Virial coefficients of hard spheres

The first seven virial coefficients of hard spheres are positive and no Boyle temperature exists for hard spheres. 

The first term is, of course, the second virial coefficient of hard spheres of radius a. The second and third terms give the correction. 

Table 3.1 State-of-the-art values for the second up to the tenth virial coefficient of hard spheres [16] in comparison with the Camahan-Starling result (3.3)...

This leads to the third virial coefficient for hard spheres. In general, the nth virial coefficient of pairwise additive potentials is related to the coefficient7) in the expansion of g(r), except for Coulombic systems for which the virial coefficients diverge and special teclmiques are necessary to resiim the series. 

The next step is whether the right side can also be transformed in a scaled form. Two observations encourage such an expectation. The first has long been known and states that all higher virial coefficients for hard spheres can be expressed in terms of the second one [74,75]... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

The difficulty of evaluating or even stud)dng the as5nnptotic dependence of the /3 as increases limits the usefulness of Eq. (7) to sufficiently dUute fluids (gases). Only in the case of hard sphere molecules have the first four virial coefficients been calculated correctly analytically. Not even the sign is known, with certainty, for a virial coefficient higher than the fifth of a three-dimensional non-ideal fluid. Still, in the hands of Mayer and his collaborators and others - these methods have been used as powerful tools for the investigation of phase transitions. An alternative powerful power series treatment of phase transitions has been developed by Yang and Lee. ... 

IV being the number of component ft hard spheres in the volume V, where = NJN, ft = The second and third virial coefficients of this mixture have been evaluated exactly by McLellan and Alder. In their notation, one can again write, P, = NJV. 

An open problem is the asymptotic behavior (with increasing n) of the virial coefficients of the hard sphere fluid. Since the only forces in this system are repulsive, it was conjectured that all the virial coefficients are necessarily positive. While this classical conjecture has neither been disproven nor verified the basis for it is untenable. Thus, the sixth virial coefficient of a fluid composed of three-dimensional hard cubes in which the forces are all repulsive is negative. ... 

Equation (15.21) is quite revealing. The second term of i 2 (v2), which expresses the deviation from hard sphere behaviour due to the addition of free polymer, is negative in sign. Accordingly, as the volume fraction of dissolved polymer V2 increases, so the second virial coefficient of the particles is diminished. If V2 and/or A are sufficiently large, B2 (v2) can clearly decrease to zero or even become negative. The latter corresponds to a net attraction between the colloidal particles. 

The Ree-Hoover series is really not understood very well, and as a consequence it is difficult to estimate the errors associated with its partial sums. However, the kind of accuracy to be expected may be judged to some extent from the following table, which shows the breakdown of the known virial coefficients for hard disks and hard spheres into contributions from the first, second, and remaining terms of the Ree Hoover series [21] ... 

The temperature dependence of several first virial coefficients is calculated for the Lennard-Johns (12,6) model potential (Equation 1.7 0). Figure 1.41 contains reduced virial coefficients (where 6 is the excluded volume of hard spheres, see... 

However, the structure and composition of microdroplets in which the reaction takes place are not the only parameters controlling polymerisation. We shall see later that particles in the initial microemulsion cannot be considered as independent reactors since interparticle interactions play an important role. Small angle light and neutron scattering experiments have shown that these interactions are attractive [6.20]. There is a clear increase in these attractive interparticle forces as the proportion of acrylamide is raised. In particular, this has two consequences the second virial coefficient of osmotic pressure takes negative values the peak in the structure factor which characterises a hard sphere system is no longer present. 

Obtain a formula for the second virial coefficient of a hard-sphere gas. Solution... 

Evaluate the second virial coefficient of helium, using the value of the hard-sphere diameter in Table A. 15 of the appendix. Compare your value with those in Table A.4 of Appendix A. 

The virial coefficient of the hard-sphere gas is equal to a constant... 

The second virial coefficient of a hard-sphere gas is positive, illustrating the fact that repulsive forces correspond to a raising of the pressure of the gas over that of an ideal gas at the same molar volume and temperature. The second virial coefficient of the square-well gas has a constant positive part that is identical with that of the hard-sphere gas, and a temperature-dependent negative part due to the attractive part of the potential, illustrating the fact that attractive forces contribute to lowering the pressure of the gas at fixed volume and temperature. 

Here is the low-concentration hmit of s, A2 is the second virial coefficient, and M is the polymer molecular weight. The concentration dependence of Dm is driven by the competition between an osmotic term 2A2M and a hydrodynamic term ks. Except that the concentration gradients are taken to be linear rather than sinusoidal in space, Eq. 11.6 is little different from the concentration expansions for Dm of hard spheres discussed in Chapter 10. 

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