_{1}

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.

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 [

An attempt to answer this question was undertaken by Ben-Naim in the middle of the second half of the twentieth century [

μ ∗ = G c + G i = − k B T ln c , (1)

is called pseudo chemical potential. It is not connected with any standard state. The term G c is free energy of creating the cavity and G i 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. [

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 [

k p a c r = ( V 0 ) 1 / 3 (2)

Here k_{pac} is packing factor. It equals to 1.88 for molecules whose shape does not sufficiently differ from spherical (

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:

Δ ν = Δ ν d i s p + Δ ν e l s t + Δ ν c h e m (3)

Here Δ ν is the shift of the spectral band expressed in wave numbers, Δ ν d i s p is contribution from dispersion interactions of the solute with solvent molecules, Δ ν e l s t is that part of the shift which is determined by interaction with sources of constant electric fields (ions, dipoles,…) in the solvent, and Δ ν c h e m is the term which corresponds to those chemical interactions including hydrogen bonding that do not change individuality of the solute molecule.

Minimum of free energy of dispersion interaction take place when the solute touches the cavity border. The share of the shift stipulated by this type of interactions is

Δ ν d i s p = − C φ ( R , r ) f ( n ) , (4)

where C is a positive coefficient depending on properties of the transition in consideration, φ ( R , r ) = R 3 / [ r 3 ( 2 R − r ) 3 ] is geometrical factor where R is radius of the cavity containing the solute molecule whose radius is r, f ( n ) = ( n 2 − 1 ) / ( n 2 + 2 ) , n is refraction index [

It was shown in ref. [

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 [_{1} − S_{0} benzene transition is approximately expressed as

Δ ν ( 10 ) = − C 1 | Δ ν 10 | 1.910 , (5)

where C_{1} is the positive coefficient, Δ ν 10 is the shift in the absence of electronic-vibrational coupling of the S_{1} electronic state with other ones, and index 1.910 is a correction for this coupling [

Solvents whose aromatic molecules contain oxygen make exciplexes with high-energy states of aromatic solutes [

Experimental details including purification of substances, recording and measuring spectral shifts were done in refs. [

The average size of cavities in the solvent able to participate in the dissolution process can be found from balance of volumes at dissolution:

V 1 u v = V a v = V 1 u + V b v + V E (6)

Here V 1 u v is average volume of that cavity in the solvent, (superscript v) which contains one solute molecule (subscript 1u), V a v is the same volume in the solvent obtained after removal a of its molecules, V 1 u is average volume per one solute molecule in the solute substance, V b v is average volume of those fluctuating cavities in the solvent which participate in the dissolution process, and V E is excess volume of mixing the solution components.

Solutions of benzene in different solvents can be considered as an instructive example. One obtains from Equation (5)

| Δ ν d i s p ( 10 ) | 1 / 910 = − k f ( n ) φ ( R , r ) (7)

where k = 2 824.9695422 cm − 1 and r = 2.72 × 10 − 10 m . The results of calculation microscopic balance of volumes at benzene dissolution in different solvents are listed in

It is readily seen from the

Next interesting consequence is seen from the Table. The excess volume plays negligible role In all considered cases. This fact supports conclusion made above that dissolution realizes through fluctuation cavities even in cases of infinite solubility, as for example in alkanes.

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 [

Solvent | n | − Δ ν ( 10 ) , cm − 1 | R × 10 10 , m | V E × 10 30 , m 3 | R b v × 10 10 , m | R ′ b v × 10 10 , m | R b v / R ′ b v |
---|---|---|---|---|---|---|---|

CCl_{4 } | 1.4607 | 433 | 3.279 | 0.030037 | 0 | 0 | |

CHCl_{3 } | 1.4459 | 386 | 3.329 | 1.4447 | 1.348 | 1.409 | 0.957 |

H_{2}O | 1.33299 | 145 | 3.815 | −11.342 | 2.890 | 2.777 | 1.041 |

D_{2}O | 1.33844 | 128 | 4.011 | −10.960 | 3.209 | 3.122 | 1.028 |

CH_{3}OH | 1.3288 | 210 | 3.408 | 0.1214 | 1.760 | 1.764 | 0.998 |

C_{2}H_{5}OH | 1.3610 | 220 | 3.515 | 3.9448 | 2.032 | 2.105 | 0.965 |

C_{3}H_{7}OH | 1.3850 | 234 | 3.563 | −0.10793 | 2.235 | 2.216 | 1.009 |

(CH_{3})_{2}SO | 1.4770 | 381 | 3.431 | −3.8325 | 1.932 | 1.846 | 1.047 |

c-C_{6}H_{12 } | 1.4262 | 368 | 3.306 | 4.5826 | 1.054 | 1.237 | 0.860 |

n-C_{5}H_{12 } | 1.3575 | 218 | 3.508 | 0.7415 | 2.072 | 2.086 | 0.993 |

n-C_{6}H_{14 } | 1.3751 | 238 | 3.502 | 2.7433 | 2.016 | 2.069 | 0.974 |

n-C_{7}H_{16 } | 1.3876 | 246 | 3.524 | 4.5033 | 2.048 | 2.130 | 0.962 |

n-C_{8}H_{18 } | 1.3975 | 254 | 3.535 | 4.649 | 2.078 | 2.160 | 0.962 |

n-C_{10}H_{22 } | 1.4102 | 257 | 3.577 | 3.6221 | 2.211 | 2.268 | 0.975 |

n-C_{14}H_{30 } | 1.4293 | 270 | 3.601 | 4.2366 | 2.263 | 2.327 | 0.972 |

1,4-С_{4}О_{2 } | 1.4224 | 276 | 3.554 | −0.16771 | 2.127 | 2.128 | 1.000 |

d p = d R R e x p [ − G b v c ( R ) / k B T ] (8)

where G b v c 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 [

G c = κ γ σ . (9)

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 [

κ ≅ 1 + ( σ 1 / σ ) ( κ 1 − 1 ) , (10)

where σ 1 is the area of the surface of the cavity created in the liquid as a result of removal of one of its molecules, and

κ 1 ≅ ( γ σ 1 ) − 1 k B T ln ( k B T / P s V 1 ) . (11)

Here P s _{ }is pressure of saturated vapor and V 1 _{ }is the volume per one molecule in the liquid. For associated liquids

κ 1 ≅ ( γ σ 1 ) − 1 k B T ln ( k B T / ξ P s V 1 ) , (12)

where ξ is average degree of association of vapor molecules [

Now V_{bv} can be expressed as

( R b v ) 2 = ∫ R m ∞ d R R 3 exp [ − G c ( R ) / k B T ] ∫ R m ∞ d R R exp [ − G c ( R ) / k B T ] = ( R m ) 2 + k B T 4 π κ γ (13)

Here R m is the radius of that minimum cavity which is still good for acceptance the solute, γ = 71.95 mN / m [

Let us consider low solubility of substance consisting of big nonpolar molecules, so low solubility that solute molecules do not touch each other. Let N is full amount of solvent molecules and n is amount of cavities able to accept a solute molecule. Then solubility is determined by

μ * / k T = − ln ( n / N ) = − ln c (14)

If the similar coincidence takes place also for other nonpolar big rigid nonpolar molecules, one has

ln ( n e / n ) = ln ∫ r e ∞ d R R exp ( − 4 π γ R 2 / k B T ) ∫ r ∞ d R R exp ( − 4 π γ R 2 / k B T ) = 4 π γ ( r e 2 − r 2 ) / k B T . (15)

One obtains after substituting the right side of Equation (15) into Equation (14):

μ * ′ = μ e * + 4 π γ ( r 2 − r e 2 ) (16)

Here prim numbers the approach to evaluation the solubility. The results obtained with this approach are given in the third column of

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,

μ * ′ ′ = G 1 u i v + G 1 u c v − G 1 u i − G 1 u c (17)

Here G 1 u i v is free energy of interaction (superscript i) of one solute molecule (subscript 1u) with the solvent (superscript v), G 1 u c v is free energy of creation the cavity in the solvent (superscript c) where the solute molecule can be placed, G 1 u i is free energy of interaction between the solute and its environment in the solute substance, and G 1 u c is free energy of creation the cavity instead removed solute molecule. The detailed Equation (17) looks as

Solute | 10^{10}r, m | μ / k T | 10^{10}δr, m | ΔN_{1} | |||
---|---|---|---|---|---|---|---|

Calculation | Experiment | ||||||

Equation (16) | Equation (22) | Equation (32) | |||||

benzene | 2.72 | - | 7.91 | - | 7.85 | - | 0 |

toluene | 2.895 | 9.85 | 8.97 | 9.41 | 9.16 | −0.04 | 3.6 |

o-xylene | 3.045 | 12.87 | 10.34 | 11.10 | 10.33 | −0.06 | 7.1 |

p-xylene | 3.055 | 12.01 | 10.03 | 11.02 | 10.31 | −0.08 | 7.4 |

m-xylene | 3.055 | 12.01 | 10.10 | 11.06 | 10.33 | −0.07 | 7.4 |

ethylbenzene | 3.04 | 11.80 | 10.00 | 10.90 | 10.26 | −0.07 | 7.0 |

cumene | 3.19 | 13.91 | 10.56 | 12.24 | 11.35 | −0.12 | 10.7 |

styrene | 2.97 | 10.85 | 10.51 | 10.68 | 9.86 | −0.01 | 5.4 |

naphthalene | 3.054 | 12.71 | 12.60 | 12.66 | 12.31 | −1·10^{-5 } | 6.4 |

μ * ′ ′ = G a v i + Δ G i + G a v c − G b v i − G b v c − G 1 u i − G 1 u c (18)

where G a v i is free energy of interaction between content of sphere which radius is R 1 u v before removal а solvent molecules out of it and the rest solvent, G a v c is free energy of this cavity surface, terms with subscripts bv and 1u describe similar characteristics of original vacancies in the solvent and components of that free energy which must be spend for removal one molecule out of the substance to be dissolved, respectively, and Δ G i is a correction which must be introduced into the process description after replacement claster of a solvent molecules for one solute molecule. According to [

G i = − 2 G c (19)

and Equation (18) becomes simplified to

μ * ′ ′ = − G a v c + G b v c + Δ G i + G 1 u c (20)

When size of the cavity which contains the solute molecule is held rather by induced electrostatic forces than by collisions at thermal movement, then equilibrium takes place at

− G a v c + G b v c + Δ G i = 0 (21)

Then

μ * ′ ′ = G 1 u c (22)

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:

μ * ′ ′ / k T = − ln C . (23)

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

When the solute molecule is transferred out of a solid phase into the solvent then Equation (23) should be specified. Zhang and Gobas [

μ * ′ ′ = k T ln ( R T / p s V 1 u ) − Δ G f (24)

Here Δ G f _{ }is a change of free energy at conversation the solute substance into the state of super cooled liquid which equals to

Δ G f = Δ H f ( 1 − T / T m ) + ∫ T T m Δ C p d T − T ∫ T T m Δ C p d T T (25)

Here Δ H f is enthalpy of solute substance fusion at melting point T m and Δ C p is change of specific heat at changing the phase state. For example, Eq. (25) may be used for evaluation that characteristic of naphthalene which is solid at the room temperature. One can evaluate function Δ C p ( T ) applying square-law extrapolation specific heat of the liquid phase taking necessary values from ref. [

Corrections to solute size changes should be introduced in both two above described approaches. Correcting term δ μ ′ to the quasi-chemical potential μ * ′ is caused by the molecular size decreasing because of pressing by reaction field forces [

μ * ′ = μ e * ′ + δ μ ′ = μ e * ′ + 4 π [ ( r + δ r ) 2 − r e 2 ] γ ≅ μ e * ′ + 4 π ( r 2 + 2 r δ r − r e 2 ) γ (26)

and

δ μ ′ ≃ 8 π γ r δ r . (27)

Such correction is the negative value because its sign coincides with the sign of δ r .

Correction to μ ″ in energetic approach looks as

μ ″ = μ * ′ ′ + δ μ * ′ ′ (28)

where δ μ * ′ ′ is correction which takes into account reversible positive work making by forces of hydrophobic (electric) repulsion which compress the solute molecule.

δ μ * ′ ′ ≈ 4 π γ ( R 1 2 − R 2 ) (29)

Here R 1 is radius of the cavity containing the solute in the case if it is not subjected to deformation, and R is the same after deformation. R can be expressed in quasi-spherical approximation as R = [ R 1 3 − r 3 + ( r + δ r ) 3 ] 1 / 3 . We get after expansion R in the Tailor series and taking into account that R ≈ 3 r / 2 , and neglecting the infinitesimal terms of decomposition, that

δ μ * ′ ′ ≈ − 8 π γ r δ r (30)

One obtains comparing Equations. (27) and (30) that

δ μ * ′ = − δ μ * ′ ′ (31)

It follows from Equation (31) that difference between values μ * ′ and μ * ′ ′ is caused only by contraction of substituent size under reaction (reactive field) of the aqueous solvent on the solute. Hence

μ * = ( μ * ′ + μ * ′ ′ ) / 2 (32)

Only deformation of the solute is taken into consideration at calculating μ * . The value of the solute size contraction is

δ r ≈ − Δ μ / ( 16 π γ r ) (33)

where Δ μ = μ * ′ − μ * ′ ′ .

The values of μ * with values of δ r are given in the fifth column of

Interaction between water molecules in hydrate shell of benzene molecule is more strong then in pure water. Really, ions K^{+} and Cl^{-} destroy water structure, i.e., weaken interaction between the molecule and other water molecules [

One can evaluate contribution of this enhancement into the μ* value comparing calculated values with measured ones. The corresponding correction equals to

Δ μ * / k T = − 0.214 Δ N 1 + 0.055 (34)

Here N 1 is amount of water molecules in the first hydrate shell of benzene and Δ N 1 is the change of this amount after transition to another solute. Approximately

N 1 = 4 π [ R + ( V 1 u ) 1 / 3 / 2 ] 2 / ( V 1 u ) 2 / 3 , (35)

where R ≈ 1.5 r . The correlation factor of dependence (34) equals to ρ = 0.943 , of root mean square deviation of coefficient at Δ N 1 σ A = 0.080 and of free term σ В = 0.028 . 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

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

Solute | 10^{10}r, m | ΔN_{1 } | μ / k T | 10^{10}δr. m | ||||
---|---|---|---|---|---|---|---|---|

Calculation | Experiment | |||||||

Equation (16) | Equation (24) | Equation (32) | With Equation (24) | |||||

α-methylstyrene | 3.145 | 9.6 | 13.30 | 11.27 | 12.29 | 11.47 | - | −0.07 |

mesitylene | 3.19 | 10.7 | 13.94 | 11.20 | 12.57 | 11.67 | 11.47; 12.72 | −0.10 |

pseudo-cumene | 3.19 | 10.7 | 13.94 | 11.10 | 12.52 | 11.62 | - | −0.10 |

hemellitene | 3.19 | 10.7 | 13.94 | 11.72 | 12.83 | 11.93 | 1152; 11.83 | −0.08 |

о-ethyltoluene | 3.19 | 10.7 | 13.94 | 11.21 | 12.58 | 11.68 | - | −0.09 |

m-ethyltoluene | 3.19 | 10.7 | 13.94 | 11.01 | 12.48 | 11.58 | - | −0.10 |

п-etyltoluene | 3.19 | 10.7 | 13.94 | 11.00 | 12.47 | 11.57 | - | −0.10 |

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. [

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

Ar’ev, I.A. (2019) Taking into Account Density Fluctuations in a Solvent in a Model of Dissolution. Open Journal of Physical Chemistry, 9, 204-215. https://doi.org/10.4236/ojpc.2019.94012