Ind. Eng. Chem. Res. 1997, 36, 4335-4341
4335
Surface Diffusion Phenomena in Reversed-Phase Liquid Chromatography with Methanol/Water and Acetonitrile/Water Mixtures Kanji Miyabe* and Shigeya Takeuchi Chemistry Section, Faculty of Education, Toyama University, Gofuku, Toyama 930, Japan
Characteristics of surface diffusion phenomena were studied under low-loading or analytical conditions of reversed-phase liquid chromatography. As mobile phases, methanol/water and acetonitrile/water mixtures were used. In both systems, several similar relationships were confirmed for the enthalpy-entropy compensation effect, the linear free-energy relation, and some correlations between the surface diffusion coefficient, Ds, and mobile-phase composition, φ, and between the ratio of Ds to molecular diffusivity and φ. Similarity of the correlations indicated that the surface diffusion mechanism was essentially identical irrespective of the type of the organic modifiers. A strong influence of φ on Ds was confirmed. A nomograph concerning corresponding mobile-phase compositions for Ds as well as for adsorption equilibrium constant was proposed for the both mobile-phase systems. On the basis of the several correlations, Ds under various conditions in low-concentration systems can be estimated. Introduction High-performance liquid chromatography is one of the most powerful tools in the field of separation chemistry. It is well-known that reversed-phase mode is most popular. A great number of studies relating to retention behavior have been made to elucidate the separation mechanism of reversed-phase liquid chromatography. As a result, much information about adsorption equilibrium has been obtained. On the other hand, a number of studies have also been made on mass-transfer phenomena in a column, which contribute to peak spreading in chromatographic processes. For example, axial dispersion and fluid-to-particle mass transfer have been studied in detail (Guiochon et al., 1994; Suzuki, 1990). For intraparticle diffusion, it has been made possible to estimate pore diffusivity and Knudsen diffusivity in porous adsorbent particles. Several studies have also been made on surface migration of adsorbate molecules in liquid-phase adsorption systems. Some pieces of information on the temperature and concentration dependence of the surface diffusion coefficient, Ds, have been reported. However, there are not so many works on the intraparticle mass transfer in reversedphase chromatographic systems in contrast to the extremely extensive studies on retention behavior. In particular, only a little information has been reported on the characteristics of surface diffusion phenomena in reversed-phase packing materials (Bogar et al., 1984; Hansen and Harris, 1995; Miyabe and Suzuki, 1992, 1993, 1994a,b, 1995; Stahlberg et al., 1988; Zulli et al., 1994). Establishment of the estimation procedure of Ds is a subject that still remains in order to simulate separation processes in reversed-phase liquid chromatography. It is essential to study mass-transfer phenomena from various viewpoints other than adsorption equilibrium to get a better understanding of separation mechanism in reversed-phase liquid chromatography. This paper is concerned with the influence of mobilephase conditions, i.e., the type and composition of the * Author to whom correspondence should be addressed. Telephone: 0764-45-6298. Fax: 0764-45-6264. E-mail:
[email protected]. S0888-5885(97)00146-2 CCC: $14.00
organic modifiers in mobile phases, on surface diffusion phenomena in reversed-phase liquid chromatography. Pulse response experiments were made by using methanol/water and acetonitrile/water mixed solvents of various compositions as mobile phases. Experimental results in both mobile-phase systems were compared with each other. Several estimation methods of Ds and a solvent-strength nomograph for Ds were proposed. Data Analysis Moment Analysis of Experimental Data. Chromatographic peaks experimentally observed were analyzed by the method of moments (Suzuki, 1973, 1990). Information about adsorption equilibrium and masstransfer rates was obtained from first and second moments of peaks, respectively. First absolute moment, µ1, and second central moment, µ2′, of chromatographic peaks are expressed as follows:
µ1 ) µ2′ )
∫Ce(t)t dt/∫Ce(t) dt ) (z/u0)δ0
(1)
∫Ce(t)(t - µ1)2 dt/∫ Ce(t) dt ) (2z/u0)(δax + δf + δd) (2) δ0 ) + (1 - )(p + FpK)
(3)
δax ) (Ez/u02)δ02
(4)
δf ) (1 - )(R/3kf)(p + FpK)2
(5)
δd ) (1 - )(R2/15De)(p + FpK)2
(6)
The first moment was analyzed by eq 7 to determine adsorption equilibrium constant K:
(µ1 - t0)/(1 - ) ) (z/u0)FpK
(7)
t0 ) (z/u0)[ + (1 - )p]
(8)
From the slope of a linear plot between (µ1 - t0)/(1 - ) and (z/u0), K was calculated. From the second mo© 1997 American Chemical Society
4336 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997
ments, Ds was determined by subtracting the contributions of some mass-transfer processes. A parameter H was calculated as follows:
H ) (µ2′/µ12)(z/2u0) ) (Ez/u02) + H0
(9)
H0 ) δf/δ02 + δd/δ02
(10)
Axial dispersion coefficient, Ez, and intraparticle diffusivity, De, can be determined from the slope and intercept of the plot of H vs 1/u0 by subtracting the contribution of fluid-to-particle mass transfer to peak spreading. The value of δf was calculated from eq 5, and its contribution to µ2′ was corrected. The fluid-toparticle mass-transfer coefficient, kf, was estimated by the equation of Wilson-Geankoplis (Wilson and Geankoplis, 1966):
Sh ) (1.09/)Sc1/3Rep1/3
(11)
Molecular diffusivity, Dm, of an adsorbate in methanol/ water mixtures was estimated by the Wilke-Chang equation (Reid et al., 1977):
Dm ) 7.4 × 10-8(RsvMsv)1/2T/(ηsvVb,a0.6)
(12)
On the other hand, Dm of the adsorbate in acetonitrile/ water mixed solvents was calculated from Dm values in acetonitrile and in water through the application of the Perkins-Geankoplis equation (Reid et al., 1977) because the value of R in the Wilke-Chang equation has not been proposed for acetonitrile:
Dmηm0.8 )
∑xiDiηi0.8
(13)
The value of Dm of the adsorbate in acetonitrile was estimated by the Scheibel equation (Reid et al., 1977): 1/3
Dm ) KcT/(ηsvVb,a ) 2/3
Kc ) 8.2 × 10 [1 + (3Vb,sv/Vb,a) ] -8
Kc ) 17.5 × 10-8
(14) (Vb,a g 2.5Vb,sv) (15)
(Vb,a < 2.5Vb,sv)
(16)
The value of Dm in water was calculated by the Hayduk-Laudie equation (Reid et al., 1977):
Dm ) 13.26 × 10-5ηsv-1.4Vb,a-0.589
(17)
The contribution of the adsorption rate at an adsorption site to µ2′ was assumed to be negligibly small. The value of De is related to both pore diffusivity, Dp, and Ds as follows:
De ) Dp + FpKDs
(18)
The value of Ds was determined by subtracting the contribution of Dp to De. According to the parallel pore model, Dp was estimated from Dm, p, and tortuosity factor, k, of pores by the following equation:
Dp ) (p/k2)Dm
(19)
The value of k was determined from pulse response experiments with inert pulses. Accuracy of Determination of Ds from µ2′. In this study, some corrections were made in order to estimate
Ds from µ2′. The influence of the corrections on the conclusion of this study is considered. As described in above equations, the uncertainty in the estimation of Dm, kf, and Dp influences the determination of Ds. The accuracy in the estimation of Dm is considered. In this study, the values of Dm were calculated by the Wilke-Chang equation and the equations of PerkinsGeankoplis, Scheibel, and Hayduk-Laudie according to the mobile-phase systems. For an example, the values of Dm for benzene in 70 vol % methanol were calculated by the two methods as 8.2 × 10-6 cm2 s-1 and 9.1 × 10-6 cm2 s-1, respectively. The relative discrepancy was about 10%. Similar values of Dm could be obtained by the two methods. An average error in the estimation of Dm on the basis of various literature correlations previously proposed was found to be less than about 10% (Reid et al., 1977). The uncertainty in the estimation of kf influences the determination of Ds. In this study, the WilsonGeankoplis equation was employed for the estimation of kf. The value of kf is proportional to the two-thirds power of Dm. If Dm is estimated with an error less than about 10% as described above, the discrepancy between the values of kf calculated from the different Dm values by the Wilson-Geankoplis equation seems to be less than about 6%. Different values of kf are estimated on the basis of various literature correlations previously proposed even if an identical value of Dm is used. For example, kf for benzene at 298 K was calculated as 1.85 × 10-2 cm s-1 according to the Wilson-Geankoplis equation under the conditions that u0 of 70 vol % methanol was 0.12 cm s-1 and Dm of benzene was 8.2 × 10-6 cm2 s-1. Another value of kf was obtained as 1.48 × 10-2 cm s-1 under the same conditions of this study by the following equation proposed by Kataoka et al. (1972):
Sh ) 1.85[(1 - )/]1/3Sc1/3Rep1/3
(20)
The resulting value of Ds calculated by taking kf as 1.48 × 10-2 cm s-1 was 3.3 × 10-6 cm2 s-1. Compared with the original value of Ds, i.e., 2.7 × 10-6 cm2 s-1, both values were of the same order of magnitude. As indicated in previous papers (Miyabe and Suzuki, 1994a,b, 1995), the contribution of axial dispersion to overall mass-transfer resistance in reversed-phase liquid chromatographic systems was found to range from 20 to 30%. The contributions of fluid-to-particle mass transfer and intraparticle diffusion were of the same order of magnitude in reversed-phase liquid chromatographic systems using the four organic compounds as adsorbates. The contribution of intraparticle diffusion was larger than or almost equal to that of fluid-toparticle mass transfer. In particular, the former contribution was larger than the latter one under the conditions that the adsorptivity of the adsorbates was low. However, almost all of experimental data could be plotted around the same lines irrespective of the adsorptivity of the adsorbates. It is concluded that appropriate values of Ds are probably obtained by subtracting the contribution of δf to µ2′. The variation in the value of kf estimated may not provide a serious influence on the determination of Ds. The contribution of Dp to De was corrected when Ds was calculated from De. The accuracy of the estimation of Dp influences the determination of Ds. In this study, Dp was calculated from Dm, p, and k according to the parallel pore model. In such a case, the uncertainty in the estimation of Dp corresponds to that of Dm. As
Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 4337
indicated in previous papers (Miyabe and Suzuki, 1992, 1993, 1994a,b, 1995), the contribution of surface diffusion to overall mass transfer in reversed-phase packing materials is usually as much as about 85-95% or above. Because surface diffusion has an extremely dominant role for intraparticle diffusion, the influence of the variation in the estimation of Dp on the determination of Ds is negligibly small. On the basis of the consideration described above, it is concluded that Ds was probably calculated with an error of several percentage points. Calculation of First and Second Moments from Chromatographic Peaks. Foley and Dorsey (1983) proposed equations for the calculation of chromatographic figures of merit for ideal and skewed peaks. The values of µ1 and µ2′ of the peaks were calculated by applying the following equations:
µ1 ) tG + τ
(21)
µ2′ ) W0.12/[1.764(B/A)2 - 11.15(B/A) + 28] (22) tG ) tR - σG[-0.193(B/A)2 + 1.162(B/A) - 0.545] (23) σG ) W0.1/[3.27(B/A) + 1.2]
(24)
τ ) (µ2′ - σG2)0.5
(25)
The peak width, W0.1, and an empirical asymmetry factor, B/A, at 10% of peak height were measured. The value of B/A represents the ratio of peak width in the rear part of a peak to that in the front part. Experimental retention time, tR,exp, includes the effects of the volume, Ve, in the pipes between an injection port and a column and that between the column and a detector. Retention time was obtained by correcting the effect of the extracolumn volume as follows:
tR ) tR,exp - Ve/v
(26)
The contribution of the extracolumn pipes to µ2′ was measured by the pulse response method without the column, and it was corrected. The effects of the first and second moments of pulses introduced at the inlet of the column were neglected because the pulse size was extremely small. Experimental Section Apparatus. A high-performance liquid chromatograph (LC-6A, Shimadzu) equipped with an ultraviolet detector was employed. A sample injector (Rheodyne Model 7125) was used in order to introduce a trace amount (a few µL) of sample solutions into a fluid flow of a carrier solvent. A thermostated water bath was employed to maintain column temperature at a constant level. Columns and Reagents. Properties of two commercial octadecylsilyl (ODS) columns (YMC) used are shown in Table 1. The ODS-silica gel particles were spherical in shape. Average diameter of the ODS particles was 45 µm. The diameter of most of the ODS particles were distributed in the range from about 20 to 70 µm like the normal distribution. The carbon content of the ODS-silica gel particles was about 17 wt %. This figure may be close to a maximum amount of carbon, which can be introduced on the surface of
silica gel by chemical bonding on an n-octadecyldimethylsilyl ligand. Mobile phases were mixtures of an organic modifier and water. As organic modifiers, methanol and acetonitrile were employed. Volumetric composition of the organic modifiers was changed in the range from 40 to 100%. As adsorbates, benzene, toluene, ethylbenzene, and naphthalene were used. As an inert substance, uracil was employed. Both porosity of the ODS-silica gel particles and void fraction of the ODS columns were determined by the pulse reponse method with the injection of various amounts of sodium nitrate (Wells and Clark, 1981). The determination of the physical properties of the ODS packing materials and columns is based on the concentration dependence of the elution volume of sodium nitrate due to an ion exclusion effect. It was suggested that, at low-electrolyte concentration in unbuffered eluents, the salt was excluded from the pores of the ODS adsorbent and that with an increase in the concentration of the electrolyte in the mobile phase, the ion exclusion effect was suppressed and the pores became accessible to the salt. The interparticular volume of the ODS column can be determined from the elution volume of sodium nitrate under the condition that an extremely small amount of sodium nitrate is injected. On the contrary, pulse response experiments with the injection of a large amount of sodium nitrate provide the information about a void volume of the ODS column. The difference between both elution volumes of sodium nitrate corresponds to the intraparticular volume of the ODS packing materials. Procedure. Experimental conditions of the liquidphase adsorption are also listed in Table 1. Pulse response experiments were made at zero surface coverage of adsorbates with varying temperature and the flow rate of the mobile phases. Small-concentration perturbation pulses were introduced into a fluid flow. All experiments of this study were carried out under infinite dilution conditions. Chromatographic peaks were measured in the temperature range from 288 to 308 K. Superficial velocity of the mobile phases was varied from 0.059 to 0.118 cm s-1. Chromatographic peaks measured were analyzed by the method of moment (Suzuki, 1973, 1990). Results and Discussion Surface Diffusion Coefficient as a Function of Mobile-Phase Composition. Figure 1illustrates almost linear correlations between the logarithm of Ds and the composition of the organic modifiers in mobile phases, φ. A strong influence of φ on Ds was observed. It seems that Ds increases with a decrease in the interaction between the adsorbate molecules and the ODS surface due to the increment of φ regardless of temperature. Though Ds is larger in acetonitrile/water mixtures than in methanol/water mixed solvents, the whole tendency of the linear correlations is similar in both mobile-phase systems. Almost parallel relationships are observed between corresponding linear correlations. Similar correlations were obtained for the four adsorbates. Enthalpy-Entropy Compensation Effect in Surface Diffusion Phenomena. Temperature dependence of Ds illustrated in Figure 1 was analyzed by the Arrhenius equation:
Ds ) Ds0 exp(-Es/RgT)
(27)
The values of Es and Ds0 were calculated from the slope
4338 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 Table 1. Properties of ODS Columns and Experimental Conditionsa
a
mobile phase system
methanol/water
acetonitrile/water
average particle diameter, dp (µm) particle density, Fp (g cm-3) pore volume (cm3 g-1) porosity, p mass of adsorbent (g) column size (mm) void fraction, tortuosity factor, k2 column temperature (K) mobile phase superficial velocity, u0 (cm s-1)
45 0.86 0.53 (100, 80, 70, 60) 0.55 (40) 0.46 (100, 80, 70, 60), 0.47 (40) 2.1 6 i.d. × 150 0.43 4.5 288-308 methanol/water: 100/0-40/60 (vol) 0.06-0.12
45 0.81 0.50 (80), 0.46 (70), 0.43 (60, 40), 0.40 (80), 0.37 (70), 0.35 (60, 40), 2.1 6 i.d. × 150 0.39 5.5 288-308 acetonitrile/water: 80/20-40/60 (vol) 0.06-0.12
The numbers in parentheses represent volumetric fraction of an organic modifier in mobile phases.
type and composition of the organic modifiers in the carrier solvents. The mechanism of surface diffusion seems to be essentially identical regardless of the type of the organic modifiers in mobile phases. The similarity of the surface diffusion mechanisms is also suggested between the gas- and liquid-phase adsorption systems. According to the compensation effect, both Ds0 and Es can be calculated from one datum of Ds estimated or experimentally determined. The value of Ds at a given temperature can be calculated by the Arrhenius equation. Linear Free-Energy Relation in Reversed-Phase Liquid Chromatography. A linear free-energy relation such as the Hammett rule and the Brensted catalytic rule can be expected to hold when the enthalpyentropy compensation effect is established. As shown in Figure 3, linear correlations were observed between ln Ds and ln K for both mobile-phase systems. They are represented by the following equations:
ln Ds ) -0.506(ln K) - 12.4 (methanol/water, 298 K) (30)
Figure 1. Surface diffusion coefficient as a function of volumetric composition of organic modifiers in methanol/water and acetonitrile/water mobile phases.
and intercept of the Arrhenius plot, respectively. Figure 2 shows the plot of ln Ds0 against Es for the two mobilephase systems. Experimental data scattered around each linear correlation:
ln Ds0 ) 0.312Es - 11.2 ln Ds0 ) 0.326Es - 11.2
(methanol/water) (28) (acetonitrile/water) (29)
The dotted line in Figure 2 represents the corresponding relationship for gaseous systems as a reference. The three straight lines are almost identical, on which all the experimental data can be plotted. As shown in the previous paper (Miyabe and Suzuki, 1994a), a number of experimental data under various reversed-phase liquid chromatographic conditions scattered around the straight lines regardless of the type of adsorbates, the length of alkyl chains of stationary phases, and methanol composition in mobile phases. It was concluded that, as well as for adsorption equilibria, the enthalpyentropy compensation effect could be observed for surface diffusion phenomena in the various gas and liquid chromatographic systems. The results in Figure 2 also indicate that the correlation between Ds0 and Es can be expressed by one straight line irrespective of the
ln Ds ) -0.509(ln K) - 11.9 (acetonitrile/water, 298 K) (31) Though the two linear correlations are not identical, they have almost the same slope irrespective of the type of the organic modifiers. The values of Ds in acetonitrile/water mobile phases were larger than those in methanol/water systems by a factor of about 1.6, even under the conditions that the value of K in both mobilephase systems were identical. The parallel correlation in Figure 3 indicates that the manner of the influence of φ on Ds differs from that on K. The role of mobilephase solvents in surface diffusion phenomena must be studied in more detail. The slope of the linear relations between ln Ds and ln K is equal to the ratio of Es/Qst. As shown in eqs 30 and 31, the ratio of Es/(-Qst) was about 0.51, which were of the same order of magnitude with those in various gaseous systems (Miyabe and Suzuki, 1993; Suzuki, 1990). By the analysis of the correlations between ln Ds and ln K in Figure 3, it is concluded that Es is smaller than (-Qst) in reversed-phase liquid chromatographic systems as well as in gas-phase adsorption. Mitani et al. (1979) also demonstrated linear correlations between the logarithm of the Henry constant and Ds for various gaseous systems. Irrespective of the type of adsorbates and adsorbents, the slope of the linear correlations were about 0.3-1.0. In the field of reversed-phase liquid chromatography, a great number of studies have been carried out for the retention behavior. On the basis of the studies, many
Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 4339
Figure 4. Ratio of surface diffusion coefficient to molecular diffusivity as a function of volumetric composition of organic modifiers in methanol/water and acetonitrile/water mobile phases.
Figure 2. Correlation of frequency factor with activation energy of surface diffusion for liquid-phase adsorption.
surface diffusion phenomena on ODS-silica gel particles. It is suggested that the surface diffusion mechanism must be studied by taking into account the role of mobile-phase solvents on mass-transfer phenomena in reversed-phase liquid chromatography. The data scattered around the straight lines represent the following correlations:
Ds/Dm ) 4.04 × 10-3φ - 6.89 × 10-3 (methanol/water, φ: vol %) (32) Ds/Dm ) 3.41 × 10-3φ + 3.53 × 10-2 (acetonitrile/water, φ: vol %) (33)
Figure 3. Correlation of surface diffusion coefficient with adsorption equilibrium constant for liquid-phase adsorption. Symbols: refer to Figure 2.
empirical and theoretical procedures have also been proposed for the estimation of capacity factor, k′. The linear free-energy relation as shown in Figure 3 makes it possible to apply the conventional studies relating to adsorption equilibrium to the estimation of Ds in reversed-phase liquid chromatography. In the cases of this study, Ds for both mobile-phase systems can be calculated from corresponding values of K by using eqs 30 and 31. Correlation between the Ratio of Ds/Dm and O. The ratio of Ds/Dm at various temperatures in both mobile-phase systems is plotted against O in Figure 4. Each plot was an average of Ds values for the four sample compounds and had a range of about (0.05. The values of Ds/Dm at each O were almost comparable irrespective of temperatures and were about 0.15-0.4 in the range of O from 40 to 100 vol %. Reversed-phase separations are quite commonly carried out in the range of O (Snyder et al., 1988). The correlations in Figures 1, 3, and 4 indicate that mobile-phase solvents influence
Estimation of Ds is required in order to analyze the adsorption mechanism of adsorbate molecules and to simulate separation processes in chromatography. However, it is difficult to accurately estimate Ds in a given experimental system because Ds significantly varies with the combination of adsorbates and adsorbents. According to eqs 32 and 33, Ds can be estimated from Dm in reversed-phase liquid chromatographic systems regardless of both the type and the composition of the organic modifiers in mobile phases. The linear correlations in Figure 4 are nearly identical, suggesting the similarity of the surface diffusion mechanisms in both mobile-phase systems. The value of Ds could be estimated by eqs 32 and 33 with an error of about (10-30%. Relative Mobile Phase Strength of Methanol and Acetonitrile. In reversed-phase liquid chromatography, a mixed solvent of water and an organic modifier, such as methanol, acetonitrile, and tetrahydrofuran, is commonly employed as a mobile phase. The substitution of one organic modifier in the mobile phase to another is frequently carried out in order to change separation selectivity in chromatography. In such cases, it is required to understand solvent compositions of the mobile-phase systems having the same elution strength. By using the isoeluotropic solvents, almost equal values of K and k′ can be obtained. These isoeluotropic correlations between a few mixed solvent systems of the organic modifiers and water can be illustrated by a nomograph. Snyder et al. (1988) already indicated a nomograph for relative mobile-phase strength of methanol, acetonitrile, and tetrahydrofuran with respect to k′. Figure 5illustrates a similar nomograph for elution
4340 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997
Nomenclature
Figure 5. Solvent-strength nomograph for adsorption equilibrium constant and surface diffusion coefficient in reversed-phase liquid chromatography.
strength of methanol/water and acetonitrile/water with regard to K. The spreading of chromatographic peaks is due to the contributions of several mass-transfer steps in a column. The change in the type of the organic modifiers influences the contributions of fluid-to-particle mass transfer and intraparticle diffusion to µ2′ and provides no influence on the contribution of axial dispersion because the contribution of molecular diffusion to Ez is negligibly small in liquid-phase adsorption. The contributions of fluid-to-particle mass transfer and pore diffusion in porous adsorbents to µ2′ can be calculated because Dm can be estimated under a given mobile-phase condition according to various correlations. On the contrary, a very few estimation procedures of Ds in reversed-phase liquid chromatography has been reported. Little information about the influence of the change in the type of the organic modifiers on Ds has been obtained. In Figure 5, a nomograph for Ds in reversed-phase liquid chromatography is illustrated. Figure 5 represents mobile-phase compositions providing the same values of Ds. For example, 80 vol % methanol has the same property as about 55 vol % acetonitrile regarding Ds as indicated by a dotted line. Simultaneously, it is also represented that the elution strength of 80 vol % methanol is almost equal to that of about 75 vol % acetonitrile. The results in Figure 5 clarify the difference between the influence of φ on Ds and that on K. When the results in Figure 5 are applied, corresponding compositions of methanol/water and acetonitrile/water mobile phases providing a same column efficiency can be predicted because the values of several parameters affecting peak spreading, i.e., Ez, kf, and Dp, other than Ds, can be experimentally determined or empirically estimated. Conclusions The influence of mobile-phase conditions on the surface diffusion on ODS-silica gel particles were studied. Pulse response experiments in reversed-phase liquid chromatography were made under infinite dilution conditions by using methanol/water and acetonitrile/ water mixtures of various compositions as mobile phases. The values of Ds in both mobile-phase systems were determined by the moment analysis and were compared with each other. Several linear correlations were demonstrated between ln Ds and φ, between ln Ds0 and Es (enthalpyentropy compensation), between ln Ds and ln K (linear free-energy relation), and between Ds/Dm and φ in both mobile-phase systems. These almost same or similar correlations indicate the similarity in the migration mechanism of surface diffusion in both mobile-phase systems. On the basis of the correlations, Ds in reversedphase liquid chromatography can be estimated.
Ce ) peak profile as a function of t D ) diffusivity, cm2 s-1 De ) intraparticle diffusion coefficient, cm2 s-1 Dm ) molecular diffusivity, cm2 s-1 Dp ) pore diffusivity, cm2 s-1 dp ) particle diameter, cm Ds ) surface diffusion coefficient, cm2 s-1 Ds0 ) frequency factor, cm2 s-1 Es ) activation energy of surface diffusion, kJ mol-1 Ez ) axial dispersion coefficient, cm2 s-1 H ) defined by eq 9 H0 ) defined by eq 10 K ) adsorption equilibrium constant, cm3 g-1 k ) tortuosity factor k′ ) capacity factor kf ) fluid-to-particle mass-transfer coefficient, cm s-1 M ) molar weight, g Qst ) isosteric heat of adsorption, kJ mol-1 R ) particle radius, cm Rep ) Reynolds number Rg ) gas constant Sc ) Schmidt number Sh ) Sherwood number T ) temperature, K t ) time, s tG ) defined by eq 23 tR ) retention time, s t0 ) defined by eq 8 u0 ) superficial velocity, cm s-1 v ) volumetric flow rate, cm3 s-1 Vb ) molar volume at normal boiling point, cm3 mol-1 Ve ) extracolumn volume, cm3 W0.1 ) peak width at 10% of height, s x ) mole fraction z ) longitudinal position in bed, cm Greek Symbols R ) association coefficient δ0 ) defined by eq 3 δax ) defined by eq 4 δf ) defined by eq 5 δd ) defined by eq 6 ) void fraction in bed p ) porosity η ) viscosity, Pa s µ1 ) first absolute moment, s µ2′ ) second central moment, s2 Fp ) particle density, g cm-3 σG ) defined by eq 24 τ ) defined by eq 25 φ ) volumetric fraction of an organic modifier in a mobile phase, vol % Subscripts a ) adsorbate exp ) experimental i ) ith component m ) mixture sv ) solvent
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Received for review February 13, 1997 Revised manuscript received June 3, 1997 Accepted June 12, 1997X IE970146F X Abstract published in Advance ACS Abstracts, August 15, 1997.
