A Thermodynamic Model on a Linear Rule between the Molal Concentration Exponential and the Osmotic Coefficients on Mole Fraction Base in Electrolyte Solutions and its Application

the molal concentration exponential (m2 ) in single electrolyte solutions. Based on this rule, a thermodynamic model is developed and successfully used to the single electrolytes with various valent types, such as uniunivalent, biunivalent, triunivalent, tetraunivalent, bibivalent, and tribivalent to predict their conventional osmotic coefficients on molal base ( ) , and also other properties, such as the relative molal vapor pressure lowering, the equivalent conductivity. Besides m2 , the x2 (the concentration exponential of mole fraction of solute), m2ln(m2 ) and x2ln(x2 ) are also linearly related to , respectively. They are all capable to be used to reproduce some properties of single electrolyte solutions. The examples with satisfied results have been given.


INTRODUCTION
The electrolyte solutions widely exist in natural world, such as molten lava erupted from the volcano, sea water, and salt lake.In industry and in daily life, a large number of electrolytes are applied.Inside all organisms exists a variety of the electrolytes.Recently, we research the Peltier heats of the single electrode reaction [1,2].It also involves some thermodynamic properties of electrolytes and ions.So the close attention to the electrolyte solutions is still of high interest.Many equilibrium and transport properties, like osmotic coefficient, activity, surface tension, conductivity, ion transport number, etc., have been measured and identified by experiments.Some of rules, such as Faraday's laws of electrolysis, Kohlrausch's law of independent migration of ions, transport number law, etc., have been presented.A variety of theoretic models are continually proposed.It includes the Debye-Hückel interionic-attraction theory [3], ionic pairing [4], short-range force [5], ionic hydration [6], semi-empirical Pitzer equation [7] for single electrolytic system, some models for multicomponent system [8][9][10][11], MSA [12], MMSA [13] for certain particular systems, such as those of miscibility gap, liquid-liquid equilibria, even computer simulation [14] etc. Harned and Owen [15], Robinson and Stokes [16], and Bassett and Melchior [17] have excellently summarized and reviewed the developments.The *Address correspondence to this author at the Chemistry and Chemical Engineering College, Central South University, Hunan 410083, China; Tel: +86-0731-88660356; E-mail: zfang@csu.edu.cnprogress in the theoretic respect of statistical physics has been also reviewed by Garcı a and Mosquera [18].But the well known and the often used are the Pitzer or the extended Pitzer model on the molality scale and the Clegg-Pitzer-Brimblecombe model on the mole fraction scale [7,19].This paper will present a new discovery that the osmotic coefficients on mole fraction base are linearly related to the molal concentration exponential in aqueous electrolytes.And apply this rule to construct a thermodynamic model predicting some properties of electrolyte solutions.

A LINEAR RULE
The mole fraction of water in single electrolytic solution on a fully ionized basis, x w is known as The conventional osmotic coefficient, is defined as [16] = n w ln(a / a 0 ) / ( m) (2) where n w is the number of moles of water, , the number of ions into which a molecule of solute dissociates, m , the molal concentration of solute, a and a 0 refer to the activity of water at x w < 1 and x w = 1 , respectively.This is a dimensional quantity and it is on molal scale.where x is the mole fraction of solute, being (1 x w ) , and k 1 is a parameter to be determined.It has been proved that the function is similar to [20], but the former is dimensionless, and the latter is just opposite, being of dimension.

Now newly define the osmotic coefficient on mole fraction scale to be
The is very good linearly related to a probabilitydistribution function for solvent around the central ion hydrated, (x w ) k 2 / ln(x w ) [20].
However, the function is also linearly related to the molal concentration exponential, m k 2 , for various valent types of single electrolytes, such as uniunivalent, biunivalent (or unibivalent), triunivalent (or unitrivalent), tetraunivalent, bibivalent, and tribivalent ones.The certain examples are given in Figure 1, where it can be seen that the correlation coefficients for different valent types of single electrolytic solutions are marvelous.

The Osmotic Coefficients for Single Electrolytes
Let For single electrolytic system, as stated above, where 1 is slope and 2 , intercept.
Combination of Eq. ( 2), Eq. ( 3), and Eq. ( 5) gives This is an equation (called as the h function model) for obtaining the osmotic coefficients for single electrolytes.Now, we give some examples for different valent types of single electrolytes.The source data are taken from Ref. [16] under no special statement and others from Refs.[21][22][23][24][25][26][27][28][29][30].The k 1 and k 2 are obtained and optimized by a gradual iteration with "Mathcad 14.0" software, and the 1 and the 2 are automatically identified subsequently.All parameters (k 2 , k 1 , 1 and 2 ) and errors are listed in Tables 1-6, respectively.In order to evaluate the errors, one gives the variance, as follows, where N is the number of datum points, and cal are, respectively, the osmotic coefficients determined experimentally and calculated by the models such as Pitzer or Eq.(6).

Reproduction of another Properties
Eq. ( 6) can be also related to some other properties which are dependent on the concentration, such as the relative molal vapor pressure lowering, , equivalent conductivity, and solubility.
Take and of KCl for example.Replace in Eq. ( 6) by or .All the original data are taken from Ref. [16].The parameters and errors are in Table 7.
The average relative deviations (ARD) of them are all much less than 10 -3 , showing the excellent reproduction.The thermodynamic properties of electrolyte, like the boiling point elevation, lowering of freezing point, etc., are related to the change in the vapor pressure, so these could be reproduced well based on this linear rule.
The present model can also be used to reproduce the solubility accurately.Take the solubility of MgCl 2 in the HCl-LiCl-MgCl 2 -H 2 O mixture at 273K for an example [31].The chemical potential of MgCl 2 in the quaternary system changes no longer owing to its saturation, not disturbing the ion distribution.The concentration of the mixture is simply considered as m mix = (2m HCl + 2m LiCl ) and x H 2 O = n w / (n w + m mix ) .The results by the present model and a BET equation improved by Ref. [32] are listed in Table 8, where it can be seen that the present results are better than the improved BET.

DISCUSSIONS
In this paper the functions and m k 2 are introduced.Similar to the conventional that is considered as a potential energy, the can also be thought as a dimensionless-thermodynamic potential [20].The h 1 = m k 2 can be seen a comprehensive effects of a variety of forces.These effects, such as the longrange, the short-range and the higher-order forces, in general, are simply summarized together to describe the total interionic interactions.Actually, they do not include all effects, e.g. the interactions between different forces.Suppose that all forces and their interactions are taken into account to be related to the ionic strength exponential, I k 2 .It could represent a whole outcome of the density and the intensity of all the ions.Therefore, we can say that the I k 2 integrates a  various effects, such as the Coulomb's force between ions, the action of ions with solvent particles, the higher-order interactions, even the deformation and polarization of ions, hydrogen bond, etc., giving a comprehensive upshot that could govern the thermodynamic behavior of electrolytes.It could be considered to be a dominant factor that represents the whole effects influencing electrolyte behavior.
Certainly, the term, I k 2 could be also seen to be a distribution of all ions in the ion atmosphere.In Debye-Hückel theory, a binomial is taken into account as the    radial distribution function of ionic atmosphere surrounding a centre ion.Pitzer extended it to a trinomial, constructing a quadratic equation of the molal concentration of solute.According to the statistic physics, the properties of system are dependent on the distribution of particles.The characteristics of electrolytes could be indentified with the aid of I k 2 .In this paper, the term, m k 2 is used instead of I k 2 , this is because the ionic strength is directly proportional to m k 2 for each kind of the single electrolyte.Therefore, m k 2 is used to replace I k 2 for all single electrolytes.
In the function, k 1 is a constant to be confirmed.In most cases, k 1 is close to one.As the valence of electrolyte or the ion strength increases, k 1 deviates from one further.For the high-dilute uniunivalent electrolytes, k 1 is more close to one.The deviation of k 1 from one could be used to judge the ideality of electrolyte solution.Due to the total effects of all forces concerning a lot of factors, the regularity of the change in k 2 is not followed.
It should be pointed out that the linear rule of against m k 2 exists only in a given concentration interval.For a certain electrolyte from the high-dilute to the saturated solution, the entire concentration range is too wide to expect one linear equation of against m k 2 to resolve all problems.Here, it needs to section whole range into two or three concentration subintervals.In every subinterval, there is a corresponding linear relation of against m k 2 .This just means that the increase in concentration would alter the total interionic interactions, and makes the electrolyte deviate ideality more far.As the concentration increases, k 1 would be gradually far from one.Take HNO 3 for example.The concentration range is from 0.001 to 28 molal, and the osmotic coefficients are taken form Ref. [21].The entire concentration range is divided into three subintervals, i.e. 0.001 to 0.1, 0.1 to 12 and 12 to 28 in molality.And construct three linear equations of against m k 2 in these subintervals, respectively.The parameters are listed in Table 9, and the reproduced results are in Figure 2. Obviously, the narrower the subintervals are taken, the more accurate the reproduced results will be.
Many properties of electrolyte solution, as shown above, can be reproduced by Eq. ( 6) as long as is replaced by corresponding properties, such as , , etc., showing the these properties are also linearly related to the ionic strength or the molal concentration exponential for single electrolytes.Because of the equivalence of m and x , construct h 2 = x k 2 .We discover that the potential is also linearly related to h 2 for a various valent type of electrolytes.If taking 'the Shannon information entropy' of h 1 or h 2 [33] i.e. h 3 = h 1 ln(h 1 ) or h 4 = h 2 ln(h 2 ) .The function is also linearly related to h 3 and h 4 , respectively.The calculation parameters and errors are listed in Tables 1-8, respectively.According to the errors, we can see that the results by the present model are better than those by Pitzer for most of uniunivalent, biunivalent, triunivalent, tetraunivalent and bibivalent electrolytes, indicating that the linear rule is feasible to predict some properties for electrolytic solutions.
It should be pointed out that, at first glance, it seems that there are four parameters k 1 , k 2 , 1 and 2 to describe the properties of system, but 1 and 2 are not dependent.They are automatically determined as k 1 and k 2 are optimized and attained.

SUMMARY
As is well-known, all thermodynamic properties of an electrolyte solution must be a comprehensive effect of the interactions of various particles in the solution.It relates to all kinds of potentials of these particles, such as those between ions, between solvent and ions, between solvent particles, and their interactions.This could be represented by a dimensionless potential .It, substantially, is an osmotic coefficient on mole fraction base.The potential can integrate the effects of various particles, and describe the macroscopic properties of the electrolyte solution well.
The h functions such as m k 2 , x k 2 and their respective 'Shannon information entropies' m k 2 ln(m k 2 ) , and x k 2 ln(x k 2 ) are all related to the micro-distribution of solute and solvent molecules.This never was taken into account before.
It is found that the dimensionless potential for expressing the integral interactions is linearly related to these h functions, respectively.Based on this pertinence, the models are established for analytical expression of some thermodynamic and transferring properties of electrolytes, such as osmotic coefficient, activity of solvent, and relative molal vapor pressure lowering, equivalent conductivity and solubility of a salt in electrolytic mixture.The models are suitable for the electrolyte solutions with different valence.

Table 1 ). Continued.
a is variance calculated on Eq. (7).b p represents Pitzer model (the same below).c This datum is according to Extended Pitzer equation.

Table 2 ). Continued.
a This datum is according to Extended Pitzer equation.

Table 7 : Parameters and Errors for Reproduction of the Relative Molal Vapor Pressure Lowering, and the Equivalent Conductivity, of KCl Solution at 298K by the h Function Model
is the number of data points, Q and Qcal are, respectively, or determined experimentally and calculated by the h function model.