Virial expansion reduced

Statistical mechanics provides physical significance to the virial coefficients (18). For the expansion in 1/ the term BjV arises because of interactions between pairs of molecules (eq. 11), the term C/ k, because of three-molecule interactions, etc. Because two-body interactions are much more common than higher order interactions, tmncated forms of the virial expansion are typically used. If no interactions existed, the virial coefficients would be 2ero and the virial expansion would reduce to the ideal gas law Z = 1). 

The equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the Hquid region. The compressibihty factor can be used to accurately determine gas volumes when used in conjunction with a virial expansion or an equation such as equation 53 (77). However, the prediction of saturated Hquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibihty factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

Equation (2.1) is written in a virial expansion form by introducing the density p = 1/V and the reduced variables for density, temperature, and pressure... 

Since the Eq must describe states ranging from the ideal gas to the dense compressed state, the G factor must reduce to the virial expansion at low density and must approach the value determined by the repulsive potential at the high density limit. Dr. Jacobs took a semiempirical approach to this problem and used the results of Monte Carlo (MC) and Lennard-Jones Devonshire (LJD) calculations to determine unknown parameters in theoretical expressions for p0(v) and G(v,T). ... 

T want to finish this introduction by a short sketch of the history of the subject. The physics of dilute polymer solutions by now has been ail active field for about TO years. Much of the early work is connected to the name of Flory and summarized in his classic books [Flo53, Fit>69], Up to about. 1970 much theoretical or experimental work concentrated on the behavior in the dilute limit, where via virial expansions the problem can be reduced to considering only a few interacting chains. The development led to the so-called two parameter theories , which essentially expand quantities like Rg or A2 in powers of z, In 1971 these developments were most carefully reviewed in a book by Yamakawa [Yam.71]. 

Although the virial expansions might seem very complicated, because they contain an infinite number of terms, their power lies in the fact that usually only a few terms must be considered. As pressure is reduced and molar volume gets very large, the higher terms in the expansion become negligible and only the first two terms need be considered. Equation (26) then becomes... 

Since the terms B/V, CfV2, etc., of the virial expansion [Eq. (3.11)] arise on account of molecular interactions, the virial coefficients B, C, etc., would be zero if no such interactions existed. The virial expansion would then reduce to... 

It is generally agreed that a virial form of isotherm equation is of greater theoretical validity than the DA equation. As explained in Chapter 4, a virial equation has the advantage that since it is not based on any model it can be applied to isotherms on both non-porous and microporous adsorbents. Furthermore, unlike the DA equation, a virial expansion has the particular merit that as p — 0 it reduces to Henry s law. 

Partial parameter, cubic equation of state 2d virial coefficient, density expansion Partial molar second virial coefficient Reduced second virial coefficient... 

The theory leads to an osmotic virial type expansion and gives a fundamental interpretation of the coefficients appeEiring in this expansion in terms of forces between the species. The expansion reduces to the Edmunds-Ogston expression only when certain assumptions are made -namely that the fluids are incompressible and that the solvent is... 

Thus the problem of calculating the thermodynamic properties and pair correlation function of a low-density gas has been reduced to that of evaluating the integrals corresponding to the graphs with certain specified numbers of field points. There is an extensive literature that makes use of this virial expansion. 

Let us first consider the osmotic pressure O. The virial expansion of II reduced by Oideai needs to be a power series of the dimensiomess concentration of the polymer chains, pR o /N ... 

As is well-known, the virial expansion, Eq. (3), provides a simple and convenient description of the EoS for dilute gases, has been widely used for both practical and theoretical studies, and represents a flrst attempt to understand the relationship between molecular interactions and the macroscopic behavior of fluids. In particular, values for the second, third, fourth and flfth virial coefficients (all in reduced L-J units) are available [16,285,286,291] for 2D L-J fluids. These values have been fltted by Reddy et al. [195] as polynomials in (T ) for the second, third, and fourth coefficients, and as powers of T y for the fifth one. The coefficients are given in Table 13. The Reddy et al. [195] results then lead to a complete description of five coefficients of the virial EoS, which has been used to obtain approximate values of the critical point see Table 11. 

Evidently, the kinetic energy T is reduced from the TF value in equation (54) by iZ2 because of the virial theorem this then is the inhomogeneity correction for the Coulomb field, in the sense of the Z-1/3 expansion. 

Reduced third virial coefficient 4th virial coefficient, density expansion... 

Moreover, by invoking the statistical mechanic expression for the second virial coefficient of the mixture written in Van Ness compact form (Van Ness and Abbott 1982), B T) = LiXiBu(T) + Li jXiXjbij(J) with 5y(r) = IBy-Bn-Bjj, the expansion coefficients in Equation 8.38 for ternary mixtures of imperfect gases reduce to... 

Usually, VPO-data are reduced to virial coefficients and not to solvent activities. Power series expansion of Equation [4.4.31] leads to the following relations ... 

The so-called VPO-specific constant contains all deviations from equilibrium state and it is to be determined experimentally. It depends on certain technical details from the equipment used and also on the temperature and solvent applied. It is assumed not to depend on the special solute under investigation and can therefore be obtained by calibration. Equation [4.4.31] can also be used if not the steady state, but the temperature difference extrapolated to a measuring time of zero is determined by the experimentator. However, the values of kypo are different for both methods. A more detailed discussion about calibration problems can be found in the papers of Bersted, or Figini. " Usually, VPO-data are reduced to virial coefficients and not to solvent activities. Power series expansion of Equation [4.4.31] leads to the following relations ... 

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