Experimental and Predicted Vapor− Liquid Equilibrium for Cyclic

Ind. Eng. Chem. Res. 2005, 44, 6981-6988

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Experimental and Predicted Vapor-Liquid Equilibrium for Cyclic Ethers with 1-Chloropentane Beatriz Giner, Santiago Martı´n, Marta Haro, He´ ctor Artigas, and Carlos Lafuente* Departamento de Quı´mica Orga´ nica-Quı´mica Fı´sica, Facultad de Ciencias, Universidad de Zaragoza, Ciudad Universitaria, Zaragoza 50009, Spain

Isobaric vapor-liquid equilibrium (VLE) measurements for mixtures containing tetrahydrofuran, tetrahydropyran, 1,3-dioxolane, or 1,4-dioxane and 1-chloropentane at 40.0 and 101.3 kPa are reported. Activity coefficients were calculated from experimental data and correlated with Wilson, NRTL, and UNIQUAC equations. The VLE measurements have been found to be thermodynamically consistent according to van Ness test. Using these VLE results the UNIFAC parameters between the CCl and c-CH2O groups were recalculated, and the results obtained with the new parameters can be considered to be satisfactory. Introduction Over the past years we have been involved in the study of isobaric vapor-liquid equilibria (VLE) of mixtures containing a cyclic ether and normal and branched chloroalkanes.1-4 In this work we report isobaric vapor-liquid equilibrium measurements for the mixtures formed by a cyclic ether (tetrahydrofuran, tetrahydropyran, 1,3-dioxolane, or 1,4-dioxane) with 1-chloropentane at 40.0 and 101.3 kPa. The VLE experimental results have been checked for thermodynamic consistency, and the activity coefficients have been calculated and correlated with the following equations: Wilson,5 NRTL,6 and UNIQUAC.7 To our knowledge, the VLE data for these mixtures have not been reported before. The group contribution methods play an important role in solving several types of problems in the chemical and petrochemical industries; among these methods UNIFAC8-10 is one of the most successful. We have used our VLE results to test the reliability of the UNIFAC predictions. In previous works involving cyclic ethers and alkanes11 or alkanols,12 the VLE predictions were quite satisfactory; that is, the corresponding UNIFAC parameters between the involved groups are accurate. On the other hand, for the kind of mixtures studied here the UNIFAC predictions are clearly unsatisfactory, especially for the mixtures containing tetrahydropyran or 1,4-dioxane. Thus, we have tried to improve the VLE predictions by recalculating the UNIFAC parameters between the CCl and c-CH2O groups using our experimental results. Experimental Section Chemicals. The liquids used were tetrahydrofuran (>99.5 mol %), 1,3-dioxolane, 1,4-dioxane, and 1-chloropentane (>99 mol %) obtained from Aldrich, and tetrahydropyran (>99 mol %) provided by Acros. No additional purification has been carried out. A comparison between experimental and literature data13-17 of densities at 298.15 K and normal boiling points is reported in Table 1. * To whom correspondence should be addressed. Tel: +34 976761195. Fax: + 34 976761202. E-mail: [email protected].

Table 1. Densities at 298.15 K, G, and Normal Boiling Points, Tb, of the Pure Compounds and Comparison with Literature Values F (kg‚m-3)

Tb (K)

compound

exptl

lit.

exptl

lit.

tetrahydrofuran tetrahydropyran 1,3-dioxolane 1,4-dioxane 1-chloropentane

882.0 878.8 1059.0 1027.89 877.0

881.9713 879.1614 1058.6615 1027.9716 877.0017

339.12 361.23 348.55 374.52 381.54

339.11516 361.1517 348.8017 374.4717 381.5417

Table 2. Isentropic Compressibility Correlation Coefficients system tetrahydrofuran tetrahydropyran 1,3-dioxolane 1,4-dioxane + 1-chloropentane

κS,2 κS,1 (TPa-1) (TPa-1) 691.8 716.8 526.9 538.6

854.4 854.4 854.4 854.4

A0

A1

A2

A3 σ(κS)

93.9 36.6 59.1 69.3 18.9 1.3 1.1 4.6 182.3 33.8 42.0 48.3 103.1 18.0 40.6 43.1

0.4 0.1 0.4 0.2

Methods. VLE experimental data were measured using a Labodest unit built by Fischer, which is an allglass dynamic recirculating still equipped with a Cottrell pump, a pressure transducer Druck PDCR 110/W, and a thermometer from Automatic System Laboratories (model F25). The accuracy of the thermometer is estimated to be (0.01 K, whereas the accuracy of the pressure transducer is (0.1 kPa. The equipment and experimental procedure have been described in a previous paper.18 The composition of both phases has been analyzed by measuring simultaneously the density, F, and the speed of sound, u, of the sample and obtaining the corresponding isentropic compressibility (κS ) 1/Fu2). Densities and speeds of sound of the samples were measured with an Anton Paar DSA-48, which was calibrated using deionized twice-distilled water and dry air; the uncertainty in the determination of the isentropic compressibility19 is (0.1 TPa-1. Prior to this, the isentropic compressibility-calibration curves were obtained at 298.15 K. For each mixture the isentropic compressibilities were

10.1021/ie0503388 CCC: $30.25 © 2005 American Chemical Society Published on Web 07/06/2005

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Table 3. Isobaric VLE Data of Cyclic Ether (1) + Chloroalkane (2): Temperature, T, Liquid-Phase, x1, and Vapor-Phase, y1, Compositions and Activity Coefficients γi T (K)

x1

y1

γ1

γ2

Tetrahydrofuran (1) + 1-Chloropentane (2) at 40.0 kPa 347.81 0.0632 0.1709 0.827w 1.001 346.40 0.0840 0.2267 0.861 1.002 342.38 0.1733 0.3999 0.833 0.994 339.43 0.2386 0.5165 0.857 0.967 336.15 0.3086 0.6069 0.865 0.978 334.38 0.3362 0.6417 0.890 0.993 330.53 0.4287 0.7472 0.924 0.944 327.66 0.5017 0.8062 0.940 0.929 326.27 0.5365 0.8323 0.952 0.914 322.28 0.6578 0.9010 0.969 0.861 319.82 0.7354 0.9320 0.980 0.848 317.84 0.8070 0.9511 0.981 0.910 315.69 0.8853 0.9756 0.994 0.839 Tetrahydrofuran (1) + 1-Chloropentane (2) at 101.3 kPa 377.50 0.0678 0.1667 0.846 0.999 376.59 0.0832 0.2100 0.888 0.988 370.93 0.1862 0.4088 0.889 0.980 368.29 0.2311 0.4746 0.890 0.996 363.86 0.3150 0.5984 0.925 0.977 362.64 0.3443 0.6341 0.926 0.966 358.78 0.4245 0.7301 0.961 0.916 355.22 0.5129 0.7871 0.947 0.958 354.50 0.5537 0.8181 0.930 0.914 350.27 0.6389 0.8784 0.978 0.869 347.22 0.7239 0.9203 0.990 0.827 344.34 0.8118 0.9500 0.994 0.842 342.15 0.8860 0.9667 0.992 1.000 Tetrahydropyran (1) + 1-Chloropentane (2) at 40.0 kPa 350.77 0.0620 0.0842 0.753 0.998 349.97 0.1363 0.1853 0.773 0.991 348.50 0.2465 0.3258 0.788 0.988 346.30 0.3801 0.4959 0.836 0.969 344.40 0.5134 0.6550 0.871 0.903 342.41 0.5782 0.7304 0.922 0.874 341.04 0.6407 0.7844 0.937 0.862 339.65 0.7047 0.8437 0.961 0.800 338.30 0.7689 0.8966 0.981 0.710 336.30 0.8604 0.9485 0.995 0.631 Tetrahydropyran (1) + 1-Chloropentane (2) at 101.3 kPa 380.46 0.0583 0.0912 0.925 0.994 379.16 0.1332 0.1989 0.913 0.986 376.99 0.2332 0.3261 0.905 0.996 374.25 0.3700 0.5000 0.940 0.972 371.00 0.5153 0.6542 0.965 0.960 369.94 0.5752 0.7175 0.976 0.923 368.30 0.6443 0.7730 0.983 0.930 367.05 0.7014 0.8221 0.995 0.901 365.74 0.7672 0.8707 1.000 0.874 363.90 0.8643 0.9284 0.997 0.878

fitted by the method of least squares to the following equation:

(1)

The values of the adjustable parameters, Ai, are collected in Table 2 together with the isentropic compressibilities of the pure compounds, κS,i. The error in the determination of liquid and vapor mole fractions is estimated to be (0.0004. The proper operation of the different devices was periodically checked and rearranged if necessary. Results Experimental vapor-liquid values, T, x1, y1, together with the calculated activity coefficients, γi, of studied

x1

y1

γ1

γ2

1,3-Dioxolane (1) + 1-Chloropentane (2) at 40.0 kPa 348.18 0.0405 0.1390 1.388 1.002 343.30 0.1227 0.3433 1.332 0.991 341.35 0.1568 0.4083 1.325 0.997 335.52 0.2944 0.5959 1.264 1.009 331.50 0.4192 0.7089 1.223 1.030 329.50 0.4986 0.7652 1.196 1.040 329.06 0.5173 0.7707 1.180 1.074 328.35 0.5563 0.7937 1.161 1.081 327.35 0.6051 0.8189 1.144 1.110 327.04 0.6197 0.8263 1.141 1.119 325.99 0.6973 0.8609 1.100 1.175 324.81 0.8161 0.9082 1.038 1.339 1,3-Dioxolane (1) + 1-Chloropentane (2) at 101.3 kPa 377.50 0.0499 0.1405 1.200 1.011 372.64 0.1238 0.3217 1.263 0.993 370.88 0.1554 0.3778 1.241 0.995 363.99 0.2969 0.5721 1.196 1.010 359.38 0.4159 0.6822 1.168 1.043 357.97 0.4768 0.7328 1.142 1.024 356.83 0.5091 0.7546 1.140 1.040 355.81 0.5501 0.7786 1.124 1.058 354.77 0.6017 0.8049 1.097 1.089 354.26 0.6184 0.8169 1.101 1.085 352.69 0.6963 0.8519 1.071 1.161 350.74 0.8186 0.9064 1.031 1.311 1,4-Dioxane (1) + 1-Chloropentane (2) at 40.0 kPa 349.55 0.1287 0.1886 1.321 0.992 348.37 0.2378 0.3038 1.201 1.013 347.9 0.28 0.3543 1.209 1.011 347 0.3831 0.4459 1.149 1.045 346.27 0.4984 0.5558 1.13 1.057 346.02 0.5776 0.6151 1.089 1.097 345.8 0.6564 0.6813 1.069 1.126 345.7 0.7308 0.7395 1.046 1.179 345.71 0.7873 0.7835 1.029 1.24 345.82 0.8337 0.8241 1.018 1.283 346.01 0.8931 0.8755 1.002 1.404 346.25 0.9413 0.9277 0.999 1.473 1,4-Dioxane (1) + 1-Chloropentane (2) at 101.3 kPa 379.00 0.1270 0.1803 1.242 1.007 377.25 0.2502 0.3220 1.185 1.018 377.38 0.2396 0.3109 1.190 1.017 375.75 0.3877 0.4578 1.136 1.041 374.88 0.4948 0.5528 1.104 1.067 374.45 0.5814 0.6215 1.069 1.103 374.21 0.6632 0.6907 1.049 1.129 373.98 0.7304 0.7430 1.032 1.180 373.99 0.7866 0.7918 1.021 1.208 373.96 0.8320 0.8316 1.015 1.242 374.10 0.9001 0.8894 0.999 1.367 374.18 0.9405 0.9373 1.005 1.299

systems are listed in Table 3. The activity coefficients have been calculated using the equations

i)p

Ai(x1 - x2)i ∑ i)0

κS (TPa-1) ) x1κS,1 + x2κS,2 + x1x2

T (K)

γi )

yiP xipi

exp 0

[

]

(Bii - Vi0)(P - pi0) + (1 - yi)2Pδij RT δij ) 2Bij - Bii - Bjj

(2) (3)

where xi and yi are the liquid- and vapor-phase molar fraction compositions, P is the total pressure, and pi0 are vapor pressures of the pure compounds calculated from the Antoine equation, for which the constants16,17,20-22 are gathered in Table 4. Bii are the second virial coefficients calculated using the RedlichKwong equation,23 and Bij are the cross second virial coefficients calculated using a suitable mixing rule. Vi0 are the molar volumes of saturated pure liquids, which

Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6983 Table 4. Constants of Antoine’s Equation for Vapor Pressures of the Pure Compounds log(P/kPa) ) A - B/(C + (T/°C)) compound

A

B

C

tetrahydrofuran16 tetrahydropyran20 1,3-dioxolane21 1,4-dioxane22 1-chloropentane17

6.12142 5.85520 6.23182 6.5564 5.93641

1203.11 1131.93 1236.7000 1554.679 1271.16

226.355 205.83 217.235 240.337 215

Table 5. Results of the Thermodynamic Consistency Test; Average Deviation in Pressure ∆P and Average Deviation in Vapor Phase Composition ∆y system

P (kPa)

∆P (kPa)

∆y

40.0 101.3 40.0 101.3 40.0 101.3 40.0 101.3

0.3 1.8 0.3 0.5 0.2 0.4 0.1 0.3

0.0026 0.0048 0.0046 0.0028 0.0020 0.0024 0.0027 0.0017

tetrahydrofuran tetrahydropyran 1,3-dioxolane 1,4-dioxane + 1-chloropentane

were calculated according to the Yen and Woods method.24 The exponential term in eq 1 was important only at diluted solutions. Thermodynamic consistency of the experimental results has been checked using the van Ness method,25 as described by Fredenslund.26 This method considers that experimental data are thermodynamically consistent if the mean absolute deviation between calculated and measured vapor-phase compositions, ∆y, is 1, that is, positive deviations from ideality.

Table 6. Correlation Parameters, Average Deviation in Temperature ∆T, and Average Deviation in Vapor-Phase Composition ∆y equation

P (kPa) A12 (J‚mol-1) A21 (J‚mol-1) ∆T (K)

∆y

Tetrahydrofuran (1) + 1-Chloropentane (2) 40.0 933.8838 -1568.4962 0.22 Wilsona 101.3 3261.7099 -3160.7303 0.20 40.0 -579.9823 -57.0121 0.22 NRTLb 101.3 -2149.1309 2012.7716 0.19 UNIQUACc 40.0 -788.9394 704.0922 0.18 101.3 -902.6049 876.5261 0.16

0.0036 0.0043 0.0036 0.0046 0.0036 0.0050

Tetrahydropyran (1) + 1-Chloropentane (2) 40.0 -3120.0170 7308.6140 0.19 101.3 -459.6099 117.8887 0.11 NRTL 40.0 3146.3659 -3369.7972 0.32 101.3 3474.8255 -2959.5546 0.08 UNIQUAC 40.0 2601.9503 -2057.3575 0.27 101.3 2513.0447 -1879.5828 0.09

0.0031 0.0042 0.0037 0.0032 0.0034 0.0026

1,3-Dioxolane (1) + 1-Chloropentane (2) 40.0 227.5679 2629.6461 0.09 101.3 174.5748 2255.0855 0.12 40.0 4630.1628 -2017.0096 0.10 101.3 4725.5847 -2315.8934 0.08 40.0 1255.6866 -441.4630 0.08 101.3 1117.0971 -416.8927 0.12

0.0025 0.0027 0.0027 0.0025 0.0027 0.0026

1,4-Dioxane (1) + 1-Chloropentane (2) 40.0 255.1795 1221.5258 0.05 101.3 915.5338 158.0679 0.08 40.0 2935.9401 -1383.5273 0.04 101.3 1702.0957 -599.3203 0.08 40.0 -365.1489 1433.4084 0.05 101.3 -740.3605 1950.7081 0.09

0.0032 0.0017 0.0033 0.0017 0.0032 0.0018

Wilson

Wilson NRTL UNIQUAC

Wilson NRTL UNIQUAC a

Aij ) λij - λii. b Aij ) gij - gii. c Aij ) uij - uii.

Figure 1. T-x1-y1 diagram for tetrahydrofuran (1) + 1-chloropentane (2): (O, b) experimental data at 40.0 kPa; (0, 9) experimental data at 101.3 kPa; (s) Wilson equation.

UNIFAC Parameters Our thermodynamically consistent vapor-liquid equilibrium results for mixtures containing a cyclic ether (tetrahydrofuran, tetrahydropyran 1,3-dioxolane, or 1,4-

dioxane) and normal and branched chloroalkanes (1chlorobutane, 2-chlorobutane, 2-methyl-1-chloropropane, 2-mehtyl-2-chloropropane, or 1-chloropentane)

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Figure 2. T-x1-y1 diagram for tetrahydropyran (1) + 1-chloropentane (2): (O, b) experimental data at 40.0 kPa; (0, 9) experimental data at 101.3 kPa; (s) Wilson equation.

Figure 4. T-x1-y1 diagram for 1,4-dioxane (1) + 1-chloropentane (2): (O, b) experimental data at 40.0 kPa; (0, 9) experimental data at 101.3 kPa; (s) Wilson equation. Table 7. Original and Recalculated UNIFAC Parameters group

group

a (K)

b

c (K-1)

CCl c-CH2O

Original UNIFAC Parameters c-CH2O -325.77 2.0412 CCl 70.075 -1.1490

0.0 0.0

CCl c-CH2O

Recalculated UNIFAC Parameters c-CH2O -125.637 -0.272 CCl 434.775 -2.822

0.00171 0.00381

The van der Waals properties of the subgroup used in the calculation routines are those of the modifiedUNIFAC method.9 The optimization of the UNIFAC parameters was based on the minimization of an objective function in terms of experimental and calculated activity coefficients27 extended to all of the systems considered i)N

F)

Figure 3. T-x1-y1 diagram for 1,3-dioxolane (1) + 1-chloropentane (2): (O, b) experimental data at 40.0 kPa; (0, 9) experimental data at 101.3 kPa; (s) Wilson equation.

have been used to recalculate the UNIFAC parameters between the groups CCl and c-CH2O.

∑ i)1

[(

) (

γ1exptl - γ1UNIFAC γ1exptl

2

+

)]

γ2exptl - γ2UNIFAC γ2exptl

2

i

(4)

where N is the number of experimental data points, N ) 478. This optimization was performed by using the simulated annealing method.28,29 In Table 7 are shown the recalculated UNIFAC parameters. The temperature and vapor-phase composition obtained experimentally were compared with the predictions provided by the original and recalculated UNIFAC parameters. When the original parameters were used to predict the behavior of our mixtures, we found that the deviations between experimental and predicted data were quite large for most of the mixtures even though the original UNIFAC predicted azeotropic points for

Figure 5. T-x1-y1 diagrams for tetrahydrofuran (THF) or tetrahydropyran (THP) + chloroalkanes at 40.0 and 101.3 kPa: (s) experimental diagram; (- - -) predicted diagram with the recalculated UNIFAC parameters.

Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6985

Figure 6. T-x1-y1 diagrams for 1,3-dioxolane (1,3) or 1,4-dioxane (1,4) + chloroalkanes at 40.0 and 101.3 kPa: (s) experimental diagram; (- - -) predicted diagram with the recalculated UNIFAC parameters.

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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6987 Table 8. UNIFAC Predictions; Average Deviation in Temperature, ∆T, and Average Deviation in Vapor-Phase Composition, ∆y original parameters system tetrahydrofuran + 1-chlorobutane 2-chlorobutane 2-methyl-1-chloropropane 2-methyl-2-chloropropane 1-chloropentane tetrahydropyran + 1-chlorobutane 2-chlorobutane 2-methyl-1-chloropropane 2-methyl-2-chloropropane 1-chloropentane 1,3-dioxolane + 1-chlorobutane 2-chlorobutane 2-methyl-1-chloropropane 2-methyl-2-chloropropane 1-chloropentane 1,4-dioxane + 1-chlorobutane 2-chlorobutane 2-methyl-1-chloropropane 2-methyl-2-chloropropane 1-chloropentane average overall

recalculated parameters

P (kPa)

∆T (K)

∆y

∆T (K)

∆y

40.0 101.3 40.0 101.3 40.0 101.3 101.3 40.0 101.3

1.42 1.31 0.66 0.93 1.30 1.67 0.34 0.72 1.09

0.0188 0.0185 0.0143 0.0148 0.0214 0.0221 0.0089 0.0132 0.0147

0.73 0.15 0.05 0.04 0.59 0.51 0.34 0.31 0.43

0.0082 0.0016 0.0015 0.001 0.0084 0.0060 0.0092 0.0038 0.0045

40.0 101.3 40.0 101.3 40.0 101.3 101.3 40.0 101.3

1.44 1.78 1.15 1.03 1.83 2.13 0.37 0.53 1.32

0.0211 0.0209 0.0159 0.0127 0.0243 0.0235 0.0059 0.0142 0.0183

0.50 0.32 0.33 0.18 0.85 0.65 0.74 1.14 0.24

0.0070 0.0039 0.0042 0.0021 0.0111 0.0069 0.0079 0.0169 0.0033

40.0 101.3 40.0 101.3 40.0 101.3 101.3 40.0 101.3

0.91 1.03 0.63 0.72 1.22 1.21 1.85 1.07 0.91

0.0245 0.0242 0.0227 0.0219 0.0266 0.0265 0.0325 0.0199 0.0184

0.06 0.53 0.72 1.34 0.34 0.40 2.85 0.35 1.08

0.0010 0.0071 0.0214 0.0179 0.0052 0.0067 0.0366 0.0058 0.0140

40.0 101.3 40.0 101.3 40.0 101.3 101.3 40.0 101.3

2.87 3.40 2.67 2.06 3.37 4.08 1.30 2.27 2.74

0.0370 0.0360 0.0319 0.0221 0.0387 0.0419 0.0111 0.0356 0.0378

1.46 1.04 1.43 0.69 1.90 1.66 0.18 0.90 0.34

0.0181 0.0108 0.0163 0.0065 0.0224 0.0166 0.0023 0.0113 0.0048

1.54

0.0226

0.70

0.0092

several mixtures such as the one formed by tetrahydropyran and 1-chlorobutane when actually it has no azeotrope. In Table 8 the average deviations in temperature and vapor-phase composition for each system calculated with the two sets of parameters are given. For most of the systems the results obtained with the recalculated parameters are far better than those obtained with the original parameters. The overall averages ∆T and ∆y with the original parameters are 1.54 K and 0.0226, respectively, whereas ∆T ) 0.70 K and ∆y ) 0.0092 are the overall average deviations with the new parameters. Note that an average deviation in temperature of ∼1 K for mixtures formed by tetrahydrofuran and 2-chlorobutane or 2-methyl-1-chloropropane leads to an exceptionally high accuracy in the prediction. When recalculated UNIFAC parameters are used to predict the VLE of the mixtures studied here, the best results are found for the tetrahydrofuran and 2-chlorobutane and the 1,3-dioxolane and 1-chlorobutane mixtures. Only for the mixtures containing tetrahydropyran or 1,3-dioxolane with 2-methyl-2-chloropropane or 1-chloropentane do the temperature and composition averages increase slightly. Finally, in Figures 5 and 6 the experimental and predicted T-x1-y1 diagrams for all of the systems are shown. It is clear that the predictions with the recal-

culated parameters are acceptable and that the general behavior of the mixtures is estimated quite well. Acknowledgment We are grateful for financial assistance from Diputacio´n General de Arago´n. B.G. thanks the Ministerio de Educacio´n y Ciencia for FPI Grant BES-2004-4957, and M.H. thanks the Ministerio de Educacio´n y Ciencia for FPU Grant AP2002-0660. Nomenclature A12, A21 ) adjustable parameters for VLE correlation equations (J‚mol-1) amn, bmn, cmn ) UNIFAC parameters Bii ) second virial coefficient of component i (m3‚mol-1) Bij ) cross second virial coefficient (m3‚mol-1) gij-gii ) parameters for NRTL equation (J‚mol-1) P ) total pressure (Pa) pi0 ) vapor pressure of component i (kPa) R ) gas constant () 8.31441 J K-1 mol-1) T ) temperature (K) Tb ) normal boiling point (K) uij-uii ) parameters for UNIQUAC equation (J‚mol-1) Vi0 ) molar volume of component i (m3‚mol-1) xi ) mole fraction of component i in the liquid phase

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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005

yi ) mole fraction of component i in the vapor phase R12 ) nonrandomness parameter in the NRTL equation (J‚mol-1) ∆ ) average deviation γi ) activity coefficient of component i λij-λii ) parameters for Wilson equation F ) density (kg‚m-3)

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Received for review March 14, 2005 Revised manuscript received May 20, 2005 Accepted June 7, 2005 IE0503388