11086
J. Phys. Chem. B 1999, 103, 11086-11097
Thermodynamic Characteristics of Surface Diffusion 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 ReceiVed: May 26, 1999; In Final Form: October 7, 1999
The thermodynamic properties of surface diffusion in reversed-phase liquid chromatography (RPLC) were analyzed in connection with the thermodynamics of surface diffusion in gas chromatography systems and of phase equilibrium in RPLC. The results suggest that the activation energy of surface diffusion in RPLC consists of only two contributions, corresponding to a hole-making and a bond-breaking process. The former is close to the activation energy of the mobile phase viscosity; the latter is correlated with the isosteric heat of adsorption (Qst), by a numerical coefficient between 0.5 and 0.6. The validity of these results was proved by the analysis of correlations between the surface diffusion coefficient (Ds) and the adsorption equilibrium constant (linear free energy relation), between Ds and the boiling point of the sample components, and between the ratio of Ds to the molecular diffusivity and Qst. These results support a restricted diffusion model, proposed earlier as an approximation of the mass transfer mechanism of surface diffusion. The restricted diffusion model provides consistent interpretations for the intrinsic thermodynamic properties of surface diffusion in RPLC.
Introduction Reversed-phase liquid chromatography (RPLC) is now the most popular mode of chromatographic separations. The fundamentals and the applications of RPLC are abundantly discussed in the literature. Numerous basic studies deal with both the thermodynamics and the mass transfer kinetics associated with the various phenomena involved in this retention mode. However, the depth of understanding achieved so far is more profound in thermodynamics than in kinetics. The elucidation of the different retention mechanisms involved (e.g., hydrophobic interactions, silanophilic activity) and of their intricate interactions was actively studied for years and is still considered as one of the most important and fundamental topics in chromatography. On the other hand, fundamental studies on the kinetics of mass transfer in RPLC have remained rather superficial. It is highly probable that systematic, more fundamental investigations in this area could provide new information regarding retention and separation mechanisms which cannot be obtained from studies on phase equilibrium alone. So far, most kinetic studies in RPLC, as in many areas of analytical chromatography, used various plate height equations, in which the HETP is derived as the sum of the individual contributions of the different mass transport processes which can be identified in a column.1,2 The analysis of the dependence of the column efficiency on the mobile phase flow rate is the usual tool to determine the values of the coefficients of each term of these plate height equations. These coefficients provide information regarding the chromatographic system, e.g., the quality of the column packing or the mass transfer resistance in the stationary phase. However, the significance of each term of a plate height equation could be analyzed further than is usually done. This analysis affords more detailed information on the mass transfer mechanisms operating in the column. For example, Giddings2 analyzed the different effects influencing
the apparent axial dispersion in a column, using the randomwalk model. He divided them into four different groups, the transchannel, short-range interchannel, long-range interchannel, and transcolumn, and derived a generalized plate height equation. Giddings also pointed out a fifth contribution, the transparticle contribution, which is usually identified with the intraparticle mass transfer. Mass transfer inside materials made of porous particles is usually assumed to consist of two mechanisms, pore and surface diffusion. The former explains the migration of the solute molecules in the mobile phase stagnant inside the particle pores. The latter accounts for the migration of the adsorbate molecules on the surface of the stationary phase while they remain in the adsorbed state. Surface diffusion takes place in the potential field of adsorption and is an activated process. It was shown to play a predominant role in the mass transfer inside the porous particles of adsorbents in both gas and liquid phase adsorption systems.3 Experimental data are available for surface and lateral diffusions in some materials used in RPLC. Bogar et al.4 determined the lateral diffusion coefficient of pyrene in the phase system consisting of a C18 silica and a methanol/water solution. Hansen and Harris5 measured the surface diffusion coefficient (Ds) of rubrene on a C18-bonded phase, using three methanol/water solutions of different compositions. Wong and Harris6 reported Ds for iodine on C1-bonded silica in contact with methanol/water solutions. Zulli et al.7 measured the lateral diffusion coefficient of acridine orange at a water/C18 interface. However, although these works make experimental values of Ds available for a few systems, no systematic interpretation of surface diffusion data in RPLC was attempted yet. Previously,8 we reviewed surface diffusion data determined in RPLC, under various experimental conditions (e.g., different compounds and stationary/mobile phase systems), and demonstrated some characteristic features of surface diffusion in RPLC. We showed that surface diffusion plays an
10.1021/jp991725w CCC: $18.00 © 1999 American Chemical Society Published on Web 11/30/1999
Thermodynamic Characteristics of Surface Diffusion in RPLC
J. Phys. Chem. B, Vol. 103, No. 50, 1999 11087
important role in intraparticle diffusion. We also proposed, as a first approximation, a restricted molecular diffusion model, formulated on the basis of the absolute rate theory. Because the contribution of surface diffusion to the mass transfer kinetics in RPLC is significant,8 it must be studied in detail for a better understanding of the separation mechanism of RPLC. The analysis of the thermodynamic properties of RPLC is probably the most effective approaches presently available for the elucidation of the migration mechanism of surface diffusion. This approach has remained ignored so far. It is expected that the thermodynamic properties of surface diffusion are closely related with its mechanism. The goal of this study is to provide an interpretation of the thermodynamic properties of surface diffusion in RPLC and to obtain new information on the mechanisms of mass transfer on the surface of C18-bonded silicas and through the particles of these packing materials. Moment Analysis Chromatographic peaks obtained experimentally were analyzed by the method of moments.9,10 This method was shown long ago to be most effective for the determination of the properties of chromatographic peaks.11,12 It is now most conventional in this kind of investigations.1 The peak profiles contain important information on the adsorption equilibrium and the mass transfer resistances. This information is derived from the first and the second moments of these peaks, respectively. The first absolute moment (µ1) and the second central moment (µ′2) of a chromatographic peak are given by the following equations.
the rate of adsorption on the actual adsorption sites to µ′2 was negligibly small. The following equations were used to derive K from µ1:
µ1 )
∫Ce(t) t dt z ) δ0 ∫Ce(t) dt u0
(1)
∫Ce(t)(t - µ1)2 dt 2z µ′2 ) ) (δax + δf + δd) u0 ∫Ce(t) dt
(2)
δ0 ) + (1 - )(p + FpK)
(3)
δax )
() DL
u0
2
δ02
(5)
Rp2 ( + FpK)2 15De p
(6)
( )
(1 - ) t0 )
)
()
()
z FK u0 p
z [ + (1 - ) p] u0
where Ce(t) is the elution profile of the peak, or concentration of the eluate in the eluent as a function of t, t is the time, z is the longitudinal distance along a column, u0 is the superficial velocity of the mobile phase, δax, δf, and δd are the contributions to µ′2 of the axial dispersion, the fluid-to-particle mass transfer resistance, and the intraparticle diffusion, respectively, is the interparticle void fraction, p and Fp are the porosity and the density of the packing material, respectively, K is the adsorption equilibrium constant, DL is the axial dispersion coefficient, Rp is the particle radius, kf is the fluid-to-particle mass transfer coefficient, and De is the intraparticle diffusivity. The value of µ1 is correlated with the adsorption equilibrium constant, K, only. On the other hand, µ′2 is obtained as the sum of the contributions of three of the basic mass transport processes in the column. In this study, we assumed that the contribution of
(7) (8)
Ds was derived from µ′2 by subtracting the contributions to peak spreading of some of the mass transfer resistances in the column. The parameter H was defined as follows:
H)
( )( ) ( ) µ′2
2
µ1
DL z ) 2 + H0 2u0 u0
H0 )
δf 2
+
δ0
δd δ02
(9)
(10)
The values of DL and De were derived from the slope and the intercept of the linear correlation between H and 1/u0. The value of δf was calculated with eq 5 and its contribution to µ′2 was subtracted. The value of kf was estimated using the WilsonGeankopolis equation.13
Sh )
Sc (1.09 )
1/3
Re1/3
(11)
where Sh, Sc, and Re are classical nondimensional parameters, the Sherwood number, the Schmidt number, and the Reynolds number, respectively. Molecular diffusivity (Dm) was estimated with the equation of Wilke-Chang.14,15
Dm )
7.4 × 10-8TxRsvMsv ηsvV0.6 b,s
(12)
The value of Ds was calculated by assuming that the contributions of surface and pore diffusions to intraparticle diffusion are parallel, i.e., that
De ) Dp + FpKDs
( )
Rp ( + FpK)2 δf ) (1 - ) 3kf p δd ) (1 - )
(4)
(µ1 - t0)
(13)
where Dp is the pore diffusivity. According to the parallel pore model, Dp was calculated from Dm by the following equation
Dp )
() p
k2
Dm
(14)
where k is the tortuosity factor of the pores. This parameter was determined by analyzing the profiles obtained in a pulse response experiment in which an unretained substance, uracil, is injected. As discussed previously,8 the value of Ds is probably determined with an error of a few percent. The values of µ1 and µ′2 are usually determined from the position and the width of an elution peak, respectively, by assuming that the peak profile is represented by a Gaussian distribution function. However, in practice, the elution peak exhibits frequently an asymmetric profile. The nonlinear behavior of the adsorption isotherm is not the only source of peak asymmetry in chromatography.1,2 Asymmetric peak profiles may originate from several other sources, such as the coexistence of fast and slow mass transfer processes, the heterogeneity of the bed of packing
11088 J. Phys. Chem. B, Vol. 103, No. 50, 1999
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TABLE 1: Properties of C18-Silica Gel Column and Experimental Conditions av particle diam, dp (µm) particle density, Fp (g cm-3) porosity, p carbon content (%) column size (mm) mass of adsorbent (g) void fraction, column temp (K) mobile phase solvent superficial velocity, u0 (cm s-1) sample substances
45 0.86 0.46 17.1 6 × 150 2.1 0.42 288-308 methanol/water, 70/30 (v/v) 0.06-0.12 alkylbenzene derivatives p-alkylphenol derivatives
material in the transverse direction of the column, and the existence of large dead volumes in the chromatographic equipment. When elution peaks are unsymmetrical, it is difficult accurately to determine µ1 and µ′2 from the mere position and width of this distorted peak without applying any correction. It is known that the phase equilibrium on C18-silica gels is usually accounted for by the simple Langmuir model, suggesting that the surface of RP packing materials can be regarded, in a first approximation, as energetically homogeneous.1,8 In this study, it was not necessary to consider the possible influence of a nonlinear adsorption isotherm on the peak shape because the pulse response experiments were carried out under linear isotherm conditions. The presence of various types of surface diffusion processes having different kinetic properties was not considered either because of the apparent energetic homogeneity of the surface of C18-silica gels implied by the validity of the Langmuir isotherm model. In previous papers,16-18 we studied the correlation between the column radial heterogeneity and the peak distortion (tailing and fronting) in linear chromatography. This provided information on the influence of the peak asymmetry on µ1 and µ′2. On the basis of these results, the appropriate values of the two moments in linear chromatography were calculated by eliminating the influence of the skewed profile of the elution peak. The contribution of the extra-column volumes of the chromatographic instrument to µ1 and µ′2 was measured by means of the results of similar pulse response experiments performed without column and corrected for. The influence on µ1 and µ′2 of the finite width of the sample pulse introduced into the chromatographic system was also corrected for by assuming a rectangular profile of the injection pulse. However, this influence is almost negligible because of the small sample volume. Experimental Section Column and Reagents. The physical properties of the C18silica gel column (YMC) used are shown in Table 1. The average diameter of the C18-silica was approximately 45 µm. The values of the surface diffusion coefficient, Ds, were derived from the second central moment of the elution peak by subtracting the contributions of the other mass transfer processes, as described earlier in this work. This justifies the selection of a packing material having rather coarse particles to pack the column. The corresponding contribution is large, and hence the values of Ds obtained are more accurate. The mobile phase was a mixture of methanol and water. Two series of homologous organic compounds (alkylbenzenes and p-alkylphenols) were used as sample substances. Uracil was used as the inert tracer. Apparatus. A high-performance liquid chromatograph (LC6A, Shimadzu) was used for the pulse response experiments
Figure 1. Van’t Hoff plot of adsorption equilibrium constant of benzene derivatives. Sample compounds: (O) benzene, (4) toluene, (0) ethylbenzene, (3) propylbenzene, (]) butylbenzene, (+) pentylbenzene, (×) hexylbenzene.
(i.e., elution chromatography). A small amount of the sample solution was introduced into the C18-silica gel column by using a valve injector. The column temperature was maintained constant by circulating temperature-controlled water through a column jacket. The concentration of the sample compounds in the eluate leaving from the column was monitored by the UV detector of the HPLC instrument. Procedure. The experimental conditions are also listed in Table 1. The pulse response experiments were made at nearzero surface coverage of the sample substances. Both the column temperature (288-308 K) and the superficial velocity of the mobile phase (0.06-0.12 cm s-1) were varied. The chromatographic peaks recorded were analyzed by the method of moments, as explained earlier. Results and Discussion Analysis of the Thermodynamic Properties of RPLC. According to the van’t Hoff equation, isosteric heat of adsorption (Qst) was determined from the temperature dependence of K.
K ) K0 exp
( ) -Qst RT
(15)
where K0 is K at T ) ∞ or Qst ) 0, T the absolute temperature, and R the universal gas constant. Figure 1 illustrates the plot of the logarithm of K against 1/T for the alkylbenzenes studied. Qst was derived from the slope of the linear correlations. The resulting values of -Qst are listed in Table 2. They are ranging from ca. 7 to 18 kJ mol-1, increasing progressively with increasing size of the alkyl substituent. These values are of the order of magnitude of others previously reported.8,19-25 The activation energies of surface diffusion (Es) for the same compounds were determined from the slopes of the Arrhenius plots illustrated in Figure 2.
Ds ) Ds0 exp
( ) -Es RT
(16)
Thermodynamic Characteristics of Surface Diffusion in RPLC
J. Phys. Chem. B, Vol. 103, No. 50, 1999 11089
TABLE 2: Thermodynamic Properties of Phase Equilibrium and Surface Diffusion in RPLC System
a
sample
-Qst (kJ mol-1)
K0 (cm3 g-1)
Es (kJ mol-1)
Ds0 (cm2 s-1)
∆Hva (kJ mol-1)
Es/-Qst
benzene toluene ethylbenzene propylbenzene butylbenzene pentylbenzene hexylbenzene p-xylene phenol p-cresol p-ethylphenol p-propylphenol p-butylphenol p-hexylphenol
6.8 8.7 9.7 11.4 13.0 15.2 17.5 10.2 8.3 8.6 9.9 11.3 13.1 16.5
1.3 × 10-1 1.1 × 10-1 1.2 × 10-1 9.8 × 10-2 8.9 × 10-2 6.1 × 10-2 4.1 × 10-2 1.0 × 10-1 1.9 × 10-2 2.7 × 10-2 2.7 × 10-2 2.6 × 10-2 2.1 × 10-2 1.6 × 10-2
21.3 22.0 22.9 22.8 24.1 24.6 27.7 22.1 20.2 19.0 20.6 21.1 22.4 25.1
1.0 × 10-2 9.5 × 10-3 1.1 × 10-2 7.7 × 10-3 9.6 × 10-3 8.7 × 10-3 2.3 × 10-2 8.1 × 10-3 5.4 × 10-3 2.9 × 10-3 4.5 × 10-3 4.4 × 10-3 5.8 × 10-3 1.0 × 10-2
33.8 38.0 42.2 46.2 51.4 b b 42.4 57.8 b b b b b
3.1 2.5 2.4 2.0 1.9 1.6 1.6 2.2 2.4 2.2 2.1 1.9 1.7 1.5
At 298 K. b No data.
Figure 2. Arrhenius plot of surface diffusion coefficient of benzene derivatives. Symbols: refer to Figure 1.
where Ds0 is the frequency factor of surface diffusion. The values of Es are listed in Table 2. They range from ca. 19 to 28 kJ mol-1, being larger than -Qst for all the compounds studied. Similar experimental results were already reported for surface diffusion in liquid phase adsorption systems.8,26-28 Under the condition that Es is larger than -Qst, it might seem to be more energetically advantageous for the sample molecules adsorbed to be desorbed from a surface to a bulk phase and diffuse in the mobile phase rather than to migrate on the surface of the stationary phase. The presence of surface diffusion is not expected in such cases. By contrast, however, there are a few experimental studies, even in liquid-phase adsorption,29-31 in which Es was found to be smaller than -Qst. The correlation between the magnitudes of Es and Qst depends probably on the properties of liquid-solid adsorption phase system investigated. As a reference, Table 3 shows some thermodynamic properties of gas adsorption systems in which a C18-silica gel was used as the adsorbent.32 The value of -Qst is slightly larger than the heat of vaporization (∆Hv). The ratio -Qst/∆Hv ranges between ca. 1.0 and 1.2. The desorption process in gas-solid adsorption resembles that of vaporization. Sample molecules leave the condensed phase on the surface of an adsorbent by
overcoming the attractive interactions between them. However, in gas-solid adsorption systems, there is an additional attractive interaction, that between the sample molecules and the surface of the adsorbent. A value of -Qst slightly larger than ∆Hv corresponds probably to this additional adsorptive interaction and is reasonable. On the other hand, the ratio Es/-Qst ranges from ca. 0.3 to 0.6. As described above, surface diffusion is considered as an activated process. The gain of the activation energy, Es, is necessary for the adsorbate molecules to surpass the boundary energy barrier between two adsorption sites. However, it is not necessary that Es exceeds -Qst because the sample molecules do not need to be completely desorbed from the surface to the bulk phase in order to diffuse. In previous works on gas phase adsorption,3,33,34 similar values of Es/-Qst smaller than unity were reported. As described earlier, the thermodynamic properties of gas adsorption on C18-silica gel seem to be reasonable. By contrast with gas-solid systems, the ratio Es/-Qst in liquid-solid ones seems to behave unreasonably, as explained earlier. It is likely that this apparent contradiction between the different thermodynamic properties of liquid-solid (RPLC) systems can be resolved by their analysis from two different viewpoints, the influence of the mobile phase on the thermodynamic properties of the phase equilibrium and the intrinsic characteristics of surface diffusion in RPLC systems. First, compared with the thermodynamic properties of gas chromatographic system, the values of -Qst in RPLC are relatively small, as shown in Tables 2 and 3. This difference of magnitude of -Qst between gas-solid and liquid-solid (RPLC) systems probably results from the influence of the mobile phase on the equilibrium thermodynamics. In previous papers,8,35 we attempted quantitatively to interpret the solvent effect on Qst in liquid-solid systems on the basis of the solvophobic theory. We concluded that Qst was influenced by the presence of the mobile phase and that smaller value of -Qst should be observed in liquid-solid systems than in gassolid ones. An interpretation of the relatively small values of -Qst in liquid-solid systems was provided. In this study, we now attempt to analyze the characteristic features of the thermodynamics of surface diffusion in liquid-solid (RPLC) systems. Finally, although the magnitudes of the values reported for Es in gas-solid and liquid-solid systems are close, as shown in Tables 2 and 3, it is still unclear whether this is a physical requirement. It does not seem that there is any theoretical proof that the values of Es in both phase systems should be similar. We attempted to provide an interpretation of the correlation observed between the values of Es.
11090 J. Phys. Chem. B, Vol. 103, No. 50, 1999
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TABLE 3: Thermodynamic Properties of Phase Equilibrium and Surface Diffusion in Gas-Solid Adsorption System sample
-Qst (kJ mol-1)
(K/T)1/Tf0 (cm3 g-1 K-1)
Es (kJ mol-1)
Ds0 (cm2 s-1)
∆Hva (kJ mol-1)
-Qst/∆Hva
Es/-Qst
benzene toluene ethylbenzene p-xylene chlorobenzene pentane hexane heptane octane cyclohexane
35.3 40.7 43.8 44.6 40.5 31.7 33.4 38.2 43.4 33.0
9.6 × 10-7 4.2 × 10-7 2.9 × 10-7 2.5 × 10-7 6.3 × 10-7 9.8 × 10-7 1.5 × 10-6 7.1 × 10-7 2.9 × 10-7 2.0 × 10-6
13.5 17.2 23.2 21.1 25.7 10.2 14.1 14.4 17.1 15.6
9.0 × 10-4 3.6 × 10-3 2.2 × 10-2 1.2 × 10-2 4.8 × 10-2 3.7 × 10-4 1.7 × 10-3 1.7 × 10-3 3.8 × 10-3 2.0 × 10-3
33.8 38.0 42.2 42.4 41.0 26.4 31.6 36.6 41.5 33.0
1.04 1.07 1.04 1.05 0.99 1.20 1.06 1.04 1.05 1.00
0.38 0.43 0.53 0.47 0.63 0.32 0.42 0.38 0.39 0.47
a
At 298 K.
Analysis of the Correlation between Qst and Es in LiquidSolid Systems. In the absolute rate theory,36 the mechanism of molecular diffusion is assumed to consist of two different processes, a hole-making step followed by a bond-breaking (or jumping) step. The activation energy of molecular diffusion (Em) is thus given by m E m ) Em h + Eb
(17)
m where Em h and Eb are the activation energies corresponding to the hole-making and the bond-breaking steps in molecular m diffusion, respectively. It was shown that Em h is larger than Eb and approximately equal to one-third of the evaporation energy (∆Ev) of the solvent, and that Em b is between 10 and 20% of Em. Previously,8,37-39 we analyzed the correlation between Ds and the parameters of the retention process, i.e., K and Qst, and developed the restricted diffusion model as an approximation of the mechanism of surface diffusion. Surface diffusion was regarded as molecular diffusion restricted by the adsorptive interactions between the adsorbate molecules and the surface of the adsorbent. In this model, a hole is first formed by removing the proper number of solvent molecules from the potential field of adsorption. Then, the sample molecule is transferred from a neighbor adsorption site into this hole. Similarly to molecular diffusion, Es is assumed to consist of two contributions, those of the hole-making step (Esh) and of the bond-breaking step (Esb):
Es ) Esh + Esb
(18)
The value of Esh is probably correlated with ∆Ev for the solvent. From considerations based on the absolute rate theory, Esh was assumed to be nearly equal to Em in previous papers.8,38 The transfer of the sample molecule into the hole is assumed to require the temporary gain of the activation energy necessary for breaking the adsorption interaction between the molecule and the surface of the stationary phase. The value of Esb should be correlated with an adsorption enthalpy, Qst. On the basis of these considerations, the following equations was derived
Es ) Esh + β(-Qst)
(19a)
Es ≈ Em + β(-Qst)
(19b)
or
Thus, Es is the sum of Esh and of the contribution of the adsorption interactions, represented by β(-Qst). It seems that
Figure 3. Activation energy of surface diffusion as a function of isosteric heat of adsorption.
the activation energy of molecular diffusion without adsorptive interactions, Em, could be approximately converted into Es by the addition of the contribution of the adsorption interactions. The value of β was determined to be close to 0.4 for surface diffusion in RPLC.8,38 Figure 3 illustrates the correlation found between the experimental values of Qst and Es in RPLC. For both homologous series, Es increases almost linearly with increasing -Qst but is not equal to zero at Qst ) 0. It is likely that the results in Figure 3 validate eq 19a. From the intercept of the linear correlation for the alkylbenzenes, Esh is estimated at 17 kJ mol-1, which is nearly 80% of Em (21 kJ mol-1) and a little more than onethird of ∆Ev (43 kJ mol-1), estimated from the activation energy of the viscosity (Evis) of the mobile phase (70% methanol). These calculation results agree with the conclusion of the absolute rate theory derived for molecular diffusion. The value of Em for the sample compounds in 70% methanol was derived from the temperature dependence of Dm, estimated by eq 12. It was reported that the organic modifier (here, methanol) of an aqueous organic solution is preferentially adsorbed onto the surface of RPLC packing materials and that the surface excess of the organic modifier may reach 1020%.40,41 The value of Evis was calculated at 16.3, 16.0, and 15.6 kJ mol-1 for 70, 80, and 90% methanol, respectively, from the temperature dependence of the viscosity of methanol/water
Thermodynamic Characteristics of Surface Diffusion in RPLC
J. Phys. Chem. B, Vol. 103, No. 50, 1999 11091 although with some scatter. If the mechanism of surface diffusion is the same in gas-solid and in liquid-solid systems, Es in gas systems (Egs ) probably consists of the sum of the activation energy of diffusion without restriction (Eg0) and of the adsorption interaction, as follows
Egs ) Eg0 + β(-Qst)
(20)
The mechanism of surface diffusion in gas-solid systems seems to be related with molecular or Knudsen diffusion. The activation energy of the molecular and Knudsen diffusions, which corresponds to Eg0, were calculated at 3.6 and 1.4 kJ mol-1, respectively, from the temperature dependence of Dm and of the Knudsen diffusivity (Dk), coefficients conventionally estimated from the following Hirshfelder and Knudsen equations.14,15
Dm ) 0.1883 Figure 4. Correlation of frequency factor with activation energy of surface diffusion. The solid, dashed, and dotted lines represent the correlations between Ds0 and Es reported in a previous paper.8
solutions.1 The value of Esh, i.e., ca. 17 and 14 kJ mol-1 for alkylbenzenes and p-alkylphenols, respectively, is close to these values of Evis. The mechanism of molecular diffusion is similar to that of viscosity. The viscosity of a solvent results from the migration of its molecules among themselves. In molecular diffusion, however, different molecules migrate among the solvent molecules. Because the solute molecules are usually larger than the solvent ones, the generation of a larger-size hole may be necessary for molecular diffusion. The slightly larger value of Em compared with Evis may be due to the difference of size of the holes needed for both types of migration. The ratio Esb/-Qst is calculated at 0.55 and 0.67 for alkylbenzene and p-alkylphenol derivatives, respectively. The contribution of the adsorptive interactions between the sample molecules and the adsorbent surface to Es is represented by Esb. The ratio Esb/-Qst in Figure 3 is larger than the value of β ()0.4) previously reported.8,38 This difference probably arises from the different first terms in the right-hand side (rhs) of eqs 19a and 19b. In our previous study,8,38 Es was analyzed by eq 19b, in which Em ()21 kJ mol-1) was taken as equal to Esh. Figure 4 shows the correlation between Es and Ds0. Previously,8 we showed that, for surface diffusion as for retention equilibrium,42-44 enthalpy-entropy compensation was observed under various RPLC conditions. Similar correlations between Es and Ds0 are observed, irrespective of the chemical nature of the sample compounds, of the type and composition of the organic modifier, of the alkyl-chain length of the RPLC packing materials, and of the type of phase system used, i.e., gas-solid or liquid-solid. The mechanism of surface diffusion seems to remain essentially the same, regardless of the experimental conditions. The three straight lines in Figure 4 (solid, dashed, and dotted lines) represent results previously reported.8 The solid and dashed lines show the correlations for nonpolar and polar compounds, respectively, in a RPLC system. The dotted line represents the correlation between Es and Ds0 for nonpolar compounds in a gas chromatographic system using C18-silica gel as the adsorbent. The experimental results of this study are shown as symbols. They group around the straight lines,
x
T3(M1 + M2) M1M2 Pσ122ΩD
x8RT πM
2 Dk ) r p 3
(21)
(22)
where M is the molecular weight, P the pressure, σ the collision diameter, ΩD the collision function, and rp the radius of the micropores. The subscripts 1 and 2 denote each of the components of a binary gas system. The values of Dm and Dk estimated for the sample compounds listed in Table 3 were of the order of 1 × 10-1 and 1 × 10-3 cm2 s-1, respectively, indicating that Knudsen diffusion is a rate-controlling process. The value of β is again assumed to be positive and less than unity. Compared with the second term in the rhs of eq 20, the first term is relatively small. Because -Qst in gas systems ranges from ca. 32 to 45 kJ mol-1, as shown in Table 3, the contribution of the second term in the rhs of eq 20 to Egs is 16-23 kJ mol-1, if we assume β ) 0.5. This value is between ca. 11 and 16 times larger than that of the first term in the rhs of eq 20, i.e., Eg0 ) 1.4 kJ mol-1. When Eg0 can be neglected, eq 20 is written as
Egs ≈ β(-Qst)
(23)
Equation 23 suggests that the ratio Egs /-Qst is approximately equal to β in such cases. On the other hand, the ratio Es/-Qst in liquid-solid (RPLC) systems is obviously larger than unity and is obviously not constant (Table 2). In conclusion, the difference between the ratio Es/-Qst in gas-solid and in liquidsolid systems is due to the difference of the contribution of Esh to Es in eq 19a (liquid-solid) and of Eg0 to Egs in eq 20 (gassolid). The values of -Qst listed in Table 2 are between ca. 7 and 18 kJ mol-1, in agreement with those reported in many other cases in RPLC.8,19-25 The contribution of the second term in the rhs of eq 19a (β(-Qst)) was calculated by taking β ) 0.5 and found to be between ca. 3.5 and 9 kJ mol-1. The value of Esh in RPLC, found to be between ca. 14 and 17 kJ mol-1 in this study, is larger than these values of β(-Qst). The unreasonable ratio of Es to -Qst larger than unity is probably a result of the large contribution of Esh to Es in RPLC. It seems that the restricted diffusion model consistently proves the similarity of the migration mechanism of surface diffusion in both gas and liquid-phase adsorption systems, as illustrated in Figure 4.
11092 J. Phys. Chem. B, Vol. 103, No. 50, 1999
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Figure 6. Plot of surface diffusion coefficient against the ratio of boiling point to experimental temperature.
about 5 times larger than that of ∆ ln K0 in most combinations of two sample compounds. Similar results are derived from experimental data presented in previous papers.8,45-47 Equation 16 accounts for the temperature dependence of Ds and a similar conclusion is derived for Ds. The difference between values of ln Ds (∆ ln Ds) principally arises from ∆Es/ RT. In the restricted diffusion model, Es for two compounds, i and j, is written as follows, according to eq 19a Figure 5. Correlation between surface diffusion coefficient and adsorption equilibrium constant: (a) alkylbenzene, (b) p-alkylphenol.
At this stage, an appropriate interpretation is provided for the apparently unreasonable situation resulting of the thermodynamic correlation between Es and Qst observed in RPLC, a ratio Es/-Qst larger than unity. In the following, we discuss the validity of this conclusion from three points of view, the correlation between Ds and K, that between Ds and the boiling point (Tb) of the sample compounds, and that between Ds/Dm and Qst. Analysis of the Correlation between Ds and K. A linear free energy relation can be expected to hold when enthalpyentropy compensation is established. Figure 5, a and b, shows the correlations between ln Ds and ln K for alkylbenzenes and p-alkylphenols, respectively. The slope of these correlations is nearly the same, irrespective of the temperature. For example, these correlations at 298 K are expressed as follows
ln Ds ) -0.57 ln K - 12.8
(alkylbenzene) (24a)
ln Ds ) -0.44 ln K - 13.6
(p-alkylphenol) (24b)
The temperature dependence of K was given by eq 15. The difference in ln K (i.e., ∆ ln K) for two compounds is mainly due to the difference between their ∆Qst/RT, if their chemical and physical properties are similar. Because the value of K0 depends on the compound studied and differs from one to the next, ∆ ln K0 is not equal to zero. However, as suggested by the results in Table 2, the contribution of ∆Qst/RT to ∆ ln K is
s Es,i ) Eh,i + βi(-Qst,i)
(25a)
s + βj(-Qst,j) Es,j ) Eh,j
(25b)
In eqs 25a and 25b, the parameters Esh,i and Esh,j seem to be almost the same if the chemical and physical properties of the two compounds are similar. In such a case, ∆ ln Ds is associated with the difference between the second terms in the rhs of eqs 25a and 25b for these two compounds, difference which can be written β∆Qst, assuming that both βi and βj have the same value ()β). Ultimately, the slope of the linear correlation between ln Ds and ln K in Figure 5a,b should be almost equal to -β. As indicated in eqs 24a and 24b, β is estimated at 0.57 and 0.44 for alkylbenzenes and p-alkylphenol, respectively. No interpretation was yet provided for differences in the values of β which may depend on the chemical properties of the sample compounds. This is another subject of investigations for the restricted diffusion model of surface diffusion. Nevertheless, these last values of β and those derived from the slope of the linear correlations shown in Figure 3, i.e., 0.6, are close. Thus, the analysis of the linear free energy relation represented in Figure 5a,b demonstrates that β is close to 0.5 but that its exact value probably depends on the compound studied. Analysis of the Correlation between Ds and Tb. In Figure 6, ln Ds for the alkylbenzene is plotted against the ratio of their boiling temperature, Tb, to the experimental temperature. Irrespective of the temperature, parallel, linear correlations are observed. For example, the correlation at 298 K is expressed
Thermodynamic Characteristics of Surface Diffusion in RPLC
Figure 7. Isosteric heat of adsorption as a function of heat of vaporization.
by
(
Ds ) 1.7 × 10-4 exp
)
-3.8Tb T
(27)
The difference of Es (i.e., ∆Es) for two compounds is
∆Es ) 3.8R∆Tb
(28)
Figure 7 illustrates the correlation between -Qst and ∆Hv in gas and in liquid-phase systems. As described earlier, the experimental data for gas systems (symbols) are located between two straight lines corresponding to values of the ratio -Qst/ ∆Hv between 1.0 and 1.2. This shows that -Qst is slightly larger than ∆Hv and is explained by the additional attraction due to adsorption interactions between the sample molecules and the surface of the adsorbent. The values of -Qst for the liquidsolid system are relatively small in comparison. This is explained by the influence of the mobile phase on Qst. Although a linear correlation (solid line) is also observed between -Qst and ∆Hv in the liquid-solid system, the properties of the linear correlation (i.e., the values of its slope and intercept) differ from those of the gas system. The slope of the linear correlation is now 0.73. The difference between Qst for two sample compounds can be written
∆(-Qst) ) 0.73∆(∆Hv)
(29)
According to Trouton’s rule, ∆Hv is related to Tb through
∆Hv ) 88Tb
(30)
From eqs 28 to 30, we can derive that the ratio ∆Es/∆(-Qst) is given by
∆Es ∆(-Qst)
)
3.8R ) 0.49 0.73 × 88
compounds, β can be derived from the ratio of ∆Es ()|Es,i Es,i|) to ∆Qst ()|Qst,i - Qst,j|), as described earlier. From eq 31, ∆Es/∆(-Qst) ) 0.49, a value close to that estimated for β from the previous results in Figures 3 and 5a. It is slightly smaller than the slope of the straight line for the alkylbenzenes in Figure 3. This straight line was drawn in the assumption that the activation energy of the hole-making step, Esh, is the same for all the alkylbenzenes. However, it is likely that Esh depends on the molecular size of the sample compounds. For instance, it can be predicted that Esh is larger for hexylbenzene than for benzene (note that the molar volume of benzene and hexylbenzene at their respective normal boiling point are respectively 96 and 229 cm3 mol-1). Although we do not have more accurate information about Esh, a value of β smaller than 0.6 might be obtained if the influence of the molecular size on Esh could be taken into account. Thus, the examination of this correlation suggests that β is probably around 0.5-0.6. Analysis of the Correlation between Ds/Dm and Qst. Similar to surface diffusion, molecular diffusion is also considered as an activated process. The temperature dependence of Dm is also described by the Arrhenius equation.
(26)
Equation 26 is equivalent to the conventional Arrhenius equation (eq 16). Comparing eqs 16 and 26 gives for Es
Es ) 3.8RTb
J. Phys. Chem. B, Vol. 103, No. 50, 1999 11093
(31)
On the other hand, eqs 25a and 25b give Es for two similar compounds. When the activation energies of the hole-making step, i.e., Esh,i and Esh,j, and β are almost the same for two
Dm ) Dm0 exp
( ) -Em RT
(32)
where Dm0 is the frequency factor of molecular diffusion. In previous papers,8,37 we pointed out the presence of a correlation between surface and molecular diffusion. The restricted diffusion model was developed for interpreting the mass transfer mechanism of surface diffusion. By combining eqs 16 and 32, the following equation is derived
[
]
Ds0 -(Es - Em) Ds ) exp Dm Dm0 RT
(33)
If Es is expressed as in eq 19b, eq 33 can be rearranged as follows
[ ( )]
Ds0 Qst Ds ) exp Dm Dm0 RT
β
(34)
According to eq 34, ln(Ds/Dm) is plotted against Qst in Figure 8a,b. The correlation of the experimental data is nearly independent of the temperature. The linear correlation between ln(Ds/Dm) and -Qst for alkylbenzenes at 298 K is the following:
( )
ln
Ds ) -0.12(-Qst) - 0.61 Dm
(35)
From the slope of the straight line, we obtain β ) 0.31 (i.e., )0.12 × R × 298). This value of β is smaller than those previously derived. Although it is not entirely in agreement with the previous results, it still demonstrates at least that β is positive and smaller than unity. As suggested by eq 34, the intercept of the linear correlation between ln(Ds/Dm) and -Qst corresponds to the ratio of Ds0 to Dm0 if Ds0/Dm0 is constant, irrespective of the compounds. From the intercept in Figure 8a,b, the ratio Ds0/Dm0 is estimated at around 0.5. On the other hand, Figure 9 illustrates the correlation between Ds0/Dm0 and Qst. The values of Ds0 and Dm0 were calculated from the temperature dependences of Ds and Dm, by applying the Arrhenius equation. Most values of the ratio Ds0/ Dm0 for the alkylbenzene and p-alkylphenol homologues seem
11094 J. Phys. Chem. B, Vol. 103, No. 50, 1999
Miyabe and Guiochon intercept of the straight lines in Figure 3. These values are smaller than Em ()ca. 21 kJ mol-1) estimated from the temperature dependence of Dm. When eq 34 was derived, the difference between Esh and Em was ignored. The difference between the values of the ratio Ds0/Dm0 calculated from the intercept of the linear correlations in Figure 8a,b and those plotted in Figure 9 probably stems from this difference between Esh and Em. For example, for the alkylbenzenes, eq 34 should be written as follows:
( )[ ( )]
Ds0 Ds Qst 4 ) exp exp Dm Dm0 RT RT
Figure 8. Correlation between the ratio of surface diffusion coefficient to molecular diffusivity and isosteric heat of adsorption: (a) alkylbenzene, (b) p-alkylphenol.
Figure 9. Correlation of the ratio of frequency factor of surface diffusion to that of molecular diffusion with isosteric heat of adsorption.
(36)
The additional term in the rhs of eq 36, i.e., exp(4/RT), is equal to nearly 5 at T ) 298 K. Although this term cannot directly explain the difference observed for the ratio Ds0/Dm0 discussed above, it is one of the causes for this difference. Figure 9 also suggests that the ratio Ds0/Dm0 tends to increase with increasing retention. This observation implies that the value of β is primarily around 0.5-0.6, as explained now. If we plot the values of Ds0/Dm0 in Figure 9 on the ordinate of Figure 8, a and b, including the contribution of the additional parameter just introduced, i.e., exp(4/RT), and if we connect the corrected value of Ds0/Dm0 ()(Ds0/Dm0) exp(4/RT)) with the corresponding value of Ds/Dm plotted in Figure 8, a and b, the slope of the straight line linking the two points, i.e., (Ds0/Dm0) exp(4/RT) and Ds/Dm, is equal to β. For example, the value of Ds0/Dm0 for benzene is 0.24 (Figure 9). On the other hand, Ds/Dm for benzene at 298 K is 0.23 (Figure 8a). If we connect the two points according to eq 36, i.e., 1.2 ()0.24 × 5) at -Qst ) 0 kJ mol-1 and 0.23 at -Qst ) 6.8 kJ mol-1, the value of β is calculated at 0.60 ()(298R/6.8) × ln(1.2/0.23)) at 298 K. Similarly, the values of Ds0/Dm0 and Ds/Dm for hexylbenzene at 298 K are respectively 0.96 and 0.063 from Figures 9 and 8a. Because -Qst for hexylbenzene is 17.5 kJ mol-1, β is calculated at 0.61 () (298R/17.5) × ln(0.96 × 5/0.063)) at 298 K. In addition, from Figures 9 and 8b, Ds0/Dm0 and Ds/Dm for p-hexylphenol at 298 K are 0.43 and 0.084, respectively. The value of β is calculated at 0.67 ()(298R/16.5) × ln[(0.43/0.084) exp(7/RT)]) at 298 K from these data and from -Qst ()16.5 kJ mol-1). These results, of course, agree with those previously derived from the plots in Figure 3. Although the calculations described above are merely based on eq 36, they show that the difference between the values of β estimated from the results in Figure 8, a and b, and from those in Figures 3, 5a, 5b, and 6 is ultimately attributed to the omission of the parameter in eq 36, exp(4/ RT), and to the assumption that the ratio Ds0/Dm0 is constant for all the compounds studied. In previous papers,8,37 we analyzed surface diffusion data in RPLC and showed that there is a correlation between Ds and Dm, Ds being almost equal to Dm under the limit conditions that the adsorption interactions between sample compound and surface of the stationary phase is negligibly small. From these results, we derived the restricted diffusion model for interpreting the mechanism of surface diffusion and proposed an estimation procedure of Ds by using the following equation.37
( )
Ds ) Dm exp to range between 0.2 and 0.3, and between 0.1 and 0.2, respectively. These values are different from those suggested by the data in Figure 8a,b. However, Esh for alkylbenzenes and p-alkylphenols was calculated at about 17 and 14 kJ mol-1, respectively, from the
β
-Er RT
(37)
where Er is the restriction energy for molecular diffusion due to the adsorption interaction between the sample compound and the surface of the stationary phase. This restriction energy is correlated with Qst by introducing an empirical parameter (γ)
Thermodynamic Characteristics of Surface Diffusion in RPLC
Er ) γ(-Qst)
J. Phys. Chem. B, Vol. 103, No. 50, 1999 11095
(38)
From eqs 36-38, the following equations are derived for the series of alkylbenzenes
Y)
( )
Ds0 -4 Dm0 +β -Qst
-RT ln
(39)
The slope of the linear correlations in Figure 8a,b represents the value of γ, which is calculated at 0.31, as described earlier. In a previous paper,37 we reported the same value of γ ) 0.32 for compounds of various kinds. Although γ is related to β as indicated in eq 39, the contribution of the parameters in eq 36, i.e., Ds0/Dm0 and exp(4/RT), to Ds/Dm is not taken into account. The meaning of γ, previously37 introduced as an empirical parameter, is provided in connection with β. The experimental data concerning Ds0 illustrated in Figure 8a,b and 9 are probably interpreted as described above. However, it is unlikely that a strict calculation and a rigorous explanation of Ds0 could be provided at this stage. The limited number of the experimental data does not allow a final conclusion. A detailed analysis of the value and characteristics of Ds0 will be the topic of further investigations. Comparison of the Thermodynamic Properties of Phase Equilibrium and Surface Diffusion. Figure 10 illustrates the linear correlations between ln K0 and Qst for the alkylbenzenes and the p-alkylphenols. It shows an enthalpy-entropy compensation in the phase equilibrium. However, the linear correlations obtained for both homologous series do not agree with each other. In addition, as suggested by the data in Tables 2 and 3, the value of K0 in gas-solid systems is probably a few orders of magnitude smaller than that in liquid-solid systems.8,32,35 These properties of the enthalpy-entropy compensation for the phase equilibrium are quite different from those obtained for surface diffusion and shown in Figure 4. The experimental values of Ds0 and Es obtained under various conditions in RPLC and in gas chromatography can be represented by nearly the same correlations, as illustrated in Figure 4. The retention behavior seems to be more extensively influenced by various factors such as the adsorption interactions between the sample molecules and the surface of a stationary phase, and the influence of the mobile phase on the attraction interactions, than surface diffusion. However, many theories and models, for instance, the lattice theory and the solvophobic theory, were already proposed for the consistent interpretation of the retention behavior in RPLC.48 On the other hand, the value of Ds depends on the combination of effects from the sample compound, the stationary and the mobile phase involved in the separation system. However, a comprehensive interpretation of the migration mechanism and the thermodynamic properties of surface diffusion may become feasible by applying the restricted diffusion model. According to the restricted diffusion model, Es is assumed to be the sum of the activation energy for the hole-making process (Esh), close to Evis for the mobile phase, and the contribution of the adsorption interactions, represented by β(Qst). As indicated in Tables 2 and 3, the values of Es determined in liquid-solid (RPLC) systems are roughly the same as those in corresponding gas-solid (gas chromatography) systems. However, as described earlier, the contributions of Esh or Eg0 and β(-Qst) to Es or Egs seem to be different in gas-solid and liquid-solid systems. It is likely that, in spite of the different contributions of each activation energy in the two types of phase
Figure 10. Correlation of K0 with isosteric heat of adsorption. Symbols: refer to Figure 1.
systems, similar values of Es were coincidentally measured. This coincidence of sets of numerical values of Es is probably not essential. The restricted diffusion model has proven its usefulness for providing consistent explanations of the intrinsic thermodynamic properties and of the mass transfer mechanism of surface diffusion in liquid phase adsorption. Conclusion Some characteristic features of the thermodynamics of surface diffusion in RPLC are now established and related to those of retention. The activation energy, Es, is the sum of two contributions. The first is the activation energy of the holemaking step (Esh), which is close to that of the mobile phase viscosity (Evis). The other contribution is due to adsorption interactions. It is correlated with Qst and represented by β(Qst). The value of β was shown to be around 0.6 from the slope of the correlation between Es and -Qst. These results substantiate the restricted diffusion model, proposed as a first approximation of the mechanism of surface diffusion in liquid phase adsorption system. In order to prove the validity of the model, experimental data regarding phase equilibrium and surface diffusion in RPLC were analyzed from different points of view. There are correlations between ln Ds and ln K (linear free energy relationship), between ln Ds and Tb/T, and between ln(Ds/Dm) and Qst. The results confirmed that β is close to 0.5-0.6. Finally, the restricted diffusion model provides an appropriate interpretation for the apparently unreasonable correlation observed between the thermodynamic properties in RPLC systems, that the ratio Es/-Qst is larger than unity. The thermodynamic properties of phase equilibrium and surface diffusion in RPLC were compared with those in similar gas chromatographic systems. Although almost the same values of Es were observed in gas-solid and in liquid-solid systems, this coincidence seemed accidental because it was predicted that the two contributions to the activation energy, Es, were different in the two-phase systems. The significance of the slight difference between the values of Es obtained in gas and in liquid phase adsorption systems could be appropriately explained by the restricted diffusion model. It is expected that this model
11096 J. Phys. Chem. B, Vol. 103, No. 50, 1999 can provide a comprehensive interpretation for the mass transfer mechanism and the thermodynamic properties of surface diffusion. Acknowledgment. This work was supported in part by Grant CHE-97-01680 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. Glossary Ce(t) De Dk DL Dm Dm0 dp Dp Ds Ds0 Eb Eh Em Er Es Evis E0 H ∆Hv H0 k K kf K0 M P Qst R Re rp Rp Sc Sh t T Tb t0 u0 Vb z
elution profile of chromatographic peak as a function of t intraparticle diffusion coefficient, cm2 s-1 Knudsen diffusivity, cm2 s-1 axial dispersion coefficient, cm2 s-1 molecular diffusivity, cm2 s-1 frequency factor of molecular diffusion, cm2 s-1 particle diameter, cm pore diffusivity, cm2 s-1 surface diffusion coefficient, cm2 s-1 frequency factor of surface diffusion, cm2 s-1 activation energy of bond-breaking (jumping) process, kJ mol-1 activation energy of hole-making process, kJ mol-1 activation energy of molecular diffusion, kJ mol-1 restriction energy for molecular diffusion, kJ mol-1 activation energy of surface diffusion, kJ mol-1 activation energy of viscosity, kJ mol-1 activation energy of diffusion without any restriction, kJ mol-1 defined by eq 9 heat of vaporization, kJ mol-1 defined by eq 10 tortuosity factor adsorption equilibrium constant, cm3 g-1 fluid-to-particle mass transfer coefficient, cm s-1 adsorption equilibrium constant at 1/T ) 0 or Qst ) 0, cm3 g-1 molecular weight pressure, Pa isosteric heat of adsorption, kJ mol-1 gas constant, J mol-1 K-1 Reynolds number pore radius, cm particle radius, cm Schmidt number Sherwood number time, s absolute temperature, K boiling point, K defined by eq 8 superficial velocity, cm s-1 molecular volume at normal boiling point, cm3 mol-1 longitudinal position in bed, cm
Greek Letters R β γ δ0 δax δd δf
association coefficient ratio of the contribution of adsorptive interaction to Es to -Qst ratio of Er to -Qst defined by eq 3 contribution of axial dispersion to µ′2, s contribution of intraparticle diffusion to µ′2, s contribution of fluid-to-particle mass transfer to µ′2, s
Miyabe and Guiochon p η µ1 µ′2 Fp σ ΩD
interparticle void fraction intraparticle porosity viscosity, Pa s first absolute moment, s second central moment, s2 particle density, g cm-3 collision diameter, cm collision function
Superscripts g m s
gas-solid-phase system molecular diffusion surface diffusion
Subscripts i j s sv 1 2
compound compound sample solvent compound compound
i j
1 2
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