ANAxYTIcA CHIMICA ACTA ELSEVIER
Analytica Chimica Acta 289 (1994) 259260
Regular mixtures Erik Hijgfeldt
’
Department of Inorganic Chemistry, The Royal Institute of TechnorogY, S100 44 Stockholm, Sweden (Received 4th January 1994)
Abstract It is shown that Guggenheim’s zeroth approximation is consistent with two kinds of solutions, one where the characteristic parameter b is independent of temperature and one dependent of temperature. The first case has tacitly been assumed to be the only one. The second one is new and is here illustrated by data on the liquid ion exchanger dinonylnaphthalene sulfonic acid. Key words: Ion exchange;
Regular
mixtures
1. Introduction The threeparameter model introduced by the present author has recently been illustrated in various kinds of ionexchange equilibria D31.
The model offers a simple way to correct for nonideality in the ion exchanger. The threeparameter model corresponds to a second degree pOlyIIOmial in the plot of log K versus X. Here K is the equilibrium quotient of the ionexchange reaction considered, and x a suitable composition variable in the ion exchanger. If the curve degenerates into a straight line, the system behaves as a regular mixture. Recently, F&land et al. [4] showed that activity coefficients for a regular mixture give rise to a straight line in the plot above, thus a confirmation of the results of the threeparameter model. For Guggenheim’s ze
roth approximation they showed that plots of the kind mentioned above should give parallel curves for different temperatures and give experimental curves illustrating this. In the present paper it is shown that straight lines with different slopes are also consistent with Guggenheim’s zeroth approximation.
2. The model The model has been discussed elsewhere [l3]. According to the model an ionexchange reaction with equilibrium quotient K is given by log
+
for correspondence: Sweden.
R5ttvikMgen
21, S191 71
K(l)X’+
2
log
log K,X(l
X)
K(o)(l
3)” (1)
model assumes that the kind of nearest neighbours influences the property studied. In a mixture of A and B it has one value when surrounded by A only, another when surrounded by
The
’ Address tillentuna,
K = log
00032670/94/$07.00 Q 1994 Elsevier Science B.V. AU rights reserved SSDI 00032670(94)00042K
260
E. H&f&t
/Analytica
Chimica Acta 289 (1994) 2S9260
B only and a third value when both components are present. This leads to Eq. 1 above. Eq. 1 is a second degree polynomial and can for simplicity be written log
K==U+bX+CX2
(2)
From Eqs. 1 and 2 the empirical parameters a, b and c be expressed in terms of the parameters of the model. We consider for simplicity ion exchange between monovalent ions. The equilibrium constant K is related to the equilibrium quotient by (3)
K=K(f2/f1)
Here f1 and f2 are the activity coefficients of the two components an the mole fraction scale and K the thermodynamic equilibrium constant of the reaction A2 +A,XA,X+A,
(4) From the GibbsDuhem equation and Eq. 3 integration gives log fi = (1/2)bx2 + (2/3)a3 log f2= (1/2)[(bi
Pa)
2c)(l x)~
(2/3)c(l
x)‘]
(5b)
For c = 0 Eqs. 5a and b reduce to log fl = (1/2)bx2 log f2  (1/2)b(l
C6a) x)’
3. The interchange energy According to Guggenheim [S] for the zeroth approximation the enthalpy of mixing can be written x)
“.W

0.00
0.20
0.40
0.60
0.80
1.00
X
Fig. 1. log K plotted vs. x~~~,~ for reaction 8 at three temperatures: 0, 273 K; A, 298 K; v 336 K.
give a set of parallel curves as illustrated in the paper by F&land et al. [4]. The possibility with b dependent upon temperature is illustrated in Fig. 1 where log K is plotted versus x (=~~~~u~n) for the reaction CH,NHg(aq) + HD(org) = CH,NHJD(org) + Hf(aq) (8) where HD is dinonylnaphthalene sulfonic acid dissolved in heptane. The reaction was studied at 273, 298 and 336 K. Three straight lines can be fitted to the data. The interchange energy could be fitted to l/T by b = 0.46  233/T (9) 4. Concluding remark
(6b)
i.e., those for a regular solution.
AH = (1/2)bx(l
0.20
(7) where b is given in Eqs. 6a and b as shown by F&land et al. [41.According to Guggenheim, b is the interchange energy per mole. For ideal solutions b  0, for the zeroth approximation b # 0. If b is independent of temperature it is seen from Eq. 2 with c  0 that plots of log K vs. x
In the systems studied here straight line plots with parallel curves are more rare than those with b dependent of temperature, as might be expected. 5. References [l] E. Hijgfcldt, J. Phys. Chem., 92 (1988) 6475. [2] E. Hiigfeldt, React. Poiym., Ion Exch., Sorbents, 11 (1989) 6475. [3] E. HGgfeldt, in J.A. Marinsky and Y. Marcus (Eds.), Ion Exchange and Sohrent Extraction, Vol. 11, 1993, pp. 109150. [4] KS. Fiirland, T. Okada and SK, Ratkje, J. Electrochem. Sot., 140 (1993) 634. [S] E.A. Guggenheim, Mixtures, Clarendon, oxford, 1952, Chap, 4.