Application of the Virial Equations

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 virial equation of state in Table 4.2 provides a sound theoretical basis for computing P-v-T relationships of polar as well as nonpolar pure species and mixtures in the vapor phase. Virial coefficients B, C, and higher can, in principle, be determined from statistical mechanics. However, the present state of development is such that most often (4-34) is truncated at B, the second virial coefficient, which is estimated from a generalized correlation. - In this form, the virial equation is accurate to densities as high as approximately one half of the critical. Application of the virial equation of state to phase equilibria is discussed and developed in detail by Prausnitz et al. and is not considered further here. 

The application of the virial equation to determine values for the virial coefficients requires a knowledge of the pressure, volume, temperatme and the number of moles of the gas. It is generally assumed that for pressures below 0.1 MPa, terms beyond the second virial coefficient can be neglected. For a fixed mass of gas, measurements are usually made of the pressure when the gas either occupies different known volumes at constant temperatme, or is heated to different known temperatmes at constant volume. 

From the chemical point of view, we must say these equations are not tractable and provide no useful information. In common, the study carried out by many authors (Salem, 1963b Byers-Brown, 1958 Byers-Brown and Steiner, 1962 Bader, 1960b Murrell, 1960 Berlin, 1951 Ben-ston and Kirtman, 1966 Davidson, 1962 Benston, 1966 Bader and Bandrauk, 1968b Kern and Karplus, 1964 Cade et al., 1966 Clinton, 1960 Phillipson, 1963 Empedocles, 1967 Schwendeman, 1966) on the force constants is based on the application of the virial and the Hellmann-Feynman or the electrostatic theorems. In particular, the Hellmann-Feynman theorem provides the expression for ki which relates the harmonic force constant to the properties of molecular charge distribution p(r), i.e., it follows (Salem, 1963b) that... 

In principle, for the calculation of VLE any equation of state can be used which is able to describe the PvT behavior of the vapor and the liquid phase, for example, cubic equations of state, further developments of the virial equation, or Helmholtz equations of state. Most popular in chemical industry are further developments of the cubic van der Waals equation of state. Great improvements were obtained by modification of the attractive part, by introducing the temperature dependence of the attractive parameter with the help of a so-called a-function and the development of improved mixing rules, the so-called g -mixing rules, which allow the applicability to asymmetric systems and systems with polar compounds. 

A main advantage of the virial equation, as compared to the other EoS, is that there is an exact relationship between B mixture) and the B values of the mixture components and their pairs, as we will see in Chapter 11. It finds, thus, extensive application in the description of the vapor phase nonideality in distillation or gas absorption design up to moderate pressures. 

Application of the virial theorem equation of state for non-ideal systems... 

Our aim here is to apply the differential virial theorem to get an expression for the Kohn-Sham XC potential. To this end, we assume that a noninteracting system giving the same density as that of the interacting system exists. This system satisfies Equation 7.4, i.e., the Kohn-Sham equation. Since the total potential term of Kohn-Sham equation is the external potential for the noninteracting system, application of the differential virial relationship of Equation 7.41 to this system gives... 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

The mutual dependence of the pressure, volume, and temperature of a substance is described by its equation of state. Many such equations have been proposed for the description of the actual properties of substances (and mixtures) in the gaseous and liquid states. The van der Waals expression is just one of these and of limited applicability. The virial equation of state ... 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The virial equation of state discussed in Section 7.2 is applicable to gas mixtures with the condition that n represents the total moles of the gas mixture that is, n = f= l n,. The constants and coefficients then become functions of the mole fractions. These functions can be determined experimentally, and actually the pressure-volume-temperature properties of some binary mixtures and a few ternary mixtures have been studied. However, sometimes it is necessary to estimate the properties of gas mixtures from those of the pure gases. This is accomplished through the combination of constants. 

For pressures above the range of applicability of Eq. (3.31) but below about SO bar, the virial equation truncated to three terms usually provides excellent results. In this case Eq. (3.11), the expansion in 1/ V, is far superior to Eq. (3.10). Thus when the virial equation is truncated to three terms, the appropriate form is... 

For an accurate description of the PVT behavior of fluids over wide ranges of temperature and pressure, an equation of state more comprehensive than the virial equation is required. Such an equation must be sufficiently general to apply to liquids as well as to gases and vapors. Yet it must not be so complex as to present excessive numerical or analytical difficulties in application. 

The application of cubic equations of state to mixtures requires that the equation-of-state parameters be expressed as functions of composition. No exact theory like that for the virial equations prescribes this composition dependence, and we rely instead on empirical mixing rules to provide approximate relation-... 

Although we have omitted an identifying subscript in the preceding equations, their application so far has been to the development of generated correlations for pure gases only. In the remainder of this section we show how the virial equation may be generalized to allow calculation of fugacity coefficients < , of species in gas mixtures. 

Equations (2.6) as well as Eqs. (2.7) were obtained by use of some approximations. The approximations of the Flory method are connected with the lattice character of his model it is difficult to estimate the degree of their accuracy. The approximations of the Onsager method are due to (a) the application of the second virial approximation and (b) the application of the variational procedure. It is rather easy to eliminate the latter approximation by solving numerically with high degree of accuracy the integral equation which appears as a result of the exact minimization of expression (2.2). This has been done in Ref.30 the results are... 

The first application of such equations to dilute solutions actually came from van t Hoff s measurements of the osmotic pressure of 1% solutions of cane sugar in water (relative to pure water), where the analogy to the virial equation of a gas expressed as a power series in the pressure is more direct. Accordingly, we will start our discussion of molecular weight measurements by considering osmotic pressure. 

Mayer achieved closure in integrals involved in his application of the McMillan-Mayer virial approach to ionic solutions by multiplying this equation by the factor... 

Equation (2-62) is the key to the application of colligative properties to polymer molecular weights. We started with Eq. (2-53), which defined an ideal solution in terms of the mole fractions of the components. Equation (2-62), which followed by simple arithmetic, expresses the difference in chemical potential of the solvent in the solution and in the pure state in terms of the mass concentrations of the solute. This difference in chemical potential is seen to be a power series in the solute concentration. Such equations are called virial equations and more is said about them on page 65. 

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