Calculation on ion exchange capacity for an ion

Calculation on Ion Exchange Capacity for an Ion Exchanger Using the Potentiometric Titration WON-KEUN SON,1SANG HERN KIM,1 TAE IL KIM2 1

Department of Chemical Technology, Taejon National University of Technology, Taejon 300-717, Korea

2

Department of Chemical Engineering, Chungnam National University, Taejon 305-764, Korea

Received 24 May 2000; revised 19 July 2000; accepted 8 September 2000

We calculated the characteristics of a phosphoric cation exchanger and studied an accurately computable method for ion exchange capacity for a type of potentiometric titration curve. The ion exchanger was prepared by phosphorylation of a styrene-divinylbenzene copolymer. The ion exchange capacity was 5.7 meq/g. The experimental pK values versus ␹ in a phosphoric cation exchanger can explain a linear equation. The ⌬pK values were obtained from the slope of a linear equation. The ⌬pK values were the differences of pK values between the apparent equilibrium constant at complete and zeroth neutralization of the ion exchanger. The experimental pK values at ␹ ⫽ 0.5 (␹:degree of neutralization of ion exchanger) showed good agreement with the theoretical data. When it was titrated with NaOH and Ba(OH)2 solutions, a good agreement between experimental and theoretical pK values for various ␹ was found in all potentiometric titration curves. The potentiometric titration curve near the inflection point in the case of divalent ions was changed more sharply than that for monovalent ions. The plot of ⭸pH/⭸g versus g (number of moles of alkali to 1 g of ion exchanger) was fitted to the Lorenzian distribution, from which ion exchange capacity was accurately evaluated. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: ABSTRACT:

3181–3188, 2000

Keywords: ion exchanger; phosphoric ion exchanger; potentiometric titration; equilibrium constant

INTRODUCTION Recently, with a rapid development of industry, removal of heavy metal ions and toxic gases in environmental pollution has received much attention. Materials for treatment of wastewater and toxic gas must be developed. Toxic heavy metal can be separated easily from wastewater by ion exchange resin.1–3 Traditional precipitation processes, such as zeolite and clay, do not always provide a satisfac-

Correspondence to: S.H. Kim Journal of Polymer Science: Part B: Polymer Physics, Vol. 38, 3181–3188 (2000) © 2000 John Wiley & Sons, Inc.

tory removal rate to meet the pollution control limit. However, ion exchange resin has various functions, such as waste treatment, separation of rare earth metals and heavy metals, separation of amino acids, purification, sorption of toxic gases, drying agents, and bleaching agents.4 Ion exchange resin can selectively separate toxic gases from the industrial resources of air pollution with simple equipment and little energy. There have been many studies on ion exchange resin by many researchers.5 The electrolytes are commonly divided into nonpolymeric and polymeric. An ion exchanger is a polymeric electrolyte. It is commonly recognized that potentiometric titration is one of the most 3181

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important methods for investigation and characterization of ion exchangers. At the same time, a great discrepancy between theory and practice of this method in application to ion exchangers exists.6 Because the inflection points of the potentiometric titration curve for strong acidic and basic ion exchangers are obvious, the capacity of the ion exchanger can be accurately determined from potentiometric titration curves. However, week acidic and basic type ion exchangers have an equilibrium constant, which is changed by the concentration of supporting electrolytes, and the inflection points of the potentiometric titration curve in this case appear ambiguous.7–12 In this work, we studied a method for computation of the main characteristics of a phosphoric cation exchanger by considering the effects of potentiometric titration curves in an ion exchanger/ solution system. The acidity parameters of ion exchangers, such as thermodynamic constant (pK0), apparent equilibrium constant, and correction for the apparent equilibrium constant (b), were used for expressing the characteristics of a carboxylic ion exchanger. For proving the theory, the phosphoric cation exchanger type was used for an experiment on potentiometric titration and potentiometric titration curves at different concentrations of the supporting electrolyte. From these results, characteristic equations related to the acidity parameters of ion exchanger systems were obtained. Also, the characteristic parameters such as solution pH, supporting electrolyte concentration (C), and degree of neutralization of ion exchanger (␹) were calculated.

Subsequently, it was washed to show neutral pH with distilled water. PI-4 (1 g) was added in 100 ml of 1 M NaCl aqueous solution with various NaOH concentrations, and was allowed to stand for 3 days at 20 ⫾ 1 °C. From the supernatant liquid in a flask, 10 ml of solution was titrated with 0.1 N HCl standard solution. In order to investigate the effects of supporting electrolytes of NaCl and BaCl2, the concentration of electrolytes was varied in the range of 1.0 ⵒ 1.0⫺3 M of NaCl and BaCl2.

EXPERIMENTAL

When a cation exchanger in H-form is placed in the contact with a solution of alkali M(OH)z, the ion exchange H⫹-Mz⫹, followed by the neutralization reaction, occurs. This can be expressed as:

Materials Phosphoric ion exchanger (PI-4) was prepared by phosphorylation to a granular type of styrene copolymer and 4% divinylbenzene and then purified by washing with HNO3 aqueous solution before use. The ion exchange capacity of PI-4 was 5.70 meq/g and the swollen volume in water was 0.59 H2O/g. HCl and NaOH aqueous solutions (0.1 N) were used as a standard solutions. Ba(OH)2 was purchased form Aldrich Co. and used without further purification. NaCl, BaCl2, buffer solution, and other chemical reagents were reagent grade and used without further purification. Potentiometric Titration PI-4 was transformed into the H-form by slow treatment with 0.5 N HNO3 aqueous solution.

Determination of Ion Exchange Capacity An error in determination of the ion exchange capacity from the titration could be unreasonably large if the titration conditions were not chosen properly. We evaluated the precision of the determination by analysis of the potentiometric titration curves near the equivalent point. The precision depends on the number of experimental points in this region, and to imitate conditions for the real experiments we have chosen 10ⵒ12 points around the inflection point. The curves were differentiated and the differential was fitted to the Lorenzian distribution curve, which appeared to be the best fit. Then, the parameters of the maximum peak in ⭸pH/⭸g were found. The g value of the maximum peak was equal to the ion exchange capacity. Theoretical Discussion

Calculation of the Acidity Function

RH ⫹

1 Z⫹ 1 M º H ⫹ ⫹ R zM z⫹ z z

(1)

The relative equilibrium constant is given as: k⫽

1/z a៮ M 䡠 aH 1/z a៮ H 䡠 a M

(2)

where a is the activity of the species in the subscript, z is the charge of counter-ion, and a៮ is the activity in the ion exchanger. However, it is practically impossible to determine a៮ M and a៮ H. Therefore, apparent equilibrium constant (K) was intro-

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duced to determine k. The K can be calculated from experimental data on the equilibrium from eq 3:

冘共i ⫺i!j兲!j! 䡠 y共i ⫺ j, j兲䡠共1 ⫺ ␹兲 j⫽i

log K ⫽

共i⫺j兲

䡠 ␹j

j⫽0

(7) K⫽

C 䡠 aH fH 1/z ⫽ k 1/z CH 䡠 aM fM 1/z M

(3)

where C is concentration, and f is the activity coefficient for the related forms of ion exchanger. It is convenient to express the apparent equilibrium coefficient via relative equivalent fractions of the counter ions in the phase of the ion exchanger, ␹:

␹⫽

zC M zC M ⫹ C H

(4)

where values CM and CH are concentrations of the metal and hydrogen form of the ion exchanger, dependant on the degree of neutralization. The K obtained from equations 3 and 4 is calculated as follows: K⫽

␹ 1/z 䡠 a H ⫺1/z 䡠共zC M ⫹ C H兲共1/z⫺1兲 1/z 䡠 z 共1 ⫺ ␹ 兲䡠 a M

(5)

After rearranging the logarithm of eq 5 we obtain the following equation:

log

␹ 1/z 1 1 ⫺ pH ⫺ log a M ⫺ log z 共1 ⫺ ␹ 兲 z z

⫹

冉冊

1 ⫺ 1 log共zC M ⫹ C H兲 ⫹ pK共 ␹ , C M兲 ⫽ 0 z

(6)

where pK (␹, CM ) ⫽ ⫺log K(␹, CM ) and CM is the concentration of the supporting electrolyte. This equation relates to the most important properties of our system: pH of the solution, degree of neutralization of the ion exchanger (i.e. the counterion sorption value), and the concentration of a background electrolyte. Therefore, we refer to it as the function of acidity of ion exchangers. The acidity parameters can be used to describe all ionic equilibria in the system “ion exchanger-solution” containing H⫹ and counter ions Mz⫹.

where i, j and y(i⫺j,j) are the Soldatov’s model constants. Their physical meaning is given as follows: i is the maximum number of neighboring exchange groups in the microenvironment of any selected exchange group in the ion exchanger. The nearest neighbors are those groups that affect the energy of interaction of counterion (Mz⫹) with a selected exchange group. The constant j is the number of neighboring exchange groups occupied with counterion (Mz⫹); y(i ⫺ j,j) are constants. In an actual case, i is not higher than three and y(i ⫺ j,j) are constants with the same order of magnitude as the log K. The pH of equilibrium solution in the potentiometric titration curve of an ion exchanger depends on the concentration of the acids or the base. The potentiometric titration curve of an ion exchanger does not imply that there are simple physico-chemical characteristics, because the curve depends on the solution volume in the system undergoing potentiometric titration. It is important to foresee how the conditions for the potentiometric titration of the ion exchangers will influence the shape of the potentiometric titration curve, and also what the best conditions for titration and the different factors in determining curve shape and position will be. Therefore, the potentiometric titration curves can be computed with knowledge of the solution pH and ␹ using the material balance condition to find the number of the alkali equivalent (g) added to 1 g of the ion exchanger with capacity E in a volume V.12,13 The relationship between g and pH in the potentiometric titration curve is as follows: g⫽

1 共E ␹ ⫹ VC OH ⫺ VC H兲 z

(8)

The equations 6,7, and 8 can be used to express the potentiometric titration curve of the ion exchanger.

RESULTS AND DISCUSSION

Calculation of Apparent Equilibrium Constant (K)

Potentiometric Titration Curve

An equation for X dependency of log K has been derived, using Soldatov’s model: 9 –11

Figure 1 shows the theoretical and experimental data of the ␹ dependency of log K in 1 M of NaCl

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These data were used to calculate the apparent equilibrium constant (K). The pK values for ␹ can be calculated from equations 10 and 12. Because the concentration of the supporting electrolyte (NaCl) was 1.0 M and the ion charge (z) is 1, eq 6 can be rewritten as follows: log

␹ ⫺ pH ⫺ log a M ⫹ pK共 ␹ , C M ⫽ 1.0兲 ⫽ 0 1⫺␹ (13)

where, because aM is the activity of counterion, the CM practically coincides with the concentration of the supporting electrolyte. Therefore, the concentration CM can replace activity aM in eq 13. Eq 13 can thus be rewritten as follows:

Figure 1. A plot of log K versus ␹ at CNaCl ⫽ 1.0 mol/l. The data points were experimental data and were fitted by different interpolation curves.

log

␹ ⫺ pH ⫺ log C M ⫹ pK共 ␹ , C M ⫽ 1.0兲 ⫽ 0 1⫺␹ (14) log

supporting electrolyte. The ␹ dependency of log K for experimental data showed the S curve [Fig. 1(a)] of a third-order equation. For sufficient accuracy, log K from eq 7 needs four parameters as follows: log K ⫽ y共3,0兲共1 ⫺ ␹ 兲 3 ⫹ 3y共2,1兲共1 ⫺ ␹ 兲 2␹ ⫹ 3y共1,2兲共1 ⫺ ␹ 兲 ␹ 2 ⫹ y共0,3兲 ␹ 3 (9) where the constants of y(3,0), y(2,1), y(1,2), and y(0,3) were ⫺0.97, ⫺3.56, ⫺1.88, and ⫺4.14, respectively, and their mean error was 0.08. It can be rewritten as follows: log K ⫽ ⫺0.97 ⫺ 7.78␹ ⫹ 12.80␹2 ⫺ 8.21␹3

(10)

The straight line [Fig. 1(b)] was obtained by the first-order equation to minimize the parameters. In order to obtain the straight line from eq 7, there needs to be at least two parameters. This can be written as follows: log K ⫽ y共1,0兲共1 ⫺ ␹ 兲 ⫹ y共0,1兲 ␹

(11)

where the constants of y(1,0) and y(0,1) were ⫺1.29 and ⫺3.95, respectively, and their mean error was 0.17. It can be rewritten as follows: log K ⫽ ⫺1.29 ⫺ 2.66 ␹

(12)

␹ ⫺ pH ⫹ pK共 ␹ , C M ⫽ 1.0兲 ⫽ 0 1⫺␹

(15)

The pH for a certain ␹ can be calculated from eq 15. In order to figure out the calculated potentiometric titration curve, the g values can be obtained by using eq 8. Because ion charge (z) is 1, eq 8 can be rewritten as follows: g ⫽ E ␹ ⫹ V共10 pH⫺14 ⫺ 10 ⫺pH兲

(16)

Therefore, the g values for a certain pH can be calculated from eq 16. The theoretical potentiometric titration curves fitted by using lines and the third-order equation in Figure 1 were presented in comparison with the experimental potentiometric titration curve in Figure 2. The (a) and (b) curves in Figure 2 were obtained by using the theoretical equation for the (a) and (b) lines in Figure 1, respectively. As Figure 2 shows, it was found that the potentiometric titration curves obtained from the thirdorder equation are more coincident with the potentiometric titration curves obtained from experimental data, than those from the linear equation. Even though the fit resulting from the third-order equation showed a better result, the expression using one parameter was adopted because of its sufficient accuracy and simplicity. Using the third-order equation can certainly improve the accuracy in describing the characteristic behavior of ion exchangers, but it leads to more


Figure 2. Potentiometric titration curves computed from different approximations and experimental data.

complicated calculations, and requires more parameters and initial information for their determination. It was found that the ␹ dependency of logK could be expressed with sufficient accuracy by approximation of a straight line. The pK value for the degree of neutralization (␹) in the titration of phosphoric ion exchanger PI-4 at various concentrations of NaCl and BaCl2 is presented in Figures 3 and 4. The plot of experimental pK values versus ␹ in phosphoric cation exchanger PI-4 can be expressed as linear eq 17.11 pK ⫽ pK ␹⫽0 ⫹ ⌬pK 䡠 ␹

3185

Figure 3. A plot of pK versus ␹ for various concentrations of NaCl.

value for all concentrations. The correlation between ⌬pK and the degree of neutralization (␹) for an ion exchanger can be expressed by the following equation obtained from an empirical equation:13,14 pK共 ␹ , C ⫽ const.兲 ⫽ pK 0 ⫹ ⌬pK共 ␹ ⫺ 1/2兲

(18)

(17)

The ⌬pK values were obtained from the slope of the straight lines in Figure 3 and 4. The slopes for various concentrations of NaCl and BaCl2 appeared to be the same values, but their slopes were not the same. In Table I, the pK values obtained from experimental data at ␹ ⫽ 0.5 are presented in comparison with the pK values calculated from eq 17. The ⌬pK values increased as the concentration of NaCl electrolytes increased, but in the case of BaCl2 electrolytes, there was no particular trend in the ⌬pK values, which were smaller than those of the NaCl electrolytes. Because the differences of the ⌬pK values at various concentrations of supporting electrolytes were not large, the ⌬pK value in the titration of NaOH and Ba(OH)2 (which was 1.968 and 1.805, respectively) was replaced by an intermediate ⌬pK

Figure 4. A plot of pK versus ␹ for various concentrations of BaCl2.

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Table I. Constants of Eq 17 for Various Concentrations

pK ⫽ 2.740 ⫺ 0.05 log C Cl ⫹ 1.805共 ␹ ⫺ 21 兲

Electrolytes

C (mol/l)

⌬pK

pK ex

pK in

NaCl

0.002 0.025 0.1 1.0 0.002 0.025 0.1 1.0

1.78 1.73 2.25 2.11 1.75 1.82 1.90 1.75

2.98 2.91 2.81 2.72 2.78 2.81 2.71 2.58

3.07 3.93 2.93 2.72 2.71 2.66 2.63 2.58

BaCl2

pK ex : experimental pK value at ␹ ⫽ 0.5. pK in : interpolation pK value of Eq 17 at ␹ ⫽ 0.5.

where the pK0 is the thermodynamic equilibrium constant in the ion exchange between a H⫹ ion of functional groups and a counterion Mz⫹. The pK can be expressed as follows:

pK 0 ⫽

冕

1

pKdx

(19)

0

This linear approximation is equal to the arithmetic average and can be rewritten as follows: 1 pK 0 ⫽ 2 共pK ␹⫽1 ⫹ pK ␹⫽0兲 ⫽ pK ␹⫽1/2

(20)

The change of pK for the various concentrations of NaCl and BaCl2 at specific ␹ is shown in Figure 5. The CM dependency of pK can be expressed by the following eq: pK共 ␹ ⫽ const., C M兲 ⫽ pK 0 ⫺ b log C M

(24)

The experimental pK values for various ␹ were presented in comparison with the potentiometric titration curves obtained from equations 23 and 24 in Figure 6 and 7, respectively. As shown in Figures 6 and 7, there is good agreement between the calculated curves and the potentiometric titration points. When the ion exchanger PI-4 was titrated with NaOH and Ba(OH)2, the character0 istic parameters (pK0,⌬pK, and b) were pKNa ⫽ 2.72, ⌬pKNa ⫽ 1.97, and bNa ⫽ 0.13 in the case 0 of NaOH and pKBa ⫽ 2.58, ⌬pKBa ⫽ 1.81, and bBa ⫽ 0.047 in the case of Ba(OH)2. Figure 8 shows the potentiometric titration curves that resulted when the ion exchanger PI-4 was titrated with NaOH and Ba(OH)2 in 0.002 M NaCl and BaCl2.When the ionic charge (z) increased from one to two, ⭸pH/⭸g increased more sharply near the inflection point. Determination of Ion Exchange Capacity The ion exchange capacity can be obtained only by potentiometric titration as a position of the inflection point. The inflection point of the potentiometric titration curve for an ion exchanger is changed by the concentration of supporting electrolytes, the volume of equilibrium solution, and the charge of titration ions. An error in determination of the ion exchange capacity from the titration

(21)

The constant b in eq 21 is the correction value of pK for various concentrations of NaCl and BaCl2, and can be obtained from the slope of the linear equation. The pK values, as a function of the degree of neutralization (␹) and the concentration of NaCl and BaCl2 (from equations 18 and 21) can be rewritten as follows: pK共 ␹ , C M兲 ⫽ pK 0 ⫹ ⌬pK共 ␹ ⫺ 12 兲 ⫺ b log C M

(22)

Therefore, when the ion exchanger PI-4 is titrated with NaOH and Ba(OH)2, the pK values for the degree of neutralization (␹) can be given as follows: pK ⫽ 3.090 ⫺ 0.13 log C Cl ⫹ 1.968共 ␹ ⫺ 21 兲

(23)

Figure 5. A plot of pK versus log C for different titration media.


Figure 6. Potentiometric titration curves for phosphoric acid ion exchanger PI-4 in various concentrations of NaCl. The data points were experimental data.

can be unreasonably large if the titration conditions are not chosen properly. In order to determine an accurate ion exchange capacity, the data around the inflection point were fitted to the Lorenz distribution curve. The error in determination of abscissa of the maximum in ⭸pH/⭸g is equal to the error in determination of the ex-

Figure 7. Potentiometric titration curves for phosphoric acid ion exchanger PI-4 in various concentrations of BaCl2. The data points were experimental data.

3187

Figure 8. Potentiometric titration curves for phosphoric acid ion exchanger PI-4 in 0.002 M solutions of NaCl and BaCl2.

change capacity in absence of experimental errors. As the ratio of the maximum height to its width at half height was increased, the error in determination of ion exchange capacity was decreased. The data around the inflection point obtained from the potentiometric titration curve in Figure 8

Figure 9. Curve fittings of a plot of ⭸pH/⭸g versus g by Lorenzian distribution for NaCl and BaCl2 solutions (0.002 M).

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Table II. Parameters of Lorenzian Equation [⭸pH/ ⭸g ⫽ f( g)] Backgrounds

g max

⌬g

H

E ⫺ g max

H/⌬g

NaCl BaCl2

5.690 5.696

0.28 0.14

17.67 51.57

0.01 0.004

63.11 369.64

pK: equilibrium constant, g max: g value of maximum peak. ⌬g: the width at half ⭸pH/⭸g value of peak. H: ⭸pH/⭸g value of peak, E: true capacity (5.7 meq/g).

were presented by Lorenz distribution curves in Figure 9. The g value for maximum ⭸pH/⭸g was equal to the inflection point of the potentiometric titration curve and the ion exchange capacity of ion exchanger PI-4. The inflection point of the potentiometric titration curve is observed clearly with increasing ⭸pH/⭸g. The parameters calculated from the Lorenz distribution curve are shown in Table II. The ratio of the maximum height of ⭸pH/⭸g to the width at half height for BaCl2 is larger than that of NaCl. The ion exchange capacity of NaCl and BaCl2 can be evaluated accurately within ⫾ 0.01 and ⫾ 0.004 meq/g, respectively. From these results, we know that the error range of NaCl was larger than that of BaCl2, and the ion exchange capacity is determined accurately in the case of titration with a divalent counterion.

CONCLUSIONS This experiment calculated the characteristics of a phosphoric cation exchanger and studied an accurately computable method for ion exchange capacity for a type of potentiometric titration curve. The conclusions obtained from these experiments are as follows: The ⌬pK values were obtained from the slope of a linear equation. The experimental pK values at ␹ ⫽ 0.5 showed good agreement with the theoretical data. Also, when the supernatant liquid from cation-exchanged solution was titrated with NaOH and Ba(OH)2 solutions, a good agreement between experimental and theoretical pK values for various ␹ was found in all the potentiometric titration curves. The data points of the potentiometric titration curve in the case of divalent ions were changed more sharply than those for monovalent ions. The plot of ⭸pH/⭸g versus g was fitted to the Lorenzian distribution curve, from which ion exchange capacity was evaluated accurately. This work was supported by the Korea Science and Engineering Foundation through the Advanced Mate-

rials Research Center for Better Environment at Taejeon National University of Technology.

NOMENCLATURE a b C f i, j g E k K pK0 ⌬pK

V ␹ z

Activity Correction for the apparent constant shift with changing supporting electrolyte Concentration of supporting electrolyte (mol/l) Activity coefficient Constant Number of moles of alkali to 1 g of ion exchanger Exchange capacity (meq/g) Equilibrium constant Apparent equilibrium constant Thermodynamic constant of ion exchanger Difference of the apparent equilibrium constant at complete and zero neutralization of the ion exchanger Solution volume, ml/g of ion exchanger Equivalent fraction Ion charge

REFERENCES AND NOTES 1. Hiroyuki, A.; Akihiko, T.; Naoki, T.; Hidefum, H. Ind Eng Res 1990, 29, 11, 2267–2272. 2. Goupil, J. M.; Hemidy, J. F.; Cornet, D. Zeolites 1982, 2, 47–55. 3. Toshima, N.; Asanuma, H.; Yamaguchi, K.; Hirai, H. Bull Chem Soc Jpn 1989, 62, 563–570. 4. Tuenter, G.; van Leeuwen, W. F.; Leo, J. M. Ind Eng Chem Prod Rev Dev 1986, 25, 633– 636. 5. Dorfoner, L. In Ion Exchangers; Walt de Gruyter Berline: New York, 1991; Chapter 1, pp 7–19. 6. Hiroyuki, A.; Naoki, T. J Polym Sci Part A: Polym Chem 1990, 28, 907–922. 7. Gustafson, R. L.; Fillius, H. F.; Kunin, R. Ind Eng Chem Fundam 1970, 9, 221–229. 8. Kunin, R.; Fisher, S. J Phys Chem1962, 66, 2275– 2277. 9. Soldatov, V. S. J Phys Chem (USSR) 1968, 42, 2287–2291. 10. Soldatov, V. S. Simple Ion Exchange Equilibrium; Nauka i Technik: Minsk, 1972; pp 220 –242. 11. Kim, T. I.; Hwang, T. S.; Son, W. K.; Choi, D. M.; Oh, I. S.; Soldatov, V. S. Polymer (Korea) 1999, 23, 747– 754. 12. Soldatov, V.S. Dokl Akad Nauk (USSR) 1990, 314, 664 – 668. 13. Soldatov, V.S. Reactive Polym 1993, 19, 105–121. 14. Soldatov, V.S. Ind Eng Chem Res 1995, 34, 2605– 2611.