Index

Abstract

This work was devoted to the formulation of thermodynamic properties of binary systems of molten salts by using an ionic model which requires three physical parameters for each salt : the crystal lattice energy, the latent heat of melting and the sum of the ionic radii. This formulation, whose adjustable parameters have been obtained by smoothing from the experimental mixing properties available in the literature, allowed us to calculate the molar enthalpies of mixing for twelve (12) binary systems of alkali nitrate and silver nitrate. The different results obtained show satisfactory general agreement between calculation and experiment with regression coefficients within the range 0.992 and 0.999 and relative uncertainties between experimental and calculated values of less than 6% throughout the compositional range for all systems studied. The present formulation more accurately determines the enthalpies of mixing of binary systems with a volume effect (without intermediate compounds) predominating over the chemical effect (formation of intermediate compounds) ; Furthermore, we were able to show that in binary systems of alkali nitrates based on lithium nitrate and sodium nitrate, the solutions are strictly regular and that the deviation from regularity increases as one goes down in the alkali metal family namely from lihium to caesium.

Keywords: Alkali nitrates , Binary systems, Ionic model, Mixing enthalpy, Silver nitrate, Thermodynamic.

Received: 13 September 2022 / Revised: 18 October 2022 / Accepted: 2 November 2022 / Published: 25 November 2022

Contribution/ Originality

This study first of all made it possible to establish a new expression for the molar enthalpy of mixing of molten salt binary systems as a function of temperature. In addition, an evolution of the regularity of solutions of binary systems of alkali nitrates was obtained.

1. INTRODUCTION

The thermodynamic properties of mixing are of considerable interest both on a theoretical level (structure-properties link, nature of interactions between particles, possible existence of species in solution, etc.) and on a practical level (estimate of phase diagrams, elaboration of materials,. . .). For example, a positive mixing enthalpy can be interpreted as a tendency to homoassociations or demixing, whereas a negative value of this one suggests a tendency to heteroassociation that can lead to the formation of intermediate compounds. 

In practice, if the thermochemical data relating to binary systems are abundant, this is not always the case for multiconstituted systems, the simplest of which are ternary systems. Due to their practical interest, the establishment of models offering possibilities of estimating their mixing quantities from experimental information obtained on lower order systems, is sought. The modeling of mixing quantities constitutes an important field of research in thermodynamics, with challenge today being the feeding of the codes used for the calculation of phase diagrams and thermochemical properties. Several works devoted to molten salts have enabled the estimation of thermochemical properties with success [1-3] .

In the specific case of alkali nitrates and silver nitrate which could be used as phase change materials (PCM) for the storage and transfer of solar energy, due to their thermal stability, their non-corrosive character and above all their low melting point [4, 5] the thermodynamic quantities of mixture of binary liquids AgNO3 – MNO3 (M = Li, Na, K, Rb, Cs) have been satisfactorily translated by the Regular Model in Modified Pair of Arms (RMMPA) [6]. This model has nevertheless shown limitations in estimating the mixing enthalpies of ternary liquids. This is probably due to the fact that this model considers each pure nitrate as constituent (neutral species). However, mixtures of molten salts are considered as an arrangement of cations and anions within which the strong Coulomb repulsions between charges of the same sign and attraction between charges of opposite signs must be taken into account during the formulation of the model. Hence the importance of considering mixtures of molten salts as systems with two ionic sublattices (anionic lattice and cationic lattice).

In the present work, the ionic model which takes into account two sublattices is used to estimate the thermodynamic quantities of twelve binary systems of alkali nitrates and silver nitrate.

2. PRESENTATION OF THE IONIC MODEL

2.1. Introduction of the Ionic Model

According to Temkin's hypothesis [1] a binary solution of molten salts of the AX-BY type (A and B being cations, X and Y being anions) consists in the liquid state of two quasi-lattices which interpenetrate . An anionic lattice and a cationic lattice, in each of which the different ions are randomly distributed between them.

The most rigorous calculation of the ionic interaction, presented by the conformal ionic solution (CIS) theory [7], led to a mixing enthalpy of the binary system that depends on the composition and a size parameter. This theory has been successfully applied to calculate the liquidus temperature as well as the miscibility gaps of phase equilibrium diagrams [8-12] . This CIS theory, although predicting satisfactorily the excess thermodynamic properties of molten salts according to the composition of the salt and the size of the constituent ions, did not give the expressions of the properties themselves. In order to obtain the expression of the excess quantities, an ionic interaction model was developed by Gal and Paligorić [13] and applied to common anion molten salt binary systems. This model requires only three physical parameters of each constituent : the energy of the crystal lattice, the latent heat of fusion and the sum of the ionic radii. The model assumes simple "hard core" interactions and ignores the effect of polarization resulting from the mixing of ions of different size and/or charge.

2.2. Expressions of Excess Quantities

According to Gal and Paligorić [13], in a binary mixture of molten salts containing monovalent ions and a common anion X- of the AX-BX type, the pairwise interaction potentials between oppositely charged ions (A+ and X-; B+ and X-) and cations ( A+ and A+; B+ and B+; A+ and B+) nearest to each other are given by the expressions in Equations 1, 2 and 3:

With :

e : elementary charge ;
ε_0 : permittivity of vacuum ;
d_AX and d_BX: interionic distances (the sum of the radii of the cation and the anion) between A+ and X- and between B+ and X- respectively; 
k: geometric factor which depends on the geometry of the quasi-lattice.

For one mole of pure molten salt AX and BX, the interaction energies as a function of the potentials of the nearest ion pairs are given by the following expressions:

With N Avogadro's number, Z and Z' are respectively the nearest numbers of anions and cations around a cation.

For one mole of mixture xAX + (1 - x)BX (x being the mole fraction of AX), assuming a random distribution of cations, the corresponding Ex energy has the expression :

In order to take into account the long-range interactions in the mixing energy, Gal and Paligorić [13] introduce a dimensionless parameter (a). Taking this parameter into account, and from Equations 2 and 3, Equation 7 becomes :

These energies U1, can be obtained with a good approximation from the standard energy of the crystal lattice (U) and the melting enthalpy of pure salt (∆Hm) according to the expression :

3. RESULTS AND DISCUSSIONS

3.1. Applicability of the Model

The expression of the excess enthalpy obtained by Gal and Paligorić [13] and presented in relation (15) was made on the basis of several assumptions, the main one being to consider the theory of regular solutions. However, considering the theory of regular solutions [15] when studying binary systems of molten salts of alkali nitrates and silver nitrate assuming that for some systems the excess enthalpy had a linear variation with temperature. This observation is confirmed by some authors [16, 17] who during their various studies obtained for the same binary systems of alkali nitrates, experimental values ​​of mixing enthalpy which vary with temperature.

From the expression of the excess enthalpy of relation (15), taking into account the observation of Vallet [15] and considering the theory of regular solutions which states that the excess enthalpy is equal to the enthalpy of mixing, expression of the latter retained in this work, for the ionic model, is as follows :

Where λ is the interaction parameter defined by Gal and Paligorić [13] and β an adjustable parameter obtained by smoothing experimental data.

3.2. Application of the Ionic Model

In this work, expression of the enthalpy of mixing obtained at relation (16), within the framework of the ionic model, is used to estimate the enthalpies of mixing of twelve binary systems of alkali nitrates and silver nitrate.

The physical parameters of the pure salts used to calculate the binary interaction parameters λ are given in Table 1.

The β parameters were obtained by smoothing the experimental data compiled and presented by Meschel and Kleppa [16]. The values ​​of the parameters λ and β obtained, in the present work, for the twelve binary systems studied are presented in Table 2.

Table 1. Physical parameters of pure salts.

Table 2. Parameters of the ionic model obtained in the present work.

In order to analyze the effect of the substitution of atoms in binary systems, we will consider the matrix AgNO3, LiNO3 et NaNO3.

3.2.1. Binary Systems AgNO3 – MNO3(M = Li, Na, K, Rb and Cs)

The mixing enthalpies calculated from the present ionic model and compared to the experimental data are presented in Figures 1 to 5.


Figure 1. Molar enthalpy of mixing of the AgNO3 – LiNO3 binary system calculated and compared to experimental data [16].

Figure 2. Molar enthalpy of mixing of the AgNO3 – NaNO3 binary system calculated and compared to experimental data [16].

Figure 3. Molar enthalpy of mixing of the AgNO3 – KNO3 binary system calculated and compared to experimental data [16].

Figure 4. Molar enthalpy of mixing of the AgNO3 – RbNO3 binary system calculated and compared to experimental data [16].

Figure 5. Molar enthalpy of mixing of the AgNO3– CsNO3 binary system calculated and compared to experimental data [16].

3.2.2. Binary systems LiNO3 – MNO3(M = Na, K, Rb and Cs)

The experimental and calculated data in the present work are compared and presented in Figures 6 to 9.


Figure 6. Molar enthalpy of mixing of the LiNO3 – NaNO3 binary system calculated and compared to experimental data [16].

Figure 8. Molar enthalpy of mixing of the LiNO3 – RbNO3 binary system calculated and compared to experimental data [16].

Figure 9. Molar enthalpy of mixing of the LiNO3 – CsNO3 binary system calculated and compared to experimental data [16].

The mixing enthalpies calculated from ionic model in LiNO3 matrix are perfect agreement with experimental data with regression coefficients 0.999 and uncertainties lower than 2.5%.

This satisfactory agreement between calculation and experiment allows us to deduce that binary systems based on lithium nitrate are strictly regular solutions. These results are in agreement with those of Guion and Brenet [18] who electrochemically had come to the same conclusion.

Moreover, when we go down in the group of alkali metals (from Na to Cs), the uncertainty between experimental and calculated values ​​increases (from 0.73% to 2.32%). We deduce that the deviation from the regularity of the solutions increases when the difference in radius between the two cations increases.

3.2.3. Binary Systems NaNO3 – MNO3 (M = Li, K, Rb and Cs)

The mixing enthalpies of these binary systems obtained in the present work and compared to the experimental data are presented in Figure 6 and Figures 10 to 12.


Figure 10. Molar enthalpy of mixing of the NaNO3 – KNO3 binary system calculated and compared to experimental data [16].

Figure 11. Molar enthalpy of mixing of the NaNO3 – RbNO3 binary system calculated and compared to experimental data [16].

Figure 12. Molar enthalpy of mixing of the NaNO3 – CsNO3 binary system calculated and compared to experimental data [16].

These figures show good agreement between calculation and experiment with regression coefficients greater than 0.992 and uncertainties less than 8%. In view of this satisfactory agreement, we can therefore say that these binary systems based on NaNO3 are strictly regular solutions. Moreover, just as in systems based on lithium nitrate, the deviation from regularity increases when the difference in radius between cations increases. Indeed, from Li to Cs the regression coefficients decrease from 0.999 to 0.992.

4. CONCLUSION

The ionic model developed in the present work was used to calculate the mixing enthalpies of twelve binary systems based on silver nitrate and alkali nitrates. A satisfactory general agreement was obtained between calculation and experiment.
This work has shown that:

  1. For binary systems based on AgNO3, the ionic model best described binary systems with a predominant volume effect over the chemical effect;
  2. For the binary systems of LiNO3 and NaNO3 matrices, the solutions are strictly regular and the deviation from regularity increases when we go down in the family of alkali metals.

The binary parameters obtained in this work will be used to estimate the thermodynamic properties of mixing of higher order systems based on silver nitrate and alkali nitrates. These data will be essential in the optimization procedures for the complete thermodynamic description of the silver nitrate and alkali nitrate systems.

Funding: This study received no specific financial support.

Competing Interests: The authors declare that they have no competing interests.

Acknowledgement: All authors contributed equally to the conception and design of the study.

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