Virial series

A2.3.2.2 EQUATIONS OF STATE, THE VIRIAL SERIES AND THE LIQUID-VAPOUR CRITICAL POINT... 

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

The virial series in temis of the packing fraction q = upa /3 is then... 

Figure A2.3.4 The equation of state P/pkT- 1, calculated from the virial series and the CS equation of state for hard spheres, as a fimction of q = where pa is the reduced density.

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]. ... 

Bjemim parameter. The virial series is an expansion in the total ionic concentration c at a fixed value of... 

Occasionally, the temperature voltage coefficient is not expressed as a simple number, but as apower series in T (we generally call it a virial series, or expansion). For example, Equation (7.19) cites such a series for the cell Pt(S) H2(g) HBr(aq) AgBr(s) Ag(s) ... 

Fig. 10. The degree of association into nearest- and next-nearest-neighbour complexes, p, versus concentration, c, at 500°C for manganese ions and cation vacancies in sodium chloride. Filled circles represent the simple association theory, open circles the Lidiard association theory, and crosses the present theory using Eq. (173) when the first term only has been retained in the virial appearing in the equation for the defect distribution function (Eq. (168)). The point of highest concentration represented by a cross may be in error due to the neglect of higher terms in the virial series, and the dotted curve has not been extended to include it.

In this section we shall explain somewhat the results which we have just presented. We are interested this time in the evolution equation for the one-particle distribution function. We write down the virial series expansion of the transport equation and we recall that every contribution to this equation is proportional to V n+d, where n is the number of particles which are involved... 

Since dilute solutions are considered we can expand the osmotic pressure in a virial series that is truncated at the second virial coefficients... 

Developing the osmotic pressure in a virial series one obtains... 

The virial series of viscosity in polyelectrolyte concentration c can be obtained from Eq. (229) by iterating S(k) to the desired order in c and then combining with Eq. (38). To the leading order in c, Eq. (229) yields... 

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

We first consider the case of a two-component solution (biopolymer + solvent) over a moderately low range of biopolymer concentrations, i.e., C < 20 % wt/wt. The quantities pm x in the equations for the chemical potentials of solvent and biopolymer may be expressed as a power series in the biopolymer concentration, with some restriction on the required number of terms, depending on the steepness of the series convergence and the desired accuracy of the calculations (Prigogine and Defay, 1954). This approach is based on simplified equations for the chemical potentials of both components as a virial series in biopolymer concentration, as developed by Ogston (1962) at the level of approximation of just pairwise molecular interactions ... 

Carbon dioxide. Collision-induced absorption in carbon dioxide shows a discernible density dependence beyond density squared, even at densities as low as 20 amagats [34]. Over a range of densities up to 85 amagats the variation of the absorption with density may be closely represented by a (truncated) virial series (as in Eq. 1.2, with I(v) replaced by a(v)) of just two terms, one quadratic and the other cubic in density. The coefficient of g3 is negative. Relative to the leading quadratic coefficient, it is,... 

The dielectric constant e of dense gases may be written in the form of a virial series, [86]... 

For low density (large Vm), the series (2.30) is expected to achieve useful accurary with only a few terms. Higher densities within the domain of convergence require additional terms to achieve a desired accuracy. For some densities, the virial series may not converge at all. 

The coefficient of the first term of the virial series in Equation I is the inverse Henry constant for the temperature considered. From the graph of the isotherm in Figure 3 of the survey paper, it follows that the linearity of the initial section of the adsorption isotherm of ethane on zeolite LiX in accordance with the calculated Henry constant is observed for adsorption values not exceeding 0.7 mmole/g. According to K. N. 

An approach based on the virial expansion suffers from the difficulty of evaluating higher coefficients for highly asymmetric particles and from the non-convergence of the virial series at the concentrations required for formation of a stable nematic phase Lattice methods therefore take precedence over the virial expansion as a basis for quantitative treatment of the liquid crystalline state. 

For some time, it was known [159,160,390,405] that the depolarization of fluids composed of isotropic atoms or molecules is nonvanishing if the particle density is sufficiently high. The density-dependent depolarization of light by isotropic fluids is now understood to be caused by the anisotropy of the collision-induced polarizability increment of two or more interacting atoms or molecules [177, 178,376, 390]. Depolarization ratios are often expressed in terms of a virial series whose nth term accounts for the n-body interactions, with n = 2, 3. . . [161, 164, 165, 167, 170, 175]. Fluids composed of optically anisotropic molecules show a variation of the depolarization ratio with density that is understood as arising from the interplay of the permanent and induced anisotropies the anisotropy of the interaction potential also plays a role [161, 188]. The literature concerned with density-dependent depolarization of light is included in Section 1.2. By contrast, CILS spectroscopy is considered in Section 1.3. 

The stabilization of the collapsing coil comes from other terms of the interaction part of the free energy. The interaction energy per unit volume is an intrinsic property of any mixture, that is often expressed as a virial expansion in powers of the number density of monomers c [Eq. (3.8)]. The relevant volume of interest here is the pervaded coil volume R. The excluded volume term is the first term in the virial series and counts two-body interactions as vc. The next term in the expansion counts three-body... 

The extended virial equations are made up of a truncated virial series followed by a closure term or terms, hi the Benedict-Webb-Rubin (BWR) Equation (4.177), the closure term is an exponential. 

In physical chemistry there are a number of applications of power series, but in most applications, a partial sum is actually used to approximate the series. For example, the behavior of a nonideal gas is often described by use of the virial series or virial equation of state. 

Those of the former school—namely Turnbull and Bagley," LeFevre," Kratky," Gordon et al., Hoare," Hiwatari, and Frenkel and McTague —either imply or explicitly assert that the virial series could be a continuous representation of the supercooled fluid and the amorphous solid, with its first singularity at the zero-temperature point of the ground-state glass. This would require that the essentially Arrhenius behavior of... 

Both Fade approximant " and the more powerful Tova approximant" predictions of the tails of the virial series are consistent with the location of the first-order pole at the crystal close-packed density, as required by the closure for the virial series. In fact, Baram and Luban" give this as a conclusion of their work. The known virial coefficients in the soft-sphere, inverse twelfth power models also imply that the virial series contains information on the crystalline phase at very high pressure, but is unrelated to the freezing transition, the glass transition, or the amorphous solid equation of state. °... 

Fig. 22. Entropy versus log-temperature diagram for the soft-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves correspond to the internally equilibrated fluid behaving in accord with (a) a six-term virial series and (fc) an alternative hypothetical equation of state giving a continuous transition to an amorphous ground state with residual entropy O.lSNk virial coefficient series with maximum. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure.

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