Spherical solution

The constants K depend upon the volume of the solvent molecule (assumed to be spherica in slrape) and the number density of the solvent. ai2 is the average of the diameters of solvent molecule and a spherical solute molecule. This equation may be applied to solute of a more general shape by calculating the contribution of each atom and then scaling thi by the fraction of fhat atom s surface that is actually exposed to the solvent. The dispersioi contribution to the solvation free energy can be modelled as a continuous distributioi function that is integrated over the cavity surface [Floris and Tomasi 1989]. 

Use the model for the size exclusion of a spherical solute molecule in a cylindrical capillary to calculate for a selection of R/a values which... 

The resolution required in any analytical SEC procedure, e.g., to detect sample impurities, is primarily based on the nature of the sample components with respect to their shape, the relative size differences of species contained in the sample, and the minimal size difference to be resolved. These sample attributes, in addition to the range of sizes to be examined, determine the required selectivity. Earlier work has shown that the limit of resolvability in SEC of molecules [i.e., the ability to completely resolve solutes of different sizes as a function of (1) plate number, (2) different solute shapes, and (3) media pore volumes] ranges from close to 20% for the molecular mass difference required to resolve spherical solutes down to near a 10% difference in molecular mass required for the separation of rod-shaped molecules (Hagel, 1993). To approach these limits, a SEC medium and a system with appropriate selectivity and efficiency must be employed. 

The exclusion effect of hard-spheres is illustrated in Figure lA., which shows a spherical solute of radius r inside an infinitely deep cylindrical cavity of radius a. Here the exclusion process can be described by straightforward geometrical considerations, namely, solute exclusion from the walls of the cavity. Furthermore, it can be shown thatiQJ... 

Solute Flux Solute partitioning between the upstream polarization layer and the solvent-filled membrane pores can be modeled by considering a spherical solute and a cylindrical pore. The equilibrium partition coefficient 0 (pore/bulk concentration ratio) for steric exclusion (no long-range ionic or other interactions) can be written as... 

Transport in a microporous biomedical membrane is described in Fig. 11. Membranes consist of cylindrical liquid-filled pores of length l and radius rp with spherical solute molecules of radius rs diffusing through the pores. The solute... 

In Figure 54, the friction coefficient of a Lennard—Jones fluid containing a spherical solute of mass fifty times that of the solvent molecules is shown following Brey and Ordonez [529]. 

The use of the Stokes-Einstein equation (2) relating the diffusion coefficient (D) of a spherical solute molecule to its radius (r), the viscosity of the medium (tj) and the Boltzmann constant (k) permits the rate coefficient ( en) to be expressed in (3) in terms of the viscosity of the medium. In this derivation, the... 

A three-zone DC model for treating electrochemical ET at a self-assembled monolayer (SAM) film-modified metal electrode surface [49] is displayed in Figure 3.27, where zones I, II, and III, defined by parallel infinite planes, correspond, respectively, to an aqueous electrolyte, a hydrocarbon film, and the metal, and the ET-active redox group is represented by a point charge shift (Aq) in a spherical solute cavity [22]. The Poisson equation has been solved for this system, and the results analyzed in terms of image charge contributions to As [22] (see below). 

For large spherical solute molecules or large spherical particles, the medium molecule has no tendency to slip at the surface of the solute molecule. Then, p becomes very large, resulting in ... 

Figure 18. A cage of water molecules surrounding a spherical solute particle (redrawn from Stillinger, 1973).

Molecular simulation methods provide an acceptable picture of the solvent structure around a solute. For small spherical solutes, the solvent structure can be represented by the radial distribution function (RDF), g(r), defined as... 

Example A certain gas has the following values of its critical constants Pc = 45.6 atm, Vm c = 0.987 dm3 moH and Tc = 190.6 K. Calculate the van der Waals constants of this gas. Also, estimate the radius of the gas molecules assuming that they are spherical. Solution ... 

Hydrocarbons in water give rise to hydrophobic solvation shells in which the water structure is thoroughly disturbed though still forming a solvation shell around a spherical solute. An example of a calculated situation of this type is shown in Fig. 2.69. 

Consider the case of a spherical solute (A) dissolved at infinite dilution in a molecular solvent (S), and show that the partial molar volume - see Eq. (4.100), p. 97 - can be expressed as... 

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. 

Figure 8.3 Convergence with number of binomial moments of hydration free energy predicted using several default models for a spherical solute with distance of closest approach 3.0 A for water oxygen atoms. Identifications are diamonds (dash-dot lines), hard-sphere default HS), crosses (short dash line), Lennard-Jones LJ) default squares (long dash line), Poisson default triangles (dotted line), cluster Poisson default and circles (solid line), flat default. For this circumstance, yth-order binomial moments are non-zero through j = 9, and the horizontal line is the prediction with all nine moments included. Among the predictions at j = 2, the best default model is the Lennard-Jones case. But with the hard-sphere model excepted, the differences are slight. See Hummer et al. (1996), Gomez et al. (1999) and Pratt etal. (1999) for details of the calculations.

Figure 8.6 shows the modelled values of for spherical solutes as a function of temperature along the saturation curve of liquid water, and compares them to the chemical potentials computed directly. The quantitative agreement between the two methods is excellent over the entire temperature range. The chemical potential increases with temperature past 400 K but eventually decreases. The maximum in chemical potential occurs at about the same temperature in each case. These curves have the same shape as the experimental ones (Harvey et al, 1991) for inert gases dissolved in water, but they are shifted upward due to the use of a hard-sphere model. 

Following the notation of Section 7.6, specifically p. 160, A is the distance of closest approach of the solvent (water) center to the hard spherical solute. The left side of Eq. (8.25) is the differential work done in expanding the solute sphere against the solvent pressure. G (A), introduced on p. 121, Eq. (5.70), is the contact value of the radial distribution function of solvent centers from the position of a hard-spherical solute. G (A) then gives molecular-scale structural information to obtain that solvent pressure, and Eig. 8.13 shows the current best information on that molecular-scale pressure (Ashbaugh and Pratt, 2004). 

We don t pursue a further detailed discussion of these results here, but confine ourselves to a few broad observations. Eirst, the theories and discussions above have focused on hard-spherical solutes of size located roughly by the maximum of G(A), Eig. 8.13. These solutes are candidates for most hydrophobic because the solvent pressure is greatest for those sizes. The location of that maximum gives a convenient size to discriminate between small and large molecule scales for these hydration problems. 

The virial coefficients reflect interactions between polymer solute molecules because such a solute excludes other molecules from the space that it pervades. The excluded volume of a hypothetical rigid spherical solute is easily calculated, since the closest distance that the center of one sphere can approach the center of another is twice the radius of the sphere. Estimation of the excluded volume of llexible polymeric coils is a much more formidable task, but it has been shown that it is directly proportional to the second virial coefficient, at given solute molecular weight. 

Standards commonly employed [5] to calibrate SEC columns do not have a well-defined size. Carefully characterized spherical solutes in the appropriate size range are therefore of considerable interest. The chromatographic behavior of carboxylated starburst dendrimers — characterized by quasi-elastic light scatter-... 

The word Targe here implies that the spherical solute molecules are large compared to the molecules of the solvent, so that it may be consid ered as a continuous medium in the hydrodynamic calculations, without serious error. 

The equilibrium partition coefficient, Keq, depends on the pore size in the porous particle as well as the size and conformation of the solute molecule. Giddings et al. [60] have shown that the characteristic size parameter for all shapes of molecules is the mean length of the molecular projection. For a rigid spherical solute this is simply the radius a. By considering the ratio of the area available for the solute molecule inside the cylindrical pore to the actual pore area, and neglecting any interaction between the solute and pore wall, we get... 

In order to evaluate the solute diffusion coefficient in the stationary phase, Ds, and the equilibrium partition coefficient, Keq, a model for the pore is required. A simple model where the pore is considered to be an infinitely long cylinder and the solute is a rigid sphere has been shown to be adequate in describing the elution process (25). The intrapore diffusivity, Ds, was estimated from the hydrodynamic theory of hindered diffusion for spherical solutes in cylindric pores (19) ... 

Solvent characteristics that influence the diffusion and extraction are found to be viscosity ( )) and polarity ( ). For spherical solutes, the diffusion coefficient depends on the solvent according to the Stokes-Einstein relation (Eq. (22)). From this, it follows that the diffusion coefficient linearly increases with T/t). Hence, the permeability increases linearly with the reciprocal viscosity of the membrane solvent [95]. Figure 2.11 shows relation of the diffusion coefficient to the solvent viscosity. 

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