The Cavity Model

In the models discussed thus far in this section, emphasis has been placed on electrostatic effects and solvent polarity. An alternative view that to some extent takes other forces into account begins with the idea that, in order to dissolve a solute molecule in a solvent, energy is required to create a cavity in the solvent the solute is then inserted into this cavity. In Section 8.2 we saw that the energy to create a cavity can be expressed as a product of the surface area of the cavity and the surface tension of the solvent. An equivalent expression is obtained as the product of the volume of the cavity and the pressure exerted by the solvent, and we now explore this concept. 

The pressure exerted by the solvent is called its internal pressure it, and it is defined by Eq. (8-34), where E is molar energy and V is molar volume. 

The internal pressure is a differential quantity that measures some of the forces of interaction between solvent molecules. A related quantity, the cohesive energy density (ced), defined by Eq. (8-35), is an integral quantity that measures the total molecular cohesion per unit volume. - p 

(8-35), Afvap is the molar energy of vaporization, and AH p is the molar heat of vaporization. In effect, -it is a measure of the energy required to break some of the solvent-solvent forces, whereas ced is a measure of the energy required to 

This approach to solution chemistry was largely developed by Hildebrand in his regular solution theory. A regular solution is one whose entropy of mixing is ideal and whose enthalpy of mixing is nonideal. Consider a binary solvent of components 1 and 2. Let i and 2 be numbers of moles of 1 and 2, 4 , and 4 2 their volume fractions in the mixture, and Vi, V2 their molar volumes. This treatment follows Shinoda.  

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We will return to the cavity model in Section 8.3, giving a description in terms of the so-called solubility parameter. 

The interpretation of these remarkable properties has excited considerable interest whilst there is still some uncertainty as to detail, it is now generally agreed that in dilute solution the alkali metals ionize to give a cation M+ and a quasi-free electron which is distributed over a cavity in the solvent of radius 300-340 pm formed by displacement of 2-3 NH3 molecules. This species has a broad absorption band extending into the infrared with a maximum at 1500nm and it is the short wavelength tail of this band which gives rise to the deep-blue colour of the solutions. The cavity model also interprets the fact that dissolution occurs with considerable expansion of volume so that the solutions have densities that are appreciably lower than that of liquid ammonia itself. The variation of properties with concentration can best be explained in terms of three equilibria between five solute species M, M2, M+, M and e ... 

The redox potential values of all metal atoms, except alkaline and earth-alkaline metals [60], are higher than that of °(H20/eaq) = —2.87 V he- However, some complexed ions are not reducible by alcohol radicals under basic conditions and thus ii°(M L/ M°L)< —2.1 Vnhe (Table 2). The results were confirmed by SCF calculations of Ag L and Ag L structures associated with the solvation effect given by the cavity model for L = CN [61] or NH3 [47], respectively. 

While the conductivity decrease may be explained adequately by the Becker, Lindquist, and Alder model (2), and while the magnetic features may be explained adequately by the cavity model proposed by Ogg (19), neither model will serve to explain all the properties of these solutions. 

We wish here to inquire if a molecular view of the basic e iv species might be taken. We shall postulate, as in the cavity model, that the solvated electron is removed from the metal cation, but we shall assume instead a molecular model—one in which several solvent molecules are bound, forming the basic unit. 

That observed volume expansion accounts for the cavity models developed by theoreticians to describe the solvated electron that is, the electron occupies a cavity or void in the solvent and is surrounded and solvated by the solvent molecules. 

T. Fox and N. Rosch,/. Mol. Struct. (THEOCHEM), 276, 279 (1992). On the Cavity Model for Solvent Shifts of Excited States—A Critical Appraisal. 

Attempts have been made to distinguish between these theories on the basis of the and A5 values anticipated for the two theories, but it may be illusory to think of them as independent alternatives. The cavity model has been criticized on the basis that it cannot account for certain observations such as the denaturing effect of urea, but it must be noted that the cavity theory includes not only the cavity term A/ly, but also a term (or terms) for the interaction of the solutes and the solvent. A more cogent objection might be to the extension of the macroscopic concepts of surface area and tension to the molecular scale. A demonstration of the validity of the cavity concept has been made with silanized glass beads, which aggregate in polar solvents and disperse in nonpolar solvents. ... 

Recently, a new category of methods, the cavity model, has been proposed to account for the solvent effect. Molecules or supermolecules are embedded in a cavity surrounded by a dielectric continuum, the solvent being represented by its static dielectric constant. The molecules (supermolecules) polarize the continuum. As a consequence this creates an electrostatic potential in the cavity. This reaction potential interacts with the molecules (supermolecules). This effect can be taken into account through an interaction operator. The usual SCF scheme is modified into a SCRF (self consistent reaction field) scheme, and similar modifications can be implemented beyond the SCF level. Several studies based on this category of methods have been published on protonated hydrates. They account for the solvent effect on the filling of the first solvation shell (53, 69), the charges (69, 76) and the energy barrier to proton transfer (53, 76). 

The cavity model of solvation provides the basis for a number of additional models used to explain retention in reversed-phase chromatography. The main approaches are represented by solvophobic theory [282-286] and lattice theories based on statistical thermodynamics [287-291]. To a lesser extent classical thermodynamics combining partition and displacement models [292] and the phenomenological model of solvent effects [293] have also been used. Compared with the solvation parameter model all these models are mathematically complex, and often require the input of system variables that are either unknown or difficult to calculate, particularly for polar compounds. For this reason, and because of a failure to provide a simple conceptual picture of the retention process in familiar chromatographic terms, these models have largely remained the province of the physical chemist. 

In this section, the features of the various models proposed for metal-ammonia solutions, ever since their discovery by Weyl in 1864, will be considered. As test of the different models, only the binding energy of the electrons as predicted by different models will be compared with experiment. Comparison of the theoretical predictions concerning the optical, electrical, magnetic and other properties with experiment will be postponed until Section III. In Part II-A we shall very briefly consider some of the earliest theories and in Part II- B the cavity model, which led to the more refined polaron model. The polaron and cluster models are currently the most successful ones and they shall be discussed in Parts II-C and II-D while in Part II-E, we shall consider the unification of the polaron and cluster models which has been recently proposed by Symons and others. However the importance of the unified picture can be fully appreciated only after a consideration of the other properties in Section III. 

Alternatively, one could assume that the atoms are completely dissociated at all concentrations and the electrons behave like a free-electron gas. This would seem attractive from a consideration of Muster s data on saturated sodium-ammonia solutions. Mow-ever the observation by Muster in dilute sodium-ammonia solutions and by Freed and Sugarman in dilute potassium-ammonia solutions that the atomic susceptibility tends to (ji jkT (and not 2/3 (i jkT as expected from free-electron gas model) is evidence against the free electron gas model. The strongest point against the free-electron gas model is the finite photoelectric threshold observed for these solutions by Masing and Teal, which indicates that the electron is not free but is bound to some center. The next step is the cavity model to be discussed in the next section. 

The polaron model is properly an extension of the primitive cavity model. In both, the electron is considered to be solvated by a number of ammonia molecules. However, in the cavity model one considers the localization or solvation of the electron as described by a cavity of some shape whose boundaries act as limiting points for the potential or the electronic wave function. In the polaron model, on the other hand, the electron is considered to polarize the surrounding ammonia molecules in such a way as to provide a trapping potential for itself. The potential is derived from the laws of electrostatics adapted to the quantum mechanical description of the electron density in terms of the electronic wave function. In the final development of the theory one would of course, require self-consistency between the wave function of the electron and the potential in which it moves. It is possible that the end result may indicate that the electronic wave function is in fact almost localised within a definite volume of certain shape. However, no such assumption is made a priori as in the cavity model. 

For the purposes of this discussion, the primitive cavity model and the polaron models will both be considered as one and referred to as the cavity model. However, we shall point out at various points instances where a certain property depends sensitively on the detailed difference between the two types of cavity model. One can loosely say that those properties of the solutions which seem to be insensitive to the nature of the metal support the cavity model while those which depend on the metal provide support to the cluster model. However, a truly consistent explanation of all the observed properties requires the unified model described in Part (2-E). The importance of the unified model will become clear as we proceed successively with the explanation of the properties listed in Section I. 

The explanation of the optical band is not easy using the cavity model. Jortner showed that excited states higher than 2p are stable for the polarons in metal-ammonia solutions and deduced the following approximate relation for the energy values of higher hydrogen-like states with n > 2,... 

The explanation of the line-width of the resonance and its variation with concentration and temperature and, what amounts to the same thing, of the relaxation times, was first proposed by Kaplan and Kittel and later modified by Poliak. They base their explanation on the cavity model. Considering the h5q>erfine interaction of the unpaired electron with the nuclei in the immediate neighborhood of the cavity, Kaplan and Kittel deduced the follow-... 

The qualitative explanation of the observed relaxation time data in dilute metal-ammonia solutions in terms of the cavity model is thus quite satisfactory. One does not have to consider the monomers in solutions with concentration equal to and less than O.OIM because the monomer concentration is lower by about a factor of... 

Explanation of Nuclear Magnetic Resonance Data. The observation of a finite Na resonance shift by McConnell and Holm is one of the strongest points proving the existence of cluster monomers. The shift can on the other hand be explained by both the cavity and cluster models. The lack of any appreciable shift in the proton resonance has been explained quantitatively using the cluster model alone. No attempt has yet been made to explain the proton resonance shift with the cavity model. We shall now consider these explanations in detail. 

For the cavity model, the Na+ ions are completely separated from the electron cavities, and so there should not be any unpaired electron density at the Na nuclei. So no shift in the Na resonance is to be expected. In the cluster monomer the electron moves around the Na+ ion albeit in a more expanded orbital - > than the sodium atom 3s orbital. An earlier semi-quantitative explanation of the Na shift was attempted at by McConnell and Holm using ex-... 

For the cavity model, a careful analysis of the two-electron polaron is necessary to test the correctness of the conclusion that it is unstable. Also, for the one-electron polaron, the orthogonality of the unpaired electron wave function to the wave functions of the electrons on neighboring ammonia molecules has to be considered carefully for possible contributions to the observed and proton Knight shifts in N.M.R. measurements. 

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