The virial coefficients

For a pure vapor the virial coefficients are functions only of temperature for a mixture they are also functions of composition. An important advantage of the virial equation is that there are theoretically valid relations between the virial coefficients of a mixture and its composition. These relations are ... 

CALCULATE THE MODIFIED REDUCED DIPOLE TO BE USED IN CALCULATING THE FREE-POLAR CONTRIBUTION TO THE VIRIAL COEFFICIENT. 

If /i = 1 ahn, it is sufficient to retain only the first temi on the right. However, one does not need to know the virial coefficients one may simply use volumetric data to evaluate the integral. 

We have so far ignored quantum corrections to the virial coefficients by assuming classical statistical mechanics in our discussion of the confignrational PF. Quantum effects, when they are relatively small, can be treated as a perturbation (Friedman 1995) when the leading correction to the PF can be written as... 

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. 

Coulomb potential multiplied by -p. The graphical representation of the virial coefficients in temis of Mayer/ -bonds can now be replaced by an expansion in temis ofy bonds and Coulomb bonds ). 

Each/-bond is replaced by any -bond and the sum of one or more Coulomb bonds in parallel with or without any -bond in parallel. The virial coefficients then have the following graphical representation ... 

J. H. Dymond and E. B. Smith, The Virial Coefficients of Gasses, Ei Critical Compilation, Clarendon Press, London, 1969, p. 72. 

Although PVT equations of state are based on data for pure fluids, they are frequently appHed to mixtures. 7h.e virial equations are unique in that rigorous expressions are known for the composition dependence of the virial coefficients. Statistical mechanics provide exact mixing rules which show that the nxh. virial coefficient of a mixture is nxh. degree in the mole fractions ... 

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). 

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 ... 

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are ... 

This equation of state applies to all substances under all conditions of p, and T. All of the virial coefficients B, C,. .. are zero for a perfect gas. For other materials, the virial coefficients are finite and they give information about molecular interactions. The virial coefficients are temperature-dependent. Theoretical expressions for the virial coefficients can be found from the methods of statistical thermodynamic s. 

The virial coefficients at 190°K have been calculated from Ewald s results22 and may be combined with the measurements at room temperature of Michels and Boerboom,47 of Cottrell and his colleagues,11 12 and of Harper and Miller,32 to give the parameters for helium + carbon dioxide... 

This expression is called the virial equation. The coefficients B, C,. . . are called the second virial coefficient, third virial coefficient, and so on. The virial coefficients, which depend on the temperature, are found by fitting experimental data to the virial equation. 

Although the virial equation can be used to make accurate predictions about the properties of a real gas, provided that the virial coefficients are known for the temperature of interest, it is not a source of much insight without a lot of advanced analysis. An equation that is less accurate bur easier to interpret was proposed by the Dutch scientist Johannes van der Waals. The van der Waals equation is... 

Dymond, J. H. Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures Oxford Press New York, 1980. 

For dilute, teal gases, where ternary and higher collisions can be neglected, the angle of deflection can be employed to evaluate a number of physical properties. Of course appropriate distributions of the values of g and b must be introduced. The resulting expressions for the virial coefficients and the transport properties (viscosity, diffusion and thermal conductivity) are quite complicated. The interested reader is referred to advanced books on this subject... 

Fig. 37 The phase diagram of a polymer globule in terms of the virial coefficients A2 and A3. (Schematically reproduced from Ref. [363])... 

Various expressions for yES have been proposed in literature based on the virial coefficients and simulation data. Most of these have the following general form ... 

Equation 8.10 is notable in that it ascribes specific energetic effects to the interactions of the aqueous species taken in pairs (the first summation) and triplets (second summation). The equation s general form is not ad hoc but suggested by statistical mechanics (Anderson and Crerar, 1993, pp. 446 -51). The values of the virial coefficients, however, are largely empirical, being deduced from chemical potentials determined from solutions of just one or two salts. 

The virial coefficient A2, A3, etc. is a measure of the resultant interaction between the Polymer chains,... 

The higher the pressure, the larger the number of terms we have to consider in equations 2.17 and 2.18. Let us assume that the pressure is such that only the second term needs to be considered. Then because the virial coefficients depend only on the temperature, we have ... 

In this study the Pitzer equation is also used, but a different, more straightforward approach is adopted in which the drawbacks just discussed do not arise. First, terms are added to the basic virial form of the Pitzer equation to account for molecule-ion and molecule-molecule interactions. Then, following Pitzer, a set of new, more observable parameters are defined that are functions of the virial coefficients. Thus, the Pitzer equation is extended, rather than modified, to account for the presence of molecular solutes. The interpretation of the terms and parameters of the original Pitzer equation is unchanged. The resulting extended Pitzer equation is... 

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