Virial coefficients/series

This hard sphere dilute solution radial distribution function gives rise to a virial coefficient series in the volume fraction, cf), for the compressibility of the suspension... 

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.

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]. ... 

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. 

Parameters B, C, D,..., are density series virial coefficients, and B (V, are pressure series virial coefficients. The second virial coefficients are defined... 

Higher virial coefficients are defined analogously. AH virial coefficients depend on temperature and composition only. The pressure series and density series coefficients are related to one another ... 

Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, ATChP J., 25, pp. 931-938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B given by Eq. (4-188), is usually preferred ... 

An important series of papers by Professor Pitzer and colleagues (26, 27, 28, 29), beginning in 1912, has laid the ground work for what appears to be the "most comprehensive and theoretically founded treatment to date. This treatment is based on the ion interaction model using the Debye-Huckel ion distribution and establishes the concept that the effect of short range forces, that is the second virial coefficient, should also depend on the ionic strength. Interaction parameters for a large number of electrolytes have been determined. 

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

Express the compressibility factor Z = PV td/iRT) for a gas that follows the Redlich-Kwong equation. Convert the resulting equation into one in which the independent variable is (l/Vm)- Obtain a McLaurin series for Z as a polynomial in (1 /Vad, and express the virial coefficients for that equation in terms of the parameters of the Redlich-Kwong equation. 

Kayukawa et al. [17] studied the PVT properties of triiluoromethyl methyl ether, because it is a possible refrigerant with zero ozone depletion potential and low global-warming potential. One series of their data is shown in Table 5.6. Calculate Z, the compressibihty factor, and the molar volume in mol m from the given data, and fit the data for Z as a function of 1 /Pm to both a linear and a quadratic equation to see whether a third virial coefficient is warranted by the data. 

The alternative value, which describes the polymer-solvent interaction is the second virial coefficient, A2 from the power series expressing the colligative properties of polymer solutions such as vapor pressure, conventional light scattering, osmotic pressure, etc. The second virial coefficient in [mL moH] assumes the small positive values for coiled macromolecules dissolved in the thermodynamically good solvents. Similar to %, also the tabulated A2 values for the same polymer-solvent systems are often rather different [37]. There exists a direct dependence between A2 and % values [37]. 

Tables VI and VII give results corresponding to two series of lignin fractions obtained with a flow-through reactor (3). (The units for dn/dc and A2 are respectively ml.g-1 and mole.ml.g-2). These results show that LALLS allows the determination of low Mw values. The dn/dc values differ from sample to sample but vary only slightly for a given set of fractions. The second virial coefficient exhibits no definite trend. Negative values indicate perhaps some association effects but light scattering alone is not sufficient to ascertain this point.

In other words, the equation of state may be written as a series in powers of density, with first, second, third,. .., virial coefficients given by... 

As pointed out in Chapter III, Section 1 some specific diluent effects, or even remnants of the excluded volume effect on chain dimensions, may be present in swollen networks. Flory and Hoeve (88, 89) have stated never to have found such effects, but especially Rijke s experiments on highly swollen poly(methyl methacrylates) do point in this direction. Fig. 15 shows the relation between q0 in a series of diluents (Rijke assumed A = 1) and the second virial coefficient of the uncrosslinked polymer in those solvents. Apparently a relation, which could be interpreted as pointing to an excluded volume effect in q0, exists. A criticism which could be raised against Rijke s work lies in the fact that he determined % in a separate osmotic experiment on the polymer solutions. This introduces an uncertainty because % in the network may be different. More fundamentally incorrect is the use of the Flory-Huggins free enthalpy expression because it implies constant segment density in the swollen network. We have seen that this means that the reference dimensions excluded volume effect. 

The determination of gas-solid virial coefficients can be a useful technique to explain the interaction between an adsorbed gas and a solid surface. The terms are defined so that the number of adsorbate molecules interacting can be readily ascertained. For example, the second order gas-solid interaction involves one adsorbate molecule and the solid surface the third order gas-solid interaction involves two adsorbate molecules and the surface, and so on. The number of adsorbed molecules under consideration is expanded in a power series with respect to the density of the adsorbed phase. 

Defect thermodynamics, as outlined in this chapter, is to a large extent thermodynamics of dilute solutions. In this situation, the theoretical calculation of individual defect energies and defect entropies can be helpful. Numerical methods for their calculation are available, see [A. R. Allnatt, A. B. Lidiard (1993)]. If point defects interact, idealized models are necessary in order to find the relations between defect concentrations and thermodynamic variables, in particular the component potentials. We have briefly discussed the ideal pair (cluster) approach and its phenomenological extension by a series expansion formalism, which corresponds to the virial coefficient expansion for gases. 

Where R is the gas constant, T the absolute temperature, and M the molecular weight of the polymer. This series is usually called the osmotic virial expansion, with A (i = 2, 3,...) being referred to as the i-th virial coefficient of the... 

Solution In order to set the second virial coefficient equal to zero, we must put the van der Waals equation into the virial form (i.e., a power series in 1 / Vm). The second term in the van der Waals equation, being proportional to (1/Vm)2, is already in this form. In order to put the term RT/(Vm — b) into the virial form, we write it as... 

This series is known as the virial expansion, where Bn are the virial coefficients. In principle, these last can be calculated if the potential is known. In practice, however, the calculation is feasible only for the first few coefficients. In this framework, Eq. (6) is only applicable to low density regimes. Obviouly, a complete theory is expected to confidently calculate the highest possible number of known virial coefficients. Such a calculation is one of the benchmarks of its overall accuracy. 

Here the quantity PV/nRT is often called the virial and the quantities 1, B(T), C(7T), etc., the coefficients of its expansion in inverse powers of the volume per mole, F/n, are called the virial coefficients, so that B(T) is called the second virial coefficient, C(T) the third, etc. The experimental results for equations of state of imperfect gases are usually stated by giving B(T), C(T), etc., as tables of values or as power series in the temperature. It now proves possible to derive the second virial coefficient B T) fairly simply from statistical mechanics. 

Equation (1-251) can also be used to derive the semiclassical expansion of the second dielectric virial coefficient. Indeed, one may hope that at intermediate temperatures an expansion of Be(T) as a power series in h2 will give sufficiently accurate results, making full quantum-statistical calculations unnecessary. Thus, one can approximate Be(T) as,... 

Before comparing theory and experiment let us discuss the convergence of the semiclassical expansion of the dielectric second virial coefficient. In Table 1-15 the classical dielectric virial coefficient the first and second quantum corrections, and the full quantum result are reported. An inspection of this table shows that the quantum effects are small for temperatures larger than 100 K, and /it(/) can be approximated by the classical expression with an error smaller than 2.5%. At lower temperatures the dielectric virial coefficient of 4He starts to deviate from the classical value. Still, for T > 50 K the quantum effects can be efficiently accounted for by the sum of the first and second quantum corrections. Indeed, for T = 50, 75, and 100 K the series (7) + lli 1 (7) + (7) reproduces the exact results with errors... 

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