J . Chem. SOC., Faraday Trans. I , 1982, 78, 3519-3527 Water Solubility in Molten-salt Mixtures A Theory for Selective Ionic Hydration BY GIUSEPPE A. SACCHETTO* Institute of Analytical Chemistry of the University, Via Marzolo 1, 35100 Padova, Italy AND ZDENfiK KODEJS Institute of Inorganic Chemistry of the Czechoslovak Academy of Sciences, Majakovskiho 24, 160 00 Prague 6, Czechoslovakia Received 1st March, 1982 A statistical thermodynamic treatment based on the quasi-lattice model of molten salts is developed to explain the non-linear dependence of the enthalpy of transfer of water from liquid water to an infinitely dilute solution of water in a binary molten-salt mixture on the molar composition of the mixture. The derived equation is used to fit previously determined data for the enthalpy of transfer of water from liquid water to (undercooled) liquid LiNO, + NH,NO, mixtures in the whole composition range; a satisfactory fit is obtained for a reasonable choice of the values for the quasi-lattice coordination numbers of LiNO, and NH,NO,.Literature data for the enthalpy of dissolution of water in molten LiNO,+NaNO, and LiNO, + KNO, mixtures are in qualitative agreement with the predictions of the present theory. Investigations of the dissolution of simple gases (e.g. inert gases, hydrogen and oxygen) and of some more complex molecular compounds (e.g. water and dimethyl- sulphoxide) in molten salts are currently receiving increased attention,l? as they can throw light on the different types of interactions between neutral molecules and ions in highly ionic liquid environments and can also have technological applications in such fields as molten-salts-based systems in nuclear reactors, electrowinning of metals from ionic melts and the utilization of geothermal energy sources.Two main methods have been adopted for the investigation of the dissolution of gases in molten salts. One method is based on direct measurements of the solubility of gases in molten ~ a l t s ~ - ~ and the other is based on vapour-pressure measurements for highly concentrated solutions of salts in molecular liquids (such as water or dimethylsulphoxide) when a suitable equation relating the activity and composition of a molecular compound in highly concentrated solutions (e.g. the activity isotherm of Stokes and Robinson6) is used in this case to calculate the activity coefficient of the molecular compound at infinite dilution in the liquid (or undercooled liquid) salt.2* This method is particularly suitable for those molecular compounds which are not thermally stable and/or chemically inert with respect to molten salts.It can be used to determine solubility data in undercooled molten salts and the data can then be extrapolated for temperatures above the melting point of the salt. The dissolution of water in molten nitrates has been extensively investigated by both method^.^-^ Some of these studies were performed using binary mixtures of metal nitrates, whose cationic composition was varied over a wide range (e.g. LiNO, + NaNO,, LiNO, + KN0,4 and LiNO, + NH,NO, mixtures2).The enthalpy of dissolution, AHsoln, of gaseous water in molten nitrates and the 3519 114-23 520 WATER SOLUBILITY IN MOLTEN SALTS enthalpy of transfer, AH,,, of water from liquid water to infinite dilution in molten nitrates were calculated from the temperature dependences of solubility data1* * and of vapour pressure data,2* respectively; their dependences on the composition of some binary nitrates mixtures have also been given.lT To our knowledge, there exists no interpretation of the non-linear dependences of the dissolution enthalpyl and of the transfer enthalpy2 on the molar composition of the binary nitrate mixtures. Indeed, the only suggestion is that of 'a non-random distribution of the water molecules between the hydration spheres of the two cations', given in ref.(2). In the present paper a statistical thermodynamic treatment is developed to explain the main features of the transfer (and solution) properties, assuming as the starting points the basic concepts of the quasi-lattice model of molten salts, as previously developed 6y BlanderlO and by Bombi and Sacchetto.ll THEORETICAL We must first make a reasonable assumption about the type of quasi-lattice sites occupied by neutral molecules entering the molten salt; this has previously been discussed by Sacchetto et For water-containing systems, extensive investigations were carried out by Braun- stein and coworkers7 on hydrous nitrate melts, where they adopted the anionic site occupancy hypothesis for the water molecules. In the following treatment we shall assume the same kind of occupancy, by postulating that water molecules compete with nitrate ions for occupancy of the sites of the anionic sub-lattice, which is equivalent to assuming that only cations are hydrated.We make also the simplifying assumption that in the dilute solutions considered here only simple cation-water pairs (M+-H,O), and not more complex groups such as H,O-M+-H,O, are formed. According to a formalism similar to those previously adopted in treatments based on the quasi-lattice model, the transfer of water from liquid water to an infinitely dilute solution of water in a single molten salt of the type AY can be expressed by the following simplified scheme A+-Y- + H,O A+-H,O + Y- (1) which represents the replacement of one Y- anion in the anionic sub-lattice of the AY molten salt with one water molecule from liquid water.The displaced anion can, for instance, be associated with a vacant cation site [see e.g. ref (7), p. 2441.t Assuming that the interaction energies for any given pair of particles (ions, neutral molecules and formally also cationic vacancies) involved in the process, i.e. the so-called pairwise ' bond' energies E , are mutually independent, the energy change per mole, AE&, associated with the process considered above can be expressed as follows where EA-H2P [ = N(cAPHpO + E ~ - , . ~ . - corresponds to the energy involved in the formation of one mole of A+-H,O pairs in the particular framework expressed by reaction (I), and EHzO [= J1TEHz0-H20] is the energy for breaking one mole of water-water bonds (electrical charges are omitted for simplicity).Similarly we can write for the salt BY B+-Y-+H,OSB+-H,O+Y- (3) t As a consequence of the choice of only anionic-sites occupancy by water molecules the possible anionic vacancies are not taken into consideration.G . A. SACCHETTO AND z. KODEJS 352 1 where the energy change per mole, AEE, can be expressed as in eqn (2) The energy parameters AE& and AEE are to be considered as temperature- independent, because they are defined as true internal energy changes.” In a similar way the transfer of water from liquid water to infinite dilution in a binary molten-salts mixture of the type [xAY + (1 - x) BYI can be expressed by the following scheme (A+, B+)-Y-+H,O=(A+, B+)--H,O+Y-. Both reactions (1) and (3) are superimposed to yield the overall process reaction (9, the contributions from each of them depending on the different relative probability of either reaction pathway.The energy change per mole associated with reaction ( 5 ) , AEtFjx, can thus be calculated according to the following combination of AE& and AE,s, A e i X = WA AE& + (1 - w,) AE,B, (6) where wA is a probability factor which is evaluated by means of a statistical mechanical calculation. The relative probabilities of the two possible reaction pathways (1) and (3) are related to the equilibrium distribution of the water molecules between the two kinds of available anionic sites, those near the A+ and B+ cations, respectively. Thus wA and (1 - w,) are equal to the fractions of A+-H,O and B+-H,O pairs, respectively, over the total amount of cation-water pairs formed at equilibrium.We now calculate, using the assumptions introduced above, the equilibrium distribution of the water molecules in the system AY+BY+H,O. Let n A and nB denote the numbers of A+ and B+ ions, respectively, and nHZO be the number of water molecules interacting with the cations from anionic sites. If we denote the quasi-lattice coordination numbers of molten AY and BY by the symbols 2, and Zz, respectively, and define Zmix as a formal quasi-lattice coordination number of the molten mixture AY+BY, then Z,n, and Z2nB are the total number of sites around A+ and B+ cations, respectively, available to accommodate water molecules, and Zmix nH20 is the total number of cationic sites around the water molecules dissolved in the melt.If a’ and b’? are used to denote the fractions of the positions adjacent to A+ and B+, respectively, occupied by the water molecules in any arbitrary distribution of the species, then 2, nA a’ and Z , nB b’ are the numbers of A+-H,O and B+-H,O pairs, respectively, and their sum gives the total number of cation-water pairs in the melt, The energy associated with the arbitrary distribution is given by the following (8) expression E = 2, nA Q’EA-H,~ + 2, nB b’EBdH,-,. The number of configurations, a’, of the ternary system AY + BY + H,O is defined (9) as a’ = R+‘a-‘ = R+’Q-’LR-’ where a+’ refers to the distribution of the cations on the cationic sites and f&’ and R,’ refer to the distributions of anions and water molecules on the anionic sites around A+ and B+ cations, respectively. A B t Primed symbols will be used to indicate an arbitrary distribution of the ions and unprimed symbols will denote the equilibrium distribution.3522 WATER SOLUBILITY IN MOLTEN SALTS Random mixing of the cations in the cationic sub-lattice gives The numbers of configurations arising from the distribution of the anions (Y- and H,O molecules) are as follows ZInA! Z,n,a’![Z,n,(l -a‘)]! and The most probable (equilibrium) distribution is found by maximizing the probability of an arbitrary distribution, P‘{ = In [a’ exp ( - E‘/AT)]), with respect to a’, at constant composition, i.e.by means of the following operation d In [a’ exp (- E‘/kT)] da‘ = 0. (13) By replacing a’ And E’ from eqn (8)-(12) and by using Stirling’s approximation, In n! = n In n - n, we obtain for the equilibrium distribution a 1-a - = -exp [- (EA-H,O - ~R--H,O)/RTl. b I - b Since we consider here only very dilute solutions of water in molten salts, we can take a 6 I and b 6 1.Eqn (14) then reduces to (15) a - = exp [ - (AE& - AE,B,)/RT] b where AES - AE,B, is written in place of EAeHZ0 - EBdHPO, according to eqn (2) and (4). The statistical factors W , and (1 - w,) in eqn (6) are the relative probabilities of the two possible reaction pathways expressed by reactions (1) and (3) and thus are equal to the fractions of A+-H,O and B+-H,O pairs, respectively, over the total number of cation-water pairs (see above); for the equilibrium condition we can write and z, nB b 1-W - A - ~ , n , a + ~ , n , b ‘ Dividing eqn (1 6) by eqn (1 7), and substituting for the a / b ratio from eqn (1 5), we obtainG .A. SACCHETTO AND z. KODEJS 3523 in which nA and nR are expressed as functions of xAY, according to the well known relationship: xAY = nAY/(nAy + n g y ) = nA/(nA + nB). Putting for simplicity (19) p = exp[-(AE,A,-AEtB,)/RT] (20) 21 2 2 @ = - and and solving for wA, we obtain Substituting eqn (21) into eqn (6) we finally have DISCUSSION From eqn (22) it is seen that AQix is not a simple mole-fraction-based linear combination of the two AE,, contributions arising from the two single salts AY and BY. Each contribution is in fact 'weighted' by a factor which depends on the energy difference AE& - AEtB,, the temperature, the spatial (or configurational) Z J Z , factor and the molar composition of the salt mixture.For instance, at constant temperature A m i x is a non-linear function of the molar composition, except for the trivial case AEk = AEtB,. It can also be easily demonstrated that the curvature of A Q i x as a function of xAY is such that the values of A m i x are nearer thc more negative of the two AE,, values for single salts than expected on the basis of a simple linear interpolation between the two AE,, values. At constant xAY, on the other hand, A G i x increases with increasing temperature and its limiting value for T + co is given by A e i x = XAY AE& + (1 - X A ~ ) AEE. (23) Thus, the linear dependence of A Q i x on molar composition is only a limiting behaviour for very high temperatures and/or small energy differences AE& - AEtB,.The above features of the quantity A G i X are shown in a graphical simulation in fig. 1. The choice of the values for AE$ and AEtB, used for the simulation is quite arbitrary and is essentially conditioned by the need to have AE& different from AEZ in order to enhance the curvature and sensitivity of the plots with respect to the varied quantities, such as the temperature and the ratio of quasi-lattice coordination numbers. The curves (a), (b) and ( c ) refer to three different values of the temperature, for 8 = 1 , i.e. 2, = 2,. Note that the shapes of the curves are influenced only by the difference AE,A,-AE,B, and not by the individual AEtr values. As regards the influence of the ZJZ, ratio, we must first recall that similar salts cannot have their quasi-lattice coordination numbers too different from one another [see, e.g.the average coordination numbers for liquid salts in ref. (1 3)]. 2 values in the range 4-6 were assumed in many applications of the quasi-lattice model of molten salts.' The influence of changing the Z J Z , ratio in the range from 6/4 to 4/6 can be seen in fig. 1, where curves (b), ( d ) and ( e ) are drawn for the same temperature (340 K). The effect is rather significant; the importance of the choice of proper values for 2, and 2, will be reconsidered below.3 524 0.0 -1.0 - I - 0 E - .Y . 5 -2.0 -3.0 WATER SOLUBILITY IN MOLTEN SALTS 0.0 0.5 XAY 1 .o FIG. I.-Graphical simulation of the dependence of the energy change A@'" on the molar composition of the AY +BY salt mixture (x,,), as derived by eqn (22).AE& = -3.0 kJ mol-I, AEE = 0.0 kJ mol-l. (a) T = 310K, 8 = l;(b) T = 340K, 8 = l;(c) T = 370K, 8 = 1; ( d ) T = 340K,8= 6/4;(e) T = 340K, B = 416. A relationship between the energy change involved in the process represented by reaction (l), AEP,, and the experimentally measurable enthalpy of transfer of water from liquid water to the liquid salt AY, AH&, can be suggested on the basis of the following considerations. When a molecule of water is transferred from liquid water to an anionic site in a liquid salt AY, it must form Z , water-cation 'bonds' to the A+ cations. If the energy change involved in the formation of the 2, successive water-cation bonds were not affected by saturation effects,14 we could write the following simple relationship7 AH&.= Z , AE&. (24) However, as saturation effects could not be excluded in every real system, we should write: AH& d Z , AE& (25) as Z , AE& represent an upper limiting value for the experimental AH& quantity. In verifying the ability of eqn (22) to represent the dependence of the experimental AHEix values on salt mixture composition, we shall use, as a first approximation, eqn (24) to calculate AEP,. As only condensed phases are involved in the transfer process, internal energy changes can be considered equal to enthalpy changes.G. A. SACCHETTO AND z. KODEJS 3525 In the same approximation we assume and AH,R, = 2,A.E: AHgix = ZmixAKix where Zmix, the formal quasi-lattice coordination number of the molten mixture AY + BY, can be calculated as a function of Z, and 2, according to the following reasoning.Eqn (7) applied to the equilibrium condition gives The factors n A a/nHzO and nR b/nHZ, represent the fractions of the total number of cation-water 'bonds' in the AY+BY+H,O system which are A+-H,O and B+-H,O bonds, respectively. As shown above these two factors are equal to wA and 1 - wA, respectively, so we can write where wA is given by eqn (21), together with the definitions of 8 and j? [eqn (19) and (20), respectively]. In the definition of p [eqn (20)] the energy terms AE& and AE,B, can be expressed as a function of AH& and AHE, respectively, through the use of eqn (24) and (26); we then obtain By combining eqn (22), (24), (26) and (27) and solving for Amix, the equation is obtained where Zmix, p and 0 have the meanings defined above [eqn (29), (30) respectively].(30) following (31) and (19), A test of the validity of the equation for AHgix can be performed by fitting it to the data for the enthalpy of transfer of water from liquid water to infinite dilution in the (undercooled) liquid LiNO, + NH,NO, mixtures, which were determined from vapour pressure measurements by Sacchetto et all5 (for the LiNO, + H,O and for the NH,NO,+H,O solutions) and by Bombi et a/., (for the LiNO,+NH,NO,+H,O mixed-salt systems). All the data were obtained using the same dew-point apparatus and can be considered as a consistent set in the temperature range 320-380 K. The values of the enthalpy of transfer [as plotted against x ~ , ~ ~ ~ ~ in fig.2 of ref. (2)] are given in table 1 . In view of the procedure adopted fGr their derivation,, these values must be regarded as average transfer enthalpies in the temperature range and may be considered to be subject to an uncertainty of kO.2 kJ mol-l. A first trial to fit the reported AHt,. data by means of eqn (3 1) was made by assuming that the quasi-lattice coordination numbers of LiNO, (Z,) and of NH,NO, (2,) have TABLE VA VALUES OF THE ENTHALPY OF TRANSFER XLiN03 0.00 0.25 0.50 0.75 1 .00 AH,,./kJ mol-l 4.1 - 1.2 - 4.4 -7.1 - 7.33 526 4.0 0.0 - I - z ;z --- - 4 . 0 -8.0 WATER SOLUBILITY IN MOLTEN SALTS 0.0 0.5 LiNO, 1.0 FIG. 2.-Comparison between experimental and calculated values of the enthalpy of transfer of water from liquid water to LiNO, +NH,NO, (undercooled) liquid mixtures, AHtYix, at 350 K.(0) Experimental data from ref. (2) (vertical bars indicate the estimated uncertainty). Calculated curves for (a) 2, = Z , = 3; (6) 2, = Z, = 4; (c) Z , = Z , = 5 ; ( d ) 2, = 2, = 6. the same value. Fig. 2 shows a plot of the experimental AH,, data together with the least-squares optimized curves, obtained by fitting the data at the average temperature of 350 K with the following sets of 2 values: 2, = 2, = 3,Z, = Zz = 4, 2, = 2, = 5 and 2, = 2, = 6.t The reported curves agree with the general trend of the experimental data, although the point at x L ~ N O , = 0.75 appears to be too low to be fitted by eqn (3 1) with the above assumption about the coordination numbers.The best fit is, however, obtained with the set 2, = 2, = 4, a value which appears to be quite reasonable in respect of the usual 4-6 range (see above). On the other hand, note that the 2 values derived by this procedure are presumably lower than the real ones, as a consequence of the fact that the above-mentioned saturation effect could not be taken into account [see eqn (25)]. A further trial was made to refine the fit by using sets of different 2, and 2, values; in view of previous considerations, however, the difference between 2, and 2, was not allowed to exceed f 2 . Under this assumption the best fit was obtained with the set 2, = 4 and 2, = 6. The corresponding fitting curve (not shown) has a distinct, although small, asymmetry, so that its curvature is stronger in the high xL~NO, range.This corresponds to the trend shown by the experimental points. Note, however, that in view of the many approximations of the model proposed and especially owing to the approximate character of eqn (24), (26) and (27), not too t It is well known that the average coordination numbers of molten salts are real, but integers are generally assumed in the applications of the quasi-lattice model.G. A. SACCHETTO AND z. KODEJS 3527 much importance can be given to the resulting values of Z , except for the fact that they fall into a range which compares well with that proposed in previous papers on this subject. Data for the enthalpy of dissolution of water in molten LiNO,+KNO, and LiNO, + NaNO, mixtures were calculated by Field1 mainly from the solubility data of Bertozzi4 and Peleg.5 However, as can be seen in fig.8 of ref. (I), although the dependence of AHsoln on xLiNO3 is non-linear and the curvature is as expected on the basis of eqn (31) applied to dissolution enthalpies, these data are not very useful for a quantitative test owing to their uncertainty and dispersion. We conclude that the most important and general features of the dependence of the enthalpy of transfer (or dissolution) of water on the composition of the salt mixtures are correctly predicted by the present theory, although for a more fruitful and detailed test further experimental data are required. 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