Ind. Eng. Chem. Res. 1995,34, 2605-2611
2605
Quantitative Presentation of Potentiometric Titration Curves of Ion Exchangers Vladimir S. Soldatovt Institute of Physical Organic Chemistry of the Belarus Academy of Sciences, 13 Surganov St., Minsk 220072, Republic of Belarus
Neutralization of ion exchangers in H+ or OH- forms by alkali M(OH), or acid H,A was considered as a specific case of ion exchange H+-Mz+ or OH--Az- accompanied by the neutralization reaction. An equation connecting pH and the external electrolyte concentration with the ion exchanger neutralization degree was derived. It can be used for presentation of the potentiometric titration curves in mathematical form or their calculation at different (not necessarily constant) concentrations of external electrolyte. Applicability of the modified HendersonHasselbach equation as one of the possible means for description of potentiometric titration curves of ion exchangers was also discussed. The range of its applicability and limitations were established. The physical meaning of its constants was clarified, and the equations of their relations with the apparent ion exchange equilibrium constant have been obtained. The theoretical considerations are illustrated with the example of potentiometric titration curves of a carboxylic acid ion exchanger obtained in a wide range of degrees of neutralization and external electrolyte concentrations.
Introduction The potentiometric titration curve (PTcurve) is the most important primary characteristic of an ion exchanger [ I ] . It is applied to determine the number of different types of ionized groups in an ion exchanger and their quantity as well as to estimate their dissociation constants. Nevertheless, it does not mean that problems connected with their interpretation and application have already been solved. Starting from the first works in this area 12-81 it appeared clear that PT curves of polyelectrolytes and ion exchangers have peculiarities distinguishing them from those for the nonpolymeric electrolytes. The PT curves of the latter allow one to find their dissociation constants knowledge of which makes it possible to compute most of the important properties of the individual solutions and mixtures of weak acids and bases as function of their concentration and degree of neutralization. Being true constants, independent of pH, neutralization degree, and concentration of the other electrolytes in the solutions, the dissociation constants appear to be the basic properties of weak electrolytes and are essential for practical calculations in the physical and analytical chemistry of weak electrolyte solutions. The situation is different with polyelectrolytes and ion exchangers. Values of "dissociation constantsnavailable from their PT curves appeared t o be strongly variable with degree of neutralization and type and concentration of the neutral electrolyte in the solution. The mathematical form of these dependencies expressed with the help of measurable parameters is not known at present. This makes it impossible to use dissociation constants for practical calculation of properties of ion exchange systems. In the present paper the interaction of a cation (anion) exchanger with alkali (acid)is considered as a particular case of ion exchange [9,IO]. This allows one to establish quantitative relations between the main properties of
the ion exchange system: the equilibrium solution pH, the concentration of the external electrolyte, and the degree of neutralization of the ion exchangers for systems with different degrees of deviation from the ideal behavior and different types of counterions formed by dissociation of the alkali or acid used for the potentiometric titration.
Equilibrium Description When a cation exchanger in H+ form or an anion exchanger in OH- or free-base form is placed in contact with a solution of an electrolyte Mz-2+4+L- the ion exchange H+-Mz+ or OH--Az- followed by the neutralization reaction occurs. This can be formulated as
+ l/zMz+= H+ + l/zR,MZ+ ROH + l/zA"- = OH- + ~/ZR$~RH
(2)
The relative equilibrium constants are
where a is the activity of a related component, z is the charge of the counterion, and the overbar represents the phase of the ion exchanger. The situation is slightly more complicated in the case of weak base anion exchangers whose functional groups are present mainly in the form of a free base, R, e.g., RNH2, and only to a small extent as RNH3+0H- groups. In this case the equilibrium can be regarded as a result of an addition reaction R
+ H++ l/zA*- = l/z(RH+)J"-
(4)
whose equilibrium constant equation after combination with the water ionic product K, = a H + a O H - gives the equilibrium constant equation in the form
~
E-mail: soldatov%
[email protected]~. Also at Technical University of Lublin, 20-618, Lublin, Poland.
0888-588519512634-2605$09.0010
t 1)
0 1995 American Chemical Society
2606 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
where = KaddKw
(6)
The same equilibrium equation as ( 5 ) is obtained if equilibrium is regarded as a result of ion exchange OH- A'- preceded by the group protonization with water molecules. Since the equilibrium constant equation for cation exchange is completely convertible into the equation for anion exchange after replacing the OH- activity with that of H+, the further consideration will be given, for certainty, t o cation exchangers. A n apparent equilibrium constant, W, can be calculated from experimental data on the equilibria from
(7) where C is the concentration (molarity, molality, or full mole fraction),f i s the activity coefficient of the related form of the ion exchanger. It is convenient to express the apparent constant via relative equivalent fractions of the counterions in the phase of ion exchanger, f
After taking the logarithm and making the usual substitutions we obtain the equation connecting the solution pH, the concentration of the counterion (via a d , its properties (char e z and affinity t o the ion exchanger via the pW = -log value), and the degree of neutralization of the ion exchanger which is identical in our case to the f value:
B
It is known that K is only slightly dependent on the total electrolyte concentration a t constant f values-(see ref 1, p 1511, which makes it possible to use the K values determined a t one concentration for the other concentrations in a certain range. Practical interest presents a dependence of pH on the concentration of electrolyte initially present in the solution, connected with the co-ion concentration CA. This value is present as a term in the total concentration of the counterion in the equilibrium solution CM:
The apparent equilibrium constant can vary with the ionic composition of an ion exchanger, and eq 10 caq be used for practical calculation only if the dependence K(x) is known. In our previous paper [I31 an equation expressing log l? as a function of f was derived. In the general case it has the form 1=1
11
where i,j , and y(i-j,j) are constants having the following physical meaning for our case: i is the maximum number of neighboring exchange groups participating in the exchange;j is the number of neighboring groups occupied with the counterion M; y(i-jj) is the logarithm of the equilibrium constant of the elementary equilibria related to different combinations of i and j . In the real cases i is not higher than 3 and they(i-jj) are-constants of the same order of magnitude as log K. These parameters can be found as empirical constants for one of the fixed concentrations of the external electrolyte. Once the form of eq 12 is established, any of the values in eq 10 can be calculated at the other chosen values. This can be used for computing the PT curves which can be compared with the experimental ones. In the literature on ion exchange the PT curves are expressed as dependencies of pH of the equilibrium solution on the number of moles of alkali or acid added to one gram of the ion exchanger (g) in a certain ionic form, pH = fi'g). Such dependencies are not simple physicochemical characteristics of an ion exchanger since they depend on the volume of the solution. Since these dependencies are widely used, it is important to have a method for their computation from the properties of the ion exchanger and a solution. This would allow one to see the importance of different factors in determining the shape and position of the PT curves and favor the reliability of deduction followed from interpretation of the experimental curves. The PT curves, pH = fi'g) can be computed using the material balance condition
(13) where E is the exchange capacity and V is the volume of the equilibrium solution. When all values in the right part of eq 13 at a chosen pH are taken as initial data, the PT curves can be obtained. The key problem in this calculation is evaluation of the f value which should be found from the dependence log K = ff) (eq 12) using eqs 10 and 11.If the ion exchanger is polyfunctional, then the first term in the right part of (13) must be replaced with the sum of the terms related to all types of the functional groups. Thus, eqs 10,12, and 13 can be used for quantitative presentation of potentiometric titration curves of ion exchangers. It is necessary to find out hoy many parameters are needed to express the log K = f(f) function to achieve the desired accuracy in computing the PT curves. This can be evaluated by considering the PT curves for typical ion exchangers. An example of a practical calculation is given below.
Quantitative Presentation of the PT Curves for a Carboxylic Acid Ion Exchanger We are taking as an example a set of potentiometric titration curves of industrially produced carboxylic acid ion exchanger KB-4, Russian production. The resin is a gel type co-polymer of methacrylic acid and 6% divinylbenzene. Its exchange capacity was 10.60 mg eq/g and its water uptake 0.86 g HzO/g in H-form. In spite of the fact that this and the other resins of this type have been extensively studied and described in many papers, it appeared impossible to find in the literature the experimental data necessary for all required calculations. The potentiometric titration curves were obtained using the batch method. The resin samples were
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2607
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n, alkali m-eq/g Figure 1. Potentiometric titration data for the carboxylic acid ion exchanger KB-4 with NaOH at different concentrations of external NaC1: (1) 1, (2) (3) (4) (5) mom.
Figure 3. Comparison of experimental data on potentiometric titration with computed data obtained using different approximations: (1)p k = constant; (2) k i s a linear function corresponding to curve 2 in Figure 3; (3)piinterpolated by symmetric third power eq 14,corresponding to line 3 in Figure 2; (4)p k is the best fit eq 14,curve 1 in Figure 2.
-4.00
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-6.00
-6.50
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ojo
1
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7
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8 I 1 I , t I II
0.80
3 8 I I
0.BO
Z, equiv. fraction Figure 2. An example of log k = f fx) (CNaC1
I
t
+
parameters t o provide the desired accuracy in the computed PT curves. Therefore different approximations of the set of log k = f f ) curves were examined t o see their influence on the computed pH values. An example of such a consideration is given below. In Figure 2 the experimental data on log K = f f ) for CN~CI = 0.01 m o m are given together with three lines, corresponding to different accuracies of interpolation and different numbers of empirical parameters. Curve 1is obtained by the best fit of the data to the third power equation in the form of (12) (four-parameter equation):
l.dO
= 0.01 m o m )
dependence approximated by different interpolation curves. Points are calculated from the experimental data. The lines correspond to different interpolation equations: (1)best fit curve corresponding eq 14;(2)best fit straight line, eq 15; (3)symmetric third power curve, eq 14.(Interpolation interval f = 0.03-0.95).
equilibrated at 20 f 1 "C with solutions of NaOH and NaC1. The concentration of the latter was constant for each of the curves. After the equilibrium was reliably achieved (4 days), the solution was separated from the resin and the pH measured with a glass electrode calibrated with buffers or (at low and high pH) with a standard solution of HC1 or NaOH in presence of NaCl a t concentrations used as background. The experimental PT curves are presented in Figure 1.
These data were used t o calculate the apparent equilibrium constants using eq 9. The log K = f 3 dependencies appeared to be S-shaped curves that correspond in terms of eq 12 t o an equation of a t least the third power and require four parameters for their quantitative presentation. We need to find dependencies log K = f f ) and substitute them into eq 10 used for calculation of f at the chosen pH with further application of eq 13 to obtain g values needed for construction of the calculated PT curves. For practical purposes it is nesessary to find a way to express the dependence log K = RZ) with a minimal number of
logRi'=y(3,0)(1- Z)3
+ 3~(2,1)(1- Z)'Z + 3~(1,2)(1 - ZE' + y(0,3)ji3 (14)
with the following values of the coefficients: y(3,O) = -4.39; y(2,l) = -6.30; y(1,2) = -4.04; y(0,3) = -6.44. The mean interpolation error e = 0.08. Straight line 2 is obtained by approximation of the data by the first-power equation (12) (two parameters) in the form 1 o g a = y(1,0)(1 - 3)
+ y(0,lE
(15)
with coefficients y(1,O) = -4.67, y(0,l) = -5.90, and e = 0.17. Curve 3 is obtained by approximation of the data with an equation similar to (14) having the common point with straight line 2 a t f =1/2 and being symmetric relative to this point (three parameters). Its coefficients arey(3,O) =y(1,2) = -4,20;y(2,1) = (0,3) = -6.28. The third parameter is the value of log Ky at 3 = 1/2 equal to -5.23; e = 0.09. Equations 13-15 were substituted into eq 10, and the f values were calculated by its numeric solution. The latter were used to calcul_ateg from eq 13. A constant log k value, equal to log K at f = 112 (one parameter), was also used in the calculations. In Figure 3 the experimental data are compared with the computed data in different approximation PT curyes. It is seen that a linear approximation of the log K = fi9 depen-
2608 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
strai ht lines necessary for the following calculations: log if = ff), CA = constant and log K = fllog CA),3c = constant. Most often, ion exchangers are characterized by PT curves obtained at moderate (10-2-1 mol&) concentrations of an external electrolyte with univalent ions (NaOH, HC1). For these conditions eqs 10-12 can be combined to form a simple and convenient equation
A
-
2
9.00
0.20
0.40
- 0.60
0.80
1.00
X
Figure 4. log k = f (i)dependences approximated by a set of straight lines expressed by eq 16 with coefficients given in the text. Concentrations of external NaCl are (1)1,(2) (3) (4)10-3, ( 5 ) mom.
dence gives an acceptable accuracy for practical purposes in the description of the experimental data. Use of one parameter does not reflect the peculiarities of the ion exchanger behavior. Use of more than two parameters improves the accuracy but leads to much more complicated calculations and requires more initial information. In view of these circumstances, we suppope that the precision of the linear approximation of log K = /Pi?)was sufficient and it was used for the data at the other concentrations of the background electrolyte. In Figure 4 the log k values for all studied systems are presented. It appeared that they can be described with acceptable accuracy by a family of parallel straight lines at concentrations of the background electrolyte mol/L. The interpolation equation is above
where KOand K1 are the apparent equilib_riumconstants at f = 0 and f E 1, rtspectively, KO*is KOat CA= 1,b is constant, ApK = pK1 - pko is independent of CA,In our case logKo* = -5.140, b = -0.213, and ApK = -1.230. The dependences computed with this equation are presented as lines in Figure 4. Equation 16 together with eqs 13 and 10 has been used to compute the PT curves at different concentrations of the background electrolyte. The agreement of the computed and experimental data is good at CA 2 mom, as seen from Figure 1. Thus, eqs 13,10,and 16 form a system which can be used for quantitative presentation of the PT curves of ion exchangers o r calculation of any of the following parameters: pH, CA,% or g in an “ion exchanger solution” system if the other parameters are chosen. The equations account for the influence of the charges of the counter- and co-ions, the exchange capacity and the swelling of the ion exchanger, and the affinity of the ion exchanger t o the counterion. The minimum initial information required for practical calculation is three experimental points forming two pairs: one is related to the same electrolyte concentration at significantly different Z values; the other one is related t o the same 3 values at significantly different CA. This is enough t o obtain coefficients of the two
(log y+ as a small and constant value can be neglected or joined to pKo*). Further simplification occurs at a constant external electrolyte concentrations: pH = constant
+ log f + A(pmf
(18)
(1 - 3 ) where constant = pko* - (1 - b ) log cA- log yk;.
Applicability of the Modified Henderson-Hasselbach Equation for Quantitative Presentation of PT Curves of Ion Exchangers Most often acido-basic properties of ion exchangers are characterized by constant PKHof the linear equation (19) where a is the degree of dissociation of the ion exchanger functional groups, signs and ’-” are related to cation and anion exchangers, respectively, and the PKHand n are empirical constants. This equation, in principle, can also be a base for quantitative presentation of the PT curves of the ion exchangers. In spite of the fact that the physical meaning of its parameters PKH and n is not clearly defined, it is so often used and is so important for theory and practice of ion exchange that it requires special consideration. If n = 1then eq 19 is a logarithmic form of the formal dissociation constant of the ion exchanger functional groups. In the application to nonpolymeric electrolytes this equation is known as the Henderson-Hasselbach equation. Investigation of the dependencies of pH on log(d(1 - a)) for polyacids has shown that the linear dependence also exists but the slope n is different from 1[2-83. In the work by Katchalsky and Spitnik [31 the linear dependence of the above values for polyacids with a slope different from 1was confirmed, and the equation was named the Generalized Henderson-Hasselbach equation. Later this equation was applied to ion exchangers [5-83 under the name modified HendersonHasselbach equation. The physical meaning of constants PKH and n has been often discussed in the literature (see for example refs 8 , 7 , and 12). The constant PKHis often named the dissociation constant index with addition of some restrictive definitions (at half neutralization degree, mean, etc.). Rigorous connection of this constant with physically understandable properties was not established. Nevertheless, it is used as a measure of the acidity of ion exchangers implying that it has much in common with the dissociation constant.
“+”
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2609 The other constant, n, is often not mentioned a t all as a characteristic of ion exchanger acidity. No certain connections of it with the physical properties of ion exchangers were reported either. It was noticed that n is dependent in some way on the structure of the ion exchanger and its capacity. The physical meaning of both constants can be established by relating PKHand n to the apparent ion exchange equilibrium constant. First of all it is necessary to note that the ion exchanger dissociation degree a was never used in practical calculations and there is no way to determine these values. i values are always used instead. It means that eq 19 can be applied only to weakly dissociating ion exchangers at sufficiently high loading of the ion exchanger with the counterion where f = a. Contrary to that, eq 10 does not have this limitation. It is also important to note that in usual procedure for determination of the PKHvalues, when it is stated that PKH = pH at a = 1/2 (in fact, 3 = 1/21 the dissociation constant index is not determined, but a value of -log fir log U M is, which simply coincides with the negative logarithm of the formal "dissociation constant" k d = C U X H-/ (a~) a t U M = 1. Contrary to pK, this value is strongly dependent on the electrolyte concentration in the solution. One more difference between eqs 10 and 19 is substantial. Equation 10 includes the influence of the counterion charge on the PT curves, while eq 19 does not. If z # 1 the indices of the apparent and the formal dissociation constant do not coincide a t a = 1/2. Straightforward application of the modified Henderson-Hasselbach equation leads to different PKHvalues for the counterions with different charges, which contradicts the physical meaning of the dissociation constant. In view of this, it can be concluded that eq 19 with E in place of a has to be considered only as a convenient empirical equation establishing a connection between the solution pH and the degree of neutralization of the ion exchanger at a constant concentration of the background electrolyte. Considering its traditionally wide use for characterization of ion exchangers it is highly desired t o clearly define the physical meaning of its constants. Comparing eqs 10, 12, and 19 at f = 1/2 we find
+
This equation relates the PKH and constants of elementary ion exchange equilibria (see ref 11 for details). At constant U M the PKH is always some combination of the elementary equilibrium constants. An especially simple relation between the PKH and K(i-jj) exists for the most important case, when log fir = ff) is a straight line. ChoosingCZM = 1we obtain from eq 20 at i = 1 or i = 2:
The value K(0,i) corresponds to H+-M+ exchange on the functional group surrounded only by Mt counter1 and K(0,i) = fir^. Hence, ions, Le., f
-
(22) and under these conditions the PKH is equal t o the arithmetic mean of the apparent ion exchange equilibrium constant indices at the zero and complete neutralization of the ion exchanger. The physical meaning of the constant n can be established in the following way. It follows from eq 10 that a linear dependence between pH and log(i/(l - 3)) exists only if the apparent equilibrium constant is independent of the neutralization degree, which, as will be shown later, is possible only if n = 1 and eq 19 is the true HendersonHasselbach equation. It also follows from eq 10 that the dependence pH = fllog(d(1 - 5))) at U M = constant has two asymptotes:
at f
at 3
-
-
-
0, log(d(1 - 3 ) )
pH = log? - log U M
-
l,log(i?/(l - 3 ) )
log i and
+ PRO
(23)
-log(l - 3 ) and
pH = -log( 1 - 2 ) - log a M
+ pR1
(24)
Since these asymptotes are parallel, the pH = fllog(i/(l - 3)))curve must have an inflection point. If the curve log K = flf) is symmetric relative to the point i = 112 (for instance linear as described above), then the second derivative d2(pH)/d(log(i/(l- i ) ) )is2equal to 0 a t this point as it follows from eqns 10 and 12. The constant n is usually determined as the slope of a close t o linear part of the graph pH = fllog(f41 - 3 ) ) ) around f = 1/2. It should be close to the derivative d(pH)/dlog(f/(l - 3)) at f = 112 and tends to this value if the j;. interval narrows to zero around thjs point. After differentiation of eq 10 with log K expressed through eq 12 under conditions U M = constant, z = 1 we have
- i)Q1OyZ") (25) where 2
W(x) = log 1-3
Q
(21) The K(i,O) is a constant of elementary H+-M+ exchange on a functional group having only H+ neighboring counterions. The number of such neighboring counterions i can be either 1 or 2. This value is equal to the apparent equilibrium constant a t f 0, denoted above as KO,so K(i,O) = KO.
-
= 1+
For 3 =1/2:
The value of n depends on the 3 interval. It changes
2610 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 10.00 9.00
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Figure 5. Henderson-Hasselbach slope for the model systems with linear dependences of log k = fl?) with parameters given in Table 1.
-
between the value of the derivative at f = 1/2 and 1for widening the interval when f 0 and 1. Consider several important cases. If i = 0 then n = 1. i = 1corresponds to the absence in the nearest vicinity of the functional group, participating in the ion exchange process, of the other functional groups. It relates t o the ideal equilibrium, and eq 19 becomes the classic Henderson-Hasselbach equation. If i = 1or 2 then n has the same meaning defined by n =1
+ In 10 2
PRO)
(27)
It is seen that n depends on the variation of the pkvalue with the degree of neutralization f from f = 0 to f = 1, i.e., on the degree of nonideality of the H+-M+ ion exchange process. Hence, n is a measure of the deviation of the system from the ideal behavior: n = 1 corresponds t o the ideal process, n > 1 to a nonideal one; the larger n , the stronger the deviation from ideality is. The deviation of n from unity is practically possible only in the direction of larger values that correspond to a decrease of the apparent ion exchange equilibrium constant with increasing degree of neutralization. Two useful formulas, connecgng ~ K H and n and allowing one to find pK0 and pK1 from the modified Henderson-Hasselbach equation, can be obtained from eqns 22 and 27: (28)
(29)
If log K = f(f) is not linear (i > 21, n is also completely determined by some combination of constants of the elementary equilibria which can be found for any particular case from eq 26. In this case the average slope can be rather different from that determined in a narrow f range around f = 112. The said is illustrated by the example considered below. In Figure 5 the Henderson-Hasselbach slope is given for two model cases corresponding to a linear change of
Figure 6. Henderson-Hasselbach slope for the carboxylic acid ion exchanger KB-4 titration in presence of 0.01 m NaC1. Table 1. Parameters of Eq 21 and the Modified Henderson-Hasselbach Equation (Eq 19) curve no. in Figure 5 pK (1.0) pK (0.1) n PKH interval 1 2
5.5 4.5
6.5 7.5
1.576 2.728
6 6
0.10-0.90 0.11-0.89
the p k with the same value equal to 6 at f = 1/2. The value is chosen equal to 1. The degree of nonideality of the model systems is chosen different only: in one of the cases the p k changes between 5.5 and 6.5, in the other one between 4.5 and 7.5. Thus the PKH in the both cases is equal t o 6. The straight lines are asymptotes computed from eqns 23 and 24. It can be seen that the Henderson-Hasselbach slope curves have extended linear parts around f = 1/2 which are usually described by the modified Henderson-Hasselbach equation. The limits of its applicability can be regarded as a cross section of points between the tangent and the asymptotes. These points can be found by joint solution of eq 19 and the asymptote equations 23 and 24:
UM
and
It is seen from Table 1that the interval of applicability of eq 19 remains almost unchanged with the p j ? ~- p j ? ~ values and is equal to approximately 0.1-0.9. It means that the linearity of the Henderson-Hasselbach slope in a somewhat more narrow f range, say 0.2-0.8, must be a quite good approximation of the experimental data which are reported in many works. At the same time, it is important t o note that in most of the works the number of experimental points and the f interval studied is not sufficient to observe deviations of the pH = fllog(f/(l - 3 ) ) )dependence from linearity. In Figure 6 experimental data on the titration of the KB-4 resin are presented in the Henderson-Hasselbach coordinates for the system with 0.01 m o m NaC1, exhibiting marked deviation from linearity in the log K =ff) slope (see Figure 2). Lines AB and CD are asymptotes. The points in the interval of log(fl(1 - f)) = -0.9-0.9, corresponding to f = 0.1-0.9 are well approximated by a straight line with n = 1.31 and PKH = 7.26 (line EF). If the experimental data in the whole range of concentrations studied (f = 0.05-0.95) are
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2611 taken into consideration then the best fit straight line has the following constants: n = 1.59 and PKH= 7.35. The experimental points approach the asymptotes at f < 0.1 a n d f > 0.9. An S-shaped curve for log fi = ff) causes the appearance of three inflection points in the pH = fllog(f/(l - 53))curves, and the slope n, determined in different f intervals, can be substantially different, as it was observed in the example considered. At the same time, the pH = fllog(f/(l - 5)))function has very low sensitivity and the real curve oscillation around the interpolation straight line is rather small and can be perceived as the experimental points scattering. In some works (e.g., refs 8 and 13) deviations of the pH = fllog(f/(l - a)))dependence from linearity and its S-shaped form were noticed. Nevertheless, a linear approximation of this function provides the same accuracy in the computation of the PT curves as a linear approximation of the log a = ff) function described above (curve 2 in Figure 3).
Conclusion Potentiometric titration curves of ion exchangers can be presented by a set of three equations: an equation of the ion exchange equilibrium constant of the H+Mz+or OH--Az- exchange, an equation of the apparent ion exchange equilibrium constant dependence on the degree of neutralization, derived in ref 11,and the mass balance condition in the “ion exchanger - solution’’ system. They allow one to calculate one of the following properties of this system, if the rest are chosen: solution pH, ion exchanger loading with the counterion, and neutral electrolyte concentration in the solution. The effect of the counter- and co-ion charges, the ion exchanger capacity and swelling and the affinity of the resin to the counterion is accounted for. The equations can be used for calculation of the potentiometric titration curves at different (not necessarily constant) concentrations of the external electrolyte. Acceptable accuracy is obtained if two-parameter equations are used for the calculation of the degree of neutralization at different pH values: either an equation with a linear dependence of the logarithm of the apparent ion exchange equilibrium constant on the degree of ion exchanger neutralization or the modified HendersonHasselbach equation. The latter equation has serious limitations restricting its use for the quantitative presentation of the potentiometric titration curves of ion exchangers and practical calculations: it is not applicable for strongly dissociating ion exchangers, and it does not account for the concentration of the external electrolyte, the charge of the ions, or the affinity of the ion exchanger, its capacity, and swelling. The physical meaning of the constants in the modified HendersonHasselbach equation was clarified, and equations of their relations to the apparent ion exchange equilibrium constants were obtained.
Organic Chemistry Dr. Z. I. Sosinovich and Kim Te-I1 for the permission to use their experimental data.
Nomenclature a = activity C = concentration f = activity coefficient i, j = constants g = number of moles of alkali added to 1g of ion exchanger K = equilibrium constant
k = apparent equilibrium constant
n = parameter of the modified Henderson-Hasselbach equation PKH= parameter of the modified Henderson-Hasselbach equation V = solution volume, mug of ion exchanger f = equivalent fraction y(i-jj) = constants in this work equal to logarithm of equilibrium constants of elementary ion exchange equilibria z = ion charge a = degree of neutralization of ion exchanger
Literature Cited (1)Helfferich, F. Ion Exchange; McGraw-Hill Book Co: New York, 1960;pp 81-84. (2)Kern, W. Z. Titration of Polyvalent Macromolecular Acids. Biochem. 2.1939,301,338-356. (3) Katchalsky, A.; Spitnik, P. Potentiometric Titrations of Polymeric Acids. J. Polym. Sci. 1947,4 , 432-446. (4)Gregor, H. P.; Juttinger, L. B.; Loebl, E. M. Titration of Polymeric Acid with Quatenary Ammonium Bases. J.Am. Chem. SOC.1954,76, 5879-5880. (5) Gregor, H. P.; Hamilton, M. J.; Becher, J.; Bernstein, F. Studies on Ion Exchange Resins. XIV. Titration, Capacity and Swelling of Methacrylic Acid Resins. J . Phys. Chem. 1955,59, 874-881. (6)Gustafson, L.; Fillius, H. F.; Kunin, R. Basicities of Weak Base Ion Exchange Resins. Ind. Eng. Chem. Fundam. 1970,9, 221-229. (7)Kunin, R.;Fisher, S. Effect of Cross-Linking on the Properties of Carboxylic Polymers. I1 Apparent Dissociation Constants as a Function of the Exchanging Monovalent Cation. J. Phys. Chem. 1962,66,2275-2277. ( 8 ) Srtobel, H. A.; Gable, R. W. Titration Studies as a Means of Characterizing Anion-Exchange &sins. J.Am. Chem. SOC.1954, 76, 5911-5915. (9)Soldatov, V.S. On Thermodynamic Description of Equilibria between Weakly Dissociating Cation Exchangers and Aqueous Solutions. J. Phys. Chem. (USSR), 1968,42,2287-2291. (10) Soldatov, V. S. Simple Ion Exchange Equilibria (in Russian). Nauka Tech (Minsk) 1972,162-170. (11)Soldatov, V. S.Mathematical Modelling of Ion Exchange Equilibria on Resinous Ion Exchange. Reactive Polym. 1993,19, 105-121. (12)Libinson, G. S. Physico-Chemical Properties of Carboxylic Ion Exchangers (in Russian). Nauka (Moscow) 1969,112-121. (13)Ushakova, G. I.; Znamensky, Yu. P.; Artiushin, G . A. Influence of Polyfunctionality of Weakly Dissociating Anion Exchangers onto Their Equilibrium Properties. J. Phys. Chem. (USSR), 1978,52,2393-2395. Received for review October 26,1994 Revised manuscript received February 14, 1995 Accepted March 13,1995@
Acknowledgment This work was supported by the foundation for fundamental research of the Republic of Belarus, Grant No. F-43-119. I thank co-workers of the Laboratory of Ion Exchange and Sorption of the Institute of Physical
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Abstract published in Advance ACS Abstracts, J u n e 15,
1995.
