Ind. Eng. Chem. Res. 2003, 42, 5305-5311
5305
Sulfuric Acid Extraction with TrialkylaminesEffect of Xylene and n-Octanol as Modifiers Jaroslav Procha´ zka,* Alesˇ Heyberger, and Eva Volaufova´ Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova´ 135, 165 02 Prague 6 - Suchdol, Czech Republic
The equilibrium of sulfuric acid extraction with trialkylamine was measured at 25 °C. Xylene, n-octanol, and their mixtures with a paraffinic solvent were used as diluents. The effects of the diluent composition and of the amine content in the organic phase on extraction equilibrium were investigated. An increase of the xylene content in the diluent shifted the limit of thirdphase formation to higher concentrations of acid in the organic phase. No third-phase formation was observed with pure xylene. With n-octanol, the third phase formed only at the highest amine and lowest modifier contents used. The shapes of the amine loading curves indicated preferential formation of the 1:2 acid/amine complex at lower acid concentrations and of the 1:1 complex and its dimer (2:2) in the high-concentration range. At lower acid contents, a positive synergetic effect of the two modifiers on the equilibrium was observed. In systems with n-octanol, the physical extraction with the modifier contributed to the overall acid extraction. The experimental results were correlated with a mathematical model incorporating chemical reactions of acid/ amine complex formation and nonspecific interactions of species present in the organic phase. Introduction The extraction of sulfuric acid with solutions of highmolecular-weight amines in organic diluents can be applied for its separation from waste aqueous streams of various industrial processes. Amine salts of sulfuric acid are also often used as liquid anion exchangers in various hydrometallurgical processes, e.g., in the production of tungsten and molybdenum.1,2 Tertiary aliphatic amines are most often proposed as extractants; pure aromatic hydrocarbons, or mixtures of various polar modifiers with kerosene, as diluents. The most frequently used tertiary amines are trioctylamine; tridecylamine; tridodecylamine; and Alamine 336, a commercial product with C8-C10 alkyl chains. As modifiers, n-octanol and iso-decanol have been applied. Allen3 and Wilson4 presented data on sulfuric acid extraction from aqueous solutions with solutions of tri-n-octylamine (TOA) in benzene. In addition to formation of the sulfate and hydrogen sulfate of TOA, the dimer and trimer of the latter were also identified. Aggregation was not observed with normal sulfate. Kim and Chiola5 obtained similar results with solutions of tri-n-caprylamine (TCA) in benzene. Their data were used by Yun,6 who developed a mathematical model of sulfuric acid extraction with trialkylamine. In the model, formation of the normal amine sulfate and of the dimer of hydrogen sulfate were incorporated. The nature and composition of the diluent has a significant effect on the extraction equilibrium. In addition to decreasing the viscosity and density of the organic phase, the diluent has to prevent formation of a third phase (second organic phase) at higher concentrations of the acid/amine complexes in the organic phase. Instead of pure active diluents, their mixtures with inert hydrocarbon diluents have often been used. The influence of the modifier content in a mixed diluent on third-phase formation was studied on the amine * To whom correspondence should be addressed. Tel.: (420) 20390236. Fax: (420) 20920661. E-mail:
[email protected].
extraction of citric acid.7 In ternary systems of modifier/ inert diluent/amine salt, the equilibrium composition of the two organic phases and the critical point were determined. The modifier can also enhance the extraction equilibrium. Frolov and Sergievski8 investigated the effect of the n-octanol content in mixtures with benzene on the extraction of sulfuric acid with TOA. They found that the addition of n-octanol enhanced the formation of sulfate and hindered the formation of hydrogen sulfate of TOA. The result they explained in terms of the formation of bisolvate (R3N)2SO4‚2ROH at low acid concentrations. They used cryoscopy for identification of that species and determination of its stability constants. Various quantitative expressions of the solvating power of diluents, based on a linear solvating energy relationship, have been proposed.9-11 They allow for the expression of the effect of the diluent on the logarithm of the extraction equilibrium constant as a linear function of coefficients characterizing the interaction forces. In a previous work,12 sulfuric acid was extracted with trialkylamine dissolved in a mixture of de-aromatized kerosene and n-octanol as the diluent. In most experiments, the mass fraction of n-octanol in the diluent was 0.1, and the range of amine molality in the solvent was 0.1-0.5 mol/kg of diluent. In the range of acid concentrations used, no third-phase (second organic phase) formation was observed. The experimental data were correlated with a mathematical model based on the formation of amine sulfate and bisulfate, as well as a dimer of the latter, in the organic phase. The nonideality of the two phases was taken into account. The model was also successfully used for correlation of the results3 of sulfuric acid extraction with TOA in benzene. From the literature cited, the properties of aromatic hydrocarbons as modifiers in the amine extraction of sulfuric acid are not clear. With trialkylamine in benzene, no formation of a third phase was reported. In contrast, the formation of a second organic phase was observed when kerosene was used as the diluent, even
10.1021/ie020864o CCC: $25.00 © 2003 American Chemical Society Published on Web 09/06/2003
5306 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
though kerosene usually contains various proportions of aromatic hydrocarbons. In the present work, the extraction equilibria of sulfuric acid with TAA dissolved in mixtures of xylene with de-aromatized kerosene, as well as in pure xylene, were investigated in a broad range of aqueous-phase acid concentrations. Analogous measurements were made with mixtures of kerosene and n-octanol and with pure n-octanol. In both systems, the content of amine in the organic phase was also varied. The conditions of third-phase formation and the effect of the diluent on the extraction equilibrium were determined. The data were correlated with a mathematical model similar to that used in the previous work.12 It also includes the effect of modifier content in the diluent and the physical extraction with pure diluent. Mathematical Model The model comprises dissociation of the acid in the aqueous phase, extraction of the ions, and formation of the acid/amine complexes. According to the literature,3-5 ammonium sulfate and bisulfate complexes, as well as the dimer of the latter, are assumed to form. +
-
diluent, xj, and the loading of amine with acid, Z ) ma/ m0e .
K′ij ) Kij(γ()b exp(Aijm0e + Bijxj + CijZ), b ) 0, 2, 3 (10) The linear combination of these factors corresponds formally to the linear solvation energy relationship.9-11 The model contains 12 empirical parameters to be determined: Kij, Aij, Bij, and Cij. The values of γ( as a function of the aqueous-phase acid molality were calculated with the relation
γ( ) 0.3 log(0.66/ma + 0.34) + 0.13ma
(11)
obtained by interpolation of the data from Atkins.13 In the case of n-octanol as the modifier, the physical extraction with the modifier cannot be neglected at low amine content, high modifier fraction in the diluent, and high acidity of the aqueous phase. The respective data were treated with a modification of the model including an empirical relation expressing this effect
map ) Kp1xjma exp(Kp2ma)
(12)
H2SO4 f H + HSO4
(1)
HSO4- ) H + + SO42-
(2)
The values of the constants, Kp1 ) 0.0025 and Kp2 ) 0.885, were determined from the measurements of acid extraction with pure n-octanol (Figure 5).
2R3N + 2H+ + SO42- ) (R3NH)2SO4
(3)
Experimental Section
(R3NH)2SO4 + H+ + HSO4- ) 2R3NH‚HSO4 (4) 2R3NH‚HSO4 ) (R3NH‚HSO4)2
(5)
Here, R3N denotes the tertiary amine, and the bar denotes a species in the organic phase. The respective thermodynamic equilibrium constants, K, and massaction law quotients, K′, are
mhma2 γ( ) K′Dγ( ma1
KD )
K12 )
K11 )
m12
γ12
γ12 ) K′12 mh ma2me (γ() γe (γ()3γe2 2
2
m112
3
2
γ112 ) K′11 mhma1m12 (γ()2γ12 (γ()2γ12 K22 )
γ112
m22 γ22 m112
γ112
γ22 ) K′22 γ112
(6)
(7)
(8)
(9)
where m is the molality and γ is the activity coefficient. For the second dissociation constant, the value KD ) 1.2 × 10-2 at 25 °C was used.13 The model does not include the formation of solvates or hydrates of the reaction products. They are taken as part of the nonspecific interactions between the species in the aqueous and organic phases. The ratios of activity coefficients in the organic phase in eqs 7-9 are expressed as exponential functions of the initial amine molality, m0e , the mass fraction of the modifier in the
The initial organic phase was a solution of trialkylamine in the diluent. The trialkylamine used, TAA, was a Russian commercial product with C7-C9 alkyl chains. Before use, it was consecutively washed with aqueous HCl and NaOH and with distilled water. Its average molar mass is 363.3 g/mol. The diluent was a mixture of xylene or n-octanol as modifiers and de-aromatized kerosene, BDR, as inert diluent. BDR is a commercial product of Chemopetrol, a.s. (bp 135-200 °C). Xylene (98.5% of isomers) was purchased from Lachema, a.s., and n-octanol (99% purity) from Aldrich. The aqueous phase was a solution of chemically pure sulfuric acid (Lachema, a.s.) in distilled water. The phases were contacted by shaking in separating funnels immersed in a thermostated bath for 15 min and were separated after several hours of settling. All experiments were performed at 25 ( 0.2° C. The initial acid content was 0.1-5.0 mol/kg of water, and the volumetric aqueous/organic phase ratio varied from 1:3 to 3:1. The samples of separated phases were filtered before analysis. The range of equilibrium acid molality in the aqueous phase was (1.2 × 10-4)-(4.5 × 100) mol/ kg of water; in the organic phase, it was (2 × 10-2)(1.3 × 100) mol/kg of diluent. The acid content in the equilibrium water phase was determined by potentiometric titration with aqueous NaOH. In the organic phase, sodium methanolate with addition of acetone was used to ensure homogeneity. Titration with perchloric acid in anhydrous acetic acid was used to determine the molality of amine in the initial organic phase. The relative uncertainty of these measurements was 1 can form. For sulfuric acid extraction, in the region of predominating formation of ammonium sulfate, the following balance of TAA in the organic phase approximately holds
m0e ) me + 2m12
(13)
Figure 1. Equilibrium loading curves, Z vs log ma, of sulfuric acid extraction with TAA in diluent with xylene as the modifier. m0e ) (a) 0.1, (b) 0.5, (c) 1.0 mol/kg of diluent.
Substituting into eq 7, one obtains
Z ) K′12mh2ma2m0e (1 - 2Z)2
(14)
5308 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
Figure 2. Effect of amine molality on loading at constant xylene content in diluent, xj ) 0.5.
Figure 3. Effect of amine molality on loading at constant xylene/ amine ratio rj ) 1.0 kg/mol.
Equation 14 predicts that, at constant aqueous-phase acid molality, the loading of amine will increase with increasing initial amine molality. Similarly it can be shown that the formation of the 2:2 dimer might cause an increase in the loading with increasing m0e , as long as the maximum loading, Z ) 1, has not been reached (see Figure 2). Equation 14 was derived and interpreted under the assumption of constancy of the mass-action law quotient K′12. However, according to eq 10, the equilibrium constant K′12 can vary with m0e as a result of nonspecific interactions of the species in the organic phase. As mentioned earlier,12 one of these effects is a drop in the constant caused by the decreasing modifier-to-amine ratio, (rj ) xj/m0e kg/mol). In Figure 3, where the loading curves at constant r are compared, an enhancement of the effect of the amine content is apparent. Experimental Results: n-Octanol. The results of experiments with n-octanol as the modifier are depicted in Figure 4a-c. The data sets for
m0e
) 0.1 mol/kg of
diluent, xj ) 0.1, and m0e ) 0.5 mol/kg of diluent, xj ) 0.1
Figure 4. Equilibrium loading curves, Z vs log ma, of sulfuric acid extraction with TAA in diluent with n-octanol as modifier. m0e ) (a) 0.1, (b) 0.5, (c) 1.0 mol/kg of diluent.
and 0.5, were obtained from previous work.12 In contrast to the results with xylene, third-phase formation was observed only at the highest amine molality and the lowest fraction of n-octanol in the diluent.
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5309
Figure 5. Physical extraction of sulfuric acid with pure octanol. Line calculated with eq 9.
Figure 6. Effect of amine molality on loading at constant n-octanol content in diluent, xj ) 0.5.
The general form of the loading curves is similar to that for the systems with xylene. Again, a significant effect of the modifier can be observed in the region of low aqueous-phase acid molality, where formation of the 1:2 complex dominates. With n-octanol, the region of rapid ascent of the loading curves is shifted to much lower aqueous-phase acid concentrations, and the curves tend to form a plateau at Z ≈×93 0.5, corresponding to the predomination of the 1:2 complex in the organic phase. Thus, n-octanol exhibits a strong stabilizing effect on the 1:2 complex because of its proton-donating character. In contrast to xylene, with n-octanol, an increase in acid extraction with increasing modifier content in the diluent appears also in the high acid concentration range. Here, also, a tendency toward overloading (Z > 1) can be observed. These effects were attributed to the acid solubility in n-octanol. The results of physical
Table 1. Optimum Parameter Valuesa for Systems with Xylene as the Modifier
m0e
) 0 mol/kg of diluent, extraction with pure octanol, xj ) 1, are depicted in Figure 5. Unlike the systems with xylene as the modifier, those with n-octanol do not show a distinct positive effect of amine molality on loading at constant xj in the low acid concentration range (Figure 6). A similar observation was made earlier12 for a smaller range of m0e variation and was explained by the decreasing modifier/amine ratio rj. Data Correlation with the Model. The experimental data depicted in Figures 1 and 4 were correlated using the model described above. A nonlinear leastsquares algorithm15 was used for model solution and optimization of the parameter values. As the objective function, the sum of squares of relative differences between the experimental and calculated values was used. Both the aqueous-phase and the organic-phase equilibrium acid molalities were taken as subject to error. N
S)
∑ Wk k)1
[(
1-
ma(cal) ma(exp)
) ( 2
+ 1-
ma(cal) ma(exp)
)]∑ 2
N
k
N
k)1
Wk
(15)
complex i:j 1:2 1:1 2:2 a
Kij
Aij
Bij
Cij
1.149 × -3.886 × 2.308 × 4.550 × 100 2.957 × 101 3.089 × 100 -6.342 × 10-3 2.504 × 10-1 2.397 × 10-1 8.429 × 10-3 -1.838 × 10-2 -3.332 × 10-1 10-1
106
100
N ) 114, s ) 6.19 × 10-2.
where N is the number of experimental points and Wk is the weight of point k. The relative standard deviation, s, of the measured and calculated equilibrium molalities in the aqueous and organic phases was also determined.
s)
x
1
N
∑
N - nk)1
[(
1-
ma(cal) ma(exp)
) ( 2
+ 1-
m j a(cal) m j a(exp)
)] 2
k
(16)
Here, n is the number of model parameters used (12 in the present case). The significance of the individual values of parameter Kij, Aij, Bij, and Cij was assessed by the F-test.16 The ratio of variances for a particular parameter P
Fp ) sP2/s2, P ) Kij, Aij, Bij, Cij
(17)
was evaluated for the model with P ) 0 and for the full model and was compared with the tabulated 5% confidence limit F0.05. For the systems with xylene, the optimum parameter values are presented in Table 1, and the calculated loading curves are depicted as lines in Figure 1a-c. The low value of the relative standard deviation s indicates a good fit of the measured and calculated results. The significance of the individual parameters and their effects on the extraction process can be assessed from the results of the F-test presented in Table 2. The respective F values indicate a high significance of formation of the 1:2 and 1:1 complexes and a low significance of dimer formation. The effect of the modifier on sulfate formation, B12, is quite significant; that for bisulfate formation, B11, is insignificant, in accord with Figure 1a-c. However, a low significance estimate of a particular reaction, based on the variance of the whole data set, does not exclude its local significance
5310 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 Table 2. F-Testa of Parameter Values for Systems with Xylene as the Modifier complex i:j
Kij
Aij
Bij
Cij
1:2 1:1 2:2
1.19 × 102 1.57 × 101 1.03 × 100
1.51 × 100 3.41 × 100 1.00 × 100
1.23 × 101 1.06 × 100 1.00 × 100
5.58 × 100 1.01 × 100 1.00 × 100
a
F0.05 ) 1.40 × 100.
Figure 8. Contributions of individual complexes, and of physical extraction, to overall loading of amine. System with n-octanol: m0e ) 1.0 mol/kg of diluent, xj ) 0.5.
Figure 7. Contributions of individual complexes to overall loading of amine. System with xylene: m0e ) 1.0 mol/kg of diluent, xj ) 0.5. Table 3. Optimum Parameter Valuesa for Systems with n-Octanol as the Modifier complex i:j 1:2 1:1 2:2 a
K
A
B
C
3.658 × 109 -4.896 × 100 1.004 × 101 -4.903 × 100 3.672 × 101 2.914 × 100 1.085 × 100 -1.116 × 100 1.991 × 10-1 -1.515 × 100 7.300 × 10-4 8.257 × 10-2
N ) 167, s ) 6.09 × 10-2.
Table 4. F-Testa of Parameter Values for Systems with n-Octanol as the Modifier complex i:j
Kij
Aij
Bij
Cij
1:2 1:1 2:2
1.24 × 102 1.16 × 101 1.02 × 100
6.96 × 100 2.22 × 100 1.02 × 100
1.28 × 101 1.29 × 100 1.00 × 100
2.73 × 100 1.21 × 100 1.00 × 100
a
F0.05 ) 1.27 × 100.
in a limited part of the acid concentration range measured. This is illustrated in Figure 7, where the calculated contributions of individual complexes, Zij, to the overall loading of amine are depicted. The figure shows the location of the individual loading curves and their partial overlapping. It can be seen that the contribution of the 2:2 complex is significant in the high acid concentration range. For the systems with n-octanol as the modifier, the calculated loading curves are depicted as lines in Figure 4a-c. In Table 3, the optimum parameter values are presented. The higher values of the equilibrium constants K12 and K11 in comparison with those in Table 1 confirm the higher effectiveness of n-octanol as a modifier. The results of the F-test are reported in Table 4, which gives a picture similar to that presented by Table 2. In Figure 8, the contributions of individual complexes and of physical extraction to the overall loading are
Figure 9. Comparison of overall and “true” amine loading in systems with n-octanol and m0e ) 0.1 mol/kg of diluent. Z1, Z′1 - xj ) 0.5; Z2, Z′1 - xj ) 1.
depicted. It can be seen that, with n-octanol, the 1:2 complex starts to form at about one-order-lower aqueous acid molality than that with xylene, and that formation of the dimer is largely suppressed. The contribution of physical extraction appears still less significant at the amine molality of m0e ) 0.5 mol/kg of diluent used. Figure 9 shows that, at m0e ) 0.1 mol/kg of diluent and xj ) 0.5 and 1, this effect is important in the range of higher acid concentrations. Here, the overall amine loading, Z, is compared with the “true” loading, Z′ ) Z - m j ap/m0e , corrected for the physical extraction. The corrected loading curves asymptotically approach Z′ ) 1, thus confirming that the apparent overloading in Figure 4a is due to physical extraction with the modifier. Conclusions The measurements of extraction equilibria of sulfuric acid with trialkylamine in mixtures of xylene and a
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5311
paraffinic diluent, as well as in pure xylene, presented xylene as a moderately active modifier. As its fraction in the paraffinic diluent was increased, the limit of third-phase formation was shifted to higher acid concentrations; with pure xylene, third-phase formation did not occur. In systems using n-octanol as the modifier, third-phase formation was observed only at the highest aqueous-phase acid molality, the highest amine content in the organic phase, and the lowest fraction of modifier in the diluent. Hence, n-octanol is more effective in preventing third-phase formation than xylene. The measurements of extraction as a function of the amine molality and the fraction of modifier in the diluent confirmed formation of the 1:2, 1:1, and 2:2 acid/ amine complexes as the main mechanisms of extraction of sulfuric acid with trialkylamine. In the low aqueous acid concentration range, the acid extraction increased with increasing modifier content in the diluent; in the systems with n-octanol, the extraction was significantly higher than in those with xylene. In this region, the loading of amine rose with increasing amine content in the organic phase in systems with xylene, but it was almost independent of this parameter in systems with n-octanol. In the higher acid concentration range with predominating 1:1 and 2:2 complexes, the loading of amine was almost independent of both the fraction of modifier and the molality of amine. With n-octanol, an effect of physical extraction with the modifier was observed. It was significant with pure octanol, at low amine content, and at high aqueous-phase acid concentration. The experimental data were correlated with a mathematical model comprising formation of three acid/ amine complexes, 1:2, 1:1, and 2:2. For systems with octanol, physical extraction with the modifier was included. A good fit was obtained for systems with both modifiers. A statistical test of the significance of the model parameters indicated low significance of 2:2 complex formation. With the model, the contributions of the individual complexes to the overall loading of amine were simulated. Acknowledgment This work was supported by Grant 104/02/1108 of the Grant Agency of Czech Republic. Nomenclature A, B, C ) model parameters in eq 10 K, K′ ) thermodynamic equilibrium constant and massaction law quotient, respectively KD, K′D ) second dissociation constant and mass-action law quotient, respectively Kp1, Kp2 ) parameters in eq 12 m ) molality in the aqueous phase, mol/kg of water m j ) molality in the organic phase, mol/kg of diluent m j ap ) molality of physically extracted acid, mol/kg of diluent m0e ) initial amine molality in the organic phase, mol/kg of diluent n ) number of model parameters N ) number of experiments p ) number of acid molecules in the acid/amine complex q ) number of amine molecules in the acid/amine complex rj ) xj/m0e ) ratio of modifier to amine in the organic phase, kg of modifier/mol of amine
s ) relative standard deviation of measured and calculated equilibrium molalities in the aqueous and organic phases S ) objective function in least-squares algorithm W ) statistical weight of an experimental point xj ) mass fraction of modifier in the diluent Z ) loading of amine with sulfuric acid Z′ ) “true” loading of amine (corrected for physical extraction) γ( ) mean ionic activity coefficient in the aqueous phase γ j ) activity coefficient in the organic phase Subscripts a ) sulfuric acid a1, a2 ) hydrogensulfate and sulfate anions e ) trialkylamine ij ) related to i:j acid/amine complex k ) kth experimental point Superscript b ) exponent in eq 10
Literature Cited (1) Ritcey, G. M.; Ashbrook, A. W. Solvent Extraction; Elsevier: Amsterdam, 1979. (2) Lassner, E.; Ortner, H.; Fichte, R. M.; Wolf, H. U. Ullmann’s Encyclopaedia of Technical Chemistry; Verlag Chemie: Weinheim, Germany, 1983 (in German). (3) Allen, K. A. The equilibria between tri-n-octylamine and sulfuric acid. J. Phys. Chem. 1956, 60, 239-245. (4) Wilson, A. Extraction of sulfuric acid with solutions of trin-octylamine in benzene. In Solvent Extraction Chemistry; Dyrssen, D., Liljenzin, J. O., Rydberg, J., Eds.; North-Holland Publishing Co.: Amsterdam, 1966; p 369. (5) Kim, T. K.; Chiola, V. Extraction of sulfuric acid in the trin-caprylamine-benzene system. Sep. Sci. 1968, 3, 455-465. (6) Yun, C. K. A mathematical description of the extraction isotherms of tungsten and sulfuric acid in a tertiary amine. Hydrometallurgy 1984, 12, 289-298. (7) Heyberger, A.; Procha´zka, J.; Volaufova´, E. Extraction of citric acid with tertiary aminesThird-phase formation. Chem. Eng. Sci. 1998, 53, 515-521. (8) Frolov, Iu. G.; Sergievski, V. V. Effect of n-octyl alcohol on sulfuric acid extraction with tri-n-octylamine. J. Inorg. Chem. 1965, 10, 697-702 (in Russian). (9) Hansen, C. M. The universality of the solubility parameter. Ind. Eng. Chem. Prod. Res. Dev. 1969, 8, 1-11. (10) Shmidt, V. S.; Smelov, V. S.; Rybakov, K. A.; Kondratev, B. A. Effect of diluent parameters VR and VR* on extraction equilibrium of various types of extraction systems. Radiokhimia 1983, 38, 191-197 (in Russian). (11) Kamlet, M. J.; Abboud, Jose´-L. M.; Abraham, M. H.; Taft, R. W. Linear solvation energy relationships. 23. A comprehensive collection of the solvatochromic parameters π*, R, and β, and some methods for simplifying the generalized solvatochromic equations. J. Org. Chem. 1983, 48, 2877-2887. (12) Vachtova´, J.; Heyberger, A.; Mrnka, M.; Procha´zka, J. Extraction of sulfuric acid with trialkylamine in a mixed diluent. Ind. Eng. Chem. Res. 1999, 38, 2028-2035. (13) Atkins, P. W. Physical Chemistry, 4th ed.; Oxford University Press: New York, 1992. (14) Tamada, J. A.; Kertes, A. S.; King, J. C. Extraction of carboxylic acids with amine extractants. 1. Equilibria and law of mass action modeling. Ind. Eng. Chem. Res. 1990, 29, 1319-1326. (15) Marquardt, D. W. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 1963, 11, 431-441. (16) Davies, O. L. Stastical Methods in Research and Production, 3rd ed.; Oliver and Boyd: London, 1958.
Received for review October 30, 2002 Revised manuscript received July 17, 2003 Accepted August 4, 2003 IE020864O
