Solvent Strength Parameters and Retention Factors in Pure Water

It is well-known in reversed-phase liquid chromatography that the solute retention factor (k) in binary eluents is well modeled as a quasi-linear func...
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Ind. Eng. Chem. Res. 2003, 42, 6320-6330

Solvent Strength Parameters and Retention Factors in Pure Water Using UNIFAC-Predicted Activity Coefficients Jung Hag Park,*,† Yun Jung Jung,† Mark F. Vitha,*,‡ and Peter W. Carr*,§ Department of Chemistry, Yeungnam University, Kyongsan 712-749, South Korea, Department of Chemistry, Drake University, Des Moines, Iowa 50311, and Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455

It is well-known in reversed-phase liquid chromatography that the solute retention factor (k) in binary eluents is well modeled as a quasi-linear function of the eluent composition (Φ) by the equation log k ) log ks/w - SΦ, where the solute parameters ks/w and S are the retention factor in pure water and the solvent strength parameter, respectively, and Φ is the volume fraction of the organic component of the eluent. S is related to the free energy of the solute transfer from water to bulk organic liquid, which in turn is related to the infinite-dilution activity coefficients (γ∞) of the solute in water and bulk organic liquid. We report that the γ∞ values obtained from a γ∞ prediction model, UNIFAC, can be used to predict S directly and also to aid in the determination of best-fit ks/w values for acetonitrile-water mobile-phase systems. More specifically, UNIFAC-based S values combined with experimentally determined ks/w values for a limited set of solutes can be used to predict the variations in retention as a function of mobile-phase composition in acetonitrile-water systems within the calibration range for solutes that were not in the original data set. 1. Introduction The UNIFAC (UNIQUAC functional group activity coefficient) model is an activity coefficient estimation scheme that has found widespread applicability in chemical engineering.1-3 There have been a number of studies on the application of UNIFAC for estimation of retention in gas and liquid chromatography.4-11 We showed that UNIFAC is generally applicable as a heuristic guide for understanding the magnitude of solute-solvent interactions in gas chromatography and normal and reversed-phase liquid chromatography (RPLC).12,13 It is fitting in this issue to explicitly recognize the significant contributions made by Eckert et al. to the analysis of UNIFAC and to the measurement, collection, and prediction of limiting activity coefficients.14-23 UNIFAC combines the UNIQUAC model of solutions24 and the so-called analytical solution of group concept.25 UNIQUAC (universal quasi-chemical) is a model of liquid mixtures developed by application of Guggenheim’s quasi-chemical lattice model of liquid mixtures26 through the use of a component’s local area fraction as the main concentration variable. Activity coefficients of solutes in binary and multicomponent liquid mixtures are computed using two descriptive parameters. One parameter is a structural parameter24 derived from the van der Waals surface areas and volumes of the functional groups27 and the other parameter, which is derived from banks of experimental phase equilibrium data,28 is a binary group-interaction parameter that characterizes the energy of mutual interaction of functional groups present in the mixture. * To whom correspondence should be addressed. Tel.: +82 53 810 2360. E-mail: [email protected]. Tel.: (515) 271-2596. E-mail: [email protected]. Tel.: (612) 624-0253. Email: [email protected]. † Yeungnam University. ‡ Drake University. § University of Minnesota.

Bastos et al. developed a set of UNIFAC parameters based solely on infinite-dilution activity coefficient (γ∞) data.29 It is well-known that the very large errors in predicted γ∞ values for nonpolar solutes in associating solvents are mainly due to errors in group-interaction parameters. This is a consequence of the fact that the parameters in the original version of UNIFAC were derived primarily from data at high concentrations.30 Thus, in principle, the use of infinite-dilution interaction parameters should improve the prediction accuracy. However, it was observed that the use of infinitedilution interaction parameters does not improve the predictive accuracy and gives worse predictions than the original UNIFAC.8,31 Two different modified UNIFAC methods, both of which show substantially improved accuracy for the prediction of γ∞, have been reported in this journal.32,33 However, where the data allow, three interaction parameters between two interacting groups are used in the modified approach, whereas only one parameter was used in the original version of UNIFAC. Thus, the applicability of the modified method is reduced because all three values must be known for any given set of functional groups, and currently, the number of available group-interaction parameters in these modified UNIFAC methods is smaller than that in the original version of UNIFAC. We also note that the threeparameter model becomes equivalent to the oneparameter version if the two additional parameters are set to zero. Given the limited predictive accuracy of the parameters derived from infinite-dilution activity coefficients and the limited number of available group-interaction parameters for the three-parameter model, we have chosen to use the original UNIFAC method2,28 in this work, combined with the revised functional group parameters from vapor-liquid equilibria by Hansen et al.34 UNIFAC and the Prediction of RPLC Retention. As we have shown, UNIFAC-based models are not

10.1021/ie0209125 CCC: $25.00 © 2003 American Chemical Society Published on Web 05/23/2003

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accurate enough, at this stage of its development, to be useful for quantitative prediction of retention.12,13,30 Because chromatography is able to distinguish between two types of solute molecules that differ in the strength of their interactions with the mobile and stationary phases by only a few calories per mole, there is as yet no general predictive method that suffices for use in chromatography. Nonetheless, UNIFAC is sufficiently accurate for predicting the relative strengths of solvents, the relative retention of solutes, and other important chromatographic properties.7,8,12,13 Several studies on the application of UNIFAC to predicting retention in RPLC have appeared in the literature.9-13 Retention prediction and selectivity optimization are important for rapid method development in RPLC.35 However, retention in RPLC is a complicated process36-40 and depends on many physical and chemical properties of the system such as temperature,41-43 solute molecular properties,44 stationary-phase characteristics,45 and mobile-phase properties.46 Although an accurate retention model for RPLC has not yet been developed, a number of practical retention models46-50 such as linear solvent strength theory and linear solvation energy relationships have been developed and widely used. In binary eluents on a RPLC column, the retention factor (k) of a single solute can be modeled as a quasi-linear function of the volume fraction of the organic component of the eluent (Φ) over a limited yet useful range of compositions:48-50

where ks/org denotes the retention factor in a purely organic eluent (i.e., solute transfer from a pure organic solvent to the stationary phase). From eqs 1, 4, and 5 and still assuming that the stationary-phase volume is the same when it is in equilibrium with either pure water or a pure organic solvent, the relationship between S and the free energy of solute transfer from water (w) to pure organic (org) solvent (∆G°org/w) can be shown to be

log k ) log ks/w - SΦ

∆G°org/w ) -2.3RTS ) -2.3RT log Korg/w

(1)

where ks/w is the hypothetical retention factor of the solute when pure water is used as the eluent and S is a solvent strength parameter that reflects the degree to which the retention factor of each solute changes for a given change in the mobile-phase composition. The parameters ks/w and S can be obtained by measuring the retention of a solute as a function of the mobilephase composition followed by a linear analysis based on eq 1. The approximate nature of eq 1 must be understood. Jandera et al. pointed out that simple solubility parameter theory requires a quadratic relationship.51,52 Dorsey et al.53-55 and others56-58 substantiate the fact that eq 1 is never exact over the entire range of mobile-phase compositions. Further evidence of the limitations of eq 1 is provided by the fact that it is linear over only a limited range in composition and the fact that the value of ks/w obtained by linear regression analysis varies substantially with the type of mobile-phase modifier.47,59,60 This should not be the case because ks/w should represent the retention factor of the solute in pure water independent of any mobile-phase modifiers. Derivations. Putting aside the approximate nature of eq 1, we consider the fundamental meaning of the two model parameters, log ks/w and S. First, ks/w is the hypothetical retention factor that the solute would have in a purely aqueous eluent. In general, for any partition process, k is related to the equilibrium constant (Ks/m) for the process as follows:

log k ) log(Vs/Vm) + log Ks/m

(2)

where Vs and Vm are the volumes of the stationary phase (s) and mobile phase (m), respectively. Ks/m is the partition coefficient for the solute transferring from the mobile phase to the stationary phase and can be

calculated from the solute activity coefficients in those two phases. Furthermore, it is related to the free energy of transfer of the solute from the mobile phase to the stationary phase via the relationship

∆G°s/m ) -2.3RT log Ks/m ) -2.3RT log(kVm/Vs) (3) If the mobile phase is pure water, the relationship in eq 3 becomes

∆G°s/w ) -2.3RT log Ks/w ) -2.3RT log(ks/wVm/Vs) (4) For a pure organic solvent as the mobile phase (Φ ) 1), assuming that the phase ratio, Vs/Vm, is the same as it is in the presence of pure water, combining eqs 1-4 yields the relationship

S ) log ks/w - log ks/org

(5)

(6)

where Korg/w is the partition coefficient for solutes transferring from pure water to a pure organic solvent. Finally, Korg/w can be related to the infinite-dilution activity coefficients in pure water and a bulk organic phase via the equation

Korg/w ) γ∞wV h w/γ∞orgV h org

(7)

where V h denotes the molar volume. Equation 6 shows that S is directly proportional to log Korg/w if the model is perfect and S is independent of the stationary phase. Combining eqs 6 and 7 results in eq 8.

h w/γ∞orgV h org) S ) log(γ∞wV

(8)

Thus, if γ∞w, γ∞org, and γ∞s can be predicted, then S can be directly computed from eq 8 and best-fit values for log ks/w can be obtained by correlating experimental log ks/w values with UNIFAC-based Ks/w values according to the relationship in eq 4. Furthermore, eq 6 shows that there should be a direct correlation between S and log Korg/w that can be tested by regressing experimentally determined S values against UNIFAC-based Korg/w values. To these ends, in this work we do the following: (1) Correlate experimentally measured retention factors with the mobile-phase composition according to eq 1 to obtain experimentally determined S and log ks/w values. (2) Use UNIFAC to calculate infinite-dilution activity coefficients and use them to generate UNIFAC-based log Korg/w and S values according to eqs 7 and 8. (3) Correlate experimentally determined S values in acetonitrile (ACN), methanol (MeOH), and tetrahydrofuran (THF) systems with log Korg/w values according to eq 6.

6322 Ind. Eng. Chem. Res., Vol. 42, No. 25, 2003 Table 1. List of Solutes and Their Assigned Numbers no.

solute

no.

solute

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

diethyl ether ACN 2-propanol MeOH 1-butanol cyclohexanol acetone 2-butanone cyclopentanone 2-hexanone n-propyl formate n-butyl acetate ethyl propionate ethyl butyrate n-propionitrile n-nitropropane n-valeronitrile butyraldehyde 2,2,2-trifluoroethanol methylene chloride chloroform dibromomethane N,N-dimethylformamide N,N-diethylformamide dimethyl sulfoxide N,N-dimethylacetamide N,N-diethylacetamide dioxane benzene toluene benzaldehyde

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 53 55 56 57 58 59 60 61

acetophenone propiophenone benzonitrile m-toluenenitrile nitrobenzene m-nitrotoluene anisole methyl benzoate ethyl benzoate phenol m-cresol benzyl alcohol 2-phenylethanol 3-phenylpropanol N-benzylformamide methyl phenyl sulfoxide fluorobenzene chlorobenzene bromobenzene benzophenone benzyl cyanide benzyl bromide p-nitrobenzyl bromide p-nitrobenzyl chloride o-nitrotoluene p-nitrotoluene p-cresol o-cresol p-ethylphenol p-chlorophenol

(4) Use octane and octane modified with organic solvents as model stationary phases, coupled with UNIFAC infinite-dilution activity coefficients in these model phases, to generate UNIFAC-based log Ks/w values. (5) Correlate experimentally determined log ks/w values with UNIFAC-based log Ks/w values according to eq 4, and from these correlations, generate best-fit values for log ks/w. (6) Use the best-fit log ks/w values combined with the UNIFAC-based S values to ultimately predict log k as a function of the mobile-phase composition according to eq 1. (7) Compare these UNIFAC-based predictions of log k values with experimentally determined log k values for solutes included in the original training set and for solutes not in the original data set in order to assess the value of this approach for predicting solute retention in RPLC. By doing the above, we show that infinite-dilution activity coefficients predicted using UNIFAC can be combined with experimental data to semiquantitatively predict the change in solute retention as a function of the percent organic modifier, even for solutes not included in the original data set of retention factors. However, as we will show, while this approach works for ACN systems, it does not yield useful information for MeOH or THF systems. 2. Experimental Section Solutes were judiciously chosen to span a wide range in solute properties in terms of size, dipolarity, and hydrogen-bonding strength. Solute names with identification numbers are listed in Table 1. The same set of solutes could not be used with all mobile-phase modifiers because of extremely short or long retention times or because of the lack of parameters for some solutesolvent pairs for UNIFAC calculations. However, the same set of solutes was used for each specific type of mobile phase at all compositions.

The retention data used in eq 1 to obtain S and log ks/w were taken from the literature61 and were all obtained using the same octylsilane RPLC column. The retention data were collected at 25 °C ((0.1 °C) using a circulating water bath with eluents at four volume-tovolume ratios (20%, 30%, 40%, and 50%) for ACN and THF and at five volume-to-volume ratios (10%, 20%, 30%, 40%, and 50%) for MeOH on Zorbax-C8 (particle size, 5 µm; pore size, 10 nm) as the stationary phase. The retention factors were correlated with mobilephase composition according to eq 1 to obtain experimentally determined S and log ks/w values. UNIFAC was used to calculate infinite-dilution activity coefficients that are then used to compute log Korg/w and log Ks/w values. UNIFAC-based S values were computed from eq 8. Experimentally determined S and log ks/w values were then correlated with UNIFAC-based log Korg/w and log Ks/w, respectively, to find the expected linear relationships based on eqs 4 and 6. The linear relationships based on eq 4 are then used to obtain best-fit log ks/w values. Finally log k values for solutes were computed according to eq 1 by using UNIFAC-based S values and best-fit log ks/w values. These computed log k values were then compared with experimentally determined log k values to allow comment on the approach of combining UNIFAC with a limited set of experimentally obtained retention data to predict solute retention as a function of the mobile-phase composition. 3. Results and Discussion 3.1. Correlations of Experimental S Values with Literature Korg/w Values. The partition coefficient, Korg/w, can be calculated from values of γ∞ for the solute in pure organic modifier and water phases according to eq 7.62 Equation 5 shows that S should be directly proportional to log Korg/w with a slope of unity and an intercept of zero if the model (eq 1) is perfect, S is independent of the stationary phase, and UNIFACcomputed γ∞ values are error-free. Presented below are correlations between the S values obtained by employing eq 1 to analyze experimental RPLC retention data and log Korg/w values obtained using eq 7. In these regressions, the slopes are allowed to be nonunity and the intercepts are allowed to be nonzero. The results for the three aqueous mobile-phase systems studied, ACN (a), MeOH (m), and THF (t), are as follows:

Sa/w ) 0.31((0.15) + 0.85((0.06) log Ka/w (n ) 30, r ) 0.936, s.d. ) 0.33)

(9)

Sm/w ) 1.40((0.15) + 0.63((0.08) log Km/w (n ) 29 r ) 0.841, s.d. ) 0.36)

(10)

St/w ) 0.94((0.24) + 0.60((0.10) log Kt/w (n ) 36, r ) 0.773, s.d. ) 0.77)

(11)

In these equations, n is the number of solutes used, r is the correlation coefficient, and s.d. is the standard error of the fit. The data for 2,2,2-trifluoroethanol and chlorobenzene in ACN-water, methylene chloride, chloroform, and dimethyl sulfoxide in MeOH-water, and diethyl ether, p-dioxane, chlorobenzene, p-ethylphenol, and p-chlorophenol in THF-water were obvious outliers based on Student’s t test and Cook’s distance63 and thus

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Figure 1. Plots of experimental S vs log Korg/w: (a) ACN-water, (b) MeOH-water; (c) THF-water. The numbers in the figure are the solute numbers. For solute names, see Table 1.

Figure 2. Plots of residuals vs solute numbers for correlations of S with log Korg/w: (a) ACN-water; (b) MeOH-water; (c) THFwater.

were excluded in the regressions. Only the correlation for the ACN-water system is deemed satisfactory. Figure 1 shows plots of S vs Korg/w for the three mobilephase systems. The solid lines are least-squares lines. Figure 2 is a plot of residuals vs solute numbers for correlations of S vs log Korg/w. Residuals for different solutes are virtually randomly distributed, and no classspecific, systematic deviations are observed. We note that all of the slopes in the above fits are nonunity and that the slopes and standard deviations decrease in the order ACN > MeOH > THF. This order is the reverse of that for the degree of solubility of an organic modifier in hexadecane (a bulk organic solvent model of the stationary phase).64 A possible explanation for the different behavior of the three systems begins by noting that in order to obtain eq 5 it is necessary to assume that S is only dependent upon log Korg/w and

thus should be independent of the stationary phase. However, it is well-known that stationary phases are substantially modified by sorption of the organic modifier. Furthermore, the extent of modification is greater in THF-water and MeOH-water than in ACNwater.65,66 It must also be considered that the poor correlations could result from inaccuracies in the activity coefficients estimated by UNIFAC. However, it has been shown that UNIFAC can predict elution sequences for various solutes and even curvatures in plots of log Ks/m vs Φ when the entire mobile-phase composition range is considered,12,13 which is in a good agreement with the quadratic relationship of log k (and K) with Φ proposed by Schoenmakers et al. and accepted by many practicing chromatographers.35,59,60 One can thus safely assume that UNIFAC can give activity coefficients that are good enough to be used to predict retention variations and that should, therefore, be generally well-correlated to experimental S values. However, while UNIFAC can accomplish these tasks, it is not good enough to predict the absolute retention of solutes as a function of the mobile-phase composition. 3.2. Correlation of Best-Fit log ks/w with UNIFAC-Based log Ks/w. There have been a number of studies on the retention mechanism in RPLC, and three main theories have evolved. These include the solvophobic theory by Horvath et al.36-38 in which retention is thought to occur through an adsorption-like process rather than a partitioning-like process, the statistical mechanical partitioning model by Martire and Boehm,67 and the interphase partitioning model of Dorsey, Dill, and co-workers.68-70 The last two models argue for and demonstrate both theoretically and experimentally that partitioning is the relevant model of RPLC retention. Although chromatographic retention cannot be very well-modeled by liquid-liquid partitioning,65 a pure partitioning model is adopted in this work for the sake of simplicity. Given this approximation, partition coefficients can be used for the assessment of retention factors because retention factors are directly proportional to partition coefficients (k ) KVs/Vm).36

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Figure 3. Plots of log ks/w vs log Koct/w: (a) ACN-water; (b) MeOH-water; (c) THF-water.

We model the stationary phase as octane, which can be regarded as a bulk analogue of the C8-bonded silica used for the retention measurements. However, recognizing that MeOH, ACN, and THF are soluble in octane64 and thereby modifying the properties of this model stationary phase, we modeled the C8-bonded phase as octane with 1 mol % ACN, 1 mol % MeOH, and 2 mol % THF because THF is more soluble in octane than are ACN and MeOH.64 We note that, even if a considerable amount of organic modifier is present in octane, modeling the stationary phase as octane with no or low concentrations of sorbed modifier is still a good approximation because the change in the activity coefficient of the solute caused by the presence of an organic modifier in octane is very small71 and thus does not alter partition coefficients appreciably. Yonker et al.65,66 have shown that chromatographic stationary phases in contact with aqueous organic mobile phases are solvated not only by association of the organic modifier but also by water associated with residual silanol groups on the silica substrate. While each of the three organic modifiers will have different effects on the stationary phase, we make the approximation that, because the same column was used for all of the retention measurements, the extent of solvation by water is essentially equal in the three systems. Therefore, we do not worry about the small amount of water that could exist in the model octane phase but rather focus only on the changes induced by sorption of the organic modifiers. On the basis of eq 4, we attempted to correlate log ks/w with two different sets of UNIFAC-computed Ks/w values: specifically, Koct/w for solute transfer from water to pure octane and Koct+org/w for solute transfer from water to octane modified by the organic additives (99 mol % octane + 1 mol % ACN and MeOH, 98 mol % octane + 2 mol % THF), which we

calculated as follows:

Koct/w ) γ∞wV h w/γ∞octV h oct

(12)

∞ h w/γoct+org V h oct+org Koct+org/w ) γ∞wV

(13)

where V h denotes the molar volume and V h oct+org is computed as XoctV h oct + XorgV h org, with X denoting the mole fraction. Pure Octane as the Model Stationary Phase. Results of linear regressions of log ks/w vs log Koct/w for ACN-, MeOH-, and THF-water systems are as follows:

log ks/w(a/w) ) 0.55((0.11) + 0.43((0.06) log Koct/w (n ) 31, r ) 0.810, s.d. ) 0.44)

(14)

log ks/w(m/w) ) 0.97((0.12) + 0.37((0.07) log Koct/w (n ) 27, r ) 0.712, s.d. ) 0.54)

(15)

log ks/w(t/w) ) 0.47((0.12) + 0.46((0.07) log Koct/w (n ) 33, r ) 0.780, s.d. ) 0.59)

(16)

The data for ethyl propionate, ethyl benzoate, chlorobenzene, and p-ethylphenol in ACN-water systems, ACN, acetone, N,N-dimethylformamide, phenol, benzyl alcohol, and 2-phenylethanol in MeOH-water systems, and diethyl ether, cyclopentanone, chloroform, chlorobenzene, and p-ethylphenol in THF-water systems are obvious outliers based on Student’s t test and Cook’s distance and thus were excluded in the regressions. Correlation coefficients for log ks/w vs Koct/w for the three mobile systems are much worse than and residuals

Ind. Eng. Chem. Res., Vol. 42, No. 25, 2003 6325

Figure 4. Plots of log ks/w vs log Koct+org/w: (a) ACN-water; (b) MeOH-water; (c) THF-water.

much greater than those for the S vs Korg/w correlations presented in eqs 9-11. Poor correlations between log ks/w and log Koct/w indicate that pure octane is not a good model for the octyl silica stationary phase. Plots of log ks/w vs log Koct/w for the three mobile-phase systems (Figure 3) show large scatter around the least-squares lines, and the extent of scatter is greatest in the THF-water and MeOHwater systems, where the solubility of these organic solvents in octane is greater. This suggests that log ks/w should be better correlated with Koct+org/w. Solvated Octane as the Model Stationary Phase. Results of linear regressions of log ks/w vs log Koct+org/w for the three systems are as follows:

log ks/w(a/w) ) -1.74((0.20) + 0.51((0.04) log Koct+a/w (n ) 27, r ) 0.947, s.d. ) 0.26)

(17)

log ks/w(m/w) ) -0.84((0.25) + 0.42((0.05) log Koct+m/w (n ) 31, r ) 0.838, s.d. ) 0.44)

(18)

in Figure 4. From these correlations the best-fit log ks/w values can be calculated and plotted against experimentally obtained values. These plots are shown in Figure 5. It seems both from the statistics of the fits and from Figures 4 and 5 that only the correlation for the ACN-water system is satisfactory. We would like to emphasize that modeling the stationary phase as solvated octane gives better correlations for log ks/w vs log Kstat+org/w but that the exact composition of the solvated octane is not required to model the stationary phase. To see if using more representative amounts of organic modifier that actually solvate octane improves the correlations, we computed γ∞ and then log Koct+org/w for solutes in 5 mol % ACN in octane, 9 mol % MeOH in octane, and 10 and 20 mol % THF in octane. We found that there were no remarkable improvements in the regressions for log ks/w vs log Koct+org/w for the three eluent systems as follows:

log ks/w(a/w) ) -1.85 + 0.53 log Koct+a/w r ) 0.950 (5 mol % ACN) vs 0.947 (1 mol % ACN) (20) log ks/w(m/w) ) -1.15 + 0.47 log Koct+m/w

log ks/w(t/w) ) -1.35((0.33) + 0.43((0.06) log Koct+t/w

r ) 0.838 (9 mol % MeOH) vs 0.865 (1 mol % MeOH) (21)

(n ) 35, r ) 0.783, s.d. ) 0.55)

log ks/w(t/w) ) -1.35 + 0.43 log Koct+t/w

(19)

The data for 2,2,2-trifluoroethanol, p-dioxane, and chlorobenzene in ACN-water systems, ACN, dichloromethane, p-dioxane, phenol, and m-cresol in MeOHwater systems, and p-dioxane, o- and p-cresol, chlorobenzene, p-chlorophenol, and p-ethylphenol in THFwater systems are obvious outliers based on Student’s t test and Cook’s distance and thus were excluded in the regressions. Plots of these correlations are shown

r ) 0.797 (20 mol % THF) vs 0.783 (2 mol % THF) (22) These results indicate that using compositions closer to actual solvated octane systems does not improve the modeling, which supports our assumption that the use of exact compositions of the solvated octane is not required to model the stationary phase.

6326 Ind. Eng. Chem. Res., Vol. 42, No. 25, 2003

Figure 5. Plots of calculated log ks/w vs experimental log kw values: (a) ACN-water, log ks/w,calcd ) 0.11((0.080) + 0.90((0.061) log ks/w,exptl (n ) 27, r ) 0.947, s.d. ) 0.25); (b) MeOH-water, log ks/w,calcd ) 0.21((0.107) + 0.80((0.072) log ks/w,exptl (n ) 26, r ) 0.914, s.d. ) 0.29); (c) THF-water, log ks/w,calcd ) 0.34((0.107) + 0.65((0.088) log ks/w,exptl (n ) 34, r ) 0.793, s.d. ) 0.43).

A Note about Outliers. The outliers in the correlations presented in this paper likely arise both from experimental uncertainties for some solutes and from modeling errors. For example, ethyl ether is weakly retained in THF-water systems and therefore susceptible to large measurement errors. This solute appeared as an outlier in the correlations of S vs log Kt/w. Retention of p-dioxane is also minimal in THF-water eluent and susceptible to large measurement errors. This solute appeared as outliers in the correlations of S vs log Kt/w and for ks/w vs log Koct+org/w in all three eluent systems. Other solutes are believed to appear as outliers because of a failure of the UNIFAC model, especially in regards to positional isomers. For example, o-, m-, and p-cresol appear as outliers in all of the correlations of ks/w vs log Koct+org/w in MeOH-water and THF-water. UNIFAC gives exactly the same activity coefficients for these three solutes while experimental S and log ks/w values are quite different in a given eluent system (e.g., S values for m- and p-cresol are 1.90 and 3.38 in the THF-water system, respectively). Finally, the model is built upon extrapolations of retention data to obtain partition coefficients of the solutes partitioning between the stationary phase and pure water. These extrapolations, however, yield different values for the same solute depending on the organic modifier being used, whereas there should only be a single value for each solute. This is another source of modeling errors inherent in the approach taken here. 3.3. Prediction of Variation of log k with Φ from UNIFAC-Based S and Best-Fit log ks/w. Our interest lies in predicting retention factors as a function of the mobile-phase composition. To do so, we combine UNIFAC-based S values with best-fit log ks/w values. The best-fit log ks/w values are determined through correlations of experimental log ks/w values with UNIFACbased partition coefficients using solvated octane as a

model stationary phase. More specifically, the challenge is to predict the retention of solutes not used in the original training set of data, and to do so for a variety of mobile-phase modifiers, so as to demonstrate the generality of the approach. However, the previous correlation analyses showed that UNIFAC-based partition coefficients correlate best with experimental results for ACN systems. Therefore, we restrict our treatment in this final section to the prediction of solute capacity factors as a function of the mobile-phase composition in ACN systems only. However, the challenge to predict the retention of solutes that were not in the original training set remains. Using UNIFAC-based S values from eq 8 and bestfit log ks/w values from eq 17, we have predicted the retention of solutes as a function of the mobile-phase composition according to eq 1. Figure 6 shows plots of predicted log k values vs Φ for eight representative solutes. Although this approach does not yield quantitative predictions of solute retention, deviations from the experimental values are not large. Furthermore, the variation of predicted retention factors with the volume fraction of ACN agrees quite well with the experiment. In order for this approach to be useful as a method for predicting the variation of solute retention as a function of the percent ACN, it must be able to predict log k values for solutes whose retention data were not used in the derivation of UNIFAC-derived S and log ks/w values. To test this, predicted log k values for six solutes whose retention data were not used in the derivation of best-fit log ks/w values are plotted against experimental values72-74 in Figure 7. While the number of data used for comparison is small, the agreement is on the whole satisfactory and shows that the method can be extended to solutes not in the training set. Thus, while it seems that this approach using UNIFAC is not good enough for predicting absolute solute retention, it can

Figure 6. Plots of log k for eight solutes with volume percent of ACN. Solutes: (a) 2-phenylethanol, (b) 1-butanol, (c) toluene, (d) chloroform, (e) 2-hexanone, (f) anisole, (g) ethyl benzoate, (h) n-propionitrile. Symbols: open symbols, experimental values; solid symbols, UNIFAC-calculated values. Retention data for these solutes were used for derivation of UNIFAC-based S and log kw values.

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Figure 7. Plots of log k for eight solutes with volume percent of ACN. Solutes: (a) 1-butanol, (b) benzene, (c) chlorobenzene, (d) benzyl alcohol, (e) 2-phenylethanol, (f) toluene. Symbols: open symbols, experimental values; solid symbols, UNIFAC-calculated values; (O) data from ref 70; (0) data from ref 71; (3) data from ref 72. Retention data for these solutes were not used for derivation of UNIFACbased S and log kw values.

be used to semiquantitatively predict variation of retention as a function of the organic volume fraction with useful accuracy for ACN-water systems for the solute types studied in this work. While we are encouraged by the utility of blending UNIFAC with experimental data for the prediction of the variation of solute retention in RPLC, the current restriction to ACN systems limits the general applicability of the approach. Future work should focus on finding the sources of nonlinearity in MeOH and THF systems in the hopes of modifying and extending this approach to those mobile-phase systems. 4. Conclusions Infinite-dilution activity coefficients predicted using UNIFAC can be used in combination with experimental RPLC data to compute S and log ks/w values. Correlations of UNIFAC-based partition coefficients with experimental S and log ks/w values are more accurate for ACN-water than for the MeOH-water and THFwater systems. Although this approach is not accurate enough for quantitative prediction of S and log ks/w values in RPLC, UNIFAC-computed S and best-fit log ks/w values determined by blending experimentally derived parameters with UNIFAC-based partition coefficients are useful for estimating variations in solute retention as a function of the mobile-phase composition in ACN-water systems.

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Received for review November 14, 2002 Revised manuscript received March 24, 2003 Accepted March 31, 2003 IE0209125