Thermodynamics of the Electrolyte Solutions

The article considers a statistical model of binary solutions of electrolytes to describe the concentration dependence of the thermodynamics properties of solution in the entire area of their existence, including separated solutions.


Introduction
The coincidence of the formulations of Van Hoff 's law for dilute solutions and the equation of state for ideal gases reflects the general statistical regularities that determine the properties of these systems [1]. Moreover, the ideal gas itself is quite similar to a system consisting of particles of solute occupying a volume equal to the volume of the solution [2]. This analogy is also evident in systems with the interaction of components, for example, the Debye-Huckel theory is applicable for both electrolyte solutions and classical plasma in the state of a fully ionized gas. These facts characterizing the observed generality in the behavior of these systems can be taken as a basis for statistical description of the properties of electrolyte solution where the simplest formulations for the ideal gas are used as starting points.

Theory
According to Landau LD et al. [1], the Helmholtz free energy of an ideal gas containing N molecules in volume V can be represent as where k is Boltzmann constant, T is temperature, χ is a certain function that depends only on temperature. In a solution that contains interacting components, free energy can be represented as [3] ln ln eV A A N kTN NkT Z w N χ = + − − (2) where A w is free energy of solvent, Z is partition function. Eq.(2) can be generalized for solution containing particles of different grades. So, for a binary electrolyte solution containing N p cations and N n anions (N=N p +N n ), we can write where Z i is the ionic constituent of the partition function. Differentiating Eq.(3) it is possible to determine the pressure in the system ln where P w refers to the pure solvent. Osmotic pressure is defined as P P P s w = − gas and is applicable to ideal solutions regardless of the nature of both the solute and the solvent. Here instead of the pressure of gas there is osmotic pressure, instead of the volume of gas there is volume of solution, instead of the number of particles in the gas there is the number of particles of the dissolved substance.
The degree of deviation from the ideal solution is defined as , where φ is the osmotic coefficient, determined according to Eq. (4) and (6) as In dilute solutions we can put N<<N w and V=N w v w , where N w is the number of solvent molecules and v w is the molecular volume of the solvent. Then Eq. (7) can be represent as [3] ln and Eq. (3) as Differentiating Eq. (9), we can determine the chemical potential of solvent in the electrolyte solution ln , , where μ w 0 is the chemical potential of a pure solvent. Using Eq.(8), Eq.(10) can be transformed to On the other hand, the chemical potential of the solvent can be determined as where a w is the solvent activity. Comparing Eq.(11) and Eq.(12), we have ln N w a w N ϕ = − (13) Eq.(13) extends to the entire field of existence of electrolyte solution and is the basis for calculation the osmotic coefficient from experimental data. In aqueous electrolyte solutions, the solvent activity can be defined as By limiting the exponential expansion to a linear term for dilute solutions, where one can put φ=1, Eq.(14) can be represented as Thus, the relative decrease in vapor pressure during dissolution is equal to the concentration of the solution (Raoul's law). Eq. (15) contains the relative concentration of the solute x=N/N w . This value is related to the molality (m) ratio where m w is the number of moles of the solvent in 1kg (for water m w =55.51), ν is stoichiometric coefficient (ν = ν p + ν n ). Assuming the statistical independence of the ionic constituent of the partition function can determine the chemical potential of each ion 0 ln ln ln , where γ i is the ion activity coefficient The mean chemical potential can be defined as Thus, the partition function of the solution can be determined experimentally. Turning to the theoretical estimation of the partition function of the solution, we assume that the various processes in the solution are independent. Then this value can be represented in form [4] Z Z Z Z a e h = where Z h ,Z a ,Z e are the partition functions corresponding to the processes of hydration, association and electrostatic interaction of ions, respectively. In this case, the osmotic coefficient of the solution Consider the process of hydration of ions, for example, an anion coordinating n water molecules around itself. The equilibrium constant of this process can be expressed as where N 0 *,N n * is the number of free and hydrated ions (hydrate complex), respectively. In the formation of various forms of hydrated ions, we can write the equation of material balance where, according Eq.(33), K 0 =1.
The distribution function for a hydrated ion can be expressed as [3] where ε n is the difference of the energy between the standard states of the hydrate complex and its constituent components. For a free ion w n makes the form where ε 0 =0 is assumed, since this energy level is taken as the reference point. Comparing equations (34) and (36) , the partition function of the ion hydration process can be determined as 0 n n Z K a n n w = ∑ To determine the activity coefficient, the partition function mast be normalized. In an infinitely dilute solution, Z n takes value This normalization condition is obviously not essential for determining the osmotic coefficient, since constant Z n0 disappears when function ln Z n * is differentiated.
The mean hydration number an ion is determined as For another ion, we can write similar equations with a complete replacement of subscripts.
The mean hydration number of the electrolyte can be defined as Eq.(56) shows the dependence of the hydration constituent of the osmotic coefficient on the solution concentration.
Turning to the description of the process of ion association in a solution, we restrict ourselves to considering the formation of molecular complexes with ratio of unlike ions 1:1, which is typical for most electrolyte solutions. The equilibrium constant of this process can by determined in the form where N a * ,N p * , N n * are the number of associates and free cations and anions, respectively. For each ion species, for example cation, the equation of material balance can be written down Then from Eq. (57) and (58) we can determine the distribution function of free ions where, when writing the last equality, the approximation is accepted N n *≈N n . Using an equation similar Eq.(36) we have For symmetric I-I and II-II valence electrolytes q p =q p and Z p =Z n =Z a , where Z a is the partition function of the process of ion association. Hence according Eq.(32) we obtain In general case, for electrolytes of various types, in a good approximation, we can use equation As is known from electrostatics, the energy of the electrical interaction of a system of charged particles can be defined as where D=4πε (ε is the dielectric constant of the medium) and χ has a dimension of inverse length The omitted terms at r=0 turn to zero. Then, the potential ψ i can be represented as Eq.(69) is the basic equation of Debye-Huckel theory, which represents an amendment to the energy due the interaction of ions [6]. However, this correction is found in the variables T,V,N, which does not correspond to the proper variables of the potential U e . For these variables the thermodynamics potential is the Helmholtz free energy. Therefore, using the well-known thermodynamics equation  The scope of Eq.(69) is limited to dilute solutions. In this approximation N≪N w ions can be considered as point charge. However, at high concentrations, the physical size of the ions plays an important role; therefore, the electric field is bounded by a spherical surface at some distance r 0 from the center of symmetry where the selected ion is located. In this case, determining ψ i for r=r 0 and substituting the result in Eq.(63), the energy of the Coulomb interaction can be determined as [7] Copyright © G R Allakhverdov The contribution of the Coulomb interaction to the osmotic pressure of the solution can be determined by differentiating Eq.(73) and further determine corresponding value In an infinitely dilute solution assuming as before V=N w v w , the Eq.(77) can be represented as  , which determines the correlation scope, it can be notes that with a decreases in the concentration of the solution (c<1), the ratio r D / r m increases, so that the correlation captures an increasing number of particles, but the solution becomes ideal because c⤑0. In the opposite case, with an increase in the concentration of the solution (c˃1), the ratio r D /r m decreases and a smaller number of particles appear in the correlation sphere, and despite the increase in the solution concentration, the osmotic pressure tends to the ideal value. To confirm this reasoning, we will give some examples. In dilute solutions, when using approximation ( ) This formula differs from the result of the Debye-Huckel theory, but fully corresponds to the second virial coefficient in the Pitzer theory [8]. Using Eq.(32), we can determine the partition function corresponding to Coulomb interaction as where Λ 0 is equivalent electric conductivity in an infinitely dilute solution, c is molarity. This equation described well the experimental data in both dilute and highly concentrated solution, and parameter β is completely consistent with the result of calculation according the Onsager theory [9]. In other cases, when, on the contrary, the Coulomb interaction can be neglected, can be found equation for describing the density of electrolyte solutions [10] and describing thermodynamics characteristics of binary solution of nonelectrolytes [11], and also calculating the separated factor of inorganic substances during crystallization from solutions [12]. However, when calculating the osmotic coefficient of electrolyte solutions, all interaction processes must be taken into account.
Let's imagine Eq.(13) in a more convenient form for analysis, from which it is clear that at a given concentration the osmotic coefficient reflects the total interaction of the dissolved substance with the solvent, so the higher its value, the lower the activity of the solvent. Thus, comparing the solvent activity in solutions of various salts at the same concentration, their relative degree of hydration can be estimated, since ions hydration leads to an increase in the osmotic coefficient, and other interaction act in the opposite direction. For example, aqueous solution of alkaline halides can see the growth of the osmotic coefficient in the series of compounds with a common ion (Figure 2): for Cs + is I<Br<Cl<F, for Rb + is Br<I<Cl<F , for K + is Cl<Br<I<F , for Na + is F<Cl<Br<I , for Li + is Cl<Br<I. Using a similar construction for cations, all ions can be arranged in order to increase hydration: I≤Cs<Rb<Br<Cl<K<F≤Na<Li. Location in this series of both cations and anions corresponds to a change in the surface density of their charge, so ions of the same species having a smaller size distinguish a greater degree of hydration. From this point of view, the inversion of the osmotic coefficient in a series of anions from Fto Ican be explained when cesium ion is replaced by a sodium ion. Large cesium ion are slightly hydrated; therefore, when replaced with a smaller sodium ion, an increase in the osmotic coefficient is observed. On the contrary, in the series from Ito F -, when anions are replaced in solutions containing sodium ions, hydration is weakened due to more powerful counteraction from the side of a smaller ion. In the series of alkali metal compounds, when passing from chlorides to hydroxides, hydration is more pronounced, the greater the difference in the sizes of the ions forming this compound ( Figure 3). Otherwise, hydration is weakened up to its complete disappearance in solution LiOH, where it is possible to put φ=1+φ e (Table 1). Thus, we can formulate a model of competitive hydration, where only one is hydrated, creating a more powerful electric field.  where h 1 * and h 2 * can belong to one of the ions or to both ions. These values include stoichiometric coefficients, so hydration number of the select ion is determined as h i 0 =ν/ν i h i *. For example, for an aqueous solution of sulfuric acid, the most stable form corresponds to hydration of the proton and the hydration number in infinitely diluted solution is determined as h p 0 =ν/ν p h 1 *=3/2h 1 *≈1. This value corresponds to formula H 3 O + , and even with solution concentration 76m fourth hydrogen ion is in this form, while the probability of existence of the second form is less than 1/400. This form should also be attributed to the hydrogen ion because of the ion radii r(H 3 O + )=0,113nm and r(SO 4 2-)=0.24nm [13] it follows that the surface charge density of the ion H 3 O + more than twice as high. A similar pattern can be observed in aqueous solution ZnI 2 . (Figure  4) shows the contribution of two hydrated forms, from which it can be seen that at high concentration only the most stable form exist. In the series of zinc halides, there is an increase in hydration during the transition from chloride to zinc iodide due to the weakening of the competitive ability of anions in this series, therefore, the second hydration number in (Table 1) should also be attributed to zinc ion. As shown in (Table 1) the above statistical model of binary solution of electrolytes describes well the known experimental data [14,15], with the exception of cadmium halide solutions, where it is possible to form a number of associates, for example, CdI + , CdI 2 , CdI 3 -, CdI 4 2-, that requires the use of supplementary parameters. In addition, this model allows us to calculate the osmotic coefficient of dilute solutions based on data from concentrated solution ( Table  2), and thus covers the entire area of existence of solution.

Conclusion
The statistical model of solution includes the main factors of the deviation of the thermodynamics functions of binary solution of electrolytes from the ideal values-Coulomb interaction, hydration and ion association, and allows us to calculate the osmotic coefficient of solution in the entire area of the existence of solution. This model provides the basis for calculation the separation factor of inorganic substances during crystallization from solution, as well as for calculation the density and conductivity of solutions required for the control of mineral processing technology.