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Excluded Volume of a Sphere

In the above example of permeable spheres, we can define then an excluded volume of a sphere (which is actually the interaction volume ) as Vex = t2 of those spheres and thus replace the above equation for c by c = 1 q>( Bet/Vex). The... [Pg.152]

Excluded Volume of a Sphere The excluded volume makes the real chains nonideal. The dimension of the real chain is different from that of the ideal chain of the same contour length, for instance. Before considering the excluded volume effect in a chain molecule, we look at the effect in a suspension of hard spheres of diameter d. In Figure 1.33, the center-to-center distance between spheres A and B cannot be less than d. In effect, sphere B is excluded by sphere A. The space not available to the center of sphere B is a sphere of radius d indicated by a dashed line. Thus the excluded volume (Ve) is eight times the volume of the sphere. [Pg.34]

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... [Pg.554]

Inserting the geometrical factor 4 r/3 for the volume of a sphere we find that s 1 corresponds to a fictitious system filled with excluded volume coils ... [Pg.236]

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. [Pg.67]

The method is based on the use of a - probe given by the excluded volume of two spheres with different radii and an identical centre corresponding to the barycentre of the molecule. The probe is layered like an onion, each layer being the excluded voliune. The first sphere, i.e. the component of the probe, is an atom (e.g. an iodine atom with van der Waals radius of 2.05 A, a carbon atom with van der Waals radius of 1.52 A, or a hydrogen atom with van der Waals radius of 1.08 A) whose volume defines the first layer, then 60 atoms of the same type (e.g. iodine atoms) construct the second sphere whose surface is like a fullerene and which shares the same centre as the first sphere. The excluded volume between the first and second spheres defines the second layer. In the same way the subsequent spheres and layers are also defined. [Pg.464]

The excluded volume of a solute molecule is the volume that is not available (because of exclusion forces or for other reasons) to the centers of mass of other similar solute molecules. As an example, let us consider the excluded volume of a spherical particle of radius, R. The position of a sphere is fully described by coordinates of its center. It is apparent from Fig. 3.7(a) that the center of one solid sphere cannot approach the center of another solid sphere closer than two radii (2i2). Hence, the volume excluded by one sphere equals 7r(2ii), that is, eight times its actual volume. The excluded volume of asymmetric particles cannot be calculated so easily. This is because the distance between their centers of mass when they are in contact depends on their orientation. Nevertheless, it has been... [Pg.178]

Pressure is expressed as the difference between a repnlsive pressure (the first term on the right-hand side) and an attractive pressure (the second term). The repnlsive pressnre is dne to the forces of molecules that resist overlapping of volnmes of different molecnles. The volnme open and free for molecular motion, called free volume, is reduced from the bulk fluid volume. In the vdW eos, the free volume is expressed by (v - b), where b is the excluded volume representing the volnme that is unavailable to molecular motion dne to the molecnlar hard cores. For the collision between two molecules assumed to be spherical, the excluded volume is a sphere at the center of either of the colliding molecules and of a radius equal to the sum of the radii of both molecules in contact,... [Pg.296]

By interpreting b as the excluded volume of a mole of spherical molecules, we can obtain an estimate of molecular size. The centers of spherical particles are excluded from a sphere whose radius is the... [Pg.10]

Here, w r) is the attractive (or repulsive) tail of the potential, a is the diameter of spheres, and we have assumed that the fluid is uniform, therefore translational invariance is implied. The first equality in the above equation embodies the physical requirement that the center of a sphere can not penetrate the excluded volume of other spheres. The second equality is just obtained from (1.25) by linearizing the entire exponential factor. Actually, it is the asymptote of the direct correlation function at the infinite separation. The approximation is known to be superior for describing the critical phenomena. The radial distribution function, however, shows an ill-behavior for a Coulombic system, similar to those from the PY closure. [Pg.8]

The depletion force depicted in Fig. 2.24 jumps firom negative (attractive) at h = a to positive (repulsive) at Iz = <7+. The key idea behind the origin of the repulsive part of the depletion force is that for small A the mutual repulsion of spheres is substantially reduced due to the fact that the excluded volumes of the spheres are hidden behind the depletion zones of the walls. In the limit h = cr+, the spheres behave eflfeetively thermodynamically ideal. To match the chemieal potential (2.67) of the spheres in the bulk the number density inside the gap must be... [Pg.83]

Fig. 5.10 Schematic picture of the excluded volume between a sphere with diameter a and a scaled spherocylinder with length AZ, and diameter ID... Fig. 5.10 Schematic picture of the excluded volume between a sphere with diameter a and a scaled spherocylinder with length AZ, and diameter ID...
Figure 9.18 The Excluded Volume of a Pair of Hard Spheres. Figure 9.18 The Excluded Volume of a Pair of Hard Spheres.
Strategy (a) Use the formula for volume of a sphere to calculate the volume of a sphere of radius 2r. This is the excluded volume defined by two molecules, (b) Multiply the excluded volume per molecule by Avogadro s number to determine excluded volume per mole. [Pg.479]

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... [Pg.181]

The solvent-excluded volume is a molecular volume calculation that finds the volume of space which a given solvent cannot reach. This is done by determining the surface created by running a spherical probe over a hard sphere model of molecule. The size of the probe sphere is based on the size of the solvent molecule. [Pg.111]

Rigid particles other than unsolvated spheres. It is easy to conclude qualitatively that either solvation or ellipticity (or both) produces a friction factor which is larger than that obtained for a nonsolvated sphere of the same mass. This conclusion is illustrated in Fig. 9.10, which shows the swelling of a sphere due to solvation and also the spherical excluded volume that an ellipsoidal particle requires to rotate through all possible orientations. [Pg.625]

Here ps is the biopolymer immobilization density A2us = 2%D /3 is the second virial coefficient based on excluded volume for a biopolymer of equivalent diameter D (a sphere of equal volume) (Neal and Lenhoff, 1995) and = As/V0 is the chromatographic phase ratio. The surface area As accessible to the biopolymer in the mobile phase is available in the literature, especially for proteins (Tessier et al., 2002 Dumetz et al., 2008). [Pg.147]

Before considering how the excluded volume affects the second virial coefficient, let us first review what we mean by excluded volume. We alluded to this concept in our model for size-exclusion chromatography in Section 1.6b.2b. The development of Equation (1.27) is based on the idea that the center of a spherical particle cannot approach the walls of a pore any closer than a distance equal to its radius. A zone of this thickness adjacent to the pore walls is a volume from which the particles —described in terms of their centers —are denied entry because of their own spatial extension. The volume of this zone is what we call the excluded volume for such a model. The van der Waals constant b in Equation (28) measures the excluded volume of gas molecules for spherical molecules it equals four times the actual volume of the sphere, as discussed in Section 10.4b, Equation (10.38). [Pg.120]

This equation acknowledges that real molecules have size. They have an exclusion volume, defined as the region around the molecule from which the centre of any other molecule is excluded. This is allowed for by the constant b, which is usually taken as equal to half the molar exclusion volume. The equation also recognizes the existence of a sphere of influence around each molecule, an interaction volume within which any other molecule will experience a force of attraction. This force is usually represented by a Lennard-Jones 6-12 potential. The derivation below follows a simpler treatment (Flowers Mendoza 1970) in which the potential is taken as a square-well function as deep as the Lennard-Jones minimum (figure 2a). Its width x is chosen to give the same volume-integral, and defines an interaction volume Vx around the molecule, which will contain the centre of any molecule in the square well. This form of molecular pair potential then appears in the Van der Waals equation as the constant a, equal to half the product of the molar interaction volume and the molar interaction energy. [Pg.13]

The ideal gas free energy functional is defined exactly from statistical mechanics, dropping the temperature-dependent terms that do not affect the fluid structure. Free energy functional contribution due to the excluded volume of the segments is calculated from Rosenfeld s (1989) DFT for a mixture of hard spheres. The functional derivatives of these free energy functional contributions, which are actually required to solve the set of Euler-Lagrange equations, are straightforward. [Pg.138]

Figure 3.3 illustrates the idea of excluded volume. It shows two protein molecules as two adjacent spheres of the same radius R. Because molecules are not penetrable by each other, the volume of a solution occupied by a macromolecule is not accessible to other macromolecules. A minimal distance between two adjacent spherical molecules of a globular protein equals the sum of their radii, or the diameter of one of them. This means that around each protein molecule there is an excluded volume U), which is 8-fold larger than that of protein molecule itself and is not accessible for centres of other protein molecules. The excluded volume is still larger for non-spherical macromolecules and depends on the flexibility of the macromolecular chain, and its configurational, rotational, vibrational properties and hydration (Tanford 1961). [Pg.30]


See other pages where Excluded Volume of a Sphere is mentioned: [Pg.115]    [Pg.129]    [Pg.185]    [Pg.329]    [Pg.115]    [Pg.129]    [Pg.185]    [Pg.329]    [Pg.451]    [Pg.564]    [Pg.32]    [Pg.225]    [Pg.146]    [Pg.127]    [Pg.12]    [Pg.411]    [Pg.451]    [Pg.12]    [Pg.411]    [Pg.12]    [Pg.1109]    [Pg.433]    [Pg.138]    [Pg.239]    [Pg.73]    [Pg.70]    [Pg.288]    [Pg.382]   


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