Calculation of the Density of Electrolyte Solution

Based on the solvate model of solutions, the relation of density with the thermodynamics functions is es-tablished and an equation for the density of binary solutions is obtained. A generalization of this equation to mixed electrolyte solutions is given.


Introduction
The density of electrolyte solutions plays a large role in the processing mineral row materials, since in many cases it is the main source of information of the concentration of substances in solutions. The exact equation for the density of solutions is also of great theoretical importance for determining the partial molar volumes of electrolytes and the osmotic pressure of solutions. This article proposes a method for describing the density of binary and mixed solutions on the solvate model of solutions.

Theory
Let us consider a binary solution containing m moles of electrolyte and w mole solvent (e.g. water) in the volume of solution V. If we assume, without loss of generality, that positive and negative ions form hydrated complexes of a fixed composition containing n p and n n solvent molecules, respectively. than the volume of such solution can be determine as * * * * * w w p p n n hp hp hn hn where N w * , N p * , N n * , N hp * , N hn * are equilibrium amounts of solvent, ions and hydrated complexes, V w is molar volume of pure solvent, V p , V n , V hp , V hn are molar volumes of ions and hydrated complexes, respectively. Taking into account the equations of material balance where ν p and ν n are the stoichiometric coefficients (ν=ν p +ν n is stoichiometric coefficient of electrolyte), Here the hydration number of corresponding current concentration of each ions h p , h n can be defined as where K p and K n is constant of formation of hydrated complex, a w is the activity of solvent.
Using Eq. (3), Eq. (2) can be transformed to Differentiating Eq. (5), we can determine the partial molar volume of the electrolyte, which in the infinitely dilute solution takes on the value Combining Eq. (5) and Eq. (6), we have where h i 0 is the hydration number of ions in the infinitely dilute solution, which can be determined from Eq. (4) at a w ⤑1. The density of the solution is determined by the formula where ρ w is the density of pure solvent. Let us further imagine the hydration number of the electrolyte h i as a function of the concentration z. The activity of the solvent according [1] can be determined using Eq. (10) as where φ is the osmotic coefficient of the solution. Using Eq. (4), the value of h i can be expanded in a Taylor series and limited to a linear term to represent it in the form Substituting Eq. (13) into Eq. (11) we finally have ( ) Next, we use the relationship of dimensionless concentration z with molality c. Combining equality c=m/V and Eq. Eq. (14) can be extended to mixed electrolyte solutions. If we neglect the influence of the last term Eq. (14), the density of the mixed solution can be calculated by the principle of additivity and is presented in the form Further, using Eq. (15), (16), the density of the mixed solution can be expressed by molality Table 1 shows the applicability of the model used for electrolytes of various types. And in all cases, the main contribution of the linear term of Eq. (17) from which it is possible to calculate the partial molar volume of the electrolyte. To estimate this value, we present the volume of the hydrated complex as the sum of the volumes of the central ion and the surrounding water molecules where per one molecule of water accounts for a significantly smaller volume V w *, then in structure of pure water V w . Then

Discussion
and further partial molar volume as ( )  . Hence, we can for each equation determined partial molar volume and compared these values with the experimental data ( Table 2). For this purpose, we use Eq.(8) from where follows Further, determining the derived function and substituting it into Eq.(22), we find Using Eq. (24), we can determine the value of v c 0 by extrapolation data ∂ρ/∂c to the field of an infinitely diluted solution at c ⤑ 0  where z 1 and z 2 are the concentration of components in mixed solution. However, the data in Table 3 show that the use of Eq. (17), (18) is a good approximation for calculating the density of mixed solutions and have significant advantage over Eq. (26), which requires a set of experimental data that is not always available.

Conclusion
Solvate model of solution leads to equations allowing to describe the density of electrolyte solutions with an accuracy acceptable for solving not only technological but also science problems [3][4][5]. The resulting equations using different concentration are convenient for the reverse calculation of the concentration reagents in the solution according to density data, which is of interest in the control of technological processes.