J. Chem. SOC., Faraday Trans. 1, 1984,80, 823-829 A Thermodynamic Model of the Mixed-salt Cobalt(@-Hexacyanoferrate(I1) Ion Exchanger BY TATJANA S. CERANIC Institute of Physical Chemistry, Faculty of Science, University of Beograd, 1 100 1 Beograd, Yugoslavia Received 10th June, 1983 A thermodynamic model for the ion exchange of K, NH, and Cs on the ion exchanger cobalt(~r)-hexacyanoferrate(~~)-M where M = K,NH, and Cs, has been postulated. Based on the experimentally determined thermodynamic functions AGg,, AS& and AH&, and on the calculated values of excess functions, the deviations from ideality of the investigated reactions have been explained. Cobalt(Ir)-hexacyanoferrate(II)-M, M = K,NH, or Cs, has been used as an + NH,,Cs; m, +K,Cs and + K,NH,. The different ionic forms of this ion exchanger are i~ostructural~ with inorganic ion exchanger for the reactions'? an f.c.c.lattice. The chemical composition of the unit cell is [Fe(CN),Co],M, - nH,O. (1) In the ion-exchange process only M+ ions which are situated in the cases of the crystallite, denoted (CoFC), take part. The effective radius of the cage opening, dw, is 147, 149 and 160 pm for K,NH, and Cs, re~pectively.~ If the ion-exchange reaction in dilute solutions is being followed, it is seen that these ion exchangers are more selective for the ion with the larger crystal radius1 This is confirmed by the selectivity series obtained : dilute concentrated solution solution I( + NH,,Cs Cs > NH, NH, > Cs Cs + K,NH, K > NH, K x NH, However, the ion with the smaller crystal radius has priority when concentrated solutions are used.' This selectivity inversion results from the physicochemical properties of competing ions in the solution, where ion hydration is a dominant effect determining the course of the ion-exchange reaction.The experimentally determined thermodynamic quantities for the ion-exchange reaction cited indicate the overall changes in this two-phase system. In the present work an attempt is made to define and describe all processes in terms of thermodynamic quantities using the results obtained in previous work., - NH,+K,Cs C s > K K > C s - THERMODYNAMIC MODEL The ion-exchange process (CoFC)Ml + Mi(aq) (CoFC)M, + M:(aq) 823824 ION EXCHANGE ON [Fe(CN),Co],M,, M = K,NH,,Cs can be represented by a sequence of intermediate reactions in which competing ions are taking part (CoFC)M, -+ (CoFC)- + Mt (dissociation) (4 Mz(aq) -+ Mz + H20 (dehydration) (b) M:+mt+ii8t+Mt (ion exchange) (4 Mj: + H 2 0 + Mf(aq) (hydration) (4 +o- + (CoFC)M, (association). (4 Processes (b) and (d) are related to the behaviour of competing ions in the solution,l whereas processes (a), (c) and (e) take place in the ion exchanger and will therefore be considered in more detail.From the ion-exchange isotherms3 it is seen that in the reactions which were studied total exchange was not attained, i.e. the ion exchanger became a mixture of two different ionic forms, a binary system which could be an ideal or a real solid sol~tion.~. The free energy of an ion in the ion exchanger, if only one ionic species is present (NM = 1 is the chosen standard state), is given by (2) and M2) the - G@ = nMRTlni@ = 0 because the thermodynamic ion activity under these conditions is @ = 1.standard free-energy change is If the ion exchanger appears as a mixture of two ionic forms From eqn (2) and (3) it follows that - Ge-Gmixture = A c e = -AM1 RT In el +ZM, RT In 28,. (4) Since a* = N M f M (where NM is the mole fraction of the M+ ion in the ion exchanger a n d f , is its activity coefficient) and is the number of moles of the M ion per unit cell, eqn (4) takes the form AGe = (AM, +AM,) RT(xMP In N M z - R M l In NMl) + @M1 +2MP) RT(xM2 1n7Mz -xM, lnfMl>. ( 5 ) The first term on the right-hand side can easily be calculated from the ion-exchange isotherm^;^ it represents the standard free-energy change (AGid) of an ideal solid solution of two ionic forms where TM, = jMz = 1.The second term in eqn (5) is the 'non-ideality ' contribution to the free-energy change (AGE) in the mixture; therefore Ace = A@d + AGE. (6) In an ideal mixture the enthalpy of mixing is always zero, thus A P & ~ , will depend only on the temperature and entropy of the system: AGGzfgl = - TASzZ/pl. (7) The calculated AP&m1 (table 1) and ASEz,Rl (table 2) values show only a change in the randomness of the mixture relative to the ion exchanger in one given form. The question as to which of the intermediate reactions in an ion-exchange process has a predominant influence on the reaction trend is most frequently answered by an analysis of the excess thermodynamic functions.T. s.CERANIC 825 Table 1. Values - of the free energy and excess functions for the free energy for the reactions M, + M, (M = K,NH, or Cs) in the ion-exchanger phase at 298 K ~ ~~~~~ ~~~ - K + NH, 11.3 -4.6 -13.5 20.2 58.8 -38.6 38.8 20.0 K+Cs 14.6 61.9 -13.4 89.9 49.2 40.7 70.8 -21.6 m , + K 20.3 4.6 -10.2 35.1 16.5 18.6 -13.5 30.0 - NH, --* c s 110 66.5 -6.9 183.4 97.1 86.3 37.1 60.0 - Cs+K 116 -61.9 -6.4 60.5 85.7 -25.2 -47.8 133.5 Cs --+ NH, 111 -66.5 -6.0 50.5 73.4 -22.9 -74.0 147.4 - The energy of interaction, w, of two different ions situated in adjacent cages of the unit cell in the non-ideal systems has a non-zero value and is made up from the mutual repulsion and interaction with framework of the ion exchanger AW = ~ W M , M * -WM,M, - w M ~ M ~ - (8) Barrer4 correlated Aw values with the contribution to the standard free energy From eqn (9) it follows that the arrangement of M, and M, ions in the ion exchanger is controlled by the factor X M F M , , i.e.by the probability. Partial differentiation of AG,E over R, in combination with eqn (6) gives the correlation between the coefficient of activity in the ion exchanger and the energy Aw where C is Kielland’s constant.6 The values of Kielland’s constant, and thus of Aw, can be determined for all the investigated reactions since the apparent thermodynamic constant K, showed a linear dependence on the mole fraction of the competing ion in the ion-exchanger phase, i.e. it obeyed Kielland’s relation where K, is the thermodynamic constant (table 3).By using eqn (9) or eqn (5) and (10) it was possible to calculate the total contribution, AGF, to the free-energy change in the ion-exchange reaction (table 1). The experimentally determined standard free-energy change, AGgp, for the reaction studied contains terms for all the processes in both phases in which the competing ions took part (12) where Gh is the hydration free energy of the competing ions in the solution [values taken from ref. (7)]. By using eqn (12) the total contribution to the free-energy change in the ion exchanger, E A @ z / ~ l , can be calculated: AGZp = (Qdz + (GEz + GEl) - (GkP - Gh,) 28Table 2. Excess functions for the entropy change in the ideal system and enthalpy contribution to the ion-exchange reactions at 298 K M - 2 : mean TI CABE,,w,/J K-' mol-' 0 - ASk21M, Mz/M1 A H ~ p AHk2,H, ~A%2,R1 n ASid - CD M, 3 M, /J K-1 mol-1 /J K-' mol-l /J K-' mo1-I /kJ rno1-I /kJ mo1-I /kJ rno1-I eqn (20) eqn (21) h 478 K+Cs 528.0 3.6 45.0 172 57.7 229.7 487 469 478 Z K + NH, 51 1.0 - 5.5 45.3 1 64 3.8 167.8 460 495 n -3 u NH, + K 42.5 5.5 34.2 33 - 3.8 29.2 14 - 20 NH, Cs 91.0 9.1 23.2 83 54.0 137.0 77 - 156 -40 '?g Cs + K - 648.0 - 3.6 21.5 - 77 - 57.7 - 134.7 - 673 - 655 - 664 z z Cs 3 NH, - 570.0 -9.1 20.1 - 59 - 54.0 - 113.0 - 599 - 548 - 573 - U - 0 - - - -T.s. CERANIC 827 Table 3. Kielland's constant C and the interaction energy, Aw, of ions in adjacent cages of the ion exchanger for the ion-exchange reactions at 298 K - K -+ NH, - 5.3 kO.1 30.2 k 0.6 K+Cs - 4.5 f 0.1 25.4 f 0.5 NH, -+ K -2.3 kO.1 12.4 f 0.2 NH, --* CS - 21.8 f 2.0 124f 10 G - + K - 20.8 & 2.0 119+ 10 - - - - Cs -+ NH, - 19.7 f 2.0 112k10 The calculated ZAG&Rl values are presented in table 1.This contribution com- prises the effects defined by GF [eqn (9)] and also the cF term, which is related to the different hydration free energies of M, and M2 ions in the ion exchanger (table 1) AG? = ZAG&M1 - AGF. (14) Using the idealized and simplified model of Eisenman,8 the ions in an ion exchanger interact only through electrostatic forces at the moment of exchange. The Born-Lande9 relation for the electrostatic interaction of monovalent ions in the solid phase can thus be applied to calculate the free-energy change, AG$21~l, where rc is the crystal radius of the corresponding cation and r- is the radius of the ion-exchange centre3 (radius of the cage in the crystallite) with negative unit charge. The contribution to the free energy from the non-ideal electrostatic interaction between two given ions can be calculated from ACT [eqn (lo)] is related to the mutal interaction of the competing ions in the exchanger and to their interaction with the framework; thus AC," = AGF - AGF (17) represents the contribution to the free energy from the non-electrostatic forces (table 1).The contribution to the entropy, SE, and enthalpy, HE, from the non-ideal conditions in an ion exchanger composed of a mixture of two ionic forms can be (18) defined as where As&,, is defined by eqn (7) and the calculated values are presented in table 2, and since AHg?/H1 = 0.For the investigated reaction, the total contribution to the entropy change in the ideal system has been calculated first from the experimental values2 of AS& and the tabulated data for the hydration entropy of competing ions in solution7 AS&=, = ASE2,~l + ZAS~&R~ AR&gl = ZAH,E2~gl (19) ZAS&g1 = AS,$& + A S & 2 / ~ l - AS&ml (20) 28-2828 ION EXCHANGE ON [Fe(CN),Co],M,, M = K,NH,,Cs I EA GE NH&-K+ &NH*, E-K+ - 100 -60 -40 -20 ( 1 /ArJnm-* Fig. 1. Dependence of excess functions (0, CAGE, 0, AGF; a, A@; 0. AC,E> in an ideal system on the reciprocal of the difference in crystal radii of competing ions in the ion exchanger at 298 K. (a) M, < M, and (b) M, > MI. where Regardless of the method used, i.e.whether ZASE was calculated using eqn (20) or (21), the values obtained agreed within the limits of experimental error (with the exception in the reaction NH, + Cs). The values obtained using these two methods are given in table 2. DISCUSSION From the results presented above it follows that the excess function for the free energy in an ideal system depends on a number of factors ZAG&B, = AG,E+AG,E+AG,E (22) where ACF is the contribution from the difference in the hydration energy of competing ions in the ion-exchanger phase, AG,E is from the electrostatic interaction of the given ions with the framework and AGqE is related to the action of the short- range forces and the steric limitations in the exchanger. All these contributions are plotted against the reciprocal of the difference of crystal radii of the competing ions in fig.1. At first glance it appears that the favoured reactions are those in which a larger ion in the exchanger, R,, is replaced by a smaller one from the solution, M,, because X A P are less positive than if M, > M, (table 1). However, the answer as to which stage in the exchange process has the greatest influence can only be obtained after considering all the excess functions (table 1 and fig. 1).T. s. CERANIC 829 The free-energy change for the hydration of competing ions in the exchanger shows the same trend as in solution (AGF is of the same algebraic sign as AGh, table 1). In the exchanger, however, the influence of hydration is more pronounced when M, > M, than when M, < MI.In the K + NH,,Cs reactions, for example, the hydra- tion process in the exchanger is more ‘allowed’ than in solution, whereas in the m, + K,Cs and + K,NH, reactions it is more ‘forbidden’ than in solution. It is due to the more ‘open’ structure of (CoFC)K (dw > as compared with the (CoFC)NH, and (CoFC)Cs crystallites (dw M rc,NH,; dw < rc,cs).3 The electrostatic excess functions, A@, in the case where M, > m, (‘forbidden’ processes) behave in an opposite fashion to the case where M, < HI (‘allowed’ processes). This agrees with the fact that the ionic character of the competing ions is proportional to l/rc. The excess function A@ is related to the steric limitations in the exchanger phase. They can be caused by the geometry as well as the method of bonding of the competing ions to the framework.The steric limitations are pronounced if the ions in the exchanger, MI, have a larger crystal radius than the competing ions, M,, in the solution. The large positive Ac,E values for the reactions CS + K,NH, point to a ‘forbidden’ process (fig. 1) which is related to the built-in CS ions; owing to their large crystal radius they fill up the volume of the cage, superimposing the short-range forces on the interaction with the framework. Note also that the effective cage radius, dw, in (CoFC)Cs crystallite is smaller than the crystal radius of Cs. When M, > m, the steric influence is smaller, thus the excess function AG,E should be smaller than ACF and All excess functions for reactions in which NH, ions are taking part, either from the solution or from the exchanger, deviate considerably from the ACE values of the other ions (fig.1). NH, ions only formally have the properties of alkaline cations; however, they interact with the framework mainly via hydrogen bonds, thus causing the observed deviations. The obtained EAS&B~ values differ only slightly from ASg, (table 2), which means that the influence of the hydration of ions, ASh, in the solution on ASg, is very small as compared with the processes in the exchanger. We presume, therefore, that the transfer of water in the course of the exchange reaction does not influence the configurational entropy in the exchanger. The XAS&M~ values lead to the conclusion that the configurationally allowed processes are those in which a small ion is being replaced by a larger one, whereas the opposite processes are ‘forbidden’. The unexpected negative CASE value for t h e m , + Cs reaction is because of’lhe hydrogen bonding of NH, in the exchanger. The structural model described earlier3 and the thermodynamic model presented have confirmed that these ion exchangers are similar to the zeolites, which may give some ideas as to their possible applications. AG?. T. Ceranid, D. Trifunovid and R. Adamovid, 2. Naturforsch., Teil B, 1978, 33, 1099. T. S. Ceranid and R. Adamovid, Z. Naturforsch., Teil B, 1979, 34, 127. T. CeraniC, 2. Naturforsch., Teil B, 1978, 33, 1484. R. M. Barrer and J. D. Falconer, Proc. R. SOC. London, Ser. A , 1956, 236, 227. D. H. Freeman, J. Chem. Phys., 1961,35, 189. J. Kielland, J. SOC. Chem. Ind. (London), 1934, 54, 232. Publishing Company, London, 1969), p. 13. G. Eisenman, J. Biophys., 1962, 2, 259. M. Born and A. Lande, Ber. Phys. Ges., 1918, 20, 210. ’ Y. Marcus and A. S. Kertes, Ion Exchange and Solvent Extraction of Metal Complexes (International (PAPER 3/969)