Anal. Chem. 1997, 69, 2530-2536
Accuracy of Empirical Correlations for Estimating Diffusion Coefficients in Aqueous Organic Mixtures Jianwei Li† and Peter W. Carr*
Department of Chemistry, University of Minnesota, Kolthoff and Smith Halls, 207 Pleasant Street, SE, Minneapolis, Minnesota 55455
Diffusion coefficients for a homologous series of alkylbenzenes and alkylphenones have been measured by the Aris-Taylor open tube method from 30 to 60 °C over a wide range in methanol/water and acetonitrile/water compositions (10-100% by volume of organic). The measurements were compared to estimates derived from five of the most common empirical correlations. The errors for methanolic mixtures by the Wilke-Chang, Scheibel, and Lusis-Ratcliff correlations are usually less than 20%. The Scheibel, Wilke-Chang, and HaydukLaudie correlations work better than others for acetonitrile/water mixtures. Overall, the Scheibel correlation shows the smallest errors, and we recommend its use to that of the more widely used Wilke-Chang method for the systems studied here. We have also developed fitting equations for estimating viscosity so that the diffusion coefficients can be easily estimated. The sources of band broadening in chromatography include longitudinal molecular diffusion, eddy dispersion, and several types of interphase mass transfer resistances.1-2 The term mass transfer, as it is used by chromatographers, includes (1) lateral diffusion through the flowing mobile phase (from the center of the flow stream in the interparticle space to the particle surface); (2) diffusion from the mobile phase into the particles, across the stagnant film barrier; and (3) diffusion through the stagnant mobile phase, inside the particle pores. Thus, diffusion is ubiquitous in almost every event in chromatography. Because diffusion is central to chromatography, the accurate estimation of diffusion coefficients is quite important. Since it usually takes a considerable effort to measure liquid phase diffusion coefficients,3-5 chromatographers often prefer to estimate diffusion coefficients by correlation equations such as the WilkeChang equation.6 However, the accuracy of diffusion coefficients estimated by the Wilke-Chang and several other empirical equations7-10 is often questionable. This can sometimes have a
significant effect on the interpretation of chromatographic results. For example, a reduced form of the plate height equation, such as the widely applied Knox equation, is often used to evaluate column performance:11
h ) B/ν + Aν1/3 + Cν
(1)
where h is the reduced plate height (≡H/dp, H is the plate height and dp is the particle diameter), ν is the reduced velocity (≡µdp/ Dm, µ is the interstitial mobile phase velocity, Dm is the diffusion coefficient in bulk mobile phase), and A, B, and C are coefficients that are related to the quality of packing, longitudinal diffusion, and the magnitude of the mass transfer resistances, respectively. The reduced velocity is computed based on the bulk diffusion coefficient. If there is an error in the diffusion coefficient, this error will translate into errors in A, B, and C through ν. Although the A term is less affected, any error in the diffusion coefficient will affect both the B and C coefficients to at least the same extent. It is important to recognize that errors as large as 30% in the estimated diffusion coefficient are common,12,13 and thus errors as large as 30% in B and C can result. In reversed-phase liquid chromatography (RPLC) methanol (MeOH)/water and acetonitrile (ACN)/water mixtures are frequently used. Use of such mixtures further complicates the estimation of diffusion coefficients because parameters such as the solvent molar volume and molecular weight must be estimated and used in the empirical correlations. Although the Wilke-Chang equation is widely used for diffusion coefficient estimation in liquid chromatography, its accuracy under conditions common in RPLC is rarely questioned. In fact, many authors observed that the Wilke-Chang correlation is satisfactory for most cases of an organic solute diffusing in water, whereas diffusion coefficients in many organic solvents including alcohols are sometimes poorly estimated.13 In this work we compared estimated diffusion coefficients for alkylbenzenes and alkylphenones by several empirical correlations to values measured by the Aris-Taylor open tube method, and determined the accuracy of the diffusion coefficient estimates under RPLC-type conditions. The open tube method is an absolute method, requiring no calibration. It works especially well to determine diffusion coefficients near infinite dilution.5 Moreover, we developed empirical equations for estimating the viscosity
† Current address: 3M Pharmaceuticals, 3M Center, Building 270-4S-02, St. Paul, MN 55144-1000. (1) Giddings, J. C. Dynamics of Chromatography, Part 1, Principles and Theory; Marcel Dekker: New York, 1965. (2) Giddings, J. C. Unified Separation Science; Wiley&Sons: New York, 1991. (3) Wakeham, W. A. J. Chem. Soc., Faraday Symp. 1980, 15, 145-154. (4) Ouano, A. C. Ind. Eng. Chem. Fundam. 1972, 11, 268-271. (5) Atwood, J. G.; Goldstein, J. J. Phys. Chem. 1984, 88, 1875-1885. (6) Wilke. C. R.; Chang, P. AICHE J. 1955, 1, 264-270. (7) Scheibel, E. G. Ind. Eng. Chem. 1954, 46, 2007-2008. (8) Reddy, K. A.; Doraiswamy, L. K. Ind. Eng. Chem. Fundam. 1967, 6, 7779. (9) Lusis, M. A.; Ratcliff, G. A. Can. J. Chem. Eng. 1968, 46, 385-386. (10) Hayduk, W.; Landie, H. AICHE. J. 1974, 20, 611-615.
(11) Knox, J. H.; Kennedy, G. J. J. Chromatogr. Sci. 1972, 10, 549-556. (12) Guiochon, G.; Shirazi, S. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, 1994; Chapter 5, p 140. (13) Reid, R. C.; Prausnite, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977; Chapter 11, p 567.
2530 Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
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© 1997 American Chemical Society
of these binary mixtures. The results from this study will help chromatographers select empirical correlations for estimating diffusion coefficients and provide empirical equations for computing the viscosity of the mobile phases used in RPLC. Furthermore, because we deliberately choose two commonly used homolog series, our results can be extrapolated and employed in almost any reversed-phase chromatographic system. THEORETICAL SECTION Viscosity of Binary Mixtures of ACN/Water and MeOH/ Water. The viscosities of solutions that are not dilute can be quite different from those of the pure chemical species. Many models have been proposed to calculate the viscosity of a liquid mixture from the viscosities of the pure components.14 In this work, we use the Lobe correlation14 to represent the viscosity of the two solvent systems:
ηm ) φorgηorg exp(φwaterRwater) + φwaterηwater exp(φorgRorg)
Reddy-Doraiswamy Correlation.8 This correlation does not include an association factor:
DA,B (cm2/s) )
K′xMWBT ηB(VAVB)1/3
where K′ ) 10 × 10-8
VB/VA e 1.5
) 8.5 × 10-8
2
DA,B (cm /s) )
DA,B (cm2/s) ) 7.4 × 10-8
xψBMWBT ηB V 0.6 A
(3)
where A and B denote solute and solvent, respectively, VA is the molar volume (mL/mol) of the liquid solute at its normal boiling point, MWB is the molecular weight of the solvent (g/mol), ηB is its viscosity (cP), T is temperature, and ψB is a constant which accounts for solvent/solvent interactions (1 for nonassociated solvents, 1.5 for ethanol, 1.9 for methanol, 2.6 for water). VA is usually obtained from a group contribution approach.6 Scheibel Correlation.7 Scheibel modified the Wilke-Chang correlation to eliminate the association factor (ψ):
DA,B (cm2/s) )
A ST ηBV1/3 A
[ ( )] 1+
3VB
2/3
(4)
VA
where VB is the molar volume (mL/mol) of the solvent, and all other symbols have the same meaning as in eq 3. The constant As is equal to 8.2 × 10-8, and VB is also estimated by a group contribution scheme. (14) Reference 13, Chapter 9, p 457. (15) Colin, H.; Diez-Masa, J. C.; Czaykowska, T.; Miedziak, I.; Guiochon, G. J. Chromatogr. 1978, 167, 41-65.
(5)
where all symbols have the same meaning as in eqs 3 and 4. Lusis-Ratcliff Correlation.9 This correlation was developed to estimate diffusion coefficients in organic solvents:
(2)
where ηorg, ηwater, and ηm are the viscosities of the pure organic components, water, and the mixture, respectively; φorg and φwater are the volume fractions of organic components and water, respectively; and Rorg and Rwater are empirical parameters determined by fitting the viscosity data15 to eq 2. Empirical Correlations for Estimating Diffusion Coefficients. There are several empirical correlations for estimating the diffusion coefficients in binary liquids. They are briefly explained below. Wilke-Chang Estimation Method.6 This is probably the most widely used correlation in LC for estimating diffusion coefficients; in essence, it is an empirical modification of the Stokes-Einstein relation:
VB/VA g 1.5
[ () ]
8.52 × 10-8T ηBV
1/3 B
VB 1.40 VA
1/3
VB
+
(6)
VA
where all symbols have the same meaning as in eqs 3 and 4. Hayduk-Laudie Correlation.10 This correlation was developed for diffusion coefficients of nonelectrolytes in water:
DA,B (cm2/s) ) 13.26 × 10-5ηB-1.4 V-0.589 A
(7)
where all symbols have the same meaning as in eq 3. Other correlations, such as the King, Hsueh, and Mao correlation16 are also available but less useful. We will compare all the correlations here with measured values to determine their accuracy. EXPERIMENTAL SECTION Solvent Delivery Apparatus. The measurements of diffusion coefficients were carried out on a Hewlett-Packard 1090 liquid chromatograph with a DR5 solvent delivery system, an autosampler, a temperature controller, a filter photometric detector and a HP 3396 integrator (Hewlett Packard S. A., Wilmington, DE 19808). Measurement of Diffusion Coefficients. The experimental diffusion coefficients were measured by the Aris-Taylor method.4-5 We used two systems for the experiment. One was made of PEEK tubing, which had a length (L) of 1600 cm and a calibrated internal diameter (dt) of 0.02 in. (Alltech Associates, Inc., Deerfield, IL 60015). The tube was coiled into a diameter (dcoil) of 12 cm to fit into the instrument for temperature control. The other was made of stainless steel (Alltech); it had a length of 1575 cm and a calibrated internal diameter of 0.0212 in. This tube was coiled into a diameter of 13 cm. The tubes were calibrated by weighing the amount of water contained within them. The HP 1090 pumps were used to deliver the solvent. Detection was carried out at 254 nm. All other conditions are listed in Table 1. (16) King, C. J.; Hsueh, L.; Mao, K. W. J. Chem. Eng. Data 1965, 10, 348-350.
Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
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Table 1. Experimental Conditions for Measuring Diffusion Coefficientsa solute
concn (mg/mL)
injection vol (µL)
solvent composition (% Org)
temp (°C)
benzene pentylbenzene 1-phenyldecane 1-phenylpentadecane
4.5 4.5 9.0 9.0
3 3 3 3
100,b 90, 80,b 70 (for ACN) 100, 90 (for MeOH)
30, 40, 50, 60
toluene butylbenzene 1-phenylheptane 1-phenyldecane
4.5 4.5 6.0 9.0
3 3 3 3
60,b 50, 40b (for ACN) 70, 50 (for MeOH)
30, 40, 50, 60
acetophenone butyrophenone heptanophenone decanophenone
0.25 0.25 0.50 0.70
1 1 1 1
30, 10 (for ACN and MeOH)
30, 40, 50, 60
a Measured for ACN/water and MeOH/water mixtures in a stainless-steel tube of length of 1575 cm, an i.d. of 0.0212 in., and a linear velocity of 0.729 cm/s at a flow rate of 0.0993 mL/min. Replicate injections were made for each measurement of the diffusion coefficient. b Measured independently for ACN/water mixtures with a PEEK tubing of length 1,600 cm, an i.d. of 0.02 in., and a linear velocity of 0.813 cm/s at a flow rate of 0.0993 mL/min. Replicate injections were made for each measurement of the diffusion coefficient.
The diffusion coefficients were computed from the Aris-Taylor dispersion equation:4-5
σ2z ) (σt µ)2 ) 2DA,Btr +
(dtL)2 96DA,Btr
(8)
where σz and σt are the peak standard deviations in length (cm) and time (second) units, respectively, µ is the linear velocity (cm/ s), DA,B is the diffusion coefficient, and tr is the residence time of the solute. We need to make three comments on eq 8. First, the first term in eq 8 is on average about 800 fold smaller than the second; and, therefore, it can be ignored. Second, the effect of the connections is not significant, as demonstrated in a previous study.17 Third, according to Atwood and Golay,18 the effects of secondary flow in the helical tubing can be avoided if the flow rate is less than a transition flow rate (FTrans) defined as
F < FTrans )
(
)
518RcoilrtDA,BηB FB
1/2
(9)
where Rcoil is the radius of the tube’s coil, rt is the internal radius of the tube, and FB is the density of the solvent. Under our experimental conditions, eq 9 is always satisfied. It has been shown5 that, if the volume and length of the tube and the residence time of the sample in it are accurately known, the accuracy of the method is relatively insensitive to small variations in crosssectional area along the tube, to small variation of flow rate during the transit time, and to small noncircularity in the tube’s cross section. All measured diffusion coefficients are given in Tables 2 and 3. Table 3 also includes some additional experimental diffusion coefficients in MeOH/water mixtures obtained from the literature.19 The data in Tables 2 and 3 were used together to evaluate the empirical correlations in MeOH/water mixtures. Reagents. All reagents used were obtained from commercial sources and were reagent grade or better, unless noted below. The organic solvents used for liquid chromatography were (17) Li, J.; Carr, P. W. Anal. Chem. 1997, 69, 2193-2201. (18) Atwood, J. G.; Golay, M. J. E. J. Chromatogr. 1981, 218, 97-100. (19) Knox, J. H.; Scott, H. P. J. Chromatogr. 1983, 282, 297-313.
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Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
ChromAR HPLC grade acetonitrile and methanol (Mallinckrodt Chemical Co., Paris, KE 40361). DI water was passed through a 0.45 µm filter (Gelman Sciences Inc., Ann Arbor, MI 48106) and then boiled to remove carbon dioxide before use. All solvents were filtered a second time with a 0.45 µm filtration disk. Solutes were benzene, toluene, butylbenzene, pentylbenzene, 1-phenylheptane, 1-phenyldecane, 1-phenylpentadecane, acetophenone, butyrophenone, heptanophenone, and decanophenone (Aldrich Chemical Co., Milwaukee, WI 53233). Computation of Diffusion coefficients by Empirical Correlations. In computing the diffusion coefficients by the empirical correlations (eqs 3-7), we need to calculate the molecular weight, molar volume, and association factor (ψ) of solvent mixtures, respectively, and the molar volume of the solute. The viscosities of the binary mixtures were obtained from the literature.15 The molecular weight (MWB) of the solvent mixtures was computed as
MWB ) xorgMWorg + xwaterMWwater
(10)
where xorg and xwater are the mole fractions of the organic component and water in the mixtures, respectively; MWorg and MWwater are the molecular weights of the organic component and water, respectively. The association factors (ψ) of the solvent mixtures were computed as
ψ ) xorgψorg + xwaterψwater
(11)
where ψorg and ψwater are the association factors of the organic component and water VA, respectively. The molar volumes (VA) of the solutes were computed by a group contribution method.6 For the molar volumes of the solvent mixtures, we first computed the molar volumes of pure components by the group contribution method and then the molar volumes (VB) of the mixtures were computed as
VB ) xorgVorg + xwaterVwater
(12)
where Vorg and Vwater are the molar volumes of organic component and water of the mixtures, respectively.
Table 2. Measured Diffusion Coefficients for ACN/Water Mixturesa temp (°C)
DA,B (×10-5 cm2/s)
solute
solvent comp 30 40 50 60 30 40 50 60 30 40 50 60 30 40 50 60
benzene
pentylbenzene
1-phenyldecane
1-phenylpentadecane
solvent comp 30 40 50 60 30 40 50 60 a
acetophenone
heptanophenone
1.0b
0.9b
0.8b
3.73 4.20 5.04 5.47 2.83 3.15 3.78 4.27 2.03 2.40 2.81 3.19 1.75 2.08 2.42 2.79
2.82 3.46 4.02 4.10 1.93 2.39 2.85 3.23 1.57 2.01 2.27 2.59 1.32 1.67 1.93 2.21
2.32
0.3b
0.1b
0.863 1.17 1.39 1.65 0.535 0.827 0.961 1.16
0.799 1.10 1.33 1.48 0.446 0.730 0.845 0.973
DA,B (×10-5cm2/s)
solute 0.7b toluene 2.27 2.75 3.14 1.24 1.60 1.95 2.26 0.973 1.26 1.56 1.79
3.63 4.11 1.71 2.13 2.55 2.92 1.36 1.70 2.03 2.34 1.15
butylbenzene
1-phenyldecane
1-phenylheptane 1.12 1.38
2.00
butyrophenone
decanophenone
0.6b
0.5b
0.4b
1.71 2.11 2.33 2.68 1.19 1.51 1.84 2.15 0.825 1.06 1.31 1.52 0.966 1.24 1.53 1.77
1.13 1.48 1.68 2.17 0.910 1.17 1.48 1.75
1.17 1.45 1.58 1.86 0.768 0.977 1.19
0.3b
0.1b
0.715 1.00 1.17 1.42 0.494 0.703 0.828 1.01
0.665 0.932 1.10 1.27
0.722 0.882 1.10 0.737 0.962 1.20 1.44
See Experimental Section and Table 1 for conditions. b Volume fraction of ACN in ACN/water mixtures.
Table 3. Measured Diffusion Coefficients for MeOH/Water Mixturesa temp (°C)
solute
solvent comp 30 40 50 60 30 40 50 60 30 40 50 60
benzene
pentylbenzene
1-phenylpentadecane
solvent comp 30 40 50 60
1-phenyldecane
solvent comp 20 20 20 20 20 20 a
acetone phenol p-cresol 3,4-xylenol anisole phenetole
DA,B (×10-5 cm2/s) 1.0b
0.9b
2.15 2.31 3.00 3.53 1.61 1.85 2.43 2.78 1.14 1.42 1.69 1.97
1.38 1.77 2.23 2.66
DA,B(×10-5 cm2/s)
solute
toluene
butylbenzene 1.33 1.59 1.84 0.741 0.935 1.12 1.35
1.0b
0.9b
1.35 1.52 2.01 2.31
0.870 1.12 1.35 1.58
0.6c
0. 45c
0.79 0.52 0.45 0.41 0.60 0.56
0.62 0.46 0.40 0.36 0.52 0.46
1-phenylheptane
0.7b
0.5b
0.762 1.07 1.36 1.86 0.712 0.904 1.14 1.33 0.596 0.792 1.01 1.20
0.610 0.889 1.15 1.37 0.539 0.747 0.947 1.15
solute
acetophenone
butyrophenone
heptanophenone 0.727 0.855
0.7b 1-phenyldecane
0.487 0.664 0.887 1.08
DA,B(×10-5 cm2/s) 0.3b
0.1b
0.639 0.884 1.12 1.31 0.521 0.730 0.924 1.12 0.419 0.585 0.802 0.908
0.766 1.02 1.30 1.52 0.612 0.833 1.08 1.27
0.5b decanophenone
0.3b 0.340 0.491 0.611 0.760
See Experimental Section and Table 1 for conditions. b Volume fraction of MeOH in MeOH/water mixtures. c Taken from ref 19.
Estimation of Errors in the Measured Diffusion Coefficients. The imprecision of the diffusion coefficients computed from eq 8 were estimated as follows. Because the tube is so long, the uncertainty in the tube length can be neglected. The effect of instrument dead volume is also insignificant, as demonstrated in a previous study.17 The uncertainty of the tube volume was 0.01 mL (0.3%), which translates into an uncertainty in (dt/µ) of
0.0002 (0.3%), assuming that there is no error in the flow rate and tube length. Thus, the main contributions to the measurement errors are due to the imprecision in the residence time, peak width (upon duplicate injections), and the ratio (dt/µ). It was found by error propagation that the greatest imprecision in the diffusion coefficients is ∼5%, and the majority of the diffusion coefficients reported here are more precise than 3%. Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
2533
Table 4. The rorg and rwater Parameters at Different Temperaturesa temp (°C)
ηOrgb
ηWaterc
15 20 25 30 35 40 45 50 55 60
0.40 0.37 0.35 0.32 0.30 0.27 0.25 0.24 0.23 0.22
1.1 1.0 0.89 0.79 0.70 0.64 0.58 0.54 0.51 0.47
15 20 25 30 35 40 45 50 55 60 65
0.63 0.60 0.56 0.51 0.46 0.42 0.39 0.37 0.36 0.33 0.28
1.1 1.0 0.89 0.79 0.70 0.64 0.58 0.54 0.51 0.47 0.40
Rwaterd
Rorge
ACN/Water Mixtures 2.01(0.19) -0.25(0.25) 2.24(0.12) -0.75(0.23) 2.24(0.11) -0.97(0.25) 2.24(0.16) -1.06(0.38) 1.91(0.18) -0.57(0.30) 1.89(0.15) -0.48(0.24) 1.53(0.14) -0.037(0.14) 1.71(0.11) -0.28(0.15) 1.51(0.11) -0.18(0.13) 1.87(0.09) -0.64(0.17) MeOH/Water Mixtures 1.90(0.07) 1.47(0.05) 1.93(0.07) 1.36(0.06) 1.89(0.07) 1.31(0.06) 1.81(0.07) 1.32(0.06) 1.70(0.06) 1.37(0.05) 1.57(0.05) 1.39(0.04) 1.57(0.05) 1.36(0.04) 1.62(0.04) 1.28(0.03) 1.64(0.05) 1.17(0.05) 1.54(0.06) 1.29(0.05) 1.09(0.07) 1.72(0.03)
(χ2)f (×105)
SDg
190 58 32 49 48 31 16 11 6.7 5.6
0.015 0.008 0.006 0.007 0.007 0.006 0.004 0.003 0.003 0.002
180 180 130 86 53 24 20 11 16 16 8.9
0.014 0.014 0.012 0.010 0.008 0.005 0.005 0.003 0.004 0.004 0.003
Figure 1. Experimental data and fitted curves for the viscosity of acetonitrile/water mixtures at different temperatures. The data were taken from ref 15. The abscissa is the volume fraction of acetonitrile. Symbols, temperature (°C, top to bottom): 9, 15; b, 20; 2, 25; 1, 30; [, 35; 0, 40; O, 45; 4, 50; 3, 55; ], 60.
a From the fit of the viscosity data15 to eq 2. b Viscosity of pure organic component. c Viscosity of water. d Parameter Rwater in eq 2 and the error of the fit in parentheses. e Parameter Rorg in eq 2 and the error of the fit in parentheses. f χ2 of fit. g Average residual of the fit.
Modeling the Viscosity of ACN/Water and MeOH/Water Mixtures. The viscosity data taken from ref 15 were fitted to eq 2 via commercial software Origin (MicroCal Software Inc., Northampton, MA 01060). The fitted parameters (Rwater and Rorg) are shown in Table 4. RESULTS AND DISCUSSION Viscosities of ACN/Water and MeOH/Water Mixtures. The viscosities of the mixtures play an important role in the estimation of the diffusion coefficient. Although Snyder and Kirkland20 recommended that viscosity be estimated based on the work of Grunberg and Nissan,21 we found that the errors predicted by this method are too large to allow the method to be used here. Figures 1 and 2 show the result of fitting eq 2 to the viscosity data,15 and Table 4 shows the values of Rwater and Rorg parameters as a function of temperature for the two systems. The fits are extremely good, indicating that eq 2 describes the viscosity data very well. The relative errors never exceed 10%. The errors for the MeOH/water mixtures are even smaller than those for ACN/ water mixtures. Since there is little need for estimation of diffusion coefficient to better than 10%, we are satisfied with these fits. Thus, eq 2 with the parameters in Table 4 allows us to estimate the viscosity of the binary mixtures at any solvent composition and temperature. We note that the viscosity curves for the ACN/water mixtures at low temperature are rather different from those at high temperatures. This is tentatively attributed to the formation of water-rich and ACN-rich clusters, or “demixing”.22 (20) Snyder, L. R.; Kirkland, J. J. Introduction to Modern Liquid Chromatography, 2nd ed.; Wiley&Sons: New York, 1979; Appendix II, p 836. (21) Grunberg, L.; Nissan, A. H. Nature 1949, 799-800. (22) Park, J. H.; Dallas, A. J.; Chau, P.; Carr, P. W. J. Phys. Org. Chem. 1994, 7, 757-769.
2534 Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
Figure 2. Experimental data and fitted curves for the viscosity of methanol/water mixtures at different temperature. The abscissa is the volume fraction of methanol. Symbols same as in Figure 1 and +, 65 °C.
Accuracy of Diffusion Coefficients Estimated by Empirical Correlations. Tables 2 and 3 give the measured diffusion coefficients as a function of temperature, composition, and solute size. These diffusion coefficients were used to evaluate the accuracy of the empirical correlations discussed above. Figures 3-7 show how the average absolute error varies with solvent composition. We plotted the averaged error vs the solvent composition because the solvent composition is the most important variable since it affects the association factor (ψ), molar volume, viscosity, and molecular weight of the solvent. Wilke-Chang Correlation6 (Figure 3). The absolute percent errors in the diffusion coefficients computed from the WilkeChang equation are generally less than 20% for MeOH/water mixtures (the estimated values for alkylbenzenes are smaller than the measured values). In the ACN/water mixtures, the absolute errors vary from 10 to 30%. Although they are not shown here, the percent errors are positive from 10 to 50% (v/v) ACN, and usually less than 20%; however, as the volume fraction of ACN exceeds 50%, the errors become negative and grow to 40%. This indicates that the Wilke-Chang correlation does not work very well in ACN-rich aqueous mixtures. Overall, the Wilke-Chang equation works better in MeOH/water than in ACN/water
Figure 3. Absolute relative error in diffusion coefficients estimated by the Wilke-Chang correlation against the volume fraction of organic modifier. The top plot is for acetonitrile/water, and the bottom plot is methanol/water. The error bars represent the range in errors over the different solutes and temperatures examined at this solvent composition. The horizontal dotted line in each plot represents the overall average of errors over all solvent compositions.
mixtures. The overall average absolute percent errors for all solvent compositions are less than 15%. Scheibel Correlation7 (Figure 4). We can see from Figure 4 that the Scheibel correlation normally gives less than 20% error in ACN/water and usually underestimates the measured diffusion coefficient. A similar conclusion can be drawn for the MeOH/ water solvent system. We observe from Figure 4 that, in general, the Scheibel correlation yields more accurate diffusion coefficients than does the Wilke-Chang correlation (the errors are usually no more than 20%). The overall average absolute percent errors for all solvent compositions are less than 10% for ACN/water mixtures and 15% for MeOH/water mixtures. We prefer it to the much more widely used Wilke-Chang equation. Reddy-Doraiswamy Correlation8 (Figure 5). This correlation is similar to the Wilke-Chang equation. The errors with this correlation are mostly positive. The absolute errors in ACN/water mixtures are often greater than 15% and are somewhat better than for MeOH/water mixtures. The overall average absolute percent errors for all solvent compositions are less than 15% for MeOH/ water mixtures. Lusis-Ratcliff Correlation9 (Figure 6). This correlation works extremely well for MeOH/water mixtures. The relative errors are usually less than 20%(the error distribution is almost symmetric). However, for ACN/water mixtures at low ACN composition, the errors are large. The correlation improves in the 60100% ACN range. The overall average absolute percent errors for all solvent compositions are less than 10% for MeOH/water mixtures.
Figure 4. Absolute relative error of diffusion coefficients estimated by the Scheibel correlation against the volume fraction of organic modifier. Other conditions same as in Figure 3.
Figure 5. Absolute relative error of diffusion coefficients estimated by the Reddy-Doraiswamy correlation against the volume fraction of organic modifier. Other conditions same as in Figure 3.
Hayduk-Laudie Correlation10 (Figure 7). Again this correlation gives significant errors, depending on the solvent composition for MeOH/water mixtures. However, this correlation works very Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
2535
Figure 6. Absolute relative error of diffusion coefficients estimated by the Lusis-Ratcliff correlation against the volume fraction of organic modifier. Other conditions same as in Figure 3.
or Lusis-Ratcliff correlations in MeOH/water mixtures, and the Lusis-Ratcliff correlation is probably the best. For ACN/water mixtures, Wilke-Chang, Scheibel, and Hayduk-Laudie correlations offer the least errors, and the Scheibel correlation is probably the best for the ACN/water system. Overall, the Scheibel equation works better than does the Wilke-Chang correlation and has the advantage that an association factor is not needed. We also observe in Figures 3-7 that at all solvent compositions the error bands are not narrow for any correlation. This shows that the dependence of diffusion coefficient on solute size and temperature is not in agreement with the empirical correlations. We must note that the error bands are not indications of the precision of the measured diffusion coefficients; rather they indicate the spread in errors of the estimated diffusion coefficients for the various solutes and temperatures at the indicated solvent composition. This issue will be addressed in a subsequent publication, in which a new, more accurate equation is proposed. In addition to our measured diffusion coefficients, we also used the diffusion coefficients of other solutes in methanol/water mixtures from the literature19 to evaluate the empirical correlations (Table 3). Although they do not belong to the homologous series, the error distributions of these solutes agree with the trend exhibited by the homologous series solutes. We also want to point out that, although we used alkylbenzenes and alkylphenones as probe solutes, the conclusions from this study are likely applicable to other test solutes. CONCLUSIONS The principal purpose of this study was to evaluate the accuracy of several empirical correlations for computing alkylbenzene and alkylphenone diffusion coefficients in ACN/water and MeOH/water mixtures over a wide range of temperature and solvent composition. We find that the errors of the estimated diffusion coefficients in MeOH/water mixtures by the WilkeChang, Scheibel, and Lusis-Ratcliff correlations are usually less than 20%. The Scheibel, Wilke-Chang, and Hayduk-Laudie correlations offer errors often less than 20% for the ACN/water solvent system. Overall, the Scheibel correlation works better than the Wilke-Chang correlation, and we recommend its use for estimating diffusion coefficients of low molecular weight solutes in RPLC. The experimental data reported here should be useful to liquid chromatographers interested in the proper determination of reduced velocity. We recommend that our measured diffusion coefficients be used instead of any estimated values. Moreover, we modeled the viscosity of ACN/water and MeOH/water solvent systems as a function of temperature and the solvent composition. The fitted equations allow us to compute the viscosity of the two solvent systems with errors of no more than 10%. ACKNOWLEDGMENT The authors acknowledge the financial support by a Grant GM 45988-05 from the National Institutes of Health.
Figure 7. Absolute relative error of diffusion coefficients estimated by the Hayduk-Laudie correlation against the volume fraction of organic modifier. Other conditions same as in Figure 3.
well in ACN/water mixtures; the errors are usually less than 20%. If 20% relative error in the estimated diffusion coefficients is acceptable, we recommend use of the Wilke-Chang, Scheibel, 2536
Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
Received for review October 1, 1996. Accepted March 25, 1997.X AC961005A X
Abstract published in Advance ACS Abstracts, May 1, 1997.
