Thermodynamics of the Solid Solutions of Isomorphic Substances

The study of the conditions and characteristics of the formation of solid solution play an important role in research the processes of formation of mineral and the technology of processing mineral raw materials. The thermodynamics characteristics of the formation of binary solid solutions of isomorphic substances can be calculated using the classical expression for the crystal lattice energy associated with determination of the Coulomb potential and repulse potential in a form acceptable for both components. At the same time this problem can be solved in a more convenient way, using general relations characterizing the dependence of thermodynamics functions on the volume of the system, which leads to an adequate description of the experimental data.


Introduction
The study of the conditions and characteristics of the formation of solid solution play an important role in research the processes of formation of mineral and the technology of processing mineral raw materials. The thermodynamics characteristics of the formation of binary solid solutions of isomorphic substances can be calculated using the classical expression for the crystal lattice energy associated with determination of the Coulomb potential and repulse potential in a form acceptable for both components. At the same time this problem can be solved in a more convenient way, using general relations characterizing the dependence of thermodynamics functions on the volume of the system, which leads to an adequate description of the experimental data.

Theory
Let us consider binary solid solutions of substitution when an impurity component is embedded in the crystal lattice of the main component without the formation of defects. In addition, we use the so-called quasi-harmonic approximation where it is assumed that the vibrations of atoms around the middle equilibrium positions correspond to vibrations of simple harmonic oscillators and the potential energy has a minimum value in these positions. Quantum theory says that the possible levels of the oscillator are defined as where ε 0 is the potential energy in equilibrium, h is the Plank constant, ω is the vibrational frequency and n i are integers from zero to infinity. Summing Eq.(1) over all states from the lowest vibrational level at n i =0, we can determine the partition function [1] ( ) where T is a temperature and k is the Boltzmann constant (hω/k is the Einstein temperature). The total free Helmholtz crystal energy for 3νN oscillators, where N is the number of atoms and ν is a stoichiometric coefficient, can be determined as is the potential energy when all atoms are in equilibrium. Differentiating Eq. (5), we can determine the pressure where V is the volume of system and g is the Gruneisen constant is the same for all normal vibrations [2] ln ln Eq. (7) shows that the vibrational frequency varies depending on the volume as where ω i is the frequency when the volume is V i and ω 0 is the frequency when the volume is V 0 . The volume of a solid solution, according Retgers rule, can be represented as where y 1 , y 2 are the mole fractions of the components (y 1 +y 2 =1) and V 1 , V 2 are the volumes of pure components having the common atom in its structures. When a common atom is surrounded by atoms of various species, this atom is shifted towards the atom having a high charge density. It leads to a local change in volume [3] ( ) where λ is the value characterizing the displacement of atoms from ideal crystallographic positions in solid solution, subject to equality ( Differentiating Eq.(5), we can determine the entropy of system The change in entropy during the formation of a solid solution associated with a change in the vibrational frequencies of the atoms can be defined as where the subscript "1", "2" and "0" refer to pure components and the solid solution, respectively. Substitution Eq.(11) into Eq.(12), we have Given the different configurations of the surrounding of the common atom by d ifferent or identical species of atoms, for example, halogen atoms around the potassium atom in a solid solution KCl-KBr, the average vibrational frequency can be represented in the form 11 12 where ω ii, ω ij are the frequencies corresponding to the surrounding of the common atom by identical and different species of atoms and W ii , W ij are probabilities of these combinations The mechanical theory of vibrations shows that in discrete systems waves with a frequency exceeding a certain final value cannot propagate. For a selected pair of atoms this value can be determined as [ where C is the force constant and M is the reduced mass of atoms. Using Eq.(16), the vibrational frequencies for various configurations can be expressed as where M 1 , M 2 are the masses of replaced atoms and M 0 is the mass of common atom (or a group of atoms for complex compounds). For each of the individual components, we can also write Combining Eq.(8) and Eq.(16), we can determine the ratio Then, using Eq.(17)-(22), the average frequency of atomic vibration in a solid solution according Eq.(14) can be determined as [3] Using Eq.(9), (10) and expansion ( ) Here, as a first approximation, we can put λ=1/2 , whence where q j p = + .
A complete change of entropy during solid solution formation can be defined as ( ) where the first term is the entropy of ideal mixing S id and R is the gas constant R=Nk , where N is the Avogadro number.

AMMS.MS.ID.000590. 4(3).2020
At high temperature (here we consider an approximation where the remaining terms of expansion (4) can be neglected) the total energy of a solid can be determined according to Eq.(5) and Eq.(11) as where the last term is the same for both the solid solution and the components forming it, so the change in energy during the formation of a solid solution can be represented as where the functional dependence of energy on volume is emphasized. Function U 0 (V 0 ) can be represented as a hybrid function of the original components [3] ( ) ( ) ( ) Take into account the probabilities of various configurations, Eq.(32) should be written as where V 01 and V 02 are defined by Eq.(10). Substituting Eq.(33) into Eq.(31), we have The functions U i (V 0 ) and U i (V 0i ) for each of the components can be expanded into Taylor series keeping only the quadratic terms in it, since dU/dV=0 in equilibrium. Then, using Eq.(9), we can write, for example, for the first component where ( ) where ΔH eq corresponds to the equal molar ratio of components.

Discussion
One of the main controversial issues in the theory of the formation of inorganic solid solution is the existence of ordered structures [4,5]. To solve this problem, we consider a partially ordered solution, where probabilities of surrounding a common atom with the same and different atoms can be represented as [4] ( ) Using the condition of minimum Gibbs energy dΔG/dσ=0, we have [6] ( ) ( Further, to simplify the view of formulas, we consider, without loss of generality, a solid solution with an equal molar ratio of components. Then the solution of differential equation (51) can be represented as The integration constant Λ 0 can be determined from the boundary condition -for σ=0 and λ=1/2 , according to Eq. For practical calculations, we define u i as where is the compressibility and the last term is about 5-10% of the basic value [3]. Thus, the value of V, and g for a pure components are required to calculate all thermodynamics characteristics of the formation of solid solutions. This data and calculation results according Eq.(42) are given in Table 1 & 2. Figure  1 shows that the change in enthalpy during the formation of a solid solution is close to a symmetric parabolic shape. 1: NaCl-NaBr, 2: KBr-KI, 3: NaBr-NaI, 4: NaBr-KBr.
and further, using well-known ratio, the chemical potential of the component (Gibbs partial energy), for example, with a subscript "2" This value determines the difference in the chemical potential of the component in a real and ideal solid solution, from where the activity coefficient of the component can be determined as ( ) Such solutions can be called quasi-regular, their difference from regular solutions consist in taking into account the vibrational component of entropy.
Co-crystallization coefficient of any component, for example, with subscript "2", can be represented as [8]  The activity coefficient of component in a mixed solution, according Ref. [8], can be represented as  . Thus, according Eq.(65), the co-crystallization coefficient can be defined for any ratio of components, whence it follows that the minimum value of the co-crystallization coefficient D 2 * takes in the limiting case when 2 0 y → . In this method the values a i * can be determined by comparing the osmotic coefficients in binary solutions of both components, using extrapolation of data for one of components to the field of supersaturated solutions [10,11]. Therefore, as a first approximation it is possible to apply a simpler method of calculation of co-crystallization coefficients using the ratio The results of calculation according to Eq.(66) show its acceptability for the estimation of co-crystallization coefficient ( Table 3); [13].

Conclusion
The considered model of the formation of solid solutions allow to determine the thermodynamics parameters of solutions based on the properties of the individual components of solution. This model is also applicable for calculating co-crystallization coefficient in the entire composition range of ternary water-salt systems, including the area of micro-concentration of the impurity component where the co-crystallization coefficient takes a minimum value.