Anal. Chem. 1997, 69, 2550-2553
Estimating Diffusion Coefficients for Alkylbenzenes and Alkylphenones in Aqueous Mixtures with Acetonitrile and Methanol Jianwei Li† and Peter W. Carr*
Department of Chemistry, University of Minnesota, Kolthoff and Smith Halls, 207 Pleasant Street, SE, Minneapolis, Minnesota 55455
The accuracy of various empirical approximation methods of estimating the diffusion coefficients of alkylbenzenes and alkylphenones in acetonitrile (ACN)/water and methanol (MeOH)/water solvent systems is reported. Diffusion coefficients for these solutes have been measured over a wide range of solvent compositions and temperatures. A novel empirical modification of the Wilke-Chang method has been developed by correlating measured values with solute and solvent parameters. The correlation, along with the Wilke-Chang and Scheibel correlations, was examined by comparing the computed diffusion coefficients with the measured values. We find that the percent errors of the proposed correlations are no greater than 10% for both ACN/water and MeOH/water systems and that the accuracy of the correlation is 2-3-fold better than those of the other two correlations. We recommend the use of this correlation with the above homologous series of solutes for the evaluation of column performance in reversed-phase liquid chromatography. In a previous paper, we reported on the accuracy of diffusion coefficients for alkylbenzenes and alkylphenones estimated from several well-known empirical correlations. The systems chosen for study were acetonitrile (ACN)/water and methanol (MeOH)/ water mixtures, which are commonly used as solvents in reversedphase liquid chromatography (RPLC).1 We found that the Scheibel equation is the more accurate. The errors are usually less than 20%. This finding is in contrast to the widespread use of the Wilke-Chang correlation in fundamental HPLC studies.2,3 As demonstrated in the previous work, the accuracy of diffusion coefficients is important in evaluating column performance in RPLC.1 If we do not restrict the solutes used to evaluate column performance and use a correlation, such as the Scheibel method, to compute the diffusion coefficients, the errors in the B and C coefficients of the Knox equation4 can exceed 20%. Obviously, it is very time-consuming to measure the diffusion coefficient for each probe solute, for example, by the open tube method.5-8 This † Current address: 3M Pharmaceuticals, 3M Center, Building 270-4S-02, St. Paul, MN 55144-1000. (1) Li, J.; Carr, P. W. Anal. Chem. 1997, 69, 2530-2536. (2) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264-270. (3) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977; Chapter 10, p 567. (4) Knox, J. H.; Kennedy, G. J. J. Chromatogr. Sci. 1972, 10, 549-556. (5) Atwood, J. G.; Goldstein, J. J. Phys. Chem. 1984, 88, 1875-1885. (6) Ouano, A. C. Ind. Eng. Chem. Fundam. 1972, 11, 268-271. (7) Pratt, K. C.; Wakeham, W. A. J. Phys. Chem. 1975, 79, 2198-2199. (8) Grushka, E.; Kikta Jr., E. J. Phys. Chem. 1975, 79, 2199-2200.
2550 Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
difficulty can be minimized if we restrict the set of solutes used to evaluate the column efficiency in RPLC to the alkylbenzenes and alkylphenones and develop accurate empirical equations for the estimation of the diffusion coefficients of these solutes in ACN/ water and MeOH/water mixtures. In fact, the use of alkylbenzenes in the evaluation of column performance in RPLC is common.9-12 In this paper, we correlate measured diffusion coefficients1 with solvent and solute parameters and propose specific equations for the estimation of diffusion coefficients of alkylbenzenes and alkylphenones in the ACN/water and MeOH/ water solvent systems. DATA PROCESSING In this study, we will follow the approach taken by Wilke and Chang2 to process our previously reported data.1 We assume that diffusion coefficients are proportional to temperature, as suggested by many studies.13 Although solvent viscosity is also temperaturedependent, it is accounted for in a second factor. The initial empirical correlation for these diffusion coefficients is written in a form similar to the Wilke-Chang equation:
T MWsolvent Dm ) k β η V h soluteγ
R
(1)
where Dm is the diffusion coefficient, k is somewhat dependent on the solvent composition, as shown later, but presently we take it to be constant, T is the absolute temperature (in kelvin), η is the temperature- and composition-dependent viscosity (in centipoise), MWsolvent is the solvent molecular weight, V h solute is the solute molar volume, and R, β, and γ are exponents of MWsolvent, η, and V h solute, respectively. The molecular weight of solvent mixtures is computed as
MWmix ) xorgMWorg + xwaterMWwater
(2)
To determine R, β, and γ, eq 1 is rewritten as
ln
( )
Dm ) ln(k) + R ln(MWsolvent) - β ln(η) - γ ln(V h solute) T (3)
Regression of ln(Dm/T) against ln(MWsolvent), ln(η), and ln(V h solute) will establish the exponential powers. (9) Colin, H.; Diez-Masa, J. C.; Czaykowska, T.; Miedziak, I.; Guiochon, G. J. Chromatogr. 1978, 167, 41-65. (10) Vigh, G.; Varga-Puchony, Z. J. Chromatogr. 1980, 196, 1-9. (11) Grushka, E.; Colin, H.; Guiochon, G. J. Chromatogr. 1982, 248, 325-339. (12) Engelhardt, H.; Jungheim, M. Chromatographia 1990, 29, 59-68. (13) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977, Chapter 11, p 587. S0003-2700(96)01170-5 CCC: $14.00
© 1997 American Chemical Society
After these exponential powers are determined, we plot Dm × 106 (≡Y) against (T/ηβ)(MWsolventR/V h soluteγ) (≡X), and the data will be fitted to the following polynomial function:
Y (cm2/s) ) -1 + AX - BX2
(4)
where the unit of 1 is cm2/s and the units of A and B depend on those of X and X2. The constant of the quadratic is arbitrarily set to 1; otherwise, we would need one more parameter in eq 4 that did not improve the fit. The fitting results are used to determine the proposed empirical correlations. EXPERIMENTAL SECTION The experimental diffusion coefficients of alkylbenzenes and alkylphenones were reported in a previous study.1 The diffusion coefficients used in this study were measured over a wide range of solvent composition (10, 30, 50, 70, 90, and 100% organic component), temperature (30-60 °C in 10 °C increments), and solute molecular weight (70-300) (see Tables 2 and 3 in ref 1). The curve-fitting and plotting were carried out via the commercial software Origin package (MicroCal Software Inc., Northampton, MA). RESULTS AND DISCUSSION Selection of the Reported Diffusion Coefficient Data. To minimize the effect of measurement errors on the proposed empirical correlations, the data were carefully screened. As will be demonstrated, the diffusion coefficient is inversely proportional to V h solute0.6. Thus, if we plot ln(Dm) against ln(V h solute) at each combination of solvent composition and temperature, the slope of the plot is about 0.6. Accordingly, we plotted every set of ln(Dm) against ln(V h solute) and superimposed a line of slope 0.6 on each data set. If a data point had a serious deviation from the line (slope of 0.6), that point was removed. We found that most of the rejected data were those for large solutes in water-rich solvent mixtures. This probably resulted from low solubilities in these mixtures and the fact that the peaks obtained from the Taylor dispersion method5 were slightly tailed. This is particularly true for MeOH/water mixtures due to their high viscosity and low solubilizing power. The remaining data sets were used to develop the correlations (total of 75 points for ACN/water and 74 points for MeOH/water). Dependence of the Diffusion Coefficient on the Solvent Viscosity, Molecular Weight, and Solute Molar Volume. The solvent temperature, viscosity, and molecular weight can be unambiguously determined. However, there are several ways to estimate the solute volume, including the group contribution method of Wilke and Chang,2 Leahy’s intrinsic volume14 (or Bondi’s volume15), molar volume from the molecular weight and density (at 20 °C in this study), and McGowan’s characteristic volume used in the study of linear solvation energy relationships.16 We determined the exponents (R, β, and γ) and evaluated which solute volume estimation method gave the best correlations to eq 3. The results presented in Table 1 were obtained. We can see clearly from Table 1 that, when all three parameters are allowed to vary, R, β, and γ are about 0.5, 1, and 0.6, respectively, for both data sets, irrespective of the solute (14) Leahy, D. E. J. Pharm. Sci. 1986, 75, 629-636. (15) Bondi, A. J. Phys. Chem. 1964, 68, 441-451. (16) Abraham, M. H.; McGowan, J. C. Chromatographia 1987, 23, 243-246.
Table 1. Effect of Solute Volume Computational Method on Correlation Resultsa parameter valuesc solute volumeb computional method
viscosity (β)
MW (R)
volume (γ)
corr relative coeff SD (%) (R)
group contributiond MW densitye intrinsic/Bondif McGowang group contributiond MW densitye intrinsic/Bondif McGowang
ACN/Water 0.99(0.04) 0.50(0.05) 0.98(0.04) 0.55(0.05) 0.99(0.04) 0.51(0.05) 0.99(0.04) 0.50(0.05) 1h 0.48(0.02) 1h 0.53(0.02) 1h 0.49(0.02) 1h 0.48(0.02)
0.52(0.02) 0.58(0.02) 0.56(0.02) 0.57(0.02) 0.52(0.02) 0.58(0.02) 0.56(0.02) 0.57(0.02)
2.6 2.5 2.5 2.5 5.6 5.5 5.6 5.5
0.994 0.994 0.994 0.994 0.973 0.974 0.974 0.974
group contributiond MW densitye intrinsic/Bondif McGowang group contributiond MW densitye intrinsic/Bondif McGowang
MeOH/Water 0.99(0.03) 0.51(0.06) 0.99(0.03) 0.59(0.06) 1.01(0.03) 0.52(0.06) 1.01(0.03) 0.50(0.06) 1h 0.51(0.07) 1h 0.57(0.07) 1h 0.53(0.07) h 1 0.51(0.07)
0.52(0.02) 0.58(0.02) 0.56(0.02) 0.56(0.02) 0.52(0.03) 0.58(0.03) 0.56(0.03) 0.56(0.03)
2.9 3.0 2.9 2.9 9.3 9.6 9.2 9.2
0.990 0.990 0.990 0.990 0.938 0.932 0.939 0.940
a Results of linear regression fit of eq 3 to 75 and 74 diffusion coefficients from acetonitrile/water mixtures and methanol/water mixtures, respectively (same for Tables 2 and 3). b Different methods for the estimation of solute volumes. c The parameters in eq 3 and their standard errors. d Computed by a group contribution method.2 e Computed as the ratio of MW/density at 20 °C. f Computed by the estimation method of Leahy or Bondi.14,15 g Computed by the estimation method of McGowan;16 this is a very simple method. h The exponent of viscosity is fixed at 1.
volume estimation method used. The overall statistics are very good (low relative standard deviation and excellent correlation coefficient). When we fixed β at unity, the parameters of R and γ were not different from those obtained when β was allowed to vary. The statistics were somewhat poorer (relative SD is larger, R is decreased), and the relatively poor statistics are due to the very small variation in the dependent variable (Dmη/T) relative to (Dm/T). This is more obvious for ACN/water mixtures because the viscosities of ACN/water mixtures are smaller. Thus, the exponents in eq 1 are in agreement with those in the WilkeChang correlation,2 indicating that the Wilke-Chang equation can basically describe our data. Moreover, the exponents and statistics are not affected by the solute volume used; however, the intercept is. Therefore, the diffusion coefficient correlations will be developed with all four types of solute volumes so that they can be used as is convenient. Correlation of Diffusion Coefficients with the Solute and Solvent Properties. Figure 1 shows plots of the measured diffusion coefficients (Dm × 106) against (T/η)(MWsolvent0.5/ V h solute0.6), with the solute volume estimated by the group contribution method (used by the Wilke-Chang equation). Obviously, the relationship is not a straight line (convex). The short dashed line denotes the diffusion coefficients computed by the WilkeChang equation. In the water-rich region toward the left, the Wilke-Chang equation works reasonably well; as the amount of organic component is increased (toward the right), the WilkeChang equation systematically deviates from the experimental values, particularly in the pure organic solvent. To overcome the disadvantage of the Wilke-Chang equation and to obtain better correlations, we fitted eq 4 to the data in Figure 1. Table 2 summarizes the fitting results. Figure 2 shows Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
2551
Table 2. Results of Fitting to Eq 5a solute volume computational method
Ab (×10)
Bb (×105)
(χ2)c RMSEd (×106)
group contribution MW density intrinsic/Bondi McGowan solute MWe
ACN/Water 1.256(0.011) 6.162(0.303) 1.098(0.008) 3.767(0.135) 0.806(0.005) 2.305(0.064) 0.994(0.008) 3.428(0.189) 1.028(0.010) 3.545(0.229)
0.86 0.62 0.59 0.67 0.80
0.11 0.09 0.09 0.10 0.10
group contribution MW density intrinsic/Bondi McGowan solute MWe
MeOH/Water 1.531(0.014) 15.075(0.518) 1.350(0.013) 11.186(0.596) 0.989(0.008) 6.088(0.164) 1.222(0.012) 9.529(0.457) 1.257(0.012) 9.562(0.407)
0.54 0.45 0.43 0.45 0.55
0.09 0.08 0.08 0.08 0.09
a Results of fitting eq 5 (see Figure 1). b Coefficients of eq 5, standard error of coefficient in parentheses. The units of A and B parameters depend on these of X and X2. c Chi-squared of the fit. d Fit standard error, root mean square error. e Solute molecular weight is used in the correlation instead of solute volume.
Figure 1. Correlation of diffusion coefficients with solute and solvent properties. The solute molar volume is computed by the group contribution method.2 Plot A denotes the methanol/water solvent system, and plot B represents the acetonitrile/water system. The open circles are experimental data points, and the solid lines are the fits of eq 4 to the data. The short dashed lines denote the diffusion coefficients computed by the Wilke-Chang correlation.
the error frequency against the percent error. For both solvent systems, most errors are less than 5%, with the largest errors usually less than 10% (only one out of 75 for acetonitrile/water and four out of 74 for methanol/water data points have errors large than 10%). The distribution is fairly symmetrical. Note that the errors also include measurement errors in the solvent viscosity.9 Furthermore, when we used the solute volume estimated by the three other methods and solute molecular weight as a variable to describe solute size to repeat the process, the range of errors and error distribution were basically identical to those in Figure 2. Table 2 also includes these fitting results. From these results, we summarize the overall empirical correlations:
Dm (cm2/s) ) 10-6(-1 + AX - BX2)
(
) 10-6X -
1 + A - BX X
)
(5)
where X denotes (T/η)(MWsolvent0.5/V h solute0.6), and A and B are given in Table 2, depending on how the solute volume is estimated. Although it is assumed that k in eq 1 is approximately constant (k is the slope of Dm against (T/η)(MWsolvent0.5/V h solute0.6)), Figure 1 clearly indicates that this is not so. As the amount of the organic component is increased, the plot tends to be convex. However, the fit of a polynomial (eq 4) to the data should allow us to keep the obtained parameters (R, β, and γ in Table 1), because the minor nonlinearity due to the use of these parameters is compensated by the A and B coefficients in eq 4. Absolute Errors Associated with the Empirical Correlations. To evaluate the absolute errors associated with eq 5 and 2552 Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
Figure 2. Error distribution of the estimated diffusion coefficients. The ordinate is the frequency of errors, and the abscissa gives the percent errors. The diffusion coefficients were computed by eq 5 (the solute volumes were estimated by the group contribution method) and compared with the measured values. The relative error was computed as 100(Dcomp - Dmeas)/Dmeas, where Dcomp is the estimated diffusion coefficient and Dmeas is the measured diffusion coefficient. The white band denotes acetonitrile/water mixtures, while the black band represents the methanol/water mixtures.
to compare it with the Wilke-Chang and Scheibel correlations, we used the same solvent and solute parameters to compute the diffusion coefficients by the three correlations. The results are shown in Table 3. It is seen that the averages of the errors are 14% for the Wilke-Chang correlation and 8% for the Scheibel correlation in acetonitrile/water mixtures. Similarly, the averages of the errors are 13% for the Wilke-Chang correlation and 16% for the Scheibel correlation in methanol/water mixtures. However, in both solvent mixtures, the absolute errors obtained from eq 5 are only 4-5%, and the error distributions are narrow. This represents a 2-3-fold improvement in the accuracy of diffusion coefficients. The error distributions of both the Wilke-Chang and Scheibel correlations are also wide, as indicated by the difference between the mean and median errors. If we select 10% error as a limit, more than 93% of the diffusion coefficients predicted by eq 5 are within the limit in both solvent systems; however, the majority of the diffusion coefficients estimated by the Wilke-Chang and Scheibel correlations have
Table 3. Comparison of Errors of the Different Correlationsa solute volume computational method
averageb error (%)
frequency (%)c (< 10%)
mediand error (%)
SEe (%)
Wilke-Changf Scheibelg group contribution MW density intrinsic/Bondi McGowan solute MW
ACN/Water Mixtures 13.9(9.4) 40 7.7(5.1) 75 4.2(3.0) 93(0-11) 4.3(2.9) 96(0-11) 3.8(2.8) 95(0-10) 3.9(2.9) 96(0-10) 4.4(3.2) 96(0-11)
11.7 7.0 4.0 4.3 3.5 3.6 3.9
1.09 0.59 0.35 0.33 0.33 0.33 0.37
Wilke-Changf Scheibelg group contribution MW density intrinsic/Bondi McGowan solute MW
MeOH/Water Mixtures 12.5(6.5) 40 15.5(6.1) 12 4.8(3.7) 90(0-11) 4.6(3.5) 92(0-10) 4.4(3.4) 93(0-10) 4.3(3.3) 93(0-10) 4.8(3.7) 95(0-10)
12.3 16.5 4.1 4.0 3.5 3.5 4.0
0.75 0.71 0.43 0.41 0.40 0.38 0.43
a Comparision of absolute percent errors by different empirical correlations. b Average of absolute percent errors computed as 100|Dcomp - Dmeas|/Dmeas, where Dcomp is the estimated diffusion coefficient and Dmeas is the measured diffusion coefficient. The standard deviation is given in parentheses. c Frequency of errors less than 10%. The range of error with 95% confidence level estimated by Gaussian approximation is given in parentheses. d Median of the absolute percent errors. e Standard error of the mean. f The Wilke-Chang correlation;2 the solute volume is estimated by the group contribution method. g The Scheibel correlation;1 the solute and solvent volumes are estimated by the group contribution method.
errors above that imposed limit. In other words, the absolute error by eq 5 is no more than 10% with 95% confidence. Because we used two homologous series of solutes in this study, the molar volume is linearly proportional to MW. This enables us to correlate the diffusion coefficients with the MW of solutes. The use of solute MW will eventually simplify the prediction of diffusion coefficients. We should emphasize at this point that the proposed empirical correlations are only applicable to the two homologous series used
in the study. There is no guarantee of accuracy if these correlations are used to estimate the diffusion coefficients of other solutes. The work of Easteal and Woolf17 clearly shows that tracer diffusion coefficients depend significantly on both general (dipolar) and specific (hydrogen bonding) interactions between the trace diffuser and the surrounding solvent. There is no simple, purely physical correlation that can accurately predict (within 10-20% accuracy) diffusion coefficients in polar and hydrogen bond donor/ acceptor media such as those used in RPLC. Nonetheless, we feel that the correlations described here will work for nonpolar and polar aprotic species with a reasonable expectation of accurate prediction. It will probably show significant errors for proton donor solutes (e.g., alcohols and organic acids) and strong hydrogen bond acceptors (e.g., ketones and aldehydes). CONCLUSIONS In this paper, new predictive equations for diffusion coefficients were developed by correlating the diffusion coefficients for alkylbenzenes and alkylphenones in ACN/water and MeOH/water solvent systems with solute and solvent properties. The correlation was examined by comparing computed diffusion coefficients with measured values. We find that the percent error of our correlation is no more than 10%, with an average of about 4%, indicating a 2-3-fold improvement in the accuracy relative to the Wilke-Chang and Scheibel correlations. Thus, if these solutes and the correlation are used in the evaluation of column performance in RPLC, the errors in A, B, and C terms in the Knox equation attributable to errors in the diffusion coefficient estimates should be better than 10%. ACKNOWLEDGMENT The authors acknowledge the financial support by Grant GM 45988-05 from the National Institutes of Health. Received for review November 18, 1996. Accepted March 31, 1997.X AC961170Q
(17) Easteal, A. J.; Woolf, L. A. J. Chem. Soc., Faraday Trans. 1 1984, 12871295.
X
Abstract published in Advance ACS Abstracts, May 15, 1997.
Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
2553
