Analysis of Surface Diffusion Phenomena in Reversed-Phase Liquid

probably from differences in Dm. It also seems that Ds tends toward Dm with decreasing retention factor. The surface diffusion mechanism was assumed t...
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Anal. Chem. 1999, 71, 889-896

Analysis of Surface Diffusion Phenomena in Reversed-Phase Liquid Chromatography Kanji Miyabe and Georges Guiochon*

Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Chemical and Analytical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

Surface diffusion data obtained for a reversed-phase liquid chromatographic system (octadecylsilyl silica gel and a 70/30 (v/v) methanol/water solution) were analyzed in relation to their molecular diffusivity (Dm). The adsorbateadsorbent interactions between the studied compounds and the stationary phase were taken into account. The surface diffusion coefficient (Ds) depends on the mobilephase composition, especially on the nature and concentration of the organic modifiers. Differences between the values of Ds measured under various conditions stem probably from differences in Dm. It also seems that Ds tends toward Dm with decreasing retention factor. The surface diffusion mechanism was assumed to be a restricted molecular diffusion in a potential field of adsorption. A restriction energy for this diffusion (Er) was introduced to correlate Ds with Dm. The ratio of Er to the isosteric heat of adsorption (-Qst) was found to be nearly constant, irrespective of the retention factor, with an average value of 0.32 for our phase system. An estimation procedure for Ds using the enthalpy-entropy compensation effect for the adsorption equilibrium is proposed. From the adsorption equilibrium constant at 298 K only, Ds could be estimated at different temperatures with an error less than ∼50%. Mass-transfer phenomena in porous adsorbents have been abundantly studied, whether in gas or liquid phase. It was established that surface diffusion plays an important role for intraparticle diffusion in many cases.1 So far, the dependence of the surface diffusion coefficient (Ds) on the temperature and the amount adsorbed (q) is the main feature of surface diffusion to have been studied. The temperature dependence of Ds is usually analyzed by applying the Arrhenius equation. On the other hand, Gilliland et al. interpreted the concentration dependence of Ds by taking into account the change in the heat of adsorption due to a change of q in gaseous adsorption.2 As a simple approximation, the activation energy of the surface diffusion (Es) was assumed to be a fraction of the heat of adsorption. Sladek et al. reinterpreted many experimental data in a wide range, from physisorption to chemisorption, on the basis of this idea.3 (1) Kapoor, A.; Yang, R. T.; Wong, C., Catal. Rev.-Sci. Eng., 1989, 31, 129. (2) Gilliland, E. R.; Baddour, R. F.; Perkinson, G. P.; Sladek, K. J. Ind. Eng. Chem. Fundam. 1974, 13, 95. (3) Sladek, K. J.; Gilliland, E. R.; Baddour, R. F. Ind. Eng. Chem. Fundam. 1974, 13, 100. 10.1021/ac9809027 CCC: $18.00 Published on Web 01/07/1999

© 1999 American Chemical Society

Empirical concepts were also applied to the interpretation of the concentration dependence of Ds in liquid-phase adsorption, using activated carbons and macroreticular adsorbents. Komiyama and Smith correlated Ds to the interaction energy between adsorbate molecules and adsorbent surface.4,5 Suzuki and Fujii determined Ds by means of steady-state experiments for a Freundlich isotherm system. A strong concentration dependence of Ds was explained in terms of the change in the heat of adsorption resulting from the change in the surface coverage of propionic acid on an activated carbon.6 Itaya et al. studied the aqueous adsorption of phenol derivatives onto some macroreticular adsorbents.7 They determined Ds by the finite bath adsorption experiments, assuming a constant value of Ds. The concentration dependence of Ds was interpreted from a linear correlation between Es and the isosteric heat of adsorption (Qst). By applying the concepts of molecular diffusion to surface diffusion phenomena, they suggested that the slope and intercept of the linear relation correspond to the contributions of a jumping step and a hole-making step, respectively. Muraki et al. also reported a dependence of Ds on q in the case of the finite bath aqueous adsorption of benzene derivatives onto an activated carbon.8 They reported that the ratio of Es to the adsorption energy was nearly equal to 0.5 even in the liquid-phase adsorption. Miyahara and Okazaki measured Ds for benzene derivatives in aqueous adsorption onto an activated carbon by batch kinetic experiments.9,10 The concentration dependency of Ds was interpreted by applying the Eyring rate theory. They assumed that the rate-controlling step was that of making a hole in the potential field of adsorption. The value of Es was regarded as a constant fraction of the sum of the evaporative energy of the adsorbates and the adsorption potential. As described above, the surface migration mechanism in liquid-solid adsorption was most frequently studied from the point of view of the dependence of Ds on q and from correlations of Ds with the adsorption energy of the adsorbates. The concentration dependence of Ds has been interpreted by taking into account the change in the heat of adsorption resulting from the variation of q or by considering the logarithmic slope of an adsorption isotherm (d ln c/d ln q). On the basis of these results, detailed (4) Komiyama, H.; Smith, J. M. AIChE J. 1974, 20, 728. (5) Komiyama, H.; Smith, J. M. AIChE J. 1974, 20, 1110. (6) Suzuki, M.; Fujii, T. AIChE J. 1982, 28, 380. (7) Itaya, A.; Fujita, Y.; Kato, N.; Okamoto, K. J. Chem. Eng. Jpn. 1987, 20, 638. (8) Muraki, M.; Iwashita, Y.; Hayakawa, T. J. Chem. Eng. Jpn. 1982, 15, 34. (9) Miyahara, M.; Okazaki, M. J. Chem. Eng. Jpn. 1992, 25, 408. (10) Miyahara, M.; Okazaki, M. J. Chem. Eng. Jpn. 1993, 26, 510.

Analytical Chemistry, Vol. 71, No. 4, February 15, 1999 889

models were proposed to describe specifically the surface diffusion mechanism.1,11 Some studies of surface and lateral diffusion phenomena were made in reversed-phase liquid chromatography (RPLC).12-21 Bogar et al. estimated the lateral diffusion coefficient of pyrene and the microviscosity of a C18 RP by means of fluorescence measurements.12 They reported that the solvated RP was a dynamic medium in which solutes could be dissolved. Stahlberg et al. measured the mobility of pyrene on RP packing materials by a fluorescence method.13 They estimated an apparent activation energy for the diffusion of pyrene on RP-18 by analyzing its temperature dependence. It was suggested that the surface of RP18 had liquidlike properties. Harris et al. measured the surface diffusion coefficients of iodine and rubrene.14,15 They reported that the surface diffusion coefficients were 2-3 orders of magnitude smaller than the molecular diffusivities in the bulk solution and that they increased with increasing methanol concentration in the aqueous mobile phase. Zulli et al. measured the lateral diffusion coefficient of acridine orange at the water-C18 interface and compared this coefficient with the diffusivity in bulk water.16 Miyabe et al. determined Ds by the pulse response method and moment analysis and reported the predominance of surface diffusion for intraparticle migration in RP packing materials, in gas and liquid-phase systems.17-21 They determined some thermodynamic properties of surface diffusion by analyzing the temperature dependence of Ds and showed that Ds increased with increasing q in RPLC. They interpreted the positive concentration dependence of Ds in terms of the chemical potential driving force model. The intraparticle migration of the sample substances should be studied from various points of view in order to elucidate the separation mechanism of RPLC in detail. In this paper, we discuss some characteristic features of surface diffusion by analyzing correlations between Ds and either the diffusivity (Dm) or the retention factor. The surface diffusion was regarded as a restricted molecular diffusion in the potential field of adsorption. The surface diffusion data in a RPLC system were analyzed on the basis of this model. The development of a simple procedure for the prediction of Ds was also attempted. MOMENT ANALYSIS Chromatographic peak profiles were analyzed by the conventional method of moments.11,22 Information on the adsorption equilibrium and the mass-transfer kinetics was obtained from the first and second moments, respectively. The first absolute moment (µ1) and the second central moment (µ2′) of the chromatographic peaks are expressed as follows (11) Suzuki, M. Adsorption Engineering; Kodansha/Elsevier: Tokyo/Amsterdam, 1990. (12) Bogar, R. G.; Thomas, J. C.; Callis, J. B. Anal. Chem. 1984, 56, 1080. (13) Stahlberg, J.; Almgren, M.; Alsins, J. Anal. Chem. 1988, 60, 2487. (14) Wong, A. L.; Harris, J. M. J. Phys. Chem. 1991, 95, 5895. (15) Hansen, R. L.; Harris, J. M. Anal. Chem. 1995, 67, 492. (16) Zulli, S. L.; Kovaleski, J. M.; Zhu, X. R.; Harris, J. M.; Wirth, M. J. Anal. Chem. 1994, 66, 1708. (17) Miyabe, K.; Suzuki, M. AIChE J. 1992, 38, 901. (18) Miyabe, K.; Suzuki, M. AIChE J. 1993, 39, 1791. (19) Miyabe, K.; Suzuki, M. J. Chem. Eng. Jpn. 1994, 27, 785. (20) Miyabe, K.; Suzuki, M. AIChE J. 1995, 41, 548. (21) Miyabe, K.; Takeuchi, S. Anal. Chem. 1997, 69, 2567. (22) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, 1994.

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µ1 )

∫C(t)t dt/∫C(t) dt ) (z/u )δ 0

∫C(t)(t - µ ) µ′) ∫C(t) dt 1

2

2

dt

)

0

2z (δ + δf + δd) u0 ax

(1)

(2)

δ0 )  + (1 - ) (p + FpK)

(3)

δax ) (Ez/u02)δ20

(4)

δf ) (1 - ) (R/3kf)(p + FpK)2

(5)

δd ) (1 - ) (R2/15De)(p + FpK)2

(6)

The first moment was analyzed using the following equation, derived from eq 1

(µ1 - t0)/(1 - ) ) (z/u0)FpK

(7)

t0 ) (z/u0)[ + (1 - )p]

(8)

The plot of (µ 1 - t 0)/(1 - ) versus z/u0 is linear. From its slope, the adsorption equilibrium constant (K) was calculated. For the analysis of the second moment, a parameter H was calculated as follows.

H)

µ′2 z Ez ) 2 + H0 2 u µ 0 u 1

(9)

0

H0 ) δf/δ20 + δd/δ20

(10)

The intraparticle diffusivity (De) and the axial dispersion coefficient (Ez) can be determined from µ2′ by subtracting the contribution of the fluid-to-particle mass transfer to the peak spreading. The value of Ds was calculated by assuming the parallel contribution of the surface and the pore diffusion:

De ) Dp + FpKDs

(11)

According to the parallel pore model, Dp was calculated from Dm by the following equation.

Dp ) (p/k2)Dm

(12)

Details of the moment analysis of chromatographic peaks were previously described.17-20 The value of Ds are determined with a probable error of about a few percent.23-26 EXPERIMENTAL SECTION Pulse response experiments were carried out with HPLC equipment (LC-6A, Shimadzu), following the conventional procedures of elution chromatography. A small amount of the sample solution (pulse) was introduced into the column. The correspond(23) Miyabe, (24) Miyabe, (25) Miyabe, (26) Miyabe,

K.; K.; K.; K.;

Suzuki, M. Ind. Eng. Chem. Res. 1994, 33, 1792. Takeuchi, S. Ind. Eng. Chem. Res. 1997, 36, 4335. Takeuchi, S. AIChE J. 1997, 43, 2997. Takeuchi, S. Ind. Eng. Chem. Res. 1998, 37, 1154.

Table 1. Properties of the RP Columns Used and Experimental Conditions column av particle diam, dp (µm) particle density, Fp (g cm-3) porosity, p (-)

TMS 45

C4 45

C8 45

ODSa 45

ODSa 45

0.74

0.73

0.75

0.86

0.81

0.62

0.61

0.55

mass of adsorbent (g) column size (mm) void fraction,  (-) tortuosity factor, k2 (-) column temp (K) mobile phase

1.9

1.8

0.40(80), 0.37(70), 0.35(60,40) 2.1

0.44 4.1

0.42 4.6

1.8 6 × 150 0.42 4.4

0.46(100,80,70,60), 0.47(40) 2.1 0.43 4.5

0.39 5.5

methanol/water 70/30 (vol)

methanol/water 70/30 (vol)

methanol/water 80/20-40/60 (vol)

acetonitrile/water 80/20-40/60 (vol)

superficial veloc, u0 (cm s-1) sample substances a

288-308 methanol/water 70/30 (vol) 0.06-0.12

benzene derivatives, phenol derivatives, n-alcohols, n-alkanes, condensed aromatic hydrocarbons

The numbers in parentheses represent the volumetric fraction of the organic modifiers in the mobile phases.

ing elution peak was recorded, and its first two moments were calculated. The properties of the several commercial columns (YMC) used are reported in Table 1. Four kinds of stationary phases, having various alkyl ligands, were used. The number of carbon atoms in the alkyl chains chemically bonded to the surface of silica gel was changed stepwise between 1 and 18 (TMS, C4, C8, ODS). The mobile phases were mixtures of methanol/water and acetonitrile/water of various compositions. Several organic compounds were used as the sample substances. Uracil was used as the unretained compound. The experimental conditions are also listed in Table 1. The pulse response experiments were made at zero surface coverage of the sample substances, different column temperatures (288-308 K), and superficial velocity of the mobile phase (0.059-0.118 cm s-1). Measurements of the chromatographic peaks were carried out by introducing the small concentration perturbation pulses into the fluid flow using a conventional sampling device. RESULTS AND DISCUSSION Correlation between Ds and Dm. In previous papers, the surface migration of adsorbate molecules on an octadecylsilyl (ODS)-silica gel was regarded as a tracer diffusion in n-octadecane, as a first approximation.18,20,23 The influence of the mobile-phase composition on the surface diffusion was not taken into account. Chen and Chen proposed the following equation for the prediction of the tracer diffusivity in binary systems involving long-chain hydrocarbons27 1/3 109Da,svηsv/TV2/3 b,sv ) (11.96/Vb,a ) - 0.8796

(13)

The values of Da,sv derived from eq 13, i.e., of hypothetical Ds, did not sufficiently agree with the experimental data. The discrepancy between experimental data and calculated values of Ds was larger in liquid-phase adsorption than in gas-phase systems.20,23 It was suggested that the mobile-phase solvents had some influence on the surface diffusion. (27) Chen, H. C.; Chen, S. H. Ind. Eng. Chem. Fundam. 1985, 24, 183.

Figure 1. Comparison between surface diffusion coefficients and molecular diffusivities in methanol/water and acetonitrile/water mobilephase systems.

The influence of the nature and concentration of the organic modifier on the surface diffusion in RPLC was studied. Figure 1 illustrates the correlation between Ds in the two RPLC systems studied (methanol/water and acetonitrile/water) and their composition (φ). Although the experimental data were somewhat scattered, the ratio of the values of Ds in acetonitrile/water to those in methanol/water was on the average equal to 2.14, the slope of the straight line in Figure 1. The trend of these plots suggests that the ratio of the two Ds values is almost constant, irrespective of the nature of the compound studied and of φ. The results in Figure 1 indicate that the surface diffusion depends on the nature of the organic modifier and on φ. As a second approximation, the surface diffusion on the ODS-silica gel particles could be assumed to be the tracer diffusion of the sample molecules in the actual stationary phase, probably a solution of the octadecyl ligands in a mixture of the mobile-phase solvents. The important role of the mobile-phase solvents on the adsorption phenomena was already shown by the extensive studies made on adsorption equilibrium Analytical Chemistry, Vol. 71, No. 4, February 15, 1999

891

Figure 2. Surface diffusion coefficient as a function of the adsorption equilibrium constant.

in RPLC. It was reported that the dependence of the retention on the mobile-phase composition in RPLC could be explained by considering a participation of its components in the stationary phase.28 In this paper, the important role of the mobile-phase solvents on the mass-transfer kinetics at the surface of RP packing materials is substantiated. Figure 1 also shows the correlation of the Dm values in both mobile phases. The Wilke-Chang equation was used for the estimation of Dm in methanol/water.29,30 The value of Dm in acetonitrile/water was calculated from the Perkins-Geankopolis, Scheibel, and Hayduk-Laudie equations, because an association coefficient in the Wilke-Chang equation has not been proposed for acetonitrile.29 The values of Dm for the compounds studied are about twice as large in acetonitrile/water mixtures than in methanol/water solutions. The average value of the ratios Dm(CH3CN/H2O)/Dm(CH3OH/H2O) is 2.07, the slope of the dashed straight line in Figure 1. The slopes of the two straight lines, the one for the Dm values and the one for the Ds values in the same mobile phases, are very close. So, we may conclude that the difference between the surface diffusion coefficients Ds in the two mobile-phase systems results from the differences in the diffusivities, Dm. The results in Figure 1 suggest a correlation between Ds and Dm. Restricted Molecular Diffusion Model for Surface Diffusion. The influence of the alkyl chain length of the RP packing materials on the adsorption characteristics in a RPLC system was reported in a previous paper.19 It was confirmed that Ds increased with decreasing alkyl chain length of the ligand. By contrast, K decreased with decreasing alkyl chain length. By combining these two relationships, Ds was plotted against K in Figure 2. The decrease in K accompanies an increase in Ds. The extrapolated values of Ds at K ) 0 are nearly equal to the corresponding values (28) Tanaka, N.; Kimata, K.; Hosoya, K.; Miyanishi, H.; Araki, T. J. Chromatogr. 1993, 656, 265. (29) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977; Chapter 11. (30) Treybal, R. E. Mass-Transfer Operations; McGraw-Hill: New York, 1980.

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of Dm, indicated by the horizontal arrows in Figure 2. Although the extrapolation is not exact, this result suggests that Ds is approximately equal to Dm when K is negligibly small. We consider here surface diffusion as a molecular diffusion process restricted in the potential field of adsorption. Compared with the diffusive migration of the solute in the case of molecular diffusion in the bulk solvent, the migration of the solute molecule diffusing along the adsorbent surface is restricted by the additional interactions between the solute molecule and the surface of the stationary phase. The extent of adsorptive interactions is represented by K. There is no additional restriction at K ) 0. So, Ds should approach Dm when K decreases toward 0. Equation 11 suggests that De tends toward Dp when K tends toward zero. Although Dp is related to Dm as shown in eq 12, eq 11 provides no information about the correlation between Ds and Dm. As shown in eq 11, De is not equal to Ds. The influence of the coefficient FpK must be considered. Even if it is assumed that De is almost equal to Ds, Ds at K ) 0 is probably 1 order of magnitude smaller than Dm in such a case, because the ratio p/k2 is nearly equal to 0.1. Figure 2 shows that Ds tends toward Dm when K tends toward zero, regardless of the adsorptivity of the test compound. A change in the alkyl chain length brings about a change in the hydrophobic selectivity of the RP packing materials. The surface diffusion mechanism, however, may remain the same, irrespective of the alkyl chain length. The results in Figure 2 suggest that Ds is almost equal to Dm under the limiting condition that the interaction between the compound studied and the stationary-phase surface is negligibly small. The diffusivity of the molecules in the potential field of adsorption may be restricted by adsorbate-adsorbent interactions. However, from the results in Figures 1 and 2, the surface diffusion seems to be well approximated by molecular diffusion restricted by adsorption interactions. Satterfield et al. suggested that the restricted diffusion of a solute in the fine pores of an adsorbent is due to both a hydrodynamic effect and the adsorption of the adsorbate.31 They showed that the logarithm of the ratio of the effective diffusivity of the solute in the pores to the bulk diffusivity decreases linearly with increasing ratio of the solute molecular diameter to the pore diameter and that adsorption phenomena further reduce the effective diffusivity. However, they did not make a detailed analysis of the surface diffusion on the basis of a restricted diffusion model. In previous papers,25,32 the restricted molecular diffusion model for surface diffusion was formulated by applying the absolute rate theory. The value of Es was divided into the contributions of a hole-making step and a jumping step. The former was correlated with the evaporative energy of the solvent, not of the adsorbate, and the latter with the isosteric heat of adsorption, Qst. The validity of this concept was proven by applying the model to a quantitative analysis of surface diffusion data in different liquid-solid adsorption systems previously published. A consistent interpretation was provided for both the temperature and the concentration dependence of Ds in the different systems studied, exhibiting Langmuir-, Freundlich-, and Jossens-type adsorption. Irrespective of the isotherm model, the dependence of Ds on q could be interpreted by taking into account the variation of both the logarithmic slope (31) Satterfield, C. N.; Colton, C. K.; Wayne, H. P., Jr. AIChE J. 1973, 19, 628. (32) Miyabe, K.; Takeuchi, S. J. Phys. Chem. B 1997, 101, 7773.

of the adsorption isotherm (d ln c/d ln q) and the adsorption potential (Eap).25 It was also demonstrated that the correlations between Es and Qst could be quantitatively explained by the model.32 Assuming that molecular diffusion is restrained according to the intensity of the adsorbate-adsorbent interactions, the following correlation should be obtained.

Ds ) rDm

(14)

where r is the empirical restriction factor, which may be formulated as a function of the intensity of the interactions. Surface diffusion on the surface of the stationary phase and molecular diffusion in the bulk solvent are usually regarded as activated processes. By applying the Arrhenius equation, both Ds and Dm are represented as follows.

Ds ) Ds,0 exp(-(Es/RgT))

(15)

Dm ) Dm,0 exp(-(Em/RgT))

(16)

Figure 3. Temperature dependence of the ratio of the surface diffusion coefficient to the molecular diffusivity. Symbols as in Figure 2.

between surface diffusion and the intensity of adsorbentadsorbate interactions. A different formulation may be required in a detailed study of the mechanism of surface diffusion. Assuming that the restriction energy (Er) for molecular diffusion is due to adsorbate-adsorbent interactions, the following equation may be postulated

Ds ) Dm exp(- (Er/RgT)) As proposed by Gilliland et al., the following equation has been conventionally used for representing the correlation between Es and Qst.2

Es ) R(-Qst)

(17)

where R is also an empirical parameter, usually smaller than unity for surface diffusion.11 The combination of eqs 15 and 17 provides the following ordinary equation.

Ds ) Ds,0 exp[- R(-Qst)/RgT]

(18)

Equation 18 suggests that Ds approaches Ds,0 when the adsorbateadsorbent interactions become weak. The values of Ds,0 were reported to range between approximately 10-4 and 10-1 cm2 s-1 in different liquid-solid adsorption systems.4-6,17,19,20,23,33-35 This would suggest that Ds is several orders of magnitude larger than Dm at Qst ) 0 because Dm is usually estimated to be of the order of 10-5 cm2 s-1.29,30 This conclusion seems to be unreasonable. Surface diffusion is a mass-transfer process which is slowed by the potential field of adsorption in the vicinity of the surface of the stationary phase. The mobility of the solute molecules is certainly restricted by the interactions taking place between these molecules and the adsorbent surface. On the other hand, when molecular diffusion takes place in the bulk solvent, the solute molecules migrate free of the influence of the adsorptive interactions. The only interactions that take place in this case are between the solute and the solvent molecules. As illustrated in Figure 2, Ds is usually smaller than Dm under ordinary chromatographic conditions, because of the additional hindrance to migration caused by the adsorption interactions between the solute molecules and the surface. Even when the influence of these interactions is small, it may not be expected that Ds is exceptionally larger than Dm. Thus, eq 18 may not be applicable to the analysis of surface diffusion phenomena when -Qst is relatively small. This drawback of eq 18 may arise from a lack of studies on the relation

(19)

The value of Er is the increment of the activation energy of molecular diffusion when the molecules migrate in the potential field of adsorption. The following equations are obtained by combining eqs 14-16 and 19.

r)

( )

[

]

Er Ds,0 Ds Em - Es ) exp ) exp Dm RgT Dm,0 RgT Er ) Es - Em - RgT ln(Ds,0/Dm,0)

(20) (21)

The value of Er is related to both the activation energy and the frequency factors of surface and molecular diffusion. When the ratio Ds,0/Dm,0 is unity, Er is identical to the difference between Es and Em. According to eq 20, a semilogarithmic plot of the ratio Ds/Dm against 1/T is shown in Figure 3. Linear correlations are observed for each adsorbate. The intercept, ln(Ds,0/Dm,0) at 1/T ) 0, ranges between -1.9 and 1.2, indicating that Ds,0 and Dm,0 are of the same order of magnitude. The slope of the plots is nearly equal to zero or slightly negative. This slope is proportional to Em - Es according to eq 20. This result demonstrates that Es is slightly larger than or nearly equal to Em if the values of Ds,0, Dm,0, Es, and Em are all constant in the temperature range of this study. This slope decreases with increasing molecular size of the compound. We conclude that the degree of restriction of surface diffusion is related to the intensity of the adsorbate-adsorbent interactions, which is usually expressed by the heat of adsorption. A correlation between Er and Qst may be predicted. Correlation between Er and Qst. Figure 4 shows that there is a linear correlation between Er and -Qst for the various compounds and RPLC system studied here:

Er ) γ(-Qst) Analytical Chemistry, Vol. 71, No. 4, February 15, 1999

(22) 893

Figure 5. Typical van’t Hoff plots of the adsorption equilibrium constant. Symbols as in Figure 4.

be derived from eqs 19 and 22. The subject of the prediction of Ds is now replaced by that of estimating Qst. Estimation of Qst. It was reported that enthalpy-entropy compensation effects take place for surface diffusion in various RPLC systems.23 Enthalpy-entropy compensation has also been reported for adsorption equilibrium.36-38 The value of K is related to ∆H° and ∆S° as follows.

ln KT ) Figure 4. Correlation between the restriction energy for molecular diffusion and the isosteric heat of adsorption for various compounds in a RPLC system.

The average value of γ is ∼0.32 for RPLC systems. Compared to bulk solution molecular diffusivity, the mobility of sample molecules on an adsorbent surface is restricted by the adsorbateadsorbent interactions. The intensity of this restriction is equal to one-third of -Qst, irrespective of the retention factor of the compound. The values of Qst were derived from the slope of the linear correlation between ln K and 1/T, according to the van’t Hoff equation.

K ) K0 exp(- (Qst/RgT))

(23)

Several typical plots are shown in Figure 5. Each data point in the figure gives the average value of K derived from six successive measurements of the retention volume. The relative standard deviations of these data are between 0.14 and 0.56%, indicating that the experimental determination of K was made with a sufficient precision. The values of -Qst were calculated by linear regression of these data. In most cases, the standard deviation of the slope of the linear plots of ln K versus 1/T was smaller than about ∼(5%. These errors are small and should not affect significantly the conclusions of this study. As shown in eq 19, Ds can be estimated from Dm and Er. A number of equations have already been proposed for the prediction of Dm.22,29,30 The value of Er can be empirically derived from eq 22, taking γ ) 0.32 for RPLC systems (with ODS-silica gel and 70/30 (v/v) methanol/water). If Qst can be estimated, Ds can 894

Analytical Chemistry, Vol. 71, No. 4, February 15, 1999

∆H0 ∆S0 ∆G0 )+ RgT RgT Rg

(24)

The compensation may be conveniently expressed by the following relationship.

∆H0 ) β∆S0 + ∆G0β

(25)

The following equation is derived by substituting eq 24 into eq 25.

ln KT ) -

∆G0β ∆H0 1 1 - Rg T β Rgβ

(

)

(26)

The compensation temperature (β) was reported to be in the range 500-1000 K. Similarly, K at 298 K was plotted against -Qst measured around 298 K. Figure 6 illustrates the results obtained for the different compounds studied. Linear correlations are observed for all homologous compounds. From the slope of these linear correlations, the following compensation temperatures are calculated: 700 (n-alkane), 680 (condensed aromatic hydrocarbon), 1150 (n-alkylbenzene), 820 (n-alcohol), and 1520 K (p-nalkylphenol). These values are similar to other values of β in the literature. Parallel correlations were observed between n-alkane and n-alcohol, and between n-alkylbenzene and p-n-alkylphenol, (33) Awum, F.; Narayan, S.; Ruthven, D. Ind. Eng. Chem. Res. 1988, 27, 1510. (34) Ma, Y. H.; Lin, Y. S.; Fleming, H. L. AIChE Symp. Ser. 1988, 84, 1. (35) Suzuki, M.; Kawazoe, K. J. Chem. Eng. Jpn. 1975, 8, 379. (36) Melander, W. R.; Campbell, D. E.; Horvath, C. J. Chromatogr. 1978, 158, 215. (37) Melander, W. R.; Chen, B. K.; Horvath, C. J. Chromatogr. 1979, 185, 99. (38) Woodburn, K. B.; Lee, L. S.; Rao, P. S. C.; Delfino, J. J. Environ. Sci. Technol. 1989, 23, 407.

Figure 7. Comparison of the surface diffusion coefficients calculated by the correlation and experimentally measured. Symbols as in Figure 4.

Figure 6. Correlation between the adsorption equilibrium constant and the isosteric heat of adsorption for various compounds in a RPLC system. Symbols as in Figure 4. Table 2. Correlation Results and Estimation of the Surface Diffusion Coefficient sample substance

Qst vs K

accuracya

n-alkane condensed aromatic hydrocarbon n-alkylbenzene n-alcohol p-n-alkylphenol

-Qst ) 4.3(ln K) + 0.05 -Qst ) 4.4(ln K) + 3.2 -Qst ) 3.4(ln K) + 4.0 -Qst ) 3.9(ln K) + 6.7 -Qst ) 3.1(ln K) + 8.8

0.21 0.24 0.14 0.17

a

Mean square deviation ) [(1/N)∑(Ds,exp - Ds,cal)2/Ds,exp2]1/2.

indicating that the incremental contribution of a methylene group to either K or Qst is identical in these two combinations of homologous compounds. The correlations between K and Qst are summarized in Table 2. Figure 6 indicates that Qst can be estimated from K at 298 K under RPLC conditions. When Qst is estimated, Ds can be calculated using eqs 19 and 22. Estimation of Ds. On the basis of the results in Figures 4 and 6, Ds can be estimated from K. Figure 7 compares the values of Ds so estimated and the experimental data. The values of Ds at 288, 298, and 308 K were calculated from only one data point of K, at 298 K. First, Qst was calculated from K by applying the correlations in Figure 6, although the value of Qst can be accurately determined from the K values measured at different temperatures according to eq 23. Then Er was derived from the Qst values thus obtained, with γ ) 0.32 (eq 22). According to eq 19, Ds was calculated from the values of Er and Dm, the latter separately estimated by the Wilke-Chang equation. Figure 7 indicates that, in the temperature range (288-308 K) investigated, Ds can be estimated from K at 298 K with an error that is less than ∼50%. Table 2 lists the mean square deviation for each adsorbate. Overall, the mean square deviation is ∼0.24.

A few papers have already been reported for the prediction of Ds. Sladek et al. proposed the use of a simple method for gassolid systems.3 However, only the order of magnitude of Ds can be estimated by this method. Tamon et al. also proposed an estimation procedure of Ds in a gas-solid system, on the basis of the hopping model.39 On the other hand, to the best of our knowledge, there is only one paper discussing the estimation of Ds in liquid-solid system. Suzuki and Kawazoe proposed an Arrhenius-type equation based on Trouton’s rule for aqueous adsorption of some volatile organic compounds from an aqueous solution onto an activated carbon.35 Few estimation procedures have been proposed for Ds in RPLC systems. The fact that an important mass-transfer kinetic parameters, Ds, could be connected with the adsorption equilibrium parameter, K, is a major result. In the field of RPLC, a great number of studies have been made on the retention behavior. Some theoretical and empirical procedures have already been reported for the estimation of K. Figure 7 shows that it is possible to apply the results of conventional studies concerning adsorption equilibrium to the estimation of Ds. CONCLUSIONS Experimental results demonstrate a correlation between Ds and Dm under various experimental conditions, at least in RPLC. The surface diffusion tends toward Dm when the retention, i.e., the adsorbate-adsorbent interactions, becomes negligibly small. Because the surface of alkyl-bonded silica gels has a low surface energy and the adsorption layer can be considered as a solution of the adsorbate in a mixture of alkanes and the components of the mobile phase, surface diffusion can be regarded as molecular diffusion restricted by the potential field of adsorption. The restriction energy is simply correlated with the latent heat of adsorption, Qst. The average ratio γ ) Er/(-Qst) is equal to 0.32 in RPLC (ODS-silicagel and 70/30 (v/v) methanol), irrespective of the retention factor of the compound. It would be interesting to see how this correlation can be extended to other RPLC (39) Tamon, H.; Okazaki, M.; Toei, R. AIChE J. 1985, 31, 1226.

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systems, with other packing materials, other ligands, and other organic modifier or organic modifier concentration. An enthalpyentropy compensation effect similar to the one observed for the adsorption equilibrium on the same RPLC system allows the derivation of an estimation procedure of Ds. Although Er, γ, and the compensation temperature are empirical parameters, surface diffusion under the limiting conditions of weak adsorbate-adsorbent interactions can be studied using the new correlation between Ds and Dm. From only one value of K, Ds can be estimated in the practical range of temperatures used for RPLC systems with an error smaller than 50%. NOTATION

∆S

entropy change, kJ mol-1 K-1

T

temperature, K

t

time, s

u0

superficial velocity, cm s-1

Vb

molecular volume at normal boiling point, cm3 mol-1

z

longitudinal position in bed, cm

Greek Letters R

ratio of Es to -Qst

β

compensation temperature, K

γ

ratio of Er to -Qst

δ0

defined by eq 3

D

diffusivity, cm2 s-1

δax

defined by eq 4

De

intraparticle diffusion coefficient, cm2 s-1

δf

defined by eq 5

Dm

molecular diffusivity, cm2 s-1

δd

defined by eq 6

Dm,0

frequency factor of molecular diffusion, cm2 s-1



void fraction in bed

dp

particle diameter, cm

p

porosity

Dp

pore diffusivity, cm2 s-1

η

viscosity, Pa s

Ds

surface diffusion coefficient, cm2 s-1

µ1

first absolute moment, s

Ds,0

frequency factor of surface diffusion, cm2 s-1

µ2

second central moment, s2

Fp

particle density, g cm-3

φ

volumetric fraction of organic modifier in mobile phase, %

mol-1

Em

activation energy of molecular diffusion, kJ

Er

restriction energy for molecular diffusion, kJ mol-1

Es

activation energy of surface diffusion, kJ mol-1

Ez

axial dispersion coefficient, cm2 s-1

∆G

Gibbs free energy change, kJ mol-1

H

defined by eq 9

∆H

enthalpy change, kJ mol-1

K

adsorption equilibrium constant, cm3 g-1

K0

adsorption equilibrium constant at 1/T ) 0, cm3 g-1

k

tortuosity factor

kf

fluid-to-particle mass-transfer coefficient, cm s-1

Pe

Peclet number

q

amount adsorbed

Qst

isosteric heat of adsorption, kJ mol-1

R

particle radius, cm

r

restriction factor of molecular diffusion

Received for review August 12, 1998. Accepted December 3, 1998.

Rg

gas constant

AC9809027

896

Analytical Chemistry, Vol. 71, No. 4, February 15, 1999

Superscript 0

standard state

Subscripts a

adsorbate

b

boiling point

sv

solvent

T

temperature

β

compensation temperature