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Mathematical modelling of ion exchange equilibria on resinous ion exchangers
105
Reactive Polymers, 19 (1993) 105-121
Elsevier Science Publishers B.V., Amsterdam
Mathematical modelling of ion exchange equilibria on resinous ion exchangers V.S. Soldatov Institute of Physical Organic Chemistry of Belarus Academy of Sciences, Surganov Str., 13, Minsk, 220603 Belarus
(Received March 27, 1992; accepted in revised form December 4, 1992)
Abstract
A mathematical model establishing relations between selectivity, structure and ionic composition of an ion exchanger is proposed. A number of nearest neighbours in the micro-environment of the exchange site has been chosen as a structural parameter. This number together with the local composition determines the exchange site micr0-state. Different thermodynamic properties of ion exchange systems (e.g., A H and AG) can be computed by summation of the relative micro-state properties. The probabilities of the presence of a different number of nearest neighbours in the micro-environment of an exchange site in the sulphostyrene resins were estimated by the molecular dynamics method. An equation "property-composition" was derived and proved to be adequate to the experimental data for the ion exchange with inorganic and organic univalent ions on sulphostyrene resins over a wide range of cross-linkage. Keywords: ion exchange; mathematical modelling; equilibrium constants
Introduction
It is well known that the structure of ions exchangers significantly affects their selectivity. Nevertheless, in the theoretical approaches usually applied for the interpretation of ion exchange phenomena, the selectivity is not associated with any concrete structural parameters. On the other hand, the selectivity is dependent on the ionic composition of the ion exchanger. Therefore, for a correct description of the selectivity on the exchanger structure, a quantitative relation between some parameters of structure, selec-
tivity and the ionic composition of the exchangers should be established. In our recent publications [1,2] a new model describing dependences of the apparent equilibrium constant on relative mole fraction of the exchangeable ion in the resin phase was suggested. Two structural parameters (number of nearest neighbours and their probabilities) were used. In the present paper this model will be described in more detail. Its applicability will be demonstrated for the description of ion exchange equilibria between alkali metal and hydrogen ions on sulphostyrene resins over a
0923-1137/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
106 wide range of cross-linkage. Ion exchange equilibria between cations of orthotetracycline and chlorotetracycline with sodium ions will be also considered. An attempt to estimate some parameters of the model from the molecular structure of the representative fragments of the sulphostyrene resin network will be presented.
V.S. Soldatov / React. Polym. 19 (1993) 105-121
An ion exchange equilibrium is formulated as follows: Ii + 12 = I2 + I1
(1)
where the barred symbols are related to the resin phase. The apparent equilibrium constant is defined as: /(- = 2 2 a l ~e~la2
Theory In the model suggested some concepts requiring accurate definition will be used. They are given below: Exchange site - - a functional group on a polymer matrix whose charge is compensated by a counter ion. The exchange site location is defined to coincide with the centre of the sulphur atom in the functional group. Interaction radius, R - - the maximal distance between two exchange sites where the interaction between a counter ion and the exchange site is regarded to be significantly influenced by the counter ion exchange at the neighbouring exchange site. Sphere o f interaction - - a sphere with radius R drawn around an exchange site. Micro-environment - - a part of a sphere of interaction with the exception of the central exchange site. Nearest neighbours - - the neighbouring exchange sites situated at a distance less or equal to the radius of interaction from the central site. Property Y - - change in a function of state per mole of the exchanged ions in a process when infinitely small amount of ion 1 is replaced by ion 2 (e.g., AG, A H , AS, and log /(). Ionic composition - - expressed by a relative molar fraction of counter ion 2, X 2, X 1 + ) ( 2 = 1. Local ionic composition - - ionic composition in the micro-environment.
(2)
where a is the counter ion activity in the solution. The main statements of the model are as follows: The exchange sites in an ion exchanger have different amounts of nearest neighbours. In a statistically regular polymeric ion exchanger it can vary from 0 to n depending on the structure of the matrix and the choice of the interaction radius. A property y in a micro-state is assumed to be characterized by a number of nearest neighbours i and a local ionic composition. A molar property Y can be represented as a sum of properties y. Two parameters were chosen to describe the micro-environment: number of counter ions type 2 denoted as j', and that of ions 1, denoted as i - j present in the micro-environment. Then all possible states of the exchange sites with counter ion I can be expressed as I ( i - j , j ) (Scheme 1). It is assumed that the property y is dependent only on the number of counter ions 1 i, number of nearest neighbours
possible states of ion I
0
I(O,O)
1 2 3
I(1,O), I(O,1) I(2,0), I(1,1), I(0,2) I(3,0), I(2,1), I(1,2), I(0,3)
-7-
t
Scheme 1.
I(i-j,
j), j = 0 - i
V.S. Soldatov/ React. Polym. 19 (1993) 105-121
107
and 2 in the micro-environment but not of their position related to the central exchange site. A real ion exchange equilibrium may be regarded as a superposition of ideal "elementary" equilibria related to each of the states:
I1(i - j , j ) + 12 = I2(i - j , j ) + I 1 K(i-j,j)
A'2(i - j,j )a 1 = ~l(i_j,j)a 2
(3) (4)
where K ( i - j , j ) is an equilibrium constant and X(i - j , j ) is a mole fraction of the ions with a ( i - j , j ) micro-environment. In a further consideration definitions y(i - j,j) = log K ( i - j , j ) and Y = l o g I~ will be used. It should be noted, that y and Y can also stand for any molar additive property of the ion exchange system. A property Y can be obtained as a sum of properties y(i - j , j ) for the elementary equilibria proportional to the contents of the exchangeable ions in the relative states. If we denote the probability of the existence of i nearest neighbours of an exchange site in a real ion exchanger as Pi, and a property Y related to a hypothetical system containing only exchange sites with i nearest neighbours as Y/, then the total property Y can be expressed as
tions of the counter ions 1 and 2 within a given combination of i counter ions with j counter ions of type 2. If the probabilities are normalized so that their sum is equal to 1 and -~2 =-~, then i!
p(i-j,j)
= ( i - j ) ! j ! (1 -
2)(i-j)~j "
(7)
and j=i i! Y/= Y'. y(i - j , j ) ( 1 _ , ~ ) ( i - j ~ j j=0 (i-j)tjt..
(8) The general equation "property-ionic composition" for a real system is: i=n
Y=
j=i
i!
y" P,y" (i_j)!j!y(i-j,j) i=0
×(1
j=0
-x)(i-J)xj
(9)
As will be shown later, the number of nearest neighbours is rarely more than 4. Thus, eqn. (8) represents the following set of equations: Y0 =y(0,0)
(10a)
Y1 = y(1,0)(1 - X ) + y(0,1).~
(10b)
y2 =y(2,0)( 1 _ ~ ) 2 + 2 y ( 1 , 1 ) ( 1 - X ) X + y (0,2).,Y 2
(10c)
i=n
Y = Y'~ Yiei
(5)
Y 2 = Y ( 3 , 0 ) ( 1 - X ) 3 + 3y(2,1)(1 - X)2,~
i=0
In its turn, Yi can be represented as a sum of the properties y related to each of the local compositions to their probabilities p: j=i
Y= ~ y(i-j,j)p(i-j,j)
(6)
j=0 The values p ( i - j , j ) are proportional to the molar fraction of each of the counter ions raised to a power equal to the number of the nearest neighbours of a relative type. It is also proportional to a number of permuta-
+ 3y(1,2)(1 _ ~),~2 +
y(0,3)~3 (10d)
Y4 = y (4,0)(1 - X)4 + 4y(3,1)(1 - . ~ ) 3 ~ + 6y(2,2)(1 - ,~)2.,~2 + 4y(1,3)(1 _ y ) ~ 3
+ y(0,4)~4 (10e)
Since all y ( i - j , j ) values are constant, these equations are polynomial presented in a special form. Equation (8) has some features making it especially convenient if X =
V.S. Soldatov /React. Polym. 19 (1993) 105-121
108
10,11, i.e., for consideration of "propertycomposition" dependences. Coefficients of the first and the last terms of eqn. (8) independently of its power, y(i,0) and y(0,i), are equal to the values of y; at .,~ = 0 and .,~ = 1, respectively: y(i,O) = Yv-o
is that if values y(i -j,j) are in linear dependence on the local composition, X(i-j,j), then Y/-- Yi(X) is also a linear function, i.e., if y(i - j,j) = y(i,0)[1 - X( i - j,j)]
+y(O,i)X(i-j,j)
(lla)
y(O,i)= Y2= 1 (llb) If all y(i-j,j) have the same values A,
then
Y,-= Y/(x=0)(l --~) + Y/(2)=1.~
then Yi = A
If all constants y(i -j,j) are changed by a constant value A, the function Y~ is changed by the same value: j=i i! Yi+A = y ' ( i _ j ) v j v [ y ( 1 - j , j ) + A ] "
"
X(1 -X)(i-J)2 j
(13)
This opens a possibility to use Yi in the .,Y interval 10,11, or its normalized values, for theoretical considerations invariantly to the absolute values of Yi. The next important peculiarity of eqn. (8)
Y3
I
\
i
\
o 0
I a
b
C
(15)
The local compositions are expressed by the discrete sets of numbers X ( i - j , j ) according to the number of neighbouring exchange sites and the combinations of the ionic species in the micro-environment. In the case of linear dependence where Y/= Yi(X), values Y/and y(i -j,j) coincide at the total ionic compositions equal to the local compositions. Thus two "end" constants y(i,0) and y(O,i) control the value of the function Yi at pure ionic forms while the intermediate constants control its curvature. This is illustrated in Fig. 1 for the important particular case where /--3.
(12)
j=O
(14)
1
0
d
Fig. 1. Hypothetical dependences II3 -- f(X) for the following set of constants: Curve y(3,0) y(2,1) y(1,2) y(0.3) a 1 1/3 2/3 0 b 1 0 1 0 c 1 0 0 0 d 1 0 -1 0 e 1 -1 1 0
V.S. Soldatov /React. Polym. 19 (1993) 105-121
Treatment of expe~mental data with the model equafions
109 log7 1.0
A computer program was used to fit experimental data to one of the eqns. (10a-e) according to a given mean square error of interpolation or according to a given value of the polynomial power. In this way a maximal possible amount of the nearest neighbours can be found as it is equal to the power of the polynomial. Nevertheless, a good fit of one of eqns. (10a-e) to the experimental data does not mean itself that the ion exchanger does not contain exchange sites with a lower number of neighbouring sites, as can be deduced from the properties of eqns. (10a-e) and (8). Only two parameters of the appropriate equations, y(i,0) and y(O,i), have their direct physical meaning corresponding to a property Y for the elementary equilibria I,(i,O) + I 2 = I2(i,0 ) + 11 11(0,i ) + 12 = I 2 ( 0 , i ) + I 1 If no independent data exist, the rest of the parameters found by fitting, y ( i - j , j ) , are just empirical constants as the model implies the presence in the ion exchange of a set of exchange sites with a different amount of neighbours. It can also be assumed that all ion exchange sites have an equal amount of neighbours. Then parameters y ( i - j , j ) have their meaning as defined in the model for i = constant. The result of the application of eqn. (8) to the experimental data is greatly dependent on their precision, range of ionic compositions and the amount of experimental data. In order to illustrate the importance of these factors the following example is given. In Fig. 2, the experimental data on the dependence of log K = f(X) for the exchange K + - H + on a highly cross-linked (25% DVB) sulphostyrene resin are presented. The data are taken from refs. 3 and 4. In this work a dynamic method for equilibrium study was used. Solutions with a constant ionic strength
0.5
0.2
0.4
\ \ \ \
Fig. 2. Dependence log /~ = fiR), exchange K + - H ÷, ion exchanger KU-2X25, I = 0.100, T = 25.0°C, data are taken from ref. 3. The points are experimental data, the curves are computed from eqn. (10d) with parameters given in Table 1.
and desired ionic compositions were passed through small columns with the resin until equilibrium was reached. Then the resin phase was analyzed for the ionic composition. This method permits accurate results on the ends of the equilibrium curves to be obtained. The results of the application of eqn. (8) to these experimental data under different conditions are summarized in Table 1. The attempts to apply to the data the equations of the first and the second power formally lead to a rather good quantitative agreement between the experimental and the computed data. The mean square errors are practically the same ( E - - 0 . 0 4 ) and correspond to ~ 10% error in the values of /~. Nevertheless, it is quite clear that the experi-
110
V.S. Soldatov / React. Polyrn. 19 (1993) 105-121
TABLE 1 Application of eqn. (8) to experimental data on log g = f(X) for K+-H + exchangeon KU-2X25 Number exper, points 14 14 14 14
.~ interval
0.034-0.963 0.034-0.963 0.034-0.963 0.034-0.963
EquationE inter- No. of power polation curve in Fig. 2 1 0.043 5 2 0.040 3 0.022 1 4 0.022
10 10 10 10
0.082-0.904 0.082-0.904 0.082-0.904 0.082-0.904
1 2 3 4
0.018 0.017 0.014 0.010
7 7 7 7
0.223-0.775 0.223-0.775 0.223-0.775 0.223-0.775
1 2 3 4
0.017 0.006 0.006 0.006
curve does not correspond to the real experimental data over the whole range of the ionic compositions. The example presented clearly shows the importance of high quality experimental data. A meaningful and reliable conclusion on the power and the coefficients of the equation describing the experimental data can be obtained only if they cover a wide range of compositions approaching .,Y = 0 and X = 1 as close as possible and if the number of the experimental points is sufficiently large.
Applications of the model 3 4 2
mental points fall into an S-shaped curve which can be only described by a polynomial of at least the third power. It can be seen from Table 1 that eqn. (8) with i = 3 represents the shape of the experimental curve correctly and the elror of interpolation corresponds to the experimental error as evaluated in ref. 4 ( E = 0.02 corresponds to a 5% error in /(). It can also be seen that application of the equation with i - - 4 does not improve the results of the computation. That means that i = 3 should be taken as the found power of the equation in the application to the given case. If only the data in the range X = 0.0820.904 are taken for treatment, then the fit of the equation to the data becomes better ( E = 0.010-0.018). Nevertheless, it is impossible to conclude what value of i should be chosen. Probably, the straight line (i = 1) would be the best choice in this case. In the case where data in the interval X = 0.223-0.775 are analyzed, then i = 2 should be taken invariantly since a further increase in i does not improve the precision of the interpolation, which is very high in this case. At the same time, the shape of the
The first set of the data is related to the ion exchange system K + - H + on KRSm resins at a constant ionic strength of the equilibria solutions 0.100 and T--25.0°C. The KRSm resins are analytical grade sulphostyrene ion exchangers with pure m-DVB ( > 98%) as a cross-linking agent [5,6]. The apparent equilibrium constants and enthalpies were computed from independent experimental data and treated with the model equation. The enthalpies were found from calorimetric measurements of the heat effects of partial ion exchange [5,6]. The number of experimental points is not large but each of the data corresponds to the integral process whose initial and final states are shown in Fig. 3 at the beginning and the end of the arrows. The points taken for computation relate to the middle of the X interval. As can be seen from Fig. 3, the composition interval covered was 10,11. The data on apparent equilibrium constants in Fig. 4 were obtained by the dynamic method. The attempts to apply eqn. (10) of different power have led to the conclusion that the resins of low cross-linkage ( < 12% DVB) are well described by the equation of the second power. The description of the higher crosslinkage resins requires the equation of the
111
V.S. Soldatou / React. Polym. 19 (1993) 105-121
In terms of the model discussed it means that the properties of the ion exchangers with low cross-linkage can be described assuming that each exchange site in the resins has at most two nearest neighbours. In the case of high cross-linkage a significant fraction of the exchange sites have at least three nearest neighbours. This must be true for the other exchange equilibria with the same resins. The following example confirms this assumption. The apparent equilibrium constants as functions of X for the exchange of alkali
1.4 log~ 1.2 6 1.0
0
I 0
0'.2
014
01.6
0'.8
y
~I 1
Fig. 3. Dependences AH= f(.~), exchange K+-H +, ion exchangers KRSm with different percent of mDVB: 1, 1%; 2, 5%; 3, 8%, 4, 12%; 5, 16%; 6, 25%; I = 0.100, T = 25.0°C. The points are experimental data, the curves are computed from eqns. (10c) and (10d) with the parameters given in Table 2 and marked with asterisk. The data are taken from refs. 5 and 6.
0.8
~
0.4 :jk. 0.2 ~
third power. This holds true both for the enthalpy and log /~. As can be seen from Table 2, application of the third power equation to the data on the low cross-linkage resins does not improve the accuracy of interpolation. On the other hand, starting from 12% DVB the m e a n interpolation error becomes significantly lower for the equation of the third power. Besides, the shape of the curves changes. The S-shaped curves observed in this case may not be described in principle by equations of the second power.
0 -0.2 "
0
0.2
0.4
0.6
0.8
I
Fig. 4. Dependence log /~ = f-(X). See caption to Fig. 3.
I~.S. Soldatov / React. Polym. 19 (1993) 105-121
112 TABLE 2
Application of the model e q u a t i o n to description d e p e n d e n c e s for the K R S m resins. E x c h a n g e K ÷ - H +, I --- 0.100 a, T -- 25.0°C i--2 % m -DVB
Y
,~ interval
E
y(2,0)
y(1,1)
y(0.2)
1 *
log I( AH
0.05-0.95 0.10-0.84
8 4
0.008 0.001
0.152 -0.90
0.086 -0.47
0.105 -0.28
5 *
log /~
5 *
AH
0.05-0.95 0.044-0.927
9 5
0.015 0.089
0.386 - 1.74
0.218 - 1.85
0.192 - 0.66
8 * 8 *
log/~ AH
0.05-0.90 0.08-0.946
8 5
0.016 0.016
0.542 - 2.58
0.333 - 2.02
0.335 - 0.56
12 12
log /~ AH
0.05-0.95 0.066-0.830
9 4
0.045 0.046
0.686 - 3.19
0.618 - 2.29
0.157 - 0.41
16
log /~
0.05-0.95
9
0.047
0.770
0.820
25 25
log /~ AH
0.01-0.95 0.058-0.927
10 5
0.036 0.230
1.228 - 5.05
0.775 - 2.09
- 0.263 - 0.280
Y
.~ interval
Number exper. points
E
y(3,0)
y(2,1)
y(1,2)
1
log/~
0.05-0.95
8
0.008
0.154
0.101
0.099
0.104
5 5
log/~ AH
0.05-0,95 0.044-0.927
9 5
0.014 0.065
0.359 - 1.90
0.333 - 1.18
0.165 - 2.04
0.201 - 0.48
8 8
log / ( AH
0.05-0.90 0.08-0.946
8 5
0.015 0.015
0.500 - 2,60
0.475 - 2.14
0.282 - 1.60
0.350 - 0.54
12 * 12 *
log /~ AH
0.05-0.095 0.066-0.830
9 4
0.042 0.0" 7
0.874 - 3.33
0.390 - 2.05
0.629 - 2.43
0.119 0.010
16 *
log/~
0.05-0.95
9
0.035
1.361
0.160
0.904
-0.13
25 * 25 *
log/~ AH
0.01-0.95 0.058-0.927
10 5
0.013 0.093
1.448 - 5.59
0.518 - 1.12
0.759 - 3.44
-0.38 0.28
1*
Number exper. points
- 0.03
i=3 % m -DVB
y(0.3)
a Ionic strength.
TABLE 3 Application of the m o d e l e q u a t i o n to the description of log /~--f(-,Y) d e p e n d e n c e s for t h e KU-2X25. E x c h a n g e alkali m e t a l - h y d r o g e n ions, I -- 0.100, T = 25.0°C Exchange
,~ interval
Number exper. points
E
y(3,0)
y(2,1)
y(1,2)
y(0,3)
Li ÷ - H + Na ÷ - H ÷ K ÷ -H ÷ Rb+-H + Cs ÷ - H ÷
0.015-0.967 0.014-0.966 0.034-0.980 0.046-0.974 0.057-0.960
13 15 15 15 15
0.026 0.040 0.032 0.030 0.032
0.159 0.653 1.122 1.327 1.453
- 0.383 - 0.120 0.363 0.312 0.446
0.144 0.707 0.660 0.751 0.572
- 0.561 - 0.600 - 0.286 -0.464 - 0.704
V.S. Soldatov /React. Polym. 19 (1993) 105-121
113 In this case the equilibria were studied on the KU-2X25 sulphonic type ion exchanger with commercial DVB as a cross-linker. Comparing the data for K + - H ÷ exchange in Figs. 4 and 5 it can be seen that m - D V B behaves as a more efficient cross-linker than the commercial DVB mixture containing 55% of the DVB isomers. Nevertheless, the shapes of the curves are identical. The data presented show that in ;he case of exchange between inorganic ions the assumption of three nearest neighbours in the micro-environment of the exchange site is sufficient for the densest possible styrene-divinylbenzene matrixes. The assumption of two neighbours is sufficient to describe the
1.5
tog~' Cs 1.0
Na
0.5
O 2.0
log~ 1.8
Li
1.6
1.4
-0.5 1.2
0
0.2
0.4
0.6
0.8
I
Fig. 5. Dependences log /( = f(,~) for the exchange of alkali metal with hydrogen ions, ion exchanger (KU2X25, I = 0.100, T= 25.0°C. The points are experimental data, the curves are computed from eqn. (10d) with the parameters given in Table 3. The data are taken from ref. 3. metal and hydrogen ion on a highly cross-linked ion exchanger are presented in Fig. 5. It is clear that all the log K = f i X ) curves have inflection points and are well described by the equation of the third power (see also Table 3).
1.0
0.8
0
0.2
0.4
0.6
0.8
1
Fig. 6. Dependences log /(= f(,Y) for exchange of orthotetracyclinium with sodium ion, ion exchanger Dowex 50 with different percent of DVB:I, 0.5%; 2, 2%; 3, 6%; 4, 8%. The points are experimental data, the curves are computed from eqn. (10e) with the parameters given in Table 4. The data are taken from ref. 7.
114
V.S. Soldatov / React. Polym. 19 (1993) 105-121
b
i
i
0.4
i
i
2
i
O.B
i
i
0.4
i
i
0.8
i
0.4
i
i
0.8
Fig. 7. Dependences log /( = f(.~) for exchange of chlorotetracyclinium with sodium ion on sulphostyrene ion exchangers: a, SK-6 resin, 6% DVB, layer thickness of the exchanger on the surface of non-sulphonated matrix is 3 /~m; b, the same type resin, telogenated, the layer thickness is 22/~m; c, Dowex 50X5. The points are experimental data, the curves are computed from eqn. (10e) with the parameters given in Table 4. The data are taken from ref. 7.
equilibria on the resins of low and moderate cross-linkage. A larger number of nearest neighbours is possible in the case of large organic ions. This can lead to very complicated depen-
dences Y = f(,,~) and can be used as a severe test for the model. In Figs. 6 and 7 dependences l o g / ( = f(,,~) are given for exchange of ortho- and chlorotetracycline cations with sodium ion on the
TABLE 4 Application of the model equation to the description of log /( = f(.~) dependences. Exchange of ortho- and chiorotetracycline with Na ÷ on the sulphostyrene resins Resin
.~ interval
Orthotetracycline Dowex 50 x 0.5 0.070-0.910 Dowex 50 x 2 0.055-0.800 Dowex 50 x 6 0.030-0.385 Dowex 50 x 8 0.020-0.270 Chlorotetracycline SK-6 3 ~m " 0.12-0.980 SK-6 22/xm 0.12-0.995 Dowex 50 x 5 0.08-0.810
Number of exper. points
E
y(4,0)
y(3,1)
y(2,2)
y(1,3)
10
0.017
0.83
2.22
- 0.57
3.38
0.91
8
0.036
1.39
1.71
0.68
3.91
- 0.61
8
0.012
1.58
1.97
0.63
17.53
6
0.016
1.51
3.69
- 21.43
11
0.017
2.27
2.04
2.47
1.31
3.22
8
0.063
1.91
1.84
3.79
1.16
3.35
7
0.052
1.89
2.54
1.73
4.40
0.26
180
y(0,4)
- 105.1 - 1228
115
v.s. Soldatov / React. Polym. 19 (1993) 105-121
sulphostyrene resins of different structure. The data are taken from ref. 7. It appeared, that the data could only be described with the equation of the fourth power (Table 4).
Estimation of the number of nearest neighbours Formal application of the main model equation to the experimental data has shown that in the real systems with sulphostyrene ion exchangers the n u m b e r of the nearest neighbours should be chosen as 2, 3 or 4 depending on the cross-linkage of the resin and size of exchanging counter ions. W e have a t t e m p t e d to estimate the probabilities of the presence of exchange sites at different distances from the central site for representative fragments of the sulphostyrene networks. Also, the value of the radius of interaction was evaluated from comparison of the probabilities c o m p u t e d with the properties of the real systems. The following method was used. Consider that the s t y r e n e - D V B matrix can be repre-
sented as a combination of t w o types of fragments: linear chains and four-beam stars with a sulphonated p - D V B fragment in the centre (cross-shaped fragments). The fragments contain 8 sulphostyrene groups. The fragments conformation structure was analyzed by a molecular mechanics method with modified parameters of the force field of the C O S M I C system [8] and specially developed service programs (for details see ref. 9). The distances b e t w e e n the centres of the sulphonic groups for different conformations corresponding to the local energy minima were computed. Then the frequencies of the presence of these centres at different distances from each other (excluding the end groups) were found. These frequencies were assumed to be approximately equal to the probabilities. The longest possible distance b e t w e e n the neighbouring sulphonic groups was found to be about 13 nm. It was assumed that the conformation energies fall into a Boltzman distribution and the final probabilities were found as the arithmetic average of the frequencies for the different conformations multiplied by the Boltzman factor. Since
TABLE 5 The probabilities of existence of i neighbours at the distance R from the exchange site in the fragments of the sulphostyrene network.
i
R(nm)
5 Linear chain 0 0.525 1 0.475 2 0.000 3 0.000 4 0.000 5 0.000
6
7
8
9
10
11
12
13
14
0.340 0.660 0.000 0.000 0.000 0.000
0.244 0.663 0.093 0.000 0.000 0.000
0.168 0.660 0.172 0.000 0.000 0.000
0.006 0.656 0.336 0.002 0.001 0.000
0.000 0.488 0.491 0.021 0.000 0.000
0.000 0.417 0.490 0.087 0.006 0.000
0.000 0.157 0.232 0.313 0.296 0.002
0.000 0.000 0.160 0.077 0.489 0.270
0.000 0.000 0.002 0.158 0.162 0.337
Four-beam star 0 0.800 1 0.200 2 0.000 3 0.000 4 0.000 5 0.000
0.833 0.267 0.000 0.000 0.000 0.000
0.200 0.800 0.000 0.000 0.000 0.000
0.002 0.730 0.202 0.067 0.000 0.000
0.002 0.598 0.333 0.067 0.000 0.000
0.000 0.400 0.203 0.198 0.198 0.000
0.000 0.000 0.400 0.200 0.202 0.198
0.000 0.000 0.400 0.002 0.198 0.400
0.000 0.000 0.002 0.398 0.198 0.202
0.000 0.000 0.000 0.000 0.467 0.332
116 the conformations considered were optimized according to the energy minima, their structure corresponds to the expanded matrix, i.e. swollen ion exchanger. The results of the calculations are presented in Table 5. The probabilities of finding a certain number of neighbours (i = 0-5) as a function of the distance from an exchange site ( R = 5 - 1 4 nm) were found. Comparing these data with the model predictions and the experimental data the value of the interactions radius can be evaluated. The experimental data on the heat effects of ion exchange of ions Na +, K +, Cu 2+ and Fe 3+ on H + on a linear polystyrene sulphonates, obtained in our laboratory, show that they are described by the model equation of the second power at most. A similar result was reported in ref. 10. Hence, in this case two neighbours are present in the micro-environment of the exchange ion. As seen from Table 5, it corresponds to a possible radius of interaction not greater than 10 nm. On the other hand, as was shown above, properties of highly cross-linked resins as a function of the ionic composition require the equation of at least the third power for their accurate mathematical description. Comparing this observation with the data of Table 5 for the cross-shaped fragment we conclude that the radius of interaction may not be less than 10 am. Summarizing these considerations, R = 10 nm has been chosen as the radius of interaction.
Discussion
The main equation of the model allows simple systematization of the ion exchange systems according to the degree of their non-ideality, somewhat different from that in ref. 11. The systems with constant /~ can be defined as ideal. The true ideality is possible if
V.S. Soldatov /React. Polym. 19 (1993) 105-121 the exchange sites are situated far away from each other and the number of nearest neighbours is 0. The I10 value is always constant. The systems with i > 0 can demonstrate pseudo-ideal behaviour under conditions expressed by eqn. (12). If the number of nearest neighbours is 1 then Yl = Y I ( X ) is always a linear function, which corresponds to regular behaviour. It should be noted, that this definition is based on the similarity of the considered system with regular mixtures, not on the identity of their structures and interaction. Pseudo-regular behaviour is also possible for systems with i > 1 under conditions expressed by eqn. (14). If the number of neighbours is more than 1, the systems behaviour can be characterized as irregular with a different degree of irregularity according to the number of the neighbours, and, hence the power of the main equation (e.g., square-irregular, cubicirregular, etc.). The simplest irregular system is described by the second power equation suggested in refs. 12 and 13 for a liquid ion exchange extractant, and rigourously derived for resinous ion exchangers in ref. 11. Later it was often used for treatment of experimental data (e.g., refs. 14-17). The limitation in the application of this particular case of the main equation "property-composition" are quite obvious from the above consideration. Applicability of the second power equation to the description of selectivity of ion exchange can be due to either the negligibly small contribution of the higher power terms, or to a special form of the dependence of the y(i j,j) parameters on the local composition in an equation of the higher power, i.e., the system can behave like a pseudo-square. In the case of dependences Y = Y(,Y) for the ion exchangers of low cross-linkage shown in Figs. 3 and 4 the absence of the higher power terms corresponds to a very small probability of presence (i.e., low contents) of
V.S. Soldatov /React. Polym. 19 (1993) 105-121
117
the exchange sites with i > 2, as seen from Table 5. Nevertheless, accurate experimental investigation of ion exchange selectivity over a wider composition range can evidence the presence of higher order terms. In Fig. 10 the data from ref. 18 on log/< = f(.Y) dependence for Li+-H ÷ exchange on an ion exchanger with 4% DVB are presented. In order to notice such effects the .~ range must be broad enough (in this case data in the range X = 0.045-0.92 were taken for treatment). The data presented in Figs. 3 and 4 are not sufficient to notice this effect for resins of low cross-linkage. The estimation of the probability of the presence of more than 2 exchange sites in the micro-environment given in Table 5 proves that it is several times higher in the
1.6
IZI 2 >~
/K
I
/ / /
0.8
/
/ 0.4
/ / Li
"l -0.4
~'
"@. . . . 0.6
0.8
0--1.0
z
~\ 1.2
11.4
"~, 1.6 R,nm
Fig. 9. Constants of eqn. (10d) y(i - j , j ) = log K(i - j , j ) for the alkali m e t a l - h y d r o g e n ion exchange on the KU-2X25 resin as a function of the ionic radii: 1, y(3,0); 2, y(0,3).
1.6
'~1.2
7 v
>,
0.8
0.4
-0.4
-0.8 0
1/3
2/3
1 X(i-j,j)
Fig. 8. Constants of eqn. (10d) y(i - i , i ) = log K(i - j , j ) for the alkali m e t a l - h y d r o g e n ion exchange on the KU-2X25 resin as a function of local composition: 1, Cs+; 2, Rb+; 3, K+; 4, Na+; 5, Li +.
vicinity of the cross-links than in the linear chain at R = 10-11 nm. This explains the influence of cross-linkage on the shape of the selectivity curves. The most broad explanation of selectivity decrease with X is based on the idea of irregularity of the resin exchange sites, suggested for styrene-DVB ion exchangers in ref. 19. Later the increase in selectivity was reported and explained by "cooperative" interaction [20]. No ways to express or interpret these effects in quantitative terms were suggested. The new model implies irregularity of the exchange sites as a natural property of the resinous ion exchangers caused by the different possible number of nearest neighbours and different combinations of the ionic species in the micro-environment. The model parameters, P, i and y(i -j,j) give a way to express this irregularity, as well as the cooperative effect, in quantitative terms. The absolute values of the selectivity parameters log K(i-j,j) are functions of the energy of interionic interactions and depend
118
011
V.S. Soldatov /React. Polym. 19 (1993) 105-121
log~
0
-0. I
i
0
~
0.2
i
l
0.4
i
i
0.6
i
i
0.8
i
1
Fig. 10. D e p e n d e n c e log /~ = f(.~) for the L i + - H + exchange on Dowex-50X4, I = 0.100, T = 25.0°C. The points are experimental data taken from ref. 18. The curve is computed from eqn. (10d) for the points in the range X - - 0.045-0.920 with the following coefficients: y(3.0) = 0.025; y(2,1) = - 0.140; y(1,2) = - 0.043; y(0,3) = -0.139.
on the total volume of the resin, ion hydration and other factors, not expressed in the model in explicit form. They influence y(i-j,j) values which can only be found at present as empirical parameters. Nevertheless, their physical meaning is well defined and they can be used for interpretation of the Y = Y( X ) dependences. Dependences of the y(i - j , j ) for i = 3 of the I + - H ÷ exchanges on the local compositions and radii of the cations are complicated and have no explanation at present (Figs. 8 and 9): In spite of that some regularities can be summarized. The elementary equilibrium constant of the same exchange is strongly dependent on the composition of the micro-environment. The largest selectivity toward a preferably sorbed ion (H ÷ at . ~ = 1) occurs if the exchanging ion is surrounded by ions of the
other type. This corresponds to the largest free energy and enthalpy of the process. Substitution of one of the originally present in the micro-environment ions with those entering leads to significantly lower log /~ and A H values. This results in a remarkable symmetry of the y ( i - j , j ) = f ( X ( i - j , j ) ) and Y = f(X) dependences. A simple dependence of log K (3,0) on the ionic radii exists. An increase in the free energy of the ion exchange correlates well with increasing electrostatic interaction between the sulphonic group and the counter ion. The counter ion entering always has the same surrounding in its micro-environment: three hydrogen ions. On the other hand, dependence of the other "end" constant, log K(0,3), on the ionic radius is very complicated. A possible reason for that is that in this case the ion entering always has different surroundings. It seems, that in this case the effect of radius should be considered separately for the "structure forming" (Li + and Na +) and "structure breaking" (K +, Rb + and Cs +) ions [21]. A slight decrease of log K(0,3) is observed in the first case while this parameter drops sharply with the ionic radius increase for the second group of ions. Intermediate dependences occur for the other parameters. In this systems the sizes of the exchangeable ions are much smaller than the radius of interaction and all exchange sites are easily accessible for the counter ions. The situation is quite different for the case of exchange of large organic ions. The data in Figs. 6 and 7 shows the absence of a sharp decrease in selectivity at low loadings of the resins with orthotetracycline (OTC) or chlorotetracycline (CITC) ions. Two effects are remarkable in these cases: a significant increase in the log g values and their sharp drop at high resin loadings with organic ions. These results can be interpreted in the model terms in the following way. The irregularity of the exchange sites, as a universal property, must always favour
V.S. Soldatov / React. Polym. 19 (1993) 105-121
decrease of selectivity of a preferably sorbed ion with X increase. On the other hand, increase in the selectivity of the considered ions can be expected because of their intensive interaction with each other. These ions contain several functional groups able to form hydrogen bonds and have sizes exceeding the interaction radius value estimated for the inorganic ions: CH 3 0 H O H
N(CH3) 2
/
/ : (,....
y
y
OH
0
yo'.y....... OH
0
/:
................. 14.67 ,~
l
/
Orthotetracycline
/n
C1 CH 3 OH H
;
/
oH .N(CH3) 2
7 Y yo.y OH
O
OH
.......
O
....... 14.32 ,~
/
Chlorotetracycline The large size of the counter ions causes two effects. The number of the nearest neighbours of the counter ion is larger than that of the inorganic ions and that can cause more complicated dependences Y = Y ( X ) . As seen from Table 5, the probability of having more than three neighbours in a sphere with radius equal to the largest dimension of this ion is practically unity. We suppose that i can be assumed equal to 4 as a sum of three neighbouring exchange sites,
119
estimated without taking into account the sizes of the counter ions, plus one of the counter ions, situated at a distance between the exchange site radius of interaction (10 nm) and the largest counter ion dimension (14.67 nm). The other obvious result of a large counter ion size is a restriction in the accessibility of the exchange sites by the counter ions due to blocking by the neighbouring bulky organic ions. It can also cause inaccessibility of certain parts of the resin networks both on the molecular and macroscopic level, known as "sieve effect". This must result in a very low equilibrium constant of the elementary equilibria for which this effect is important. The data for the exchange OTC+-Na ÷ on resins of different cross-linkage can be interpreted by an overlap of the mentioned effects. A possible rationalisation of the observed phenomena can be done by considering the values of the elementary equilibria constants y(i - j , j ) , given in Table 4. The appearance of an OTC ÷ ion in the micro-environment of the exchange site causes an increase in the equilibrium constant of the second elementary equilibrium due to its additional interaction between the incoming ion and the presaturant ion in spite of the tendency of the selectivity drop due to the irregularity effect. The presence of two organic ions in the vicinity of the exchange site results in a marked selectivity drop probably due to the blocking phenomenon. In the case when there are three OTC ÷ ions in the sphere of interaction, the equilibrium constant becomes high again. It is supposed that the high interaction energy in this case is caused by an intensive aggregation of the incoming ion with the three already present in the micro-environment OTC ÷ ions. The low values of the last constant, progressively decreasing with the cross-linkage of the resins, certainly reflects the sieve effects. Regularities of the CITC+-Na ÷ exchange on Dowex 50X5 resin are similar to that of
V.S. Soldatov / React. Polym. 19 (1993) 105-121
120
OTC+-Na ÷. Absence of the sieve effect for the thin layer resins SK-6 evidences that this effect is caused by the inaccessibility of the central parts of the Dowex 50 resins by the bulky organic ions. In the telogenated SK-6 resin the accessibility of the exchange sites is high enough to avoid the sieve effect in a layer 22/zm thick. A more loose structure of this resin makes the irregularity effect less important which gives a sharper rise of the C1TC ÷ selectivity at low X values compared to the non-telogenated ion exchanger.
Conclusions The suggested mathematical model describes a real ion exchanger as a set of exchange sites with a different number of the nearest neighbours. The main equation of the model describes dependences "propertycomposition" with the help of the following parameters: (a) number of nearest neighbours of an exchange site, (b) the probability of its existence, and (c) equilibrium constants of the elementary equilibria related to each of the micro-states of the exchangeable ions. The ways of estimating these parameters have been considered. The model was applied for the interpretation of the ion exchange equilibria of univalent ions (alkali metal-hydrogen ions, ortho- and chloro-tetracycliniumsodium ions) on sulphostyrene resins of different cross-linkage.
Acknowledgements All the work concerning computation of the number of the exchange site nearest neighbours in the fragments of the resin networks was done by a group of my colleagues from the Institute of Physical Organic Chemistry of the Belarus Academy of Sciences: A.L. Pushkarchuk, V.I. Gogolinsky and V.M. Zelenkovsky, to whom I express my deep
gratitude and admiration for their high professionalism. This work was supported by the foundation for fundamental research of the Republic of Belarus, Grant No. F 216-181.
List of symbols a E G H I i j
K
P,p R X Y y
activity mean square error Gibbs energy enthalpy exchangeable ion total number of neighbouring exchange sites number of neighbouring sites whose change is compensated by counter ion 2 apparent equilibrium constant the probabilities interaction radius relative mole fraction of exchangeable ion an additive property of the system property Y related to a micro-state
References 1 V.S. Soldatov, Mathematical modelling of simple ion exchange equilibria, Dokl. Acad. Sci., BSSR, 34 (1990) 528. 2 V.S. Soldatov, Mathematical modelling of dependencies "ionic composition-property" in ion exchange systems, Dokl. Acad. Sci., USSR, 314 (1990) 664. 3 V.S. Soldatov, R.V. Martzinkevich and A.I. Pokrovskaya, Peculiarities of exchange on ion exchangers of high cross-linkage, J. Phys. Chem. (USSR), 43 (1969) 2889. 4 V.S. Soldatov and R.M. Sukhover, Estimation of errors of experimental and computed thermodynamic quantities in ion exchange systems, J. Phys. Chem. (USSR), 42 (1968) 1798. 5 V.S. Soldatov, R.V. Martzinkevich, A.I. Pokrovskaya, Z.I. Kuvaeva, S.B. Makarova, T.A. Aptova and E.V. Egorov, Comparative characteristics of sulfo-
V.S. Soldatov / React. Polym. 19 (1993) 105-121 styrene ion exchangers with different isomers of divinylbenzene, J. Phys. Chem. (USSR), 50 (1976) 480. 6 V.S. Soldatov, Z.I. Kuvaeva and R.V. Martzinkevich, Thermodynamics of ion exchange on sulfostyrene type ion exchangers linked with para- and meta-divinylbenzene, in M. Streat (Ed.), The Theory and Practice of Ion Exchange, Proceedings of an Internal Conference held at Churchill College, University of Cambridge, July, Soc. Chem. Indus. UK, 1976, p. 20. 7 G.V. Samsonov and A.T. Melenevskij, Sorption and
Chromatographic Methods of Physico-Chemical Biotechnology, Nauka, Leningrad, 1986, pp. 121-127 8
9
10
11
12
13
(in Russian). J.G. Vinter, A. Davis and M.R. Saunders, Strategic approaches to drug design. An integrated software framework for molecular modelling, J. Comp.-Aided Mol. Design, 1 (1987) 31. V.S. Soldatov, V.I. Gogolinsky, A.L. Pushkarchuk and V.M. Zelenkovsky, Estimation of structural parameters of polymer matrix fragments for polystyrenesulfonic ion exchangers, Dokl. Acad. Sci. Belarus, 37 (1993) in press. E. H6gfeldt, 10 Years experience of a simple three-parameter model to fit ion exchange data, React. Polym., 11 (1989) 199. V.S. Soldatov and V.I. Bichkova, Quantitative description of ion exchange selectivity in non-ideal systems, React. Polym., 1 (1983) 251. Z.I. Kuvaeva, V.S. Soidatov, F. Fredlund and E. H6gfeldt, On the properties of solid and liquid ion exchangers - VII. Application of a simple model to some ion exchange reactions on dinonylnaphthalene sulfonic acid in different solvents and for different cations with heptane as solvent, J. Inorg. Nucl. Chem., 40 (1978) 103. E. H6gfeldt and V.S. Soldatov, On the properties of solid and liquid ion exchangers - VII. A simple
121
14
15
16
17
18
19
20 21
model for the formation of mixed micelles applied to salts of dinonylnaphthalene sulfonic acid, J. Inorg. Nucl. Chem., 41 (1979) 575. Z.I. Kuvaeva, V.S. Soldatov and E. H6gfeldt, On the properties of solid and liquid ion exchangers VII. Potassium-hydrogen and calcium-hydrogen exchange in dinonylnaphthalene sulfonic acid dissolved in heptane-isooctanol mixtures, J. Inorg. Nucl. Chem., 41 (1979) 579. V.S. Soldatov and A.V. Mikulich, Free energy additivity for exchange of alkylammonium ions on a liquid cation exchanger - dinonylnaphthalene sulfonic acid, J. Phys. Chem. (USSR), 58 (1984) 889. E. H6gfeldt, R. Chiarizia, P. Danesi and V.S. Soldatov, Structure and ion exchange properties of dinonylnaphthalene sulfonic acid and its salts, Chem. Scr., 18 (1981) 13. E. H6gfeldt, The H6gfeldt three-parameter model. A useful approach to summarising data in ion exchange, in M. Abe, T. Kataoka and T. Suzuki (Eds.), New Developments in Ion Exchange, Kodansha-Elsevier, Tokyo, 1991, p. 19. L.K. Arkhangelsky, E.A. Materova, S.S. Mikhailova and G.P. Lepnev, Relations between the apparent equilibrium constants, ion exchanger composition and swelling for large contents of one of the counterions, in Thermodynamics of Ion Exchange, Nauka i Technika, Minsk, 1978, p. 60 (in Russian). D. Reichenberg, K.W. Pepper and D.J. McCauley, Properties of ion exchange resins in relation to their structure. Part II. Relative affinities. J. Chem. Soc., (1951) 493. H.P. Gregor, Ion-pair formation in ion exchange systems, J. Am. Chem. Soc., 73 (1951) 3537. J. Steigman and J. Dobrov, The interaction of antagonistic and cooperative polyelectrolytes in water at 25°C. A hypothesis concerning ion exchange resins, J. Phys. Chem., 72 (1968) 3424.
Reactive Polymers, 19 (1993) 105-121
Elsevier Science Publishers B.V., Amsterdam
Mathematical modelling of ion exchange equilibria on resinous ion exchangers V.S. Soldatov Institute of Physical Organic Chemistry of Belarus Academy of Sciences, Surganov Str., 13, Minsk, 220603 Belarus
(Received March 27, 1992; accepted in revised form December 4, 1992)
Abstract
A mathematical model establishing relations between selectivity, structure and ionic composition of an ion exchanger is proposed. A number of nearest neighbours in the micro-environment of the exchange site has been chosen as a structural parameter. This number together with the local composition determines the exchange site micr0-state. Different thermodynamic properties of ion exchange systems (e.g., A H and AG) can be computed by summation of the relative micro-state properties. The probabilities of the presence of a different number of nearest neighbours in the micro-environment of an exchange site in the sulphostyrene resins were estimated by the molecular dynamics method. An equation "property-composition" was derived and proved to be adequate to the experimental data for the ion exchange with inorganic and organic univalent ions on sulphostyrene resins over a wide range of cross-linkage. Keywords: ion exchange; mathematical modelling; equilibrium constants
Introduction
It is well known that the structure of ions exchangers significantly affects their selectivity. Nevertheless, in the theoretical approaches usually applied for the interpretation of ion exchange phenomena, the selectivity is not associated with any concrete structural parameters. On the other hand, the selectivity is dependent on the ionic composition of the ion exchanger. Therefore, for a correct description of the selectivity on the exchanger structure, a quantitative relation between some parameters of structure, selec-
tivity and the ionic composition of the exchangers should be established. In our recent publications [1,2] a new model describing dependences of the apparent equilibrium constant on relative mole fraction of the exchangeable ion in the resin phase was suggested. Two structural parameters (number of nearest neighbours and their probabilities) were used. In the present paper this model will be described in more detail. Its applicability will be demonstrated for the description of ion exchange equilibria between alkali metal and hydrogen ions on sulphostyrene resins over a
0923-1137/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
106 wide range of cross-linkage. Ion exchange equilibria between cations of orthotetracycline and chlorotetracycline with sodium ions will be also considered. An attempt to estimate some parameters of the model from the molecular structure of the representative fragments of the sulphostyrene resin network will be presented.
V.S. Soldatov / React. Polym. 19 (1993) 105-121
An ion exchange equilibrium is formulated as follows: Ii + 12 = I2 + I1
(1)
where the barred symbols are related to the resin phase. The apparent equilibrium constant is defined as: /(- = 2 2 a l ~e~la2
Theory In the model suggested some concepts requiring accurate definition will be used. They are given below: Exchange site - - a functional group on a polymer matrix whose charge is compensated by a counter ion. The exchange site location is defined to coincide with the centre of the sulphur atom in the functional group. Interaction radius, R - - the maximal distance between two exchange sites where the interaction between a counter ion and the exchange site is regarded to be significantly influenced by the counter ion exchange at the neighbouring exchange site. Sphere o f interaction - - a sphere with radius R drawn around an exchange site. Micro-environment - - a part of a sphere of interaction with the exception of the central exchange site. Nearest neighbours - - the neighbouring exchange sites situated at a distance less or equal to the radius of interaction from the central site. Property Y - - change in a function of state per mole of the exchanged ions in a process when infinitely small amount of ion 1 is replaced by ion 2 (e.g., AG, A H , AS, and log /(). Ionic composition - - expressed by a relative molar fraction of counter ion 2, X 2, X 1 + ) ( 2 = 1. Local ionic composition - - ionic composition in the micro-environment.
(2)
where a is the counter ion activity in the solution. The main statements of the model are as follows: The exchange sites in an ion exchanger have different amounts of nearest neighbours. In a statistically regular polymeric ion exchanger it can vary from 0 to n depending on the structure of the matrix and the choice of the interaction radius. A property y in a micro-state is assumed to be characterized by a number of nearest neighbours i and a local ionic composition. A molar property Y can be represented as a sum of properties y. Two parameters were chosen to describe the micro-environment: number of counter ions type 2 denoted as j', and that of ions 1, denoted as i - j present in the micro-environment. Then all possible states of the exchange sites with counter ion I can be expressed as I ( i - j , j ) (Scheme 1). It is assumed that the property y is dependent only on the number of counter ions 1 i, number of nearest neighbours
possible states of ion I
0
I(O,O)
1 2 3
I(1,O), I(O,1) I(2,0), I(1,1), I(0,2) I(3,0), I(2,1), I(1,2), I(0,3)
-7-
t
Scheme 1.
I(i-j,
j), j = 0 - i
V.S. Soldatov/ React. Polym. 19 (1993) 105-121
107
and 2 in the micro-environment but not of their position related to the central exchange site. A real ion exchange equilibrium may be regarded as a superposition of ideal "elementary" equilibria related to each of the states:
I1(i - j , j ) + 12 = I2(i - j , j ) + I 1 K(i-j,j)
A'2(i - j,j )a 1 = ~l(i_j,j)a 2
(3) (4)
where K ( i - j , j ) is an equilibrium constant and X(i - j , j ) is a mole fraction of the ions with a ( i - j , j ) micro-environment. In a further consideration definitions y(i - j,j) = log K ( i - j , j ) and Y = l o g I~ will be used. It should be noted, that y and Y can also stand for any molar additive property of the ion exchange system. A property Y can be obtained as a sum of properties y(i - j , j ) for the elementary equilibria proportional to the contents of the exchangeable ions in the relative states. If we denote the probability of the existence of i nearest neighbours of an exchange site in a real ion exchanger as Pi, and a property Y related to a hypothetical system containing only exchange sites with i nearest neighbours as Y/, then the total property Y can be expressed as
tions of the counter ions 1 and 2 within a given combination of i counter ions with j counter ions of type 2. If the probabilities are normalized so that their sum is equal to 1 and -~2 =-~, then i!
p(i-j,j)
= ( i - j ) ! j ! (1 -
2)(i-j)~j "
(7)
and j=i i! Y/= Y'. y(i - j , j ) ( 1 _ , ~ ) ( i - j ~ j j=0 (i-j)tjt..
(8) The general equation "property-ionic composition" for a real system is: i=n
Y=
j=i
i!
y" P,y" (i_j)!j!y(i-j,j) i=0
×(1
j=0
-x)(i-J)xj
(9)
As will be shown later, the number of nearest neighbours is rarely more than 4. Thus, eqn. (8) represents the following set of equations: Y0 =y(0,0)
(10a)
Y1 = y(1,0)(1 - X ) + y(0,1).~
(10b)
y2 =y(2,0)( 1 _ ~ ) 2 + 2 y ( 1 , 1 ) ( 1 - X ) X + y (0,2).,Y 2
(10c)
i=n
Y = Y'~ Yiei
(5)
Y 2 = Y ( 3 , 0 ) ( 1 - X ) 3 + 3y(2,1)(1 - X)2,~
i=0
In its turn, Yi can be represented as a sum of the properties y related to each of the local compositions to their probabilities p: j=i
Y= ~ y(i-j,j)p(i-j,j)
(6)
j=0 The values p ( i - j , j ) are proportional to the molar fraction of each of the counter ions raised to a power equal to the number of the nearest neighbours of a relative type. It is also proportional to a number of permuta-
+ 3y(1,2)(1 _ ~),~2 +
y(0,3)~3 (10d)
Y4 = y (4,0)(1 - X)4 + 4y(3,1)(1 - . ~ ) 3 ~ + 6y(2,2)(1 - ,~)2.,~2 + 4y(1,3)(1 _ y ) ~ 3
+ y(0,4)~4 (10e)
Since all y ( i - j , j ) values are constant, these equations are polynomial presented in a special form. Equation (8) has some features making it especially convenient if X =
V.S. Soldatov /React. Polym. 19 (1993) 105-121
108
10,11, i.e., for consideration of "propertycomposition" dependences. Coefficients of the first and the last terms of eqn. (8) independently of its power, y(i,0) and y(0,i), are equal to the values of y; at .,~ = 0 and .,~ = 1, respectively: y(i,O) = Yv-o
is that if values y(i -j,j) are in linear dependence on the local composition, X(i-j,j), then Y/-- Yi(X) is also a linear function, i.e., if y(i - j,j) = y(i,0)[1 - X( i - j,j)]
+y(O,i)X(i-j,j)
(lla)
y(O,i)= Y2= 1 (llb) If all y(i-j,j) have the same values A,
then
Y,-= Y/(x=0)(l --~) + Y/(2)=1.~
then Yi = A
If all constants y(i -j,j) are changed by a constant value A, the function Y~ is changed by the same value: j=i i! Yi+A = y ' ( i _ j ) v j v [ y ( 1 - j , j ) + A ] "
"
X(1 -X)(i-J)2 j
(13)
This opens a possibility to use Yi in the .,Y interval 10,11, or its normalized values, for theoretical considerations invariantly to the absolute values of Yi. The next important peculiarity of eqn. (8)
Y3
I
\
i
\
o 0
I a
b
C
(15)
The local compositions are expressed by the discrete sets of numbers X ( i - j , j ) according to the number of neighbouring exchange sites and the combinations of the ionic species in the micro-environment. In the case of linear dependence where Y/= Yi(X), values Y/and y(i -j,j) coincide at the total ionic compositions equal to the local compositions. Thus two "end" constants y(i,0) and y(O,i) control the value of the function Yi at pure ionic forms while the intermediate constants control its curvature. This is illustrated in Fig. 1 for the important particular case where /--3.
(12)
j=O
(14)
1
0
d
Fig. 1. Hypothetical dependences II3 -- f(X) for the following set of constants: Curve y(3,0) y(2,1) y(1,2) y(0.3) a 1 1/3 2/3 0 b 1 0 1 0 c 1 0 0 0 d 1 0 -1 0 e 1 -1 1 0
V.S. Soldatov /React. Polym. 19 (1993) 105-121
Treatment of expe~mental data with the model equafions
109 log7 1.0
A computer program was used to fit experimental data to one of the eqns. (10a-e) according to a given mean square error of interpolation or according to a given value of the polynomial power. In this way a maximal possible amount of the nearest neighbours can be found as it is equal to the power of the polynomial. Nevertheless, a good fit of one of eqns. (10a-e) to the experimental data does not mean itself that the ion exchanger does not contain exchange sites with a lower number of neighbouring sites, as can be deduced from the properties of eqns. (10a-e) and (8). Only two parameters of the appropriate equations, y(i,0) and y(O,i), have their direct physical meaning corresponding to a property Y for the elementary equilibria I,(i,O) + I 2 = I2(i,0 ) + 11 11(0,i ) + 12 = I 2 ( 0 , i ) + I 1 If no independent data exist, the rest of the parameters found by fitting, y ( i - j , j ) , are just empirical constants as the model implies the presence in the ion exchange of a set of exchange sites with a different amount of neighbours. It can also be assumed that all ion exchange sites have an equal amount of neighbours. Then parameters y ( i - j , j ) have their meaning as defined in the model for i = constant. The result of the application of eqn. (8) to the experimental data is greatly dependent on their precision, range of ionic compositions and the amount of experimental data. In order to illustrate the importance of these factors the following example is given. In Fig. 2, the experimental data on the dependence of log K = f(X) for the exchange K + - H + on a highly cross-linked (25% DVB) sulphostyrene resin are presented. The data are taken from refs. 3 and 4. In this work a dynamic method for equilibrium study was used. Solutions with a constant ionic strength
0.5
0.2
0.4
\ \ \ \
Fig. 2. Dependence log /~ = fiR), exchange K + - H ÷, ion exchanger KU-2X25, I = 0.100, T = 25.0°C, data are taken from ref. 3. The points are experimental data, the curves are computed from eqn. (10d) with parameters given in Table 1.
and desired ionic compositions were passed through small columns with the resin until equilibrium was reached. Then the resin phase was analyzed for the ionic composition. This method permits accurate results on the ends of the equilibrium curves to be obtained. The results of the application of eqn. (8) to these experimental data under different conditions are summarized in Table 1. The attempts to apply to the data the equations of the first and the second power formally lead to a rather good quantitative agreement between the experimental and the computed data. The mean square errors are practically the same ( E - - 0 . 0 4 ) and correspond to ~ 10% error in the values of /~. Nevertheless, it is quite clear that the experi-
110
V.S. Soldatov / React. Polyrn. 19 (1993) 105-121
TABLE 1 Application of eqn. (8) to experimental data on log g = f(X) for K+-H + exchangeon KU-2X25 Number exper, points 14 14 14 14
.~ interval
0.034-0.963 0.034-0.963 0.034-0.963 0.034-0.963
EquationE inter- No. of power polation curve in Fig. 2 1 0.043 5 2 0.040 3 0.022 1 4 0.022
10 10 10 10
0.082-0.904 0.082-0.904 0.082-0.904 0.082-0.904
1 2 3 4
0.018 0.017 0.014 0.010
7 7 7 7
0.223-0.775 0.223-0.775 0.223-0.775 0.223-0.775
1 2 3 4
0.017 0.006 0.006 0.006
curve does not correspond to the real experimental data over the whole range of the ionic compositions. The example presented clearly shows the importance of high quality experimental data. A meaningful and reliable conclusion on the power and the coefficients of the equation describing the experimental data can be obtained only if they cover a wide range of compositions approaching .,Y = 0 and X = 1 as close as possible and if the number of the experimental points is sufficiently large.
Applications of the model 3 4 2
mental points fall into an S-shaped curve which can be only described by a polynomial of at least the third power. It can be seen from Table 1 that eqn. (8) with i = 3 represents the shape of the experimental curve correctly and the elror of interpolation corresponds to the experimental error as evaluated in ref. 4 ( E = 0.02 corresponds to a 5% error in /(). It can also be seen that application of the equation with i - - 4 does not improve the results of the computation. That means that i = 3 should be taken as the found power of the equation in the application to the given case. If only the data in the range X = 0.0820.904 are taken for treatment, then the fit of the equation to the data becomes better ( E = 0.010-0.018). Nevertheless, it is impossible to conclude what value of i should be chosen. Probably, the straight line (i = 1) would be the best choice in this case. In the case where data in the interval X = 0.223-0.775 are analyzed, then i = 2 should be taken invariantly since a further increase in i does not improve the precision of the interpolation, which is very high in this case. At the same time, the shape of the
The first set of the data is related to the ion exchange system K + - H + on KRSm resins at a constant ionic strength of the equilibria solutions 0.100 and T--25.0°C. The KRSm resins are analytical grade sulphostyrene ion exchangers with pure m-DVB ( > 98%) as a cross-linking agent [5,6]. The apparent equilibrium constants and enthalpies were computed from independent experimental data and treated with the model equation. The enthalpies were found from calorimetric measurements of the heat effects of partial ion exchange [5,6]. The number of experimental points is not large but each of the data corresponds to the integral process whose initial and final states are shown in Fig. 3 at the beginning and the end of the arrows. The points taken for computation relate to the middle of the X interval. As can be seen from Fig. 3, the composition interval covered was 10,11. The data on apparent equilibrium constants in Fig. 4 were obtained by the dynamic method. The attempts to apply eqn. (10) of different power have led to the conclusion that the resins of low cross-linkage ( < 12% DVB) are well described by the equation of the second power. The description of the higher crosslinkage resins requires the equation of the
111
V.S. Soldatou / React. Polym. 19 (1993) 105-121
In terms of the model discussed it means that the properties of the ion exchangers with low cross-linkage can be described assuming that each exchange site in the resins has at most two nearest neighbours. In the case of high cross-linkage a significant fraction of the exchange sites have at least three nearest neighbours. This must be true for the other exchange equilibria with the same resins. The following example confirms this assumption. The apparent equilibrium constants as functions of X for the exchange of alkali
1.4 log~ 1.2 6 1.0
0
I 0
0'.2
014
01.6
0'.8
y
~I 1
Fig. 3. Dependences AH= f(.~), exchange K+-H +, ion exchangers KRSm with different percent of mDVB: 1, 1%; 2, 5%; 3, 8%, 4, 12%; 5, 16%; 6, 25%; I = 0.100, T = 25.0°C. The points are experimental data, the curves are computed from eqns. (10c) and (10d) with the parameters given in Table 2 and marked with asterisk. The data are taken from refs. 5 and 6.
0.8
~
0.4 :jk. 0.2 ~
third power. This holds true both for the enthalpy and log /~. As can be seen from Table 2, application of the third power equation to the data on the low cross-linkage resins does not improve the accuracy of interpolation. On the other hand, starting from 12% DVB the m e a n interpolation error becomes significantly lower for the equation of the third power. Besides, the shape of the curves changes. The S-shaped curves observed in this case may not be described in principle by equations of the second power.
0 -0.2 "
0
0.2
0.4
0.6
0.8
I
Fig. 4. Dependence log /~ = f-(X). See caption to Fig. 3.
I~.S. Soldatov / React. Polym. 19 (1993) 105-121
112 TABLE 2
Application of the model e q u a t i o n to description d e p e n d e n c e s for the K R S m resins. E x c h a n g e K ÷ - H +, I --- 0.100 a, T -- 25.0°C i--2 % m -DVB
Y
,~ interval
E
y(2,0)
y(1,1)
y(0.2)
1 *
log I( AH
0.05-0.95 0.10-0.84
8 4
0.008 0.001
0.152 -0.90
0.086 -0.47
0.105 -0.28
5 *
log /~
5 *
AH
0.05-0.95 0.044-0.927
9 5
0.015 0.089
0.386 - 1.74
0.218 - 1.85
0.192 - 0.66
8 * 8 *
log/~ AH
0.05-0.90 0.08-0.946
8 5
0.016 0.016
0.542 - 2.58
0.333 - 2.02
0.335 - 0.56
12 12
log /~ AH
0.05-0.95 0.066-0.830
9 4
0.045 0.046
0.686 - 3.19
0.618 - 2.29
0.157 - 0.41
16
log /~
0.05-0.95
9
0.047
0.770
0.820
25 25
log /~ AH
0.01-0.95 0.058-0.927
10 5
0.036 0.230
1.228 - 5.05
0.775 - 2.09
- 0.263 - 0.280
Y
.~ interval
Number exper. points
E
y(3,0)
y(2,1)
y(1,2)
1
log/~
0.05-0.95
8
0.008
0.154
0.101
0.099
0.104
5 5
log/~ AH
0.05-0,95 0.044-0.927
9 5
0.014 0.065
0.359 - 1.90
0.333 - 1.18
0.165 - 2.04
0.201 - 0.48
8 8
log / ( AH
0.05-0.90 0.08-0.946
8 5
0.015 0.015
0.500 - 2,60
0.475 - 2.14
0.282 - 1.60
0.350 - 0.54
12 * 12 *
log /~ AH
0.05-0.095 0.066-0.830
9 4
0.042 0.0" 7
0.874 - 3.33
0.390 - 2.05
0.629 - 2.43
0.119 0.010
16 *
log/~
0.05-0.95
9
0.035
1.361
0.160
0.904
-0.13
25 * 25 *
log/~ AH
0.01-0.95 0.058-0.927
10 5
0.013 0.093
1.448 - 5.59
0.518 - 1.12
0.759 - 3.44
-0.38 0.28
1*
Number exper. points
- 0.03
i=3 % m -DVB
y(0.3)
a Ionic strength.
TABLE 3 Application of the m o d e l e q u a t i o n to the description of log /~--f(-,Y) d e p e n d e n c e s for t h e KU-2X25. E x c h a n g e alkali m e t a l - h y d r o g e n ions, I -- 0.100, T = 25.0°C Exchange
,~ interval
Number exper. points
E
y(3,0)
y(2,1)
y(1,2)
y(0,3)
Li ÷ - H + Na ÷ - H ÷ K ÷ -H ÷ Rb+-H + Cs ÷ - H ÷
0.015-0.967 0.014-0.966 0.034-0.980 0.046-0.974 0.057-0.960
13 15 15 15 15
0.026 0.040 0.032 0.030 0.032
0.159 0.653 1.122 1.327 1.453
- 0.383 - 0.120 0.363 0.312 0.446
0.144 0.707 0.660 0.751 0.572
- 0.561 - 0.600 - 0.286 -0.464 - 0.704
V.S. Soldatov /React. Polym. 19 (1993) 105-121
113 In this case the equilibria were studied on the KU-2X25 sulphonic type ion exchanger with commercial DVB as a cross-linker. Comparing the data for K + - H ÷ exchange in Figs. 4 and 5 it can be seen that m - D V B behaves as a more efficient cross-linker than the commercial DVB mixture containing 55% of the DVB isomers. Nevertheless, the shapes of the curves are identical. The data presented show that in ;he case of exchange between inorganic ions the assumption of three nearest neighbours in the micro-environment of the exchange site is sufficient for the densest possible styrene-divinylbenzene matrixes. The assumption of two neighbours is sufficient to describe the
1.5
tog~' Cs 1.0
Na
0.5
O 2.0
log~ 1.8
Li
1.6
1.4
-0.5 1.2
0
0.2
0.4
0.6
0.8
I
Fig. 5. Dependences log /( = f(,~) for the exchange of alkali metal with hydrogen ions, ion exchanger (KU2X25, I = 0.100, T= 25.0°C. The points are experimental data, the curves are computed from eqn. (10d) with the parameters given in Table 3. The data are taken from ref. 3. metal and hydrogen ion on a highly cross-linked ion exchanger are presented in Fig. 5. It is clear that all the log K = f i X ) curves have inflection points and are well described by the equation of the third power (see also Table 3).
1.0
0.8
0
0.2
0.4
0.6
0.8
1
Fig. 6. Dependences log /(= f(,Y) for exchange of orthotetracyclinium with sodium ion, ion exchanger Dowex 50 with different percent of DVB:I, 0.5%; 2, 2%; 3, 6%; 4, 8%. The points are experimental data, the curves are computed from eqn. (10e) with the parameters given in Table 4. The data are taken from ref. 7.
114
V.S. Soldatov / React. Polym. 19 (1993) 105-121
b
i
i
0.4
i
i
2
i
O.B
i
i
0.4
i
i
0.8
i
0.4
i
i
0.8
Fig. 7. Dependences log /( = f(.~) for exchange of chlorotetracyclinium with sodium ion on sulphostyrene ion exchangers: a, SK-6 resin, 6% DVB, layer thickness of the exchanger on the surface of non-sulphonated matrix is 3 /~m; b, the same type resin, telogenated, the layer thickness is 22/~m; c, Dowex 50X5. The points are experimental data, the curves are computed from eqn. (10e) with the parameters given in Table 4. The data are taken from ref. 7.
equilibria on the resins of low and moderate cross-linkage. A larger number of nearest neighbours is possible in the case of large organic ions. This can lead to very complicated depen-
dences Y = f(,,~) and can be used as a severe test for the model. In Figs. 6 and 7 dependences l o g / ( = f(,,~) are given for exchange of ortho- and chlorotetracycline cations with sodium ion on the
TABLE 4 Application of the model equation to the description of log /( = f(.~) dependences. Exchange of ortho- and chiorotetracycline with Na ÷ on the sulphostyrene resins Resin
.~ interval
Orthotetracycline Dowex 50 x 0.5 0.070-0.910 Dowex 50 x 2 0.055-0.800 Dowex 50 x 6 0.030-0.385 Dowex 50 x 8 0.020-0.270 Chlorotetracycline SK-6 3 ~m " 0.12-0.980 SK-6 22/xm 0.12-0.995 Dowex 50 x 5 0.08-0.810
Number of exper. points
E
y(4,0)
y(3,1)
y(2,2)
y(1,3)
10
0.017
0.83
2.22
- 0.57
3.38
0.91
8
0.036
1.39
1.71
0.68
3.91
- 0.61
8
0.012
1.58
1.97
0.63
17.53
6
0.016
1.51
3.69
- 21.43
11
0.017
2.27
2.04
2.47
1.31
3.22
8
0.063
1.91
1.84
3.79
1.16
3.35
7
0.052
1.89
2.54
1.73
4.40
0.26
180
y(0,4)
- 105.1 - 1228
115
v.s. Soldatov / React. Polym. 19 (1993) 105-121
sulphostyrene resins of different structure. The data are taken from ref. 7. It appeared, that the data could only be described with the equation of the fourth power (Table 4).
Estimation of the number of nearest neighbours Formal application of the main model equation to the experimental data has shown that in the real systems with sulphostyrene ion exchangers the n u m b e r of the nearest neighbours should be chosen as 2, 3 or 4 depending on the cross-linkage of the resin and size of exchanging counter ions. W e have a t t e m p t e d to estimate the probabilities of the presence of exchange sites at different distances from the central site for representative fragments of the sulphostyrene networks. Also, the value of the radius of interaction was evaluated from comparison of the probabilities c o m p u t e d with the properties of the real systems. The following method was used. Consider that the s t y r e n e - D V B matrix can be repre-
sented as a combination of t w o types of fragments: linear chains and four-beam stars with a sulphonated p - D V B fragment in the centre (cross-shaped fragments). The fragments contain 8 sulphostyrene groups. The fragments conformation structure was analyzed by a molecular mechanics method with modified parameters of the force field of the C O S M I C system [8] and specially developed service programs (for details see ref. 9). The distances b e t w e e n the centres of the sulphonic groups for different conformations corresponding to the local energy minima were computed. Then the frequencies of the presence of these centres at different distances from each other (excluding the end groups) were found. These frequencies were assumed to be approximately equal to the probabilities. The longest possible distance b e t w e e n the neighbouring sulphonic groups was found to be about 13 nm. It was assumed that the conformation energies fall into a Boltzman distribution and the final probabilities were found as the arithmetic average of the frequencies for the different conformations multiplied by the Boltzman factor. Since
TABLE 5 The probabilities of existence of i neighbours at the distance R from the exchange site in the fragments of the sulphostyrene network.
i
R(nm)
5 Linear chain 0 0.525 1 0.475 2 0.000 3 0.000 4 0.000 5 0.000
6
7
8
9
10
11
12
13
14
0.340 0.660 0.000 0.000 0.000 0.000
0.244 0.663 0.093 0.000 0.000 0.000
0.168 0.660 0.172 0.000 0.000 0.000
0.006 0.656 0.336 0.002 0.001 0.000
0.000 0.488 0.491 0.021 0.000 0.000
0.000 0.417 0.490 0.087 0.006 0.000
0.000 0.157 0.232 0.313 0.296 0.002
0.000 0.000 0.160 0.077 0.489 0.270
0.000 0.000 0.002 0.158 0.162 0.337
Four-beam star 0 0.800 1 0.200 2 0.000 3 0.000 4 0.000 5 0.000
0.833 0.267 0.000 0.000 0.000 0.000
0.200 0.800 0.000 0.000 0.000 0.000
0.002 0.730 0.202 0.067 0.000 0.000
0.002 0.598 0.333 0.067 0.000 0.000
0.000 0.400 0.203 0.198 0.198 0.000
0.000 0.000 0.400 0.200 0.202 0.198
0.000 0.000 0.400 0.002 0.198 0.400
0.000 0.000 0.002 0.398 0.198 0.202
0.000 0.000 0.000 0.000 0.467 0.332
116 the conformations considered were optimized according to the energy minima, their structure corresponds to the expanded matrix, i.e. swollen ion exchanger. The results of the calculations are presented in Table 5. The probabilities of finding a certain number of neighbours (i = 0-5) as a function of the distance from an exchange site ( R = 5 - 1 4 nm) were found. Comparing these data with the model predictions and the experimental data the value of the interactions radius can be evaluated. The experimental data on the heat effects of ion exchange of ions Na +, K +, Cu 2+ and Fe 3+ on H + on a linear polystyrene sulphonates, obtained in our laboratory, show that they are described by the model equation of the second power at most. A similar result was reported in ref. 10. Hence, in this case two neighbours are present in the micro-environment of the exchange ion. As seen from Table 5, it corresponds to a possible radius of interaction not greater than 10 nm. On the other hand, as was shown above, properties of highly cross-linked resins as a function of the ionic composition require the equation of at least the third power for their accurate mathematical description. Comparing this observation with the data of Table 5 for the cross-shaped fragment we conclude that the radius of interaction may not be less than 10 am. Summarizing these considerations, R = 10 nm has been chosen as the radius of interaction.
Discussion
The main equation of the model allows simple systematization of the ion exchange systems according to the degree of their non-ideality, somewhat different from that in ref. 11. The systems with constant /~ can be defined as ideal. The true ideality is possible if
V.S. Soldatov /React. Polym. 19 (1993) 105-121 the exchange sites are situated far away from each other and the number of nearest neighbours is 0. The I10 value is always constant. The systems with i > 0 can demonstrate pseudo-ideal behaviour under conditions expressed by eqn. (12). If the number of nearest neighbours is 1 then Yl = Y I ( X ) is always a linear function, which corresponds to regular behaviour. It should be noted, that this definition is based on the similarity of the considered system with regular mixtures, not on the identity of their structures and interaction. Pseudo-regular behaviour is also possible for systems with i > 1 under conditions expressed by eqn. (14). If the number of neighbours is more than 1, the systems behaviour can be characterized as irregular with a different degree of irregularity according to the number of the neighbours, and, hence the power of the main equation (e.g., square-irregular, cubicirregular, etc.). The simplest irregular system is described by the second power equation suggested in refs. 12 and 13 for a liquid ion exchange extractant, and rigourously derived for resinous ion exchangers in ref. 11. Later it was often used for treatment of experimental data (e.g., refs. 14-17). The limitation in the application of this particular case of the main equation "property-composition" are quite obvious from the above consideration. Applicability of the second power equation to the description of selectivity of ion exchange can be due to either the negligibly small contribution of the higher power terms, or to a special form of the dependence of the y(i j,j) parameters on the local composition in an equation of the higher power, i.e., the system can behave like a pseudo-square. In the case of dependences Y = Y(,Y) for the ion exchangers of low cross-linkage shown in Figs. 3 and 4 the absence of the higher power terms corresponds to a very small probability of presence (i.e., low contents) of
V.S. Soldatov /React. Polym. 19 (1993) 105-121
117
the exchange sites with i > 2, as seen from Table 5. Nevertheless, accurate experimental investigation of ion exchange selectivity over a wider composition range can evidence the presence of higher order terms. In Fig. 10 the data from ref. 18 on log/< = f(.Y) dependence for Li+-H ÷ exchange on an ion exchanger with 4% DVB are presented. In order to notice such effects the .~ range must be broad enough (in this case data in the range X = 0.045-0.92 were taken for treatment). The data presented in Figs. 3 and 4 are not sufficient to notice this effect for resins of low cross-linkage. The estimation of the probability of the presence of more than 2 exchange sites in the micro-environment given in Table 5 proves that it is several times higher in the
1.6
IZI 2 >~
/K
I
/ / /
0.8
/
/ 0.4
/ / Li
"l -0.4
~'
"@. . . . 0.6
0.8
0--1.0
z
~\ 1.2
11.4
"~, 1.6 R,nm
Fig. 9. Constants of eqn. (10d) y(i - j , j ) = log K(i - j , j ) for the alkali m e t a l - h y d r o g e n ion exchange on the KU-2X25 resin as a function of the ionic radii: 1, y(3,0); 2, y(0,3).
1.6
'~1.2
7 v
>,
0.8
0.4
-0.4
-0.8 0
1/3
2/3
1 X(i-j,j)
Fig. 8. Constants of eqn. (10d) y(i - i , i ) = log K(i - j , j ) for the alkali m e t a l - h y d r o g e n ion exchange on the KU-2X25 resin as a function of local composition: 1, Cs+; 2, Rb+; 3, K+; 4, Na+; 5, Li +.
vicinity of the cross-links than in the linear chain at R = 10-11 nm. This explains the influence of cross-linkage on the shape of the selectivity curves. The most broad explanation of selectivity decrease with X is based on the idea of irregularity of the resin exchange sites, suggested for styrene-DVB ion exchangers in ref. 19. Later the increase in selectivity was reported and explained by "cooperative" interaction [20]. No ways to express or interpret these effects in quantitative terms were suggested. The new model implies irregularity of the exchange sites as a natural property of the resinous ion exchangers caused by the different possible number of nearest neighbours and different combinations of the ionic species in the micro-environment. The model parameters, P, i and y(i -j,j) give a way to express this irregularity, as well as the cooperative effect, in quantitative terms. The absolute values of the selectivity parameters log K(i-j,j) are functions of the energy of interionic interactions and depend
118
011
V.S. Soldatov /React. Polym. 19 (1993) 105-121
log~
0
-0. I
i
0
~
0.2
i
l
0.4
i
i
0.6
i
i
0.8
i
1
Fig. 10. D e p e n d e n c e log /~ = f(.~) for the L i + - H + exchange on Dowex-50X4, I = 0.100, T = 25.0°C. The points are experimental data taken from ref. 18. The curve is computed from eqn. (10d) for the points in the range X - - 0.045-0.920 with the following coefficients: y(3.0) = 0.025; y(2,1) = - 0.140; y(1,2) = - 0.043; y(0,3) = -0.139.
on the total volume of the resin, ion hydration and other factors, not expressed in the model in explicit form. They influence y(i-j,j) values which can only be found at present as empirical parameters. Nevertheless, their physical meaning is well defined and they can be used for interpretation of the Y = Y( X ) dependences. Dependences of the y(i - j , j ) for i = 3 of the I + - H ÷ exchanges on the local compositions and radii of the cations are complicated and have no explanation at present (Figs. 8 and 9): In spite of that some regularities can be summarized. The elementary equilibrium constant of the same exchange is strongly dependent on the composition of the micro-environment. The largest selectivity toward a preferably sorbed ion (H ÷ at . ~ = 1) occurs if the exchanging ion is surrounded by ions of the
other type. This corresponds to the largest free energy and enthalpy of the process. Substitution of one of the originally present in the micro-environment ions with those entering leads to significantly lower log /~ and A H values. This results in a remarkable symmetry of the y ( i - j , j ) = f ( X ( i - j , j ) ) and Y = f(X) dependences. A simple dependence of log K (3,0) on the ionic radii exists. An increase in the free energy of the ion exchange correlates well with increasing electrostatic interaction between the sulphonic group and the counter ion. The counter ion entering always has the same surrounding in its micro-environment: three hydrogen ions. On the other hand, dependence of the other "end" constant, log K(0,3), on the ionic radius is very complicated. A possible reason for that is that in this case the ion entering always has different surroundings. It seems, that in this case the effect of radius should be considered separately for the "structure forming" (Li + and Na +) and "structure breaking" (K +, Rb + and Cs +) ions [21]. A slight decrease of log K(0,3) is observed in the first case while this parameter drops sharply with the ionic radius increase for the second group of ions. Intermediate dependences occur for the other parameters. In this systems the sizes of the exchangeable ions are much smaller than the radius of interaction and all exchange sites are easily accessible for the counter ions. The situation is quite different for the case of exchange of large organic ions. The data in Figs. 6 and 7 shows the absence of a sharp decrease in selectivity at low loadings of the resins with orthotetracycline (OTC) or chlorotetracycline (CITC) ions. Two effects are remarkable in these cases: a significant increase in the log g values and their sharp drop at high resin loadings with organic ions. These results can be interpreted in the model terms in the following way. The irregularity of the exchange sites, as a universal property, must always favour
V.S. Soldatov / React. Polym. 19 (1993) 105-121
decrease of selectivity of a preferably sorbed ion with X increase. On the other hand, increase in the selectivity of the considered ions can be expected because of their intensive interaction with each other. These ions contain several functional groups able to form hydrogen bonds and have sizes exceeding the interaction radius value estimated for the inorganic ions: CH 3 0 H O H
N(CH3) 2
/
/ : (,....
y
y
OH
0
yo'.y....... OH
0
/:
................. 14.67 ,~
l
/
Orthotetracycline
/n
C1 CH 3 OH H
;
/
oH .N(CH3) 2
7 Y yo.y OH
O
OH
.......
O
....... 14.32 ,~
/
Chlorotetracycline The large size of the counter ions causes two effects. The number of the nearest neighbours of the counter ion is larger than that of the inorganic ions and that can cause more complicated dependences Y = Y ( X ) . As seen from Table 5, the probability of having more than three neighbours in a sphere with radius equal to the largest dimension of this ion is practically unity. We suppose that i can be assumed equal to 4 as a sum of three neighbouring exchange sites,
119
estimated without taking into account the sizes of the counter ions, plus one of the counter ions, situated at a distance between the exchange site radius of interaction (10 nm) and the largest counter ion dimension (14.67 nm). The other obvious result of a large counter ion size is a restriction in the accessibility of the exchange sites by the counter ions due to blocking by the neighbouring bulky organic ions. It can also cause inaccessibility of certain parts of the resin networks both on the molecular and macroscopic level, known as "sieve effect". This must result in a very low equilibrium constant of the elementary equilibria for which this effect is important. The data for the exchange OTC+-Na ÷ on resins of different cross-linkage can be interpreted by an overlap of the mentioned effects. A possible rationalisation of the observed phenomena can be done by considering the values of the elementary equilibria constants y(i - j , j ) , given in Table 4. The appearance of an OTC ÷ ion in the micro-environment of the exchange site causes an increase in the equilibrium constant of the second elementary equilibrium due to its additional interaction between the incoming ion and the presaturant ion in spite of the tendency of the selectivity drop due to the irregularity effect. The presence of two organic ions in the vicinity of the exchange site results in a marked selectivity drop probably due to the blocking phenomenon. In the case when there are three OTC ÷ ions in the sphere of interaction, the equilibrium constant becomes high again. It is supposed that the high interaction energy in this case is caused by an intensive aggregation of the incoming ion with the three already present in the micro-environment OTC ÷ ions. The low values of the last constant, progressively decreasing with the cross-linkage of the resins, certainly reflects the sieve effects. Regularities of the CITC+-Na ÷ exchange on Dowex 50X5 resin are similar to that of
V.S. Soldatov / React. Polym. 19 (1993) 105-121
120
OTC+-Na ÷. Absence of the sieve effect for the thin layer resins SK-6 evidences that this effect is caused by the inaccessibility of the central parts of the Dowex 50 resins by the bulky organic ions. In the telogenated SK-6 resin the accessibility of the exchange sites is high enough to avoid the sieve effect in a layer 22/zm thick. A more loose structure of this resin makes the irregularity effect less important which gives a sharper rise of the C1TC ÷ selectivity at low X values compared to the non-telogenated ion exchanger.
Conclusions The suggested mathematical model describes a real ion exchanger as a set of exchange sites with a different number of the nearest neighbours. The main equation of the model describes dependences "propertycomposition" with the help of the following parameters: (a) number of nearest neighbours of an exchange site, (b) the probability of its existence, and (c) equilibrium constants of the elementary equilibria related to each of the micro-states of the exchangeable ions. The ways of estimating these parameters have been considered. The model was applied for the interpretation of the ion exchange equilibria of univalent ions (alkali metal-hydrogen ions, ortho- and chloro-tetracycliniumsodium ions) on sulphostyrene resins of different cross-linkage.
Acknowledgements All the work concerning computation of the number of the exchange site nearest neighbours in the fragments of the resin networks was done by a group of my colleagues from the Institute of Physical Organic Chemistry of the Belarus Academy of Sciences: A.L. Pushkarchuk, V.I. Gogolinsky and V.M. Zelenkovsky, to whom I express my deep
gratitude and admiration for their high professionalism. This work was supported by the foundation for fundamental research of the Republic of Belarus, Grant No. F 216-181.
List of symbols a E G H I i j
K
P,p R X Y y
activity mean square error Gibbs energy enthalpy exchangeable ion total number of neighbouring exchange sites number of neighbouring sites whose change is compensated by counter ion 2 apparent equilibrium constant the probabilities interaction radius relative mole fraction of exchangeable ion an additive property of the system property Y related to a micro-state
References 1 V.S. Soldatov, Mathematical modelling of simple ion exchange equilibria, Dokl. Acad. Sci., BSSR, 34 (1990) 528. 2 V.S. Soldatov, Mathematical modelling of dependencies "ionic composition-property" in ion exchange systems, Dokl. Acad. Sci., USSR, 314 (1990) 664. 3 V.S. Soldatov, R.V. Martzinkevich and A.I. Pokrovskaya, Peculiarities of exchange on ion exchangers of high cross-linkage, J. Phys. Chem. (USSR), 43 (1969) 2889. 4 V.S. Soldatov and R.M. Sukhover, Estimation of errors of experimental and computed thermodynamic quantities in ion exchange systems, J. Phys. Chem. (USSR), 42 (1968) 1798. 5 V.S. Soldatov, R.V. Martzinkevich, A.I. Pokrovskaya, Z.I. Kuvaeva, S.B. Makarova, T.A. Aptova and E.V. Egorov, Comparative characteristics of sulfo-
V.S. Soldatov / React. Polym. 19 (1993) 105-121 styrene ion exchangers with different isomers of divinylbenzene, J. Phys. Chem. (USSR), 50 (1976) 480. 6 V.S. Soldatov, Z.I. Kuvaeva and R.V. Martzinkevich, Thermodynamics of ion exchange on sulfostyrene type ion exchangers linked with para- and meta-divinylbenzene, in M. Streat (Ed.), The Theory and Practice of Ion Exchange, Proceedings of an Internal Conference held at Churchill College, University of Cambridge, July, Soc. Chem. Indus. UK, 1976, p. 20. 7 G.V. Samsonov and A.T. Melenevskij, Sorption and
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