Solute-sized cavities

While formally exact, Widom s formula is of practical use only for small solutes up to around the size of methane or xenon. This limitation exists because the value of e ) is dominated by relatively few low-energy insertions where there is little or no overlap between the solute and solvent molecules. Sampling of these low energy states requires the spontaneous formation of solute-sized cavities in the solvent, an event that is common for very small solutes but becomes exceedingly rare as solute size increases. 

Figure 5 shows pn distributions for spherical observation volumes calculated from computer simulations of SPC water. For the range of solute sizes studied, the In pn values are found to be closely parabolic in n. This result would be predicted from the flat default model, as shown in Figure 5 with the corresponding results. The corresponding excess chemical potentials of hydration of those solutes, calculated using Eq. (7), are shown in Figure 6. As expected, /x x increases with increasing cavity radius. The agreement between IT predictions and computer simulation results is excellent over the entire range d < 0.36 nm that is accessible to direct determinations of po from simulation.

Obviously, the availability of large enough cavities decreases strongly with the size of the solute. We have already discussed another explanation of the relationship between permeation rate and solute size, based completely on solubility, i.e. the size-selectivity of solute partitioning in the ordered lipid-... 

Useful chemical reactions have been carried out in the nano-sized cavity, as illustrated by the in situ isolation of a labile cyclic siloxane trimer (Fig. 20.3.19). In the first step, three to four molecules of phenyltrimethoxysilane enter the cage and are hydrolyzed to siloxane molecules. Next, condensation takes place in the confined environment to generate the cyclic trimer SiPh(0H)0- 3, which is trapped and stabilized in a pure form. The overall reaction yields an inclusion complex [ SiPh(0H)0- 3 c Pt(bipy) 6L4](N03)i2-7H20, which can be crystallized from aqueous solution in 92% yield. The all-cis configuration of the cyclic siloxane trimer and the structure of the inclusion complex have been determined by NMR and ESI-MS. 

A chemical reaction can be viewed as a phenomenon dealing with molecular shape (topology) changes. Whether a particular reaction will take place will depend upon whether the product can fit within the space occupied by the reactant. The space occupied by the reactant is the reaction cavity. Since the boundaries of a reaction cavity are undefined in an isotropic solution, size matching of the reactant, products and the reaction cavity is not important in this medium. On the other hand, when the reaction cavity has a well-defined boundary, as in most organized assemblies (especially in solid state), size matching can become important and occasionally may even become the sole factor controlling the feasibility of a reaction (Fig. 8). 

The LSER approach relates a bulk property, P, to molecular parameters thought to account for cavity formation, dipole moment/polarizability, and hydrogen-bonding effects at the molecular level. The cavity term models the energy needed to provide a solute molecule-sized cavity in the solvent. The dipole moment/polarizability terms model dipole and induced dipole interactions between solute and solvent these can be viewed as related to dispersion interactions. The hydrogen-bonding terms model HBA basicity and EIBD acidity interactions. 

In Eq. [19], Vxi is the McGowan volume that models the energy needed to make a solute molecule-sized cavity in the solvent. Again, the subscript 2 denotes a solute molecule. The parameters ti and 82 account for dipolarity/ polarizability, and ai and pi model hydrogen bond (HB) acidity and basicity, respectively. This parameter set was used to correlate more than 250 biological, chemical, and physical properties successfully. ... 

In the work of Famini and Wilson,a molecular volume, Vmc, (units of 100 A ) is used to model the cavity term that measures the energy required to create a solute-molecule sized cavity in the solvent. The dipolarity/polarizability term, which attempts to account for dispersion-type interactions, is modeled by the polarizability index, tij, (unitless). This index is defined as the average molecular polarizability divided by the molecular volume, a/Vmc, and helps account for the correlation between polarizability and molecular volume. 

The information theory approach studied here grew out of earlier studies of formation of atomic sized cavities in molecular liquids (Pohorille and Pratt, 1990 Pratt and Pohorille, 1992 1993). Since we deal with rigid and spherical solutes in the discussion we will drop the explicit indication of conformational coordinates and discuss p n) = Pa n lR ). We emphasize that the overall distribution p(n) is well described by the information theory with the first two moments, (n)o and n n — 1)/2)q. It is the prediction of the extreme member p 0) that makes the differences in these default models significant. Computing thermodynamic properties demands more than merely observing typical behavior. 

B. Mechanical Relaxation Theory Berg suggested that a change in the solute size on excitation could produce a significant solvent response.[6,8] He treated the solute as a spherical cavity of radius rc ttnd the solvent as a viscoelastic continuum with time-dependent compression and shear moduli AT(t) and G(t). For a spherically symmetric change in cavity size, two time scales are relevant. 

The cavity term is a measure of the endoergic cavity-forming process, that is, the free energy necessary to separate the solvent molecules, overcoming solvent-solvent cohesive interactions, and provide a suitably size cavity for the solute. The magnitude of the cavity term depends on the —> Hildebrand solubility parameter 5na.nd volume descriptors ofthe solute. The solute volume... 

SP is some free energy related solute property such as a distribution constant, retention factor, specific retention volume, relative adjusted retention time, or retention index value. Although when retention index values are used the system constants (lowercase letters in italics) will be different from models obtained with the other dependent variables. Retention index values, therefore, should not be used to determine system properties but can be used to estimate descriptor values. The remainder of the equations is made up of product terms called system constants (r, s, a, b, I, m) and solute descriptors (R2,7t2, Stt2, Sp2 log Vx). Each product term represents a contribution from a defined intermolecular interaction to the solute property. The contribution from cavity formation and dispersion interactions are strongly correlated with solute size and cannot be separated if a volume term, such as the characteristic volume [Vx in Eq. (1.6) or V in Eq. (1.6a)] is used as a descriptor. The transfer of a solute between two condensed phases will occur with little change in the contribution from dispersion interactions and the absence of a specific term in Eq. (1.6) to represent dispersion interactions is not a serious problem. For transfer of a solute from the gas phase to a condensed phase this... 

As discussed above, solvation free energy is t3q)ically divided into two contributions polar and nonpolar components. In one popular description, polar portion refers to electrostatic contributions while the nonpolar component includes all other effects. Scaled particle theory (SPT) is often used to describe the hard-sphere interactions between the solute and the solvent by including the surface free energy and mechanical work of creating a cavity of the solute size in the solvent [148,149]. 

In the example presented here the reverse micelle was prepared by dispersion of the surfactant, sodium bis(2-ethyl hexyl) sulfosuccinate (AOT) in isooctane. The hydrolysis and sol-gel processing of titanium isopropoxide was carried out in the 5 nm size cavity of the reverse micelle to produce highly uniform Ti02 nanoparticles. These particles were redispersed in a polymer (polyimide) solution and cast as a film of the polymer composite containing Ti02 nanoparticles. 

Thus differences in gA may arise from differences in solute polarity, acting through g. But A may itself change, rather obviously as a result of solute size, but also as a consequence of change of solvent, for the solvent size and geometry will affect die shape and size of the cavity that houses the solute. 

Before elaborating on a specific example involving real solutes, it is instructive to work out a simple case. Consider a solvent of N hard spheres of diameter aw The solute particles are also hard spheres of diameter ag. Clearly, the distance between these two solute particles cannot be smaller than ag. However, if we focus our attention on the function SA(R) and since we know that this function is smooth and finite in the entire region 0 < R < oo, we can inquire about the value of SA(R) for any distance R < a. This point of view is equivalent to replacing the two solute particles by two cavities of appropriate size. While a system of two HS particles in a solvent is certainly different from a system with two cavities, from the point of view of the solvent the two systems are identical. In contrast to the two HS solutes, two cavities can be brought to any distance including R = 0. For this limiting case, expression (4.4.5) reduces to... 

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