Ten years experience of a simple three-parameter model to fit ion exchange data

Reactive Polymers, 11 (1989) 199-219

199

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

STATE-OF-THE-ART REPORT

TEN YEARS EXPERIENCE OF A SIMPLE THREE-PARAMETER M O D E L TO FIT ION E X C H A N G E DATA ERIK H O G F E L D T

Department of Inorganic Chemistry, The Royal Institute of Technology, S-I O0 44 Stockholm (Sweden) (Received February 6, 1989; accepted March 10, 1989.

The application of the Hrg[eldt three-parameter model to ion exchange data is reviewed. The few cases where it does not work are also mentioned. The model is simple and straightforward to use, and it compares favorably with more elaborate methods for treating weak-acid and complex-forming resins.

INTRODUCTION In 1972 I had the good fortune to meet Professor Vladimir S. Soldatov from the Belorussian Academy of Sciences, in Minsk, USSR. We started to co-operate in the field of ion exchange and have continued to do so ever since. During this co-operation the three-parameter model described below appeared, giving us a very simple way to summarize various data in ion exchange.

by B only. When surrounded by both A and B the property has the value Ym" According to Guggenheim's zeroth approximation [1] the number of A - A pairs is proportional to x 2 (x A = mole fraction of A in the mixture) and the number of B-B pairs is proportional to x 2 (x B = mole fraction of B in the mixture). The number of A - B ( = B-A) pairs is proportional to 2XAX w Thus: Y = y A x2 + y . x 2 + 2YmXAX a

In the literature the property Y is plotted against x A or xa, not against x 2 or (x2). It is then convenient to introduce the equation

THE M O D E L

Y = yAXA + yaXB + BXAX B The model is concerned with the three kinds of pairs appearing in a mixture of components A and B, i.e., A - A , B-B and A-B. If component A is surrounded by A only, any molar property Y has the same property as in pure A. This quantity is denoted YA" Similarly for component B, YB refers to B surrounded 0923-1137/89/$03.50

(1)

(2)

where B is an empirical constant. By fitting experimental data to eqn. (2) by least-squares methods, the parameters YA and Ya and the empirical constant B are obtained. The third parameter Ym is now obtained from Ym=½[Ya + Y , + B ]

© 1989 Elsevier Science Publishers B.V.

(3)

200 Equation (3) is obtained by setting eqn. (1) identical to eqn. (2) and using the fact that XA+XB=J.

by many workers, who have, however, disregarded the water and ended up with essentially the pseudo-binary approach, but with different standard and reference states.

S O M E STATISTICAL CRITERIA

The ion exchange reaction

In the application of the model to experimental data, the goodness of fit will be illustrated by the following statistical criteria. The residual squares sum, U, is defined by

For simplicity, let us consider cation exchange between two monovalent ions:

n

n

U = E ( Y e x p - Ycalc) 2 = E r i 2 1 1

(4)

where n = number of experimental points, r i = i th residual. The Hamilton R factor in percent, R(%), is defined by

M~- + M , X ~ M2X + M~-

For heterovalent exchange the reaction will be formulated for each system discussed in the following. The equilibrium quotient, K', of reaction (8) is defined by '

x2[Mf]Y' -

R (%) = 100i•ri2/ZYe2xp

(5)

The standard deviation, s(Y), is given by

s(Y)=¢U/(n-k)

(6)

where k = number of parameters estimated. Sometimes the standard deviation s '( Y ) is given:

s'(Y)= ¢U/(n-1)

(8)

(9)

yl[M~-] y 2

where xl,x z are the mole fractions of the two ionic forms in the exchanger, and Yl,Y2 are the activity coefficients of the two ions in the aqueous phase. In most studies the ionic strength, I, in the aqueous phase is kept constant. Then it may be assumed that the ratio Yl/Yz is practically constant and can be included in r ' , i.e.:

(7)

x = x ' ( y 2 / y ~ ) - x2 [M~] X, IM~-]

(10)

A P P L I C A T I O N T O ION EXCHANGE

Free energy The activity scale The model outlined above refers to binary mixtures. It is thus natural to choose the pseudo-binary approach first outlined by Sill4n et al. [2-4] and somewhat later by Argersinger et al. [5]. In this approach, an ion exchange system is assumed to contain a binary mixture of the two ionic forms; the third component, water, is neglected. It was shown by HSgfeldt [6,7] that this is an acceptable approximation. The three-component system was treated by Thomas [8]. This approach has been used

The logarithm of ~ is related to the free energy of reaction (8) by

AG ° = - R T In x = - 2 . 3 0 3 R T log x in the following log x or In r is used for fitting ion exchange data to the model. From eqn. (2): log x = log x(1)Y + log x(O)(1 - Y) +B~(1 - Y )

(Y=x2)

(11)

The parameter log K(1) is the limiting value of log ~ when X = 1. Similarly, log ~(0) is the limiting value of log K when X = 0. These

201

parameters are not directly available experimentally. They can, however, be estimated by using tracer techniques; see Ref. [9] for illustration. Equation (11) is fitted by least-squares methods, giving the parameters log K(0) and l__ogK(1) together with the empirical constant B. Then the third parameter, log Xm, is obtained from eqn. (3). The integral free energy of reaction (8), given as a thermodynamic equilibrium constant K, can be obtained from: log K = f011og t¢(Y) dY

According to the Gibbs-Duhem equation, Y d In f2 + (1 - Y ) d In fl = 0 (p, T constant)

From eqns. (14)-(16), using the pure components as standard and reference states, integration gives: log f, = ibm2 + 2cy3

(12) Observe that, for a straight line, B = 0, eqns. (3) and (12) give log /~m = log K = ½[log r(0) + log K(1)]

(17a,b)

log f 2 = ½(b+ 2c)(1 - y ) 2 _ 2c(1 _ y)3 The constants b and c are are related to the parameters of the model by b = 2[log x m - l o g ~(0)] c = log x(0) + log to(l) - 2 log Km

= ½[log r(O) + log r(1) + log Km]

(16)

(18a,b)

Observe that, for c = 0, i.e., log x = a + b~, the activity coefficients reduce to log

fl

=

½ bY 2

(19a,b)

log f2 = ½b(a - y)2

The expression In K = f011n K(Y) dY

(13)

was first given by Argersinger et al. [5]. It is often wrongly attributed to Thomas. An equivalent but less convenient expression for estimating K was given by Sill4n et al. [2]. By introducing eqn. (11) into eqn_. (13), integrating termwise and substituting B from eqn. (3), eqn. (12) is obtained.

Activity coefficients With knowledge of the parameters of the model, it is possible to obtain expressions for the activity coefficients of the two ionic forms in the exchanger. For simplicity let us consider uni-univalent exchange only. Equation (11) can be rewritten as log K = a + bY + cY2

(14)

From eqn. (10) and the definition of K

K= x(f2/f, )

(15)

Water uptake The number of water molecules per equivalent of ion exchanger, W, is given by W = n.2o/So

(20)

For solid resins n~20 is the number of millimoles of water in the exchanger. For liquid ion exchangers, n H 2 0 ---~ n tot - - n solv, where n so~v refers to the water extracted by the organic solvent containing the liquid ion exchanger. The capacity of the sample, So, is expressed in milliequivalents. By measuring the water content of the exchanger as a function of the composition, eqn. (2) can be used to fit the experimental data. Observe that the parameters w(0) and w(1) are experimentally available, leaving only one unknown constant, B, to be determined experimentally. Then w m is obtained from eqn (3). The water extracted during the ion exchange process thus offers a rather severe test of the model, with only one unknown constant to be determined. !

202

EXAMPLES +0.~

In the following section, some of the results obtained during the last ten years will be used to illustrate the use of the model. Cases where it does not work will also be shown.

K+H

+

I

I

Iogo~ |

+0.

--0."

Liquid cation exchangers

-0."

The model was first applied to the liquid cation exchanger dinonylnaphthalenesulfonic acid, HD, dissolved in various solvents. H D has the general structure (I). For HD, R =

-0.!

I

0

I

XK

Fig. 1. log ~ plotted against xK for system K+-H ÷ on HD in heptane. Data from Ref. [10]. o, 1° C; O, 11° C; n, 25 ° C; II, 80 ° C. Curves have been computed from model with parameters given in Table 1.

SO~ H÷ (I)

C9H19. For didodecylnaphthalenesulfonic acid, H D D N S , R = C12H25 . The commercial products of H D and H D D N S are isomer mixtures. In all experiments with H D and H D D N S , the concentration is 0.100 M in the organic phase.

log x(0) = - 1 . 0 3 7 + 3 4 5 / T [s(log x(O)) = 0.012]" log x(1) = - 0 . 4 7 7 _ 0.029

(22a-c)

log x m = - 0 . 1 1 9 _+ 0.049

Free energy

The system K +-H + on HD in heptane at different temperatures. HSgfeldt et al. [10] studied the reaction K + + HD ~ KD + H +

By applying the van't Hoff equation to the parameters, the following expressions were obtained by linear regression

(21)

However, a satisfactory fit was not obtained for 1 ° C. For that reason linear regression on the variables 1 / T and log T was applied to log x(1) and log gin, giving: log K(1) = - 1 3 . 5 1 6 + 6 1 8 / T + 4.428 log T [s(log x ( 1 ) ) = 0 . 0 3 5 ]

(23a)

log x m = - 6 3 . 8 8 3 + 3 0 0 0 / T + 21.686 log T on H D dissolved in heptane at temperatures of 1°C, l l ° C , 2 5 ° C and 80°C. In Fig. 1, logx is plotted against ~K at the four temperatures. The curves have been computed from the model using the parameters given in Table 1 and obtained by least-squares fitting to eqn. (11). The fit, as judged from eqns. (4)-(6), is acceptable.

[s(log Xm) = 0.053]

(23b)

In Table 2 parameters obtained from eqns. (22a) and (23a,b) are given, together with the fit obtained. By comparison with the data in Table 1, it is seen that an acceptable fit is obtained by reducing the n u m b e r of parameters from 12 to 8.

203

0 r.~

l=l

0 0 0 0 0 0 ~ 0 0 0 ~ 0 0 0 ~ 0 0 0 0 0

I I I I I I I

7

X X X X X X X

X

?7777~7777777~?~???? X X X X X X X X X X X X X X X X X X X X X

"0

0

IIIl

,2 °

I I I I

"r. el el)

~

~

~

~

~

~

.

l

I=

0

0

0

0

0

0

~

fill

.-.Z

"' IlO

.2o

0

0

I

l

0

0

0

0

~

I

2

=1 0..,

,20

~ N N N N

o

I=I

o o ~ 5 o o o

I

~ ~U I +1 +1

!=I

c; ~

U

0 0 0 0 0

C, 0 0 0 0 0 0 0 0 0 . 1

' ~ ,'-~ ,'~ C~ 0 0 0

+1

0

7, =, +1 +1 +1 +1

++++

~ ~

O0

z

~1 ~ I ~ I

~====LLL~m==

z

~

~

~

~

204 TABLE 2 P a r a m e t e r s f r o m eqns. (22a) a n d (23a, b) given t o g e t h e r w i t h the statistical q u a n t i t i e s in eqns. ( 4 ) - ( 6 ) ; D a t a f r o m Ref. [101 T ( ° C)

log x(0), eqn. (22a)

log K(1), eqn. (23a)

log/¢m, eqn. (23b)

log K , eqn. (12)

U

R(%)

s ( l o g K)

1 11

0.221 0.177

-0.466 -0.477

-0.070 -0.119

- 0.105 -0.140

8 . 5 0 2 x 10 - 3 1.216 x 10 - 2

16.245 16.336

0.035 0.0417

25 80

0.120 -0.060

-0.487 -0.483

-0.162 -0.132

-0.176 -0.225

8.623 × 10 - 3 6 . 3 9 0 × 10 - 4

16.193 3.878

0.0351 0.0103

TABLE 3 C o m p a r i s o n b e t w e e n e x p e r i m e n t a l a n d c o m p u t e d A H ° values for r e a c t i o n (21) at t e m p e r a t u r e s o f 1 ° C, 11 ° C, 25 ° C a n d 80 ° C u s i n g the p a r a m e t e r s given in T a b l e 4; d a t a f r o m Ref. [10]

"~K

AH ° (cal/eq) 1°C, exp

1°C, calc

11°C, exp

ll°C, calc

25°C, exp

25°C, calc

80°C, exp

80°C, calc

0.1

- 1677

- 1689

- 1426

- 1452

- 1197

- 1225

- 969

- 969

0.2 0.3 0.4

- 1563 - 1357 - 1238

- 1532 - 1383 - 1240

- 1366 - 1174 -992

- 1311 - 1175 - 1044

- 1142 - 951 - 800

- 1087 -956 - 831

- 836 -699 - 544

- 829 -697 - 572

0.5 0.6

- 1106 - 1010

- 1104 -976

-932 - 804

-919 -799

- 704 - 626

-714 -604

-454 - 361

-456 - 346

0.7 0.8 0.9

- 832 -717 - 649

- 854 -739 -632

-704 - 557 -475

-684 - 575 -471

- 507 - 388 - 320

- 500 -403 - 314

- 247 - 160 - 55

- 245 - 151 - 65

The differential enthalpy, AH °, of reaction (21) was evaluated from AH ° = - R [ 0 In

x/~(1/T)~]

(24)

These data were fitted to the model by using eqn. (2) and least-squares methods. In Table 3, experimental and computed AH ° values are compared using the parameters given in Table 4. The fit as given by s ( A H °) is around

30 cal/eq, which can be regarded as acceptable. In Table 4 the quantity AH ° computed from eqn. (12) is the integral enthalpy of reaction (21). Applications of the model to HD as well as H D D N S can be found in Refs. [11]-[17]. Muraviev and HiSgfeldt [18] showed that the model could also be applied to HD sorbed on polystyrene beads.

TABLE 4 P a r a m e t e r s o b t a i n e d b y fitting the A H ° d a t a of r e a c t i o n (21) to the m o d e l ; d a t a f r o m Ref. [10] T

A H ° (0)

( o C)

(cal/eq)

(cal/eq)

(cal/eq)

1 11 25 80

-

- 531 - 373 -231 + 13

- 1017 - 851 -627 - 359

1853 1599 1370 1116

A H ° (1)

AHm°

AH °

U

R(%)

s(AH °)

- 1134 - 941 -743 -487

4202 7340 5689 1248

1.833 2.887 3.165 2.092

26.5 35.0 30.8 14.4

205

W ~

In Fig. 2, W is plotted against X B a for the system Ba2+-H + on H D in heptane at 298 K, as studied by Soldatov [19]. The curve has been computed from the model using the parameters in Table 5. The result is typical; a curve with a more or less pronounced minim u m is obtained. In some cases the data can be fitted to a straight line. Examples are the systems C a 2 + - H ÷ on H D in 2-ethylhexanol [12] and A g + - H ÷ on H D D N S in heptane

Ba2+-H ÷ o

\o

8

"--o o

6

~o~°6"

I

0

I

I

0.2

I

0.4

I

I

~ 0.6

I

o o

I

[171.

I

0.8

Fig. 2. W plotted against equivalent fraction of Ba 2+, ~aa, for system Ba2+-H ÷ on HD in heptane. T = 298 K. Data from Ref. [19]. Curve was computed from eqn. (2) using parameters given in Table 5.

When applying the model to water extraction, it is easy to determine w(0) and w(1) for the pure ionic forms. Only one parameter then remains to be determined, w m" This has been done as follows. F r o m eqn. (2): = [W~xp - w(0)(1 - ~ ) -

Water uptake The water extraction has been studied together with the ion exchange equilibria. When necessary, the water extraction by the solvent has been corrected for by W = (nto , - nso,v)/s 0

(25)

where n tot = total n u m b e r of millimoles of water extracted, nsolv = n u m b e r of millimoles of water extracted by the solvent, and s o = n u m b e r of millimoles of H D or H D D N S .

w ( 1 ) ~ ] / ~ ( 1 - ~) (26)

Then the average B value, Bav, is taken and W calculated from Wcalc = w(1)~ q- w(0)(1 - ~) + BayS(1 - ~)

(27) The W values computed from eqn. (27) have been compared with those obtained by including the experimental limiting values in

TABLE 5 Parameters obtained by least-squares fitting to the model for the water uptake during ion exchange; all data refer to 298 K and ! = 0.100 M System

Ion exchanger

w(0)

w(1)

wm

U

R(%)

s(W)

Ref.

BaZ+-H + Me3NH ~- - H + Me3NH ~ - H + Me4N+-H + Me4N+-H + K+-H + K+-H ÷ K+-H ÷ K+-H + Cd2+-H ÷ CdZ+-H + Me4N + - H + Me4N÷-H +

HD HD HD HD HD HD HD HDDNS HDDNS HDDNS HDDNS KRS-8 KRS-8

8.865 10.173 10.30 10.197 10.30 9.606 9.80 7.416 7.50 6,470 6.45 12.885 12.76

6.097 3.964 3.74 10.679 10.57 7.472 7.60 6.234 6.20 6.624 6.67 7.119 7.17

5.376 3.857 4.160 3.628 3.616 5.013 4.620 4.276 4,155 5.262 5,237 7,291 7,284

0.1941 0.5878 0.8177 7.143×10 -2 0.1135 0.1933 0.3048 0.2390 0.2604 3.714×10 -2 4.382× 10 -z 6.990 × 10 -2 0.1012

1.805 3.584 4.923 0.9436 1.394 1.595 2.242 2.420 2.882 0.8720 1.044 0.8742 1.202

0.1328 0.2556 0.3015 0.0891 0.1123 0.1390 0.1746 0.1728 0.1804 0.0609 0.0662 0.0935 0.1125

[19] [20] [20] [20] [20] [16] [16] [16] [16] [17] [17] [20] [20]

206 10 K+... H + W \ i

\o

10

8 o

4 6 I 0

L

I

I

It

Fig. 3. W plotted against ~s~t for systems CH3NH ~-H + and (CH3)4N+-H + on HD in heptane and (CH3)4N+-H ÷ on KRS-8 at 298 K. Data from Ref. [201. o, CH3NH~--H + on HD in beptane; e, (CH3)4N÷-H + on HD in heptane; I-7,(CH3)4N÷-H ÷ on KRS-8. Curves have been computed from model with second set of parameters given in Table 5. the least-squares calculations. This has been done for a n u m b e r of systems. In Fig. 3, W is p l o t t e d a g a i n s t ~sa~t for the s y s t e m s C H 3 N H ~ - - H + and ( C H 3 ) a N + - H + studied by Mikulich [20]. In Table 5 the parameters obtained by the two methods are given, together with the fit obtained as given by eqns. (4)-(6). The curves in Fig. 3 have been computed from the values obtained with eqn. (27). In Fig. 4, W is plotted against ~K for the system K + - H + on H D (©) and H D D N S (O) in heptane at 298 K using the data in Ref. [16]. In Table 5 the parameters and the fit obtained by the two methods are given. The curves in Fig. 4 have been computed from eqn. (27). F r o m Figs. 3 and 4 it is seen that a fairly good fit is obtained. This is supported by the data in Table 5, where the statistical data do not differ much. Moreover, the estimates of w(0) and w(1) obtained by varying all data agree, within the experiment uncertainty, with the experimental values. Observe that w(0) differs from one sample to another, probably due to differences in isomer composition.

I

0

I

I

I

Re(

Fig. 4. W plotted against xK for system K+-H + at 298 K. Data from Ref. [16]. o, HD in heptane; O, HDDNS in heptane. Curves have been computed from model with second set of parameters given in Table 5.

To illustrate further the agreement between experiment and model, Table 6 compares experimental and c o m p u t e d W values for the system C d 2 + - H ÷ on H D D N S in heptane at 298 K as studied by Kuvaeva et al. [17]. F r o m these examples it can be concluded that the water extraction data strongly support the model. However, there is one case so

TABLE 6 Comparison between experimental and computed W values for the system Cd2+-H + on HDDNS in heptane at 298 K; data from Ref. [17] Xca

W, exp

IV, eqn. (2)

W, eqn. (27)

0.094 0.130 0.240 0.291 0.326 0.407 0.470 0.631 0.689 0.721 0.841

6.29 6.19 6.05 6.05 5.85 5.91 5.97 5.97 6.02 6.08 6.14

6.266 6.199 6.038 5.985 5.956 5.912 5.902 5.962 6.025 6.064 6.256

6.245 6.179 6.020 5.968 5.940 5.901 5.984 5.972 6.034 6.076 6.281

207 far identified for which the model does not work. That is the system ( C 2 H s ) 4 N + - H + on H D in heptane, studied by Mikulich [20], where the pure salt extracts nearly 26 water molecules per equivalent. This suggests a structural change from micellar form, possibly to microemulsion. When applying the model to the experimental data, w m becomes negative. A n acceptable fit is obtained by a third-degree polynomial: W = 26.037 + 10.33(1 - Y)

Iog~ Na ÷ - H +

0.4

-<

0.3

0.2

0.1

o

\

-0.1

I

t

I

I

0

- [ 2 9 . 9 7 + 29.0871Y(1 - Y) In terms of the model this could be explained by triplet interactions. However, the free energies are well described by the model, which makes such an explanation unlikely. Applications of the model to water extraction data on H D and H D D N S are given in several of Refs. [11]-[17]. The salts of H D and H D D N S form micelles. The average aggregation number (~) varies with composition and can be fitted to the model [16,17,21]. If each ionic form is monodisperse, the parameters obtained have an obvious physical meaning. However, lowangle scattering data obtained in Minsk indicate polydispersity. The parameters are thus some kind of average sizes. Solid cation exchange resins

Fig. 5. log x plotted against -~Na for system Na+-H + on Dowex 50 of different degrees of crosslinking at 298 K. Data from Ref. [21]. e, 8% DVB; o, 16% DVB. Curves have been computed from model with parameters given in Table 1. estimated from eqn. (12) is given. As judged by the statistical criteria, the fit is satisfactory. Sometimes it is possible to correlate parameters of the model with the degree of crosslinking, as shown in Ref. [23] for the system C s + - N a ÷ studied by Soldano et al. [24] and in Ref. [14] for the system C s + - H ÷ studied by Bonner [25]. The system Ca2+-Na ÷ on bleached kraft pulp. Cellulose may be included among the solid resins. Ampulski [26] studied the reaction Ca 2+ + 2 N a R ~ CaR 2 + 2Na +

Free energy The system Na + - H ÷ on Dowex 50 of various degrees of crosslinking. Bonner and Rhett [22] studied the reaction Na + + H R ~ N a R + H +

on bleached northern softwood kraft pulp. The equilibrium quotient of reaction (29) was given as

(28)

on Dowex 50 of crosslinking 8% or 16% DVB. The temperature was 298 ___1 K and the ionic strength was ---0.1 M. The data used were read from Fig. 1 in their paper. In Fig. 5, estimated data are compared with those from the model by using the parameters given in Table 1. Also, log K

(29)

r=

[CAR2] [Na+] 2 [NaR]2[Ca2+]

(30)

The concentration scale was equivalent fractions in both phases. The relation between the corrected equilibrium quotient, x', and K is given by In x ' = In r

+ ln(~/4 NaCl/3t 3:t: CaC12)

(31)

208

In~'

Ca 2.-

Na ÷

3.5

3.0

by least-squares fitting to eqn. (2) are compared with those obtained from eqns. (26) and (27). In Fig. 3, W is plotted against YUe,NR. The curve has been obtained from eqn. (27). As expected, the resin takes up more water except at high E values, where the liquid cation exchanger has a larger water uptake. Other applications of the model to water uptake are given in Refs. [14,23,29,30].

2.5_

Inorganic cation exchangers

2.01 I

0

°°~ I I

t

~

Fig. 6. In K' plotted against ~ca for system Ca2+-Na + on kraft pulp at 296 + 1 K. Data from Ref. [261. Straight line is that obtained from parameters of first set (B = 0) given in Table 1. For the total chloride concentration in the aqueous phase of 0.0035 M, In •' is plotted against the equivalent fraction ~Ca in Fi_g. 6. In Table 1, parameters are given for B = 0 and for B 4= 0. F r o m Table 1 it is evident that a satisfactory fit is obtained for a straight line, B = 0, in view of the low ionic strength and capacities studied. Other applications of the model to equilibria on organic cation exchangers m a y be found in Refs. [14,23,28, 29,30]. Water uptake Few studies of water uptake during ion exchange have been carried out. Soldano et al. [24] found the water uptake to be linear with composition for the system C s + - N a + on Dowex 50 of crosslinking ranging from 1% to 24% DVB. Mikulich [20] studied various organic a m m o n i u m ions versus H + on the Soviet resin KRS-8, which is a sulfonated polystyrene resin like Dowex 50. The degree of crosslinking is 8% DVB. The system ( C H 3 ) 4 N + - H + is chosen for illustration. In Table 5 the parameters and the fit obtained

Free energy The system Cs +-H + on semicrystalline zirconium phosphate. NancoUas and Tilak [31] studied the reaction Cs + + HX ~ CsX + H +

(32)

on semicrystalline zirconium phasphate at 298 K and ionic strength I = 0.100 M (Cs,H)C1. In Fig. 7 log ~; is plotted against Ecs. The curve has been computed from the model using the parameters given in Table 1. A n acceptable fit is obtained. The system K +-H + on semicrystalline zirconium phosphate. Nancollas and Tilak [31] also studied the reaction K + + HX ~ KX + H +

(33)

on the same zirconium phosphate under the Iog~ Cs+_H + 1.5

1.0

0.5

0

....

i

I

i

i

RCs

Fig. 7. log x plotted against ~cs for system C s + - H + on semicrystalline zirconium phosphate at 298 K. Data from Ref. [31]. Curve has been c o m p u t e d from model with parameters given in Table 1.

209

same experimental conditions as above. In Fig. 8, log x for reaction (33) is plotted against x K. The sigmoid shape of this curve can not be accommodated by the model. This is illustrated by the dashed curve in Fig. 8, which has been obtained by fitting to the model with the exclusion of the two lowest points. On the other hand, a third-degree polynomial gives an excellent fit as shown by the solid curve in Fig. 8. There are several possible explanations. The most obvious is to assume site heterogeneity [32]. However, it is then difficult to understand why the other two systems studied in Ref. [31] can be fitted to the model. Similarly, triplet interactions give rise to a third-degree polynomial [33]. Again it is difficult to understand why the other two systems do not also require triplets. The explanation must be left for the future. T h e s y s t e m M n 2 + - H + on s t a n n i c arsenate.

Dabral and HSgfeldt [34] studied the exchange of divalent cations against H ÷ on

Iog~

/i

K+_H + O,

-0.8

-0.6

S o,dd°':''6/

-0.4

2.0

Mn2"-H+

o

o/ ,7 o

°

10

0.5 I

[

I

I

0 Fig. 9. In K' plotted against ~Mn for system MnZ+-H + on stannic arsenate. Data from Ref. [34]. o, 15°C; O, 25 o C; rn, 40 o C; II, 60 o C. Curves have been computed from model with parameters given in Table 1.

stannic arsenate at temperatures of 15°C, 25 ° C, 40 ° C and 60 ° C. In order to illustrate the application of the model to the temperature dependence, the system MnZ+-H + is chosen. In Fig. 9, In x' is plotted against XMn for the four temperatures. The corrected equilibrium quotient x' is defined in the same way as for the Ca2+-Na + system studied by Ampulski; see eqn. (31). By fitting the three parameters to the van't Hoff equation, the following expressions were obtained by linear regression: In x'(0) = - 1 . 7 9 7 + 8 1 3 / T

o

[s(ln K'(O))=0.0081 In x'(1) = - 5 . 9 4 5 + 2 3 5 9 / T

-0.2

I

0

I

[

I

7K

Fig. 8. log K plotted against ~K for system K + - H + on semicrystalline zirconium phosphate at 298 K. Data from Ref. [31]. Dashed curve has been obtained by fitting to model excluding two lowest points. Solid curve has been obtained by fitting to a third-degree polynomial.

[s(ln x'(1)) = 0.0721 In x~ = - 0 . 6 4 4 + 4 3 8 / T [s(ln x" =

(34a-c)

0.109]

In Table 1 the parameters found from eqns. (34a-c) are given together with the fit. In view of the fact that natural logarithms have been used, the fit obtained is acceptable.

210

Activity coefficients In order to illustrate the use of eqns. (17) and (18), the system C s + - H ÷ studied by Nancollas and Tilak [31] is chosen. From the data in Table 1 for this system, the following activity coefficient expressions are obtained:

0

log fHX = --2.313x2 + 1.871x 3

log fcsx = 0.494(1 - ~ ) 2 _ 1.871(1 - 2)3 (35a,b) In Fig. 10, the activity coefficients obtained originally by graphical integration are compared with those from eqns. (35a,b). While the agreement is excellent for fcsx, the data for fHx differ, probably because of different estimates of the limiting values of log x. The data from the model can be expected to be the best, because here no uncertain extrapolations are needed.

L-

I

i

I

I

I

l

I

[

I

o 0.2 0.4 ~ o.e o.8 Fig. 11. log tc plotted against x1 for system I - - B r - on Dowe×-2 of various cross]inkings at 298 K. Data from

Ref. [37]. o, 4% DVB; e, 8% DVB; n, 16% DVB. Straight lines have been computed from model with parameters given in Table 1.

[37] studied various anion exchange equilibria on Dowex 2 of different degrees of crosslinking. The reaction I - + RBr ~ RI + Br-

Anion exchange resins

For anion exchange resins, only the free energy will be considered. The system I - - B r - on Dowex 2 of various crosslinking at 298 K. Soldano and Chestnut

f

Cs+_H +

.0

.8

(36)

was studied at crosslinkings of 4%, 8% and 16% DVB. In Fig. 11, log x is plotted against ~i for the three degrees crosslinking. The straight lines were obtained by least-squares fitting to eqn. (2) with B = 0. The parameters and the fit obtained are given in Table 1. It is shown elsewhere [38] that, by fitting log r(0) to w(0) by a second-degree polynomial and log x(1) to w(1) by a straight line, an acceptable fit is obtained. The system S C N - - N O 3 on a pyridinebased resin. The reaction S C N - + RNO 3 ~ RSCN + NO 3

(37)

.6

.4

.2

/

3

i 0

I

I

I

~Ca

Fig. 10. fnx and fcsx plotted against Xcs for system C s + - H + on semicrystalline zirconium phosphate at 298 K. Data from Ref. [31]. Curves computed from eqns. (35a,b).

was studied on a pyridine-based resin by G ~ t n e r [39]. From Fig. 2 in his paper, data of In r for reaction (37) were obtained as a function of composition. In Fig. 12, In K is plotted against ~scr~, and in Table 1 the parameters and fit are recorded. The curve in Fig. 12 is that from eqn. (11) with those parameters. The fit is excellent. The data in Table 1 differ slightly from those given in a recent paper by HiSgfeldt and Marcus [38]. This is because of differences in decimals

211

2.0

Free energy A weak-acid or chelating resin can be treated by the model in the same way as the systems already treated. For illustration, consider the reaction

SCN-- NO~ O

In~ 1.8

H + + NaR ~ HR + Na +

1.6 o

1.4 I

0

I

0.2

I

[

I

I

0.4

~

0.6

[

I

0.8

\

(38)

As independent variable, the fraction of sodium ions in the resin, a, is chosen.

[NaR]

I

1

Fig. 12. In ~ plotted against ~SCN for system S C N - - N O 3 on a pyridine-based resin at 298 K. Data obtained from Fig. 2 in Ref. [39]. Curve has been computed from model with parameters given in Table 1.

used, etc., and the differences are immaterial. The same is true of other data in this paper as compared to data already published or in course of publication. Other anion exchange equilibria on resins may be found in Ref. [38].

Inorganic anion exchangers Here, an example is given illustrating the application of the model to the free energy. The system S C N - - N O f on hydrous zirconia. Nancollas and Paterson [40] studied reaction (37) on hydrous zirconia at 298 K. In Fig. 13, log • is plotted against YscN for this reaction. The curve has been computed from the model with the parameters given in Table 1. The first experimental point, being an obvious outlier, has been excluded from calculations. F r o m Table 1 it is evident that an excellent fit is obtained. Other anion exchange equilibria on inorganic exchangers treated by the model are to be found in Ref. [41].

Weak-acid and chelating resins

a = [NaR] + [ H R l = XNaR In practice a is often defined as CO

a = - - - XN~R

(40)

s0

In eqn. (40), v is the volume of strong base of concentration c added to a sample with the capacity s 0. The units are chosen such that cv and s o are compatible. Equation (40) is not valid at high or low p H because H + and N a + in the aqueous phase are neglected. This deftnition of a has been used by scientists who calibrate their glass electrodes in terms of proton activity using standard buffers. They do not know the hydrogen ion or hydroxide ion concentrations to be used in the mass balance conditions. This difficulty can be overcome by using the Stockholm method of

8°.

No;

o3

02

0.1 I

For weak-acid and chelating resins the model will be applied to the free energy only. A method for the determination of capacity of such resins will also be given.

(39)

I

I

I

I

I

I

I

I

0 0.2 0.4 ~ 0.6 0.8 Fig. 13. log ~ plotted against ~SCN for system S C N - - N O 3 on hydrous zirconia at 298 K. Data from Ref. [40]. Curve has been computed from model with parameters given in Table 1.

212 d e f i n i n g p H as the h y d r o g e n ion c o n c e n t r a tion in the ionic m e d i u m used, as has b e e n d o n e for s e a w a t e r [42]. Below is s h o w n h o w the c a p a c i t y o f a n ion e x c h a n g e s a m p l e c a n b e d e t e r m i n e d using glass electrodes calibrated in terms o f c o n c e n t r a t i o n a n d the G r a n method. A p p l i c a t i o n o f the law o f mass a c t i o n to r e a c t i o n (38) gives, a f t e r taking logarithms:

pH 9

H+--Na + o

S g~

5

log r = l o g ( ~ - - ~ ) +

pH + log(Na + }

(41)

In ionic m e d i a o f c o n s t a n t ionic strength, the s o d i u m ion activity can b e r e p l a c e d b y the s o d i u m ion c o n c e n t r a t i o n , [ N a ÷ ]. In the r a n g e w h e r e eqn. (40) is applicable, [Na ÷] = I, the ionic s t r e n g t h in the a q u e o u s phase. F r o m t i t r a t i o n data, a = f ( p H ) is o b t a i n e d , a n d log x is t h e n o b t a i n e d f r o m eqn. (41). D a t a o f log ~: = f ( a ) are t h e n fitted to eqn. (11) b y least-squares m e t h o d s . T h e p H c a n n o w b e c a l c u l a t e d f r o m eqn. (41): PHcaxc = log /¢calc

-

-

('°t

log - - ~

- log I

(42)

The s y s t e m H + - N a ÷ on a carboxylate resin. S o l d a t o v a n d N o v i t s k a y a [43] studied rea c t i o n (38) o n the Soviet c a r b o x y l a t e resin K B - 4 at the following s o d i u m c h l o r i d e c o n -

4

t 0

I

I

~

Fig. 14. pH plotted KB-4 at 296 + 1 K. been obtained from using eqn. (42). o, I = 0.500 M; m, I =

against a for system H + - N a + on Data from Ref. [431. Curves have model with parameters in Table 1 I = 0.020 M; O, I = 0.100 M; n, 2.500 M.

c e n t r a t i o n s : I = [NaC1] = 0.02 M , 0.10 M , 0.50 M a n d 2.50 M. T h e t e m p e r a t u r e was 296 + 1 K. T h e p H was c a l i b r a t e d in t e r m s o f p r o t o n activity using s t a n d a r d buffers. T h u s , eqn. (40) was used to o b t a i n a. HiSgfeldt a n d N o v i t s k a y a [44] o b t a i n e d d a t a o f a = f ( p H ) b y r e a d i n g off p o i n t s o n a figure in Ref. [43]. T h e d a t a were t h e n t r e a t e d b y the

TABLE 7 Comparison between estimated and computed pH values for reaction (38) on the Soviet carboxylate resin KB-4; T = 296 + 1 K; data from Ref. [43] a

1 = 0.02

1 = 0.10

1 = 0.50

I = 2.50

pH, exp.

pH, calc.

pH, exp.

pH, calc.

pH, exp.

pH, calc.

pH, exp.

pH, calc.

0.10 0.28 0.48 0.68 0.87 0.92 0.97

5.30 6.40 6.90 7.50 8.30 8.50 9.00

5.454 6.310 6.970 7.554 8.205 8.470 8.945

4.90 5.80 6.40 6.80 7.50 7.80 8.20

4.813 5.669 6.324 6.893 7.523 7.782 8.251

4.20 5.20 5.85 6.20 6.85 7.10 7.60

4.242 5.101 5.741 6.280 6.864 7.109 7.561

3.80 4.70 5.30 5.70 6.20 6.50 6.90

3.830 4.694 5.303 5.772 6.255 6.467 6.885

U R (%) s' (pH)

0.0526 1.154 0.0936

0.0426 1.138 0.0843

0.0316 1.078 0.0726

0.0105 0.681 0.0418

213

model. It was found that the parameters could be fitted to v~ by linear regression, giving log x(0) = 4.47 + 0.31~/7 log x(1) = 5.75 + 0.04

(43a-c)

log x m = 5.47 + 0.43v~F r o m eqns. (12) and (43a-c) log K = 5.230 + 0.247v~

(44)

In Table 1 the parameters obtained from eqns. (43) and (44) are given, together with the fit obtained. In Fig. 14, p H is plotted against a for the four sodium chloride concentrations studied. In Table 7 experimental and computed pH values are compared. F r o m Table 7 it is evident that a satisfactory fit is obtained, as the accuracy of reading is 0.05 to 0.1. The

system

H +-Na ÷

on

Li gande x - L

Szabadka [45] studied the protonation equilibria of a resin containing iminodiacetic acid

groups, which was named Ligandex-I. The resin is protonated in three discrete steps. At low pH values the resin is an anion exchanger, and at higher p H values it works as a cation exchanger. For each range the water and counter-ion contents in the resin phase were determined, as well as the p H in the aqueous phase. Szabadka applied ordinary complex chemistry methods to the system. For each step the protonation can be written R° +

H + ~

RH +

K1

R - + H + ~ R°

K2

R 2- + H + ~ R -

K3

(45a-c)

These equilibria refer to the resin phase. The pH in the resin phase is related to the pH measured in the aqueous phase by pH = pH + log G -Y-log a c

(46)

Here, p H = p H in the resin phase; G = the concentration of counter-ions in the imbibed

TABLE 8 P r o t o n a t i o n equilibria of Ligandex-I; T = 298 K, I = 1.00 M KC1; data from Ref. [45]; log a ~ = - 0 . 2 1 5 F r o m the data below the following average log K i values are obtained; log K 1 = 1.427 + 0.044, log K 2 = 3.169 + 0.087, log K 3 = 9 . 1 6 6 + 0 . 0 4 3 pH, exp.

G

pH

log K i

pH, calc.

log Ki, exp.

log ri, cal.

pH, calc.

0.234 0.447 0.599 0.876 0.889 0.903

0.380 0.660 0.950 1.860 1.950 2.010

2.485 2.592 2.526 1.550 1.506 1.460

0.991 1.289 1.568 2.266 2.343 2.391

1.506 1.381 1.394 1.417 1.440 1.422

0.302 0.706 0.984 1.871 1.938 2.016

0.895 0.752 0.776 1.011 1.046 1.041

0.893 0.760 0.770 1.014 1.032 1.053

0.378 0.668 0.944 1.863 1.936 2.022

0.204 0.277 0.458 0.582 0.713 0.933

2.970 3.120 3.400 3.700 4.080 4.930

1.444 1.460 1.638 1.689 1.760 1.901

2.595 2.740 2.970 3.257 3.619 4.436

3.186 3.157 3.043 3.113 3.224 3.292

2.953 3.132 3.525 3.755 4.025 4.807

3.561 3.537 3.473 3.556 3.685 3.786

3.548 3.530 3.527 3.560 3.626 3.808

2.957 3.113 3.454 3.704 4.021 4.952

0.098 0.156 0.260 0.487 0.495 0.937

8.650 8.940 9.260 9.740 9.780 10.980

2.032 2.061 2.123 2.230 2.294 2.539

8.127 8.411 8.718 9.163 9.204 10.360

9.091 9.144 9.172 9.186 9.213 9.188

8.725 8.962 9.254 9.707 9.733 10.958

9.614 9.673 9.714 9.763 9.789 9.808

9.629 9.658 9.706 9.779 9.781 9.807

8.665 8.925 9.252 9.756 9.772 10.979

214 water of the resin (mmol/g), and a G is the activity of counter-ion in the solution. For this quantity the value log a c = -0.215 was used, corresponding to the mean activity of 1 M KC1, the ionic medium used. For anion exchange the sign in front of log G is positive, for cation exchange it is negative. The experimentally determined quantities a, pH and G have been taken from Table 1 in Ref. [45] and are given in Table 8. The pH was obtained from eqn. (46), and finally log K i was computed from [ 1-~t K,--.. + (47)

The system H +-Na + on IRC-50 with 1% and 5% DVB. Chatterjee and Marinsky [47] studied the protonation of Amberlite IRC-50 containing crosslinked poly(methacrylic acid). The degrees of crosslinking were 1% and 5% DVB, and two ionic strengths were used: I = 0.100 and 0.300 m NaCI at 298 + 0.05 K. For fitting the data the following expression was used

p H = p K - log

a

Na+}+A)

+ log {Na+}

(48)

;-)

From the data in Table 8, the average log K~ values have been taken for use in calculating p H from eqn. (47) and then p H from eqn. (46). The data in Table 8 have also been used to compute log ~ for each pH region using eqn. (41). These data were then fitted to the model, giving the parameters in Table 1. In Table 9 the fit in p H obtained by the two methods is given. Not surprisingly, the three-parameter model gives the best fit, as is to be expected with three parameters instead of one for each region. In view of that, the Szabadka method works surprisingly well. This might be fortuitous, as other systems may give a poorer fit to his model [46]. The great advantage of the three-parameter model is that it requires only a in the resin phase and pH in the aqueous phase. Few workers have carried out such a painstaking experimental study as Szabadka in determining water content and salt invasion, etc.

The quantity A is obtained from the model of Katchalsky and Lifson [48]: A-

0.4343 ( OFe I k T ~ ~, ]~

(49)

when Fe is the electrostatic free energy, defined as the difference between the free energy of the actual polyelectrolyte carrying v negatively charged groups and the free energy of the same polyelectrolyte in a hypothetical state, in which all the ions, fixed or free, carry no charge. The quantity K is the inverse Debye radius. K in eqn. (48) refers to the dissociation of the acid HR, i.e., the reverse of reaction (45b). In Table 10, average p K values estimated by the Marinsky approach have been used to compute p H values to compare with the experimental ones. Also, equilibrium quotients estimated by the three-parameter model have been obtained by least-squares fitting to eqn. (11). The parameters and the fit obtained are given in Table 1.

TABLE 9 Fit obtained for pH data by the methods of Szabadka and HSgfeldt U Szabadka

K1 K2 K 3

9.657 X 1 0 - 3 3.724× 10-2 9.927 x 10- 3

HiSgfeldt 4.530 X 10-4 7.115 × 1 0 - 3 8.350 × 10- 4

R (%) Szabadka

2.748 2.096 0.424

HiSgfeldt 0.595 0.916 0.123

s" (pH) Szabadka 0.0439 0.0863 0.0446

HiSgfeldt 0.0095 0.0377 0.0129

215 TABLE 10 Comparison between Marinsky's approach (M) and the three-parameter model (H) for IRC-50; T = 298±0.05 K; data from Ref. [47]

a

log( aNa+ ]

A

pK

pH, M

log K exp.

log K, calc.

pH, H

I % D V B , I = O.lOOmNaCl, pK= 4.941 ±0.123 0.1041 0.786 0.110 4.75 0.3024 0.855 0.278 5.68 0.5046 0.881 0.431 6.40 0.8136 0.918 0.657 7.20

4.789 4.910 5.080 4.985

4.902 5.771 6.261 7.156

4.685 5.043 5.392 5.560

4.673 5.076 5.365 5.566

4.738 5.713 6.373 7.206

I % D V B , I = 0 . 3 0 0 m N a C l , p K = 4.973 ±0.098 0.1089 0.475 0.100 4.50 0.3044 0.583 0.260 5.45 0.5075 0.550 0.386 6.00 0.8119 0.600 0.527 6.80

4.838 4.966 5.051 5.038

4.635 5.457 5.922 6.735

4.890 5.286 5.464 5.642

4.903 5.250 5.493 5.635

4.513 5.414 6.029 6.793

5%DVB, I = 0 . 1 0 0 m N a C l , p K = 5.168 ±0.210 0.1080 0.968 0.0945 5.22 0.3044 1.252 0.197 6.10 0.5092 1.284 0.290 6.70 0.8000 1.297 0.425 7.80

5.075 5.010 5.110 5.476

5.314 6.258 6.758 7.492

5.137 5.459 5.684 6.198

5.136 5.431 5.739 6.177

5.219 6.072 6.755 7.779

5%DVB, I = 0 . 3 0 0 m N a C l , p K = 5.112±0.178 0.1139 0.424 0.086 4.75 0.3117 0.837 0.192 5.63 0.5086 0.909 0.283 6.27 0.7955 0.878 0.411 7.22

5.131 4.913 5.063 5.341

4.731 5.797 6.319 6.991

5.118 5.451 5.732 6.107

5.143 5.428 5.712 6.125

4.775 5.607 6.250 7.238

aNa+ ]

pH, exp.

In Table 11 the fit obtained in the pH data is given. As expected, the three-parameter model gives a satisfactory fit. While the Marinsky approach is interesting from the insight it offers, it is somewhat elaborate for fitting titration data. Here the three-parameter model is simpler to use. Further illustrations of the model as applied to free energy as well as water uptake can be found in Refs. [28,49-51].

Determination of capacity The practice of calibrating glass electrodes in terms of proton activity using standard buffers has several drawbacks. It is better to use the Stockholm method of definition of pH and calibrate a glass electrode in terms of hydrogen ion concentration in the ionic medium used. Since most ion exchange equilibria are determined at constant ionic strength this method can be applied, and eqn.

TABLE 11 Fit in pH data obtained with the methods of Marinsky (M) and HiSgfeldt (H); IRC-50, T = 298_+ 0.05 K; data from Ref. [47] /% DVB

I

U

R(%)

M

H

1 1 5 5

0.100 0.300 0.100 0.300

4.532 X 10 - z 2.858 x 10 -2 0.1320 8.309x 10 -1

1.998 x 2.355 x 4.251 x 1.878 x

10 -3 10 -3 10 -3 10 -3

s ' (pH)

M

H

M

H

1.752 1.471 2.785 2.388

0.368 0.422 0.500 0.359

0.1230 0.0976 0.2010 0.1664

0.0258 0.0280 0.0376 0.0250

216 T A B L E 12 Sample of F I B A N AK-22 weighing 0.0891 g titrated with 0.200 M HCI; vo = 25.00 c m 3 v

v0 + v

pH, exp.

x

Y

Y, calc.

pH, calc.

1.90 2.16 2.46 2.76

26.90 27.16 27.46 27.76

2.32 2.12 1.95 1.82

0.380 0.432 0.492 0.552

0.1288 0.2060 0.3081 0.4202

0.1233 0.2116 0.3135 0.4154

2.339 2.108 1.942 1.825

U = 5.940 × 10 - 4

R(%) = 0.591

(40) can then be replaced by eqn. (39). Moreover, this approach allows determination of the sample capacity using the G r a n method [52]. For illustration, some data by Soldatov et al. [53] on the fibrous chelating resin F I B A N AK-22, which contains both carboxylate and amine groups, are used. The m e d i u m was 1.00 M NaC1. The glass electrode was calibrated in standard buffers. In view of the constant ionic strength, it can be expected that activity coefficients will be constant. Two cases will be considered. A c i d titration. Consider a sample weighing m o g titrated beyond the end-point with v cm 3 of C M HC1. The starting volume is v0 cm 3. The capacity s o is sought. The hydrogen concentration in the solution is given by [H+] = C v - moS o vo + v

(50)

F r o m the definition of p H [H +1 = 10-PH/yH +

(51)

s ( p H ) = 0.0172

Here yH ÷ is a constant relating the hydrogen ion concentration to the proton activity as defined by the calibration procedure. F r o m eqns. (50) and (51):

The following variables are introduced: Y = ( v 0 + v)10 -pH

(53a,b)

x = Cv

(54)

Y = - moSOYH+ + yH÷X

By applying linear regression to data for Y = the constants yH ÷ and moSOYH+ Can be obtained. In Table 12, data of v and p H are given for a typical sample [53]. By linear regression to these data, the following expression was obtained:

f(x)

Y = - 0 . 5 2 1 9 + 1.698x giving yH ÷ = 1.698, and moS o = 0.5219/ 1.698 =0.307 meq or So--0.5219/(1.698 x 0.0891) = 3.450 m e q / g .

T A B L E 13 Sample of F I B A N AK-22 weighing 0.0889 g titrated with 0.207 M N a O H ; v0 = 25.00 c m 3 v

vo + v

pH, exp.

x

Y ' × 102

Y ' × 102, calc.

pH, calc.

2.70 2.90 3.08 3.27 3.52 3.74

27.70 27.90 28.08 28.27 28.52 28.74

11.05 11.19 11.30 11.37 11.46 11.52

0.559 0.600 0.638 0.677 0.729 0.774

3.108 4.321 5.603 6.627 8.225 9.517

3.139 4.362 5.495 6.658 8.209 9.551

11.062 11.194 11.292 11.372 11.459 11.522

U = 2.330 × 10 - 4

R(%) = 0.0551

(52)

( v o + v ) I o - - P H = CvYH+-- moSOYH +

s ( p H ) = 0.00763

217 For a titration on the basic side, eqn. (53b) is obtained for x, but Base titration.

Y ' = (v 0 + v)10 tpn-pKw)

(55)

and Y' = - moSOYoH- + YoH-X

nomial in log r = f ( Y ) , thus supporting the model presented here.

ACKNOWLEDGEMENTS (56)

where K w is the ionic product of water and Yon is a constant relating hydroxide ion concentration to p O H . In Table 13 data for v and p H are given for a typical sample [53]. By linear regression, Y' = - 0 . 1 3 5 3 + 0.298x giving YOH-= 0.298 and m o S o = 0 . 4 5 3 8 meq or s o = 0 . 1 3 5 3 / (0.298 × 0.0889) = 5.105 m e q / g . The G r a n method thus offers a simple way of estimating ion exchange capacity.

The author is indebted to Professor Vladimir Soldatov and his group in Minsk, USSR, for a fruitful and pleasant co-operation during the last 15 years. The Swedish and Soviet Academies of Science have generously supported this research. The Royal Institute of Technology supported this work during the last six years. King Industries, Norwalk, CT, U.S.A., is thanked for providing samples of H D and H D D N S .

C O N C L U D I N G REMARKS

REFERENCES

The model implies that pair interactions suffice to describe binary systems (no terms containing XlY ~, etc., are needed). Thus it is possible to predict ternary systems from the binary ones. Examples are found in Refs. [54] and [55]. Soldatov and Bichkova [56] showed that, by applying the three-parameter model to the binary systems, an improvement in prediction of the ternary systems was obtained. The extension of the model to activity coefficients with multivalent ions has yet to be done. There are several ways of writing such equilibria [6,7]. It is interesting to note that the model also applies to linear polyelectrolytes [57] and to binary liquid [58] and gaseous mixtures [33]. Here, however, quintuplets are needed to give a satisfactory fit to the experimental data. The model has sometimes been referred to as thermodynamic. This is incorrect; the model applies to all kinds of molar properties. Recently, Marton and Inczedy [59] derived from electrolyte theory a second-degree poly-

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