff.d,f 3.82 3.73 3.99 4.82

(4.40) (4.37) (4.87) (5.84)

no.e

% diff.d

85 85 85 85

6.94 6.87 7.24 7.97

The solutes are ethylbenzene − 1-C14Ph in n-alkanes, HPMN, pristane, and squalane. bThe solutes are 1-CiPh with i = 2−6, 8, and 12. c20 D values for benzene and toluene as solutes. dAverage percentage difference between the experimental and calculated D values. eNumber of solute−solvent combinations. fThe number in parentheses is the average percentage difference when benzene and toluene as solutes are included (105 total D values). a

3. RESULTS AND DISCUSSION 3.1. Results. The diffusion constants for benzene and the alkylbenzenes in n-C9, n-C10, n-C12, n-C15, HPMN, pristane, and squalane are shown in Figure 1; the data are from Table 1

Values of d for l/d = 1.57. bValues of σ2 for σ1/σ2 = 2.17. cFrom ref 25. dFrom ref 39. eFrom ref 40. fFrom ref 26. gFrom ref 41. hFrom ref 20. a

pressures, and viscosities for those solutions are given in Table 4. The diffusion constants in refs 9−22 were reported to either Table 4. Values of the Viscosities, d for the Cylinder Model, and σ2 for the Lollipop Model for Alkylbenzene Solutes in Supercritical CO2 T (K)

P (MPa)

102η (P)a

d (Å)b

σ2 (Å)c

333 333 313 323 313

15.0 20.0 20.0 30.0 35.0

0.0476 0.0598 0.0772 0.0851 0.102

1.79 1.63 1.59 1.55 1.49

1.19 1.08 1.05 1.03 0.986

a c

The viscosities are from refs 9 and 10. bValues of d for l/d = 1.57. Values of σ2 for σ1/σ2 = 2.17.

Figure 1. Experimental D values of benzene and the alkylbenzenes in n-C9, n-C10, n-C12, n-C15, HPMN, pristane, and squalane versus the number of alkyl carbon atoms in the alkylbenzenes’ chains. Data from ref 1 are included with the new D values from this article.

two or three significant figures, and the average differences between the experimental and calculated results mentioned in the text and Tables 5 and 6 are given to two significant figures with a subscripted insignificant figure.

Table 5. Average Percentage Differences without Benzene and Toluene for Cylinder Diffusion solutes 1-CiPh

1-CiPh

1-CiPh

alkanesb

CO2c

totald

n-Ci and 1-Ci

B and Ta

all

solvents l/d

% diff.

3.08 2.00 1.57 1.00 0.770

3.65 3.48 3.59 4.33 5.05

g

% diff. 5.41 4.41 3.95 3.01 2.58

g

% diff.

n-Ci, B, Te

g

g

% diff.

4.32 3.84 3.73 3.83 4.11

3.95 3.16 2.82 2.88 3.20

allf % diff. 4.17 3.58 3.38 3.46 3.75

totala g,h

% diff.a,g

(6.54) (5.53) (5.03) (4.53) (4.55)

21.2 17.1 14.8 10.5 8.67

20 D values for benzene and toluene as solutes. b57 D values for ethylbenzene − 1-C17Ph in n-alkanes, HPMN, pristane, and squalane. c35 D values for 1-CiPh with i = 2−6, 8, and 12. dTotal of 92 D values for 1-CiPh solutes in the alkanes and CO2. e58 D values in n-alkanes, benzene, and toluene. f Total of 150 D values for 1-CiPh, n-Ci, and 1-Ci solutes. gAverage percentage difference between the experimental and calculated D values. hThe number in parentheses is the average percentage difference when benzene and toluene are included as solutes (170 total D values). a

C

DOI: 10.1021/acs.jpcb.7b10078 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

The experimental D values for ethylbenzene − 1-C17Ph are compared with the values calculated for l/d = 1.57 in Figure 2;

of this article and Tables 1−3 of ref 1. The D values decrease as the solutes’ chain lengths increase and, as shown below, are in general agreement with the predictions of cylinder and lollipop diffusion, two models developed for biological macromolecules42−44 that appear to be useful for smaller solutes; they are based on6,7

D = kBT /f

(1)

where kB is Boltzmann’s constant, T is the absolute temperature, and f is the solute’s friction coefficient. Eq 1, the Einstein relation, is a general result, independent of the friction mechanism.8 The friction coefficients for the cylinder and lollipop models are given by Stokes law with stick boundary conditions

f = 6πηϕ(r )

(2) Figure 2. Comparison of the experimental diffusion constants (in cm2 s−1) for 1-CiPh, n-Ci, and 1-Ci solutes (without benzene and toluene as solutes) with the predictions of cylindrical diffusion for l/d = 1.57.

where η is the viscosity and ϕ(r) is a model-dependent function of a solute’s shape and size. 3.2. Cylinder Diffusion Model. Hansen4 used Monte Carlo methods to calculate ϕ(r) for cylinders and gave his results as ϕ(cyl)i = (3/16)1/3 (Li /di)−2/3 Lif [ln(Li /di)]

the overall agreement (3.38%) is good and indicates that the cylinder model is useful for alkylbenzenes, n-alkanes, and 1alkenes in several types of nonpolar solvents (n-alkanes, methyl-substituted alkanes, arenes, and CO2). The fits may be insensitive to the value of l/d, but the cylinder model’s length dependence predicts 150 D values with an average difference of less than 4%. The alkylbenzenes, n-alkanes, and 1-alkenes are not rigid cylinders, but the degree of agreement suggests that their chains are relatively extended. This is consistent with molecular dynamics (MD) simulations2 for n-C8 − n-C20, which showed that they are more rodlike than spherical; the alkyl chains have a time-averaged shape that is approximately cylindrical because there is2 “little change in length between dynamic time steps”. Monte Carlo simulations45 for n-C6 − n-C12 in the liquid phase showed that they were relatively extended with trans populations of 75−80% and average end-to-end distances 87−93% of their all-trans conformations. Similar results were obtained in other experimental and MD studies.46−48 We are not aware of MD calculations for the alkylbenzenes in n-alkanes. Such calculations have been carried out for a series of alkylbenzenes (toluene through 1-C12Ph) in supercritical CO2;49 they gave diffusion constants in agreement with experiment and mean alkyl chain lengths within 5% of those calculated for the corresponding neat n-alkanes.2,45,46 Our D values would provide tests of the force fields and computer codes used in such calculations.2,45,46,49 The values of diameter d for l/d = 1.57 from the fits without benzene and toluene are given in Tables 3 and 4 and generally decrease as the solvent viscosity increases. For the alkylbenzenes, the diameters vary from 1.79 Å in CO2 (0.0476 cP) to 0.288 Å in squalene (30.3 cP) and are less than twice the common van der Waals radius of the methyl and methylene groups (2rvdW = 4.00 Å).50 The d values for the n-alkanes and 1alkenes follow the same pattern; they vary from 0.971 Å in n-C6 (0.300 cP) to 0.847 Å in n-C8 (0.508 cP) and are smaller than 2rvdW. These small values of d and their dependence on viscosity will be discussed in Section 3.4. 3.3. Lollipop Diffusion Model. An alkylbenzene also can be modeled as a diffusing lollipop. The phenyl ring is the candy, a sphere with radius σ1; the alkyl chain is the handle, a rod of length Li = 2niσ2 composed of ni smaller spheres of radius σ2.

(3)

where Li is the length of cylinder i, di is its diameter, and f [ln(Li/di)] is given by eq 16 of ref 4. The alkylbenzenes are assumed to have a common diameter di = d and ni alkyl chain elements of length l. Space-filling models suggest that the phenyl ring adds three elements, giving Li = (ni + 3)l and Li/di = (ni + 3)(l/d); eq 3 is then ϕ(cyl)i = (d)[(3/16)(l /d)(ni + 3)]1/3 f {ln[(ni + 3) (l /d)]}

(4)

where ni + 3 is replaced by ni, the number of carbon atoms for the n-alkane and 1-alkene solutes, which also are assumed to have a common diameter. Comparisons of experimental and calculated diffusion constants were made as a function of l/d for 170 D values with and 150 D values without benzene and toluene as solutes. The agreement was better when they were omitted, and the assumption of a rodlike shape for the disclike phenyl ring is a plausible reason for the larger differences. The smallest average difference when benzene and toluene were omitted was 3.38% for l/d = 1.57 (under “all” in Table 5); the difference when they were included was 5.03%. The results, however, are a weak function of l/d; the differences when benzene and toluene are omitted (Table 5) range from 3.46% (l/d = 1.00) to 4.17% (l/d = 3.08). The data for l/d = 1.57 illustrate the difference made by benzene and toluene. The average difference for these two solutes (14.8%, 20 D values) is substantially larger than 3.73%, the average difference for ethylbenzene − 1-C17Ph (92 D values). Similar results are obtained for the other values of l/d (see Table 5). The n-alkanes and 1-alkenes, the solutes for which the cylinder model would seem most appropriate, have the smallest average difference (2.82% for l/d = 1.57, 58 D values, Table 5). The agreement could be improved using the best value of l/d for each type of solvent (e.g., n-alkanes, CO2) or for individual solvents (e.g., n-C6, squalane), but our intent was to keep the number of parameters to a minimum while fitting D values over the largest possible range of viscosities. D

DOI: 10.1021/acs.jpcb.7b10078 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B Garcia de la Torre and Bloomfield5 used iterative numerical methods to calculate ϕ(r) and gave their results as ⎡ ϕ(loll)i = σ2(σ1/σ2)⎢1 + ⎢⎣

As for the cylinder model, the agreement could have been improved using the best value of σ1/σ2 for individual solvents or each type of solvent, but we were interested in fitting D values over the largest range of viscosities with the smallest number of parameters. The values of σ2 for σ1/σ2 = 2.17 from the fits for ethylbenzene − 1-C14Ph are given in Tables 3 and 4; they too are relatively small, decrease as η increases, and are discussed in the following section; the largest value (σ2 = 1.19 Å for CO2) is less than the aforementioned van der Waals radius for the CH2 and CH3 groups (rvdW = 2.00 Å).50 3.4. Modified Stokes−Einstein (MSE) Relation. The viscosity dependence found for d and σ2 is similar to that of the apparent sizes of alkenes, alkynes, polyenes, aromatic hydrocarbons, and C60 in n-alkanes, squalane, and cycloalkanes. Their values of ϕ(r) decreased as η increased and were often atypically small.23,24,37,51 The solutes’ D values could be fitted to the modified Stokes−Einstein (MSE) relation37,52

⎤

∑ ∑ bjk(Li /σ1) j (σ1/σ2)k− 1⎥⎥ j

k

⎦ (5)

where both j and k have values of 1, 2, and 3. The double summation in eq 5 depends only on ni and σ1/σ2 because Li /σ1 = 2niσ2/σ1 = 2ni(σ1/σ2)−1

(6)

and ϕ(loll)i = σ1 for benzene (ni = 0). The values of bjk in eq 5 are given in Table 1 of ref 5 and are valid for 2 ≤ σ1/σ2 ≤ 10 and Li/σ1 ≤ 16. The limits for σ1/σ2 and Li/σ1 eliminate a number of solutes from our analyses. Neither the n-alkanes and 1-alkenes, for which σ1/σ2 = 1, nor the longest alkylbenzene, 1-C17Ph, was included. Preliminary calculations indicated that the best agreement between the experimental and calculated D values was found near the lower limit of σ1/σ2 = 2, for which Li/σ1 = 17 when ni = 17 (eq 6). The upper limit of Li/σ1 is 16, and to compare a common number of solutes, the diffusion constants for 1-C17Ph were not calculated for any values of σ1/σ2. Given these eliminations, calculations and comparisons were carried out as a function of σ1/σ2 for 105 D values with and 85 D values without benzene and toluene as solutes. As for the cylinder model, the agreement was better when they were omitted, and the assumption of a spherical shape for the phenyl ring is a likely reason for the larger differences. The smallest average difference without benzene and toluene was 3.73% for σ1/σ2 = 2.17, although the results are not a sensitive function of σ1/σ2 (Table 6). The average differences for benzene and toluene (under “B & T % diff.”) are not as large as those for cylinder diffusion, but they are still larger than those for the other alkylbenzenes for all values of σ1/σ2 (Table 6). The experimental D values for ethylbenzene − 1-C14Ph are compared with the values calculated using σ1/σ2 = 2.17 in Figure 3. The degree of agreement, slightly less than that for cylinder diffusion, is still reasonably good and lends credence to the suggestion that the alkylbenzenes’ chains are relatively extended.

D/T = ASE /η p

(7)

which has been shown to follow from the Doolittle−Cohen− Turnbull free volume theory;53−55 p and ASE are constants for a given solute. The decrease in ϕ(r) as η increased gave values of p < 1, which approached p = 1, the Stokes−Einstein (hydrodynamic) limit, as the solutes’ sizes increased.23,24,37,51 This might seem to make eqs 1 and 2, with their η−1 viscosity dependence, open to question, but Zwanzig and Harrison8 gave reasons for using them, with the solvent-dependent values of ϕ(r) measuring the coupling between the solute motion and the solvent flow. Other authors have mentioned the solvent dependence of sizes. Pastor and Karplus56 did so in a discussion of the stochastic simulation of polymers, as did Paul and Mazo57 who calculated n-alkanes’ diffusion constants using a Monte Carlo generation of conformers for chains with 1.54 Å between elements (beads). They found agreement with the experiment for the n-alkanes in CCl4 with an element radius a = 0.77 Å but noted that different values of a would be needed in other solvents (see Figure 1 of ref 57). They also stated that a was indicative of solute−solvent interactions and should not be interpreted as a molecular radius. Similar conclusions can be drawn from the work of Garcia de la Torre and coworkers,58−60 who carried out calculations of diffusion constants for n-alkanes with a more complete type of Monte Carlo conformation averaging for their bead model than Paul and Mazo.57 In ref 24, we found that the MSE fits for 26 solutes in some or all of the n-alkanes n-C6 − n-C16 predicted their D values in squalane, a factor of ∼10 more viscous than n-C16, with an average difference of 12.5%; none were larger than 25%. We then suggested that if uncertainties of ≈20% were tolerable, the MSE fits for solutes in the n-alkanes and squalane could be used to estimate their diffusion constants for viscosities outside the n-C6 − squalane range (0.30−30 cP, Table 3). The D values for benzene and toluene in CO2 provided a successful test of this suggestion; the viscosities of the five solutions mentioned in Table 4 are less than that of n-C6 by factors of 2.9−6.3. The MSE fits in the n-alkanes and squalane24 gave calculated D values that differed from the experiment by averages of only 5.0% for benzene and 4.2% for toluene. We then fitted the D values for benzene, ethylbenzene, 1C5Ph, and 1-C8Ph in n-C9, n-C10, n-C12, n-C15, HPMN, pristane, squalane, and CO2 to eq 7; these were the only solutes for which data were available in all of the eight solvents. The

Figure 3. Comparison of the experimental diffusion constants (in cm2 s−1) for the alkylbenzene solutes (without benzene, toluene, and 1C17Ph as solutes) with the predictions of lollipop diffusion for σ1/σ2 = 2.17. E

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The Journal of Physical Chemistry B plots of log(D/T) versus log η for benzene and 1-C8Ph are shown in Figure 4. The fits are generally good with values of R2

As suggested in ref 1 we have determined new D values in the more viscous HPMN (3.44 cP), pristane (6.61 cP), and squalane (30.3 cP), but even larger values have been reported for micelles (50 < η < 190 cP63,64). It may be possible to estimate some of these larger values, given the agreement found in the previous section when calculating diffusion constants outside the range of MSE fits. Precautions must be taken, however, even if D values are known with a reasonable certainty. They include34,57,65 separating the probe’s internal diffusion from that of the micelle as a whole, determining the probe’s location in the micelle, which may not be hydrophobic, and realizing that micellar viscosities have been found to be probe- and methoddependent. We also noted1 that our results could be of use in chromatography. A column’s reduced plate height, h, is a measure of its efficiency and depends on the mobile and stationary phases.35,36,66−69 Alkylbenzenes’ diffusion constants have been used in determinations of h,67 and some of the solvents mentioned in this paper, supercritical CO2 in particular, have been used as mobile phases.66 Alkylbenzenes’ D values also have been used in the determination of relations between a solute’s diffusion in solution and on a stationary phase.70

Figure 4. Plots of log(D/T) vs log η for benzene (upper fit) and 1C8Ph (lower fit) in n-C9, n-C10, n-C12, n-C15, HPMN, pristane, squalane, and supercritical CO2. The units of D are cm2 s−1.

≥ 0.996. The values of p and log ASE are given in Table 7. The p values show a small increase from benzene to 1-C8Ph, as would be expected as the solute size increases incrementally in a common group of solvents.23,24,37,51

4. SUMMARY AND CONCLUSIONS The diffusion constants of benzene and a series of alkylbenzenes have been determined at room temperature in n-C15, HPMN, pristane, and squalane using a capillary flow method. They have been compared with the predictions of Hansen’s cylinder diffusion model,4 as have the D values for (a) benzene and alkylbenzenes in n-C9, n-C10, n-C12, and supercritical CO2 and (b) n-alkanes and 1-alkenes in n-C6, nC7, n-C8, benzene, and toluene. The diffusion constants for benzene and the alkylbenzenes also have been compared with D values calculated using Garcia de la Torre and Bloomfield’s lollipop diffusion model.5 Both models give overall differences of less than 4% between the experimental and calculated diffusion constants when benzene and toluene are omitted as solutes. The decreased degree of agreement for these two solutes is likely due to the apparent differences in their shapes and the shapes assumed by the models. The agreement found using the cylinder model indicates that the diffusion of the alkylbenzenes and 1-alkenes is similar to that of the n-alkanes and that they, like the n-alkanes, are relatively extended in these solutions. The alkylbenzenes’ agreement with the lollipop model is consistent with this suggestion. The viscosities for the solvents used in this study vary by a factor of more than 600, and the D values for benzene, ethylbenzene, 1-C5Ph, and 1-C8Ph in n-C9, n-C10, n-C12, n-C15, HPMN, pristane, squalane, and CO2 were fitted to the modified Stokes−Einstein (MSE) relation, D/T = ASE/ηp. The quality of these fits was good with values of R2 ≥ 0.996.

Table 7. Values of p and ASE for Benzene and Alkylbenzenes in n-Alkanes, HPMN, Pristane, Squalane, and Supercritical CO2 p

solute C6H6/C6D6 ethylbenzene 1-phenylpentane 1-phenyloctane

0.668 0.699 0.700 0.721

± ± ± ±

0.0063 0.0041 0.0037 0.0039

−log ASE

R2

± ± ± ±

0.999 0.998 0.996 0.996

8.508 8.638 8.758 8.882

0.015 0.011 0.010 0.010

Even better agreement could have been achieved by making separate fits to the data in the alkanes and CO2 but, as for the cylinder and lollipop models, we were interested in fitting D values over the largest range of viscosities while keeping the number of parameters to a minimum. 3.5. Applications. Binary mixtures of alkylbenzenes and nalkanes similar to ours are of interest because they are surrogates for petroleum- and bio-based fuels.31,39,61,62 The physical properties and combustion reactions of these solutions are being modeled in conjunction with an ongoing experimental program. The viscosities, surface tensions, densities, and bulk moduli have been determined for (a) nC16 with each of 1-C6Ph, 1-C8Ph, and 1-C12Ph31 and (b) 1C4Ph with each of n-C10, n-C12, n-C14, n-C16, and n-C17.61 Our diffusion constants and their viscosity dependence would provide additional tests of the modeling methods. The two other areas in which our results may be helpful were mentioned in ref 1 and are revisited here. We noted1 that benzene and alkylbenzenes have been solubilized in micelles32 and, if determined, their D values could be compared with ours to estimate the micelles’ internal viscosities.33,34 The smallest micellar viscosities, however, are typically 8−20 cP,33,63 and the largest viscosity in ref 1 was only 2.57 cP for n-C15.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 314-977-2837. Fax: 314977-2521. ORCID

Bruce A. Kowert: 0000-0002-4030-8670 F

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The Journal of Physical Chemistry B Notes

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The author declares no competing financial interest.

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ACKNOWLEDGMENTS The Department of Chemistry, Saint Louis University, has supported this research. The data acquisition system and detector were purchased with grants to Dr. Barry Hogan from Research Corporation and the donors of the Petroleum Research Fund, administered by the American Chemical Society. We thank Dr. Ryan McCulla for C6D6 and C6H6 and Dr. Michael Lewis for ethylbenzene, 1-C4Ph, 1-C5Ph, 1-C6Ph, and 1-C8Ph. Paul Register is thanked for his help in taking the data for 1-C11Ph in pristane and 1-C6Ph in n-C15.

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