Taking into Account Density Fluctuations in a Solvent in a Model of Dissolution

Earlier it was shown by different authors that there are cavities (vacancies, holes) in any liquid. The cavities should play a prominent role in dissolution processes. Nevertheless this fact was ignored in previous model of dissolution. The sizes of the cavities in different solvents containing benzene molecules were determined using solvent induced spectral shift method. The measurements of S1←S0 benzene transition spectral shifts permit to conclude that 1) macroscopic excess volumes play an almost negligible role in processes of benzene dissolution in very different solvents and 2) the minimal size of the cavity in water able to accommodate benzene molecule coincides with the solute size. Generalization of this conclusion to other nonpolar aromatics leads to evaluation contraction of the solutes under aqueous solvent influence permits to predict the solubility values of other aromatics in water and to evaluate effect of enhancement hydrate cell around these molecules on solubility.


Introduction
A question on quantitative prediction of solubility stands in front of scientists almost since ancient times. Nevertheless the first attempt to use quantitative parameters for qualitative prediction of solubility was done by Hildebrand in the middle of the twentieth century [1]. He introduced parameter named density of is called pseudo chemical potential. It is not connected with any standard state. The term c G is free energy of creating the cavity and i G is free energy of interaction between the solute and the solvent. The term c is concentration expressed in mole shares, k B is the Boltzmann constant, and T is temperature.
This equation looks correct. Nevertheless successes of its direct application are very modest (look for example, ref. [4] and references therein). The cause is simple enough. Still between the first and the second world wars Frenkel and slightly later Schottky [5] proved that cavities (holes, vacancies) must exist even in the most ideal crystals. Naturally, they must exist in liquids. The fluctuations, namely fluctuation cavities, should participate in the process of dissolution. Luck [6] gathered a lot of indirect evidence that some kind of cavities really should exist in any liquid. A problem was how to measure these cavities, especially those of them which participate in the dissolution. The problem can be solved at least in part using the solvent induced spectral shift method. The method is in essence one of reverse spectroscopic problems when the shift of electronic spectrum of dissolved molecules serves a basis for decision of a question: how solvent molecules are distributed around the solute.

Model
It is convenient to adopt the simplest model of the solution at least for a beginning. The solvent is considered as continual dielectric with dielectric constant ε v and refractive index n. The solute is represened by sphere of radius r which is determined according to the Dejardin et al. procedure [7]: dependence should be built of molar volume of the solute substance in liquid state on its fluidity at different temperatures 0 V , the volume at the fluidity equal to zero, is connected with r by equation Here k pac is packing factor. It equals to 1.88 for molecules whose shape does not sufficiently differ from spherical ( Figure 1).

Interactions in the Solution and the Spectral Shift
There are two sorts of interactions in a dilute aqueous solution of nonpolar substance: solute-solvent and solvent-solvent interactions. Only the first one affects directly the spectral shift whereas the second of them affects indirectly participating in organization distribution solvent molecules around the solute. An electronic spectrum can be used for study the solution structure rather than vibrational one because in contrast to vibrational spectrum it belongs to whole molecule rather than to some of its fragments. Let the simplest case of the solute will be considered, when the solute is nonpolar molecule, and let electronic absorption of the solute is far from the solvent edge of the solvent absorption and let solute electronic states are mutually independent. Then the shift of a purely electronic or electronic-vibrational (vibronic) band in the transfer of a molecule from the gas phase to the solution may be considered as sum of the different contributions: Here ν ∆ is the shift of the spectral band expressed in wave numbers, where C is a positive coefficient depending on properties of the transition in consideration, ( ) ( ) is geometrical factor where R is radius of the cavity containing the solute molecule whose radius is r, 1 2 f n n n = − + , n is refraction index [8].
It was shown in ref. [9] that there are neither electrical nor chemical interactions between the solute and the solvent in the aqueous solution of benzene. This fact will be used below at construction the simplest version of solubility model in which density fluctuations in the solute are taken into account.
This consideration is related to mutually independent electronic transitions.
When electronic states are connected by a vibration, then the low which describes the shift suffers changes [10] [11]. So the shift of S 1 − S 0 benzene transition is approximately expressed as ( ) where C 1 is the positive coefficient, is a correction for this coupling [12].
Solvents whose aromatic molecules contain oxygen make exciplexes with high-energy states of aromatic solutes [12]. This fact permits to solve reverse spectroscopy problem using only the most low-energy transitions at handling with such solvents.

Experimental Data
Experimental details including purification of substances, recording and measuring spectral shifts were done in refs. [9] [10] [11]. Data concerning solubilities are cited below.

Microscopic Balance of Volumes
The average size of cavities in the solvent able to participate in the dissolution process can be found from balance of volumes at dissolution:  Table 1.
It is readily seen from the

Minimal Radius of the Suitable Cavity
The minimal radius of the cavity yet participating in the process of benzene dissolution can be evaluated from the function of cavity size distribution in the solvent [5]: where c bv G is average free energy of the fluctuation cavity surface which participate in dissolution (the same indexes are used in Equation (6)). It may be expressed through microscopic surface tension [13] [14]: Here σ is the cavity surface area, γ is the macroscopic surface tension, and κ is the coefficient correcting the macroscopic surface tension to the microscopic one. It is expressed as [13]: where 1 σ is the area of the surface of the cavity created in the liquid as a result of removal of one of its molecules, and ( ) ( ) Here s P is pressure of saturated vapor and 1 V is the volume per one molecule in the liquid. For associated liquids ( ) ( ) where ξ is average degree of association of vapor molecules [15]. Now V bv can be expressed as Here m R is the radius of that minimum cavity which is still good for acceptance the solute, 71.95 mN m γ = [16], 1 κ ≈ [15]. Hence benzene molecular radius only about 3%. This is too low difference for our crude model of solution. Therefore we may think that the minimal size of the cavity in water able to take the benzene molecule coincides with benzene molecule size.
This conclusion is extended further to other big nonpolar solutes.

Fluctuation Approach
If the similar coincidence takes place also for other nonpolar big rigid nonpolar molecules, one has One obtains after substituting the right side of Equation (15) into Equation Here prim numbers the approach to evaluation the solubility. The results obtained with this approach are given in the third column of Table 2. They are not very significantly deviated from empirical data. Note that this approach is not connected even with phase states of solution components.

Energetic Approach
Free energy of molecular transfer out of the fixed position in the substance which will be dissolved into the fixed position in vacuum and then into the fixed position in the solvent, * µ ′′ , is considered in the second approach called energetic one. In the idealized case when solute properties do not change at these transitions, and Equation (18) In essence, * µ ′′ is pseudo chemical potential of transfere a molecule out of the condensed substance liable to dissolution into the solvent. It is described with the next equation: Thus it is the quasi chemical potential. The double prim numbers the approach to evaluation * µ . The data on calculated solubilities are given in the fourth column of Table 2. They also are close enough to empirical results.
When the solute molecule is transferred out of a solid phase into the solvent then Equation (23) should be specified. Zhang and Gobas [17] supposed that a surface molecule of a solid substance dissolving in a liquid is bound with other ones in that manner as molecules of super cooled liquid. Then ( )

United Approach
Corrections to solute size changes should be introduced in both two above described approaches. Correcting term δµ′ to the quasi-chemical potential * µ ′ is caused by the molecular size decreasing because of pressing by reaction field forces [9]. It can be found from the next expression:  The values of * µ with values of r δ are given in the fifth column of Table 2.
Firstly, it is readily seen from the

Taking into Account Solvent Shell Strengthening
Interaction between water molecules in hydrate shell of benzene molecule is more strong then in pure water. Really, ions K + and Cldestroy water structure, i.e., weaken interaction between the molecule and other water molecules [19].
Nevertheless, addition salt KCl into aqueous solution of benzene does not destroys its hydrate cell, in contrast to addition such salts as RbCl and CsCl [9] which are more actively then KCl [19].  . One can readily see from Equation (34) that interaction between water molecules in the first hydrate shell is enhanced owing to interaction with the solute and lowering the solvent free energy per one its molecule in the solvent shell equals approximately 0.2kT. The low value of the free term in the right side of Equation (34) witness that adopted approximation is correct (see Table 3).

Conclusions
The above consideration clearly shows that fluctuations of density such as vacancies (holes, cavities) in diverse solvents should be taken into account at evaluations solubility of different solutes. This fact leads to paradoxy at the first glance conclusion that the excess volume plays a very modest role in microscopic balance of volumes at dissolution process, at least, in considered cases.
The calculated values of solubility based on values of cavities obtained from spectral shifts data are close enough to empirical ones, and evaluated size  changes, and change of strength of hydrogen bonds in the solvent near the solutes look right. So, the main points of the above consideration are correct. In essence, the called conclusions are obtained owing to taking into account, evidently or not evidently, effect of electric field described in ref. [9]. Hence such effect should be taken into consideration in advanced models of solubility, including computer simulation.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.