Virial coefficients from statistical mechanics

While virial coefficients can be calculated from statistical-mechanical formulas, for practical work it is usually more convenient to employ semi-empirical correlations. Most of these correlations are based on the principle of corresponding states and as a result their applicability is limited to normal... 

However, these new mixing rules (based both to infinite- or zero pressure limit) give, for the composition dependence of the second virial coefficient, results that are inconsistent with those obtained from statistical mechanics. 

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. 

Using trajectory calculations with an ab initio pair-wise potential or an assumed Lennard-Jones pair-wise potential, we can calculate the intermoleeular dynamic global potential which can be used to calculate experimentally obtained quantities sueh as a second virial coefficient. From classical statistical mechanics one obtains the following, well known, equation for the second virial coefficient ... 

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. 

In addition to pure gases, the Taylor expansion (4.5.8) can be applied to gaseous mixtures. The resulting form is still (4.5.11), but the virial coefficients now depend on both temperature and composition. The composition dependence is rigorously obtained from statistical mechanics here we are interested only in the results. For a mixture containing n components. 

The values of the virial coefficients for a gas at a given temperature can be determined from the dependence of p on Fm at this temperature. The value of the second virial coefficient B depends on pairwise interactions between the atoms or molecules of the gas, and in some cases can be calculated to good accuracy from statistical mechanics theory and a realistic intermolecular potential function. 

The virial equation of state given by eq 5.1 applies to gases and has been discussed in Chapter 3. The composition dependence for the second and third virial coefficients are obtained from statistical mechanics and given by eqs 5.3 and 5.4. Consequently, the virial equation has formed the basis for the development of other semi-empirical equations of state capable of correlating both (p, V, T) and phase behaviour some approaches are discussed in Chapter 12. One example of this form of equation is the Benedict, Webb and Rubin (known by the acronym BWR) equation of state given by ... 

Figure 5. Second virial coefficient of COt predicted from statistical mechanics using the...

Because of its theoretical basis, i.e. its development from Statistical Mechanics, the virial equation (Section 8.9) represents the only EoS where rigorous mixing rules for the mixture coefficients are available. Thus ... 

Lets now consider application of mixing rules to the virial equation. Since there is a sound theoretical basis for the virial coefficients in terms of intermolecular interactions, we can relate the virial coefficients for mixtures in terms of intermolecular potentials via Equation (4.29) with no arbitrary assumptions that is, these mixing rules are rigorous results from statistical mechanics. 

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. 

This is a virial equation, the word virial being taken from the Latin word for force and thus indicating that forces between the molecules are having an effect. It turns out that statistical mechanical models also give equations that can be written in this form with the virial coefficients, B C > etc., being related to various interaction parameters. 

In classical statistical mechanics, the physical meaning of the virial coefficients is this A corresponds to an absence of interaction between the molecules (free molecules), B takes account of pairwise interaction (the interaction of two molecxdes at a time), and the higher coefficients contain multiple interactions. Thus the deviation of a real gas from perfect-gas behaviour is primarily due to intermolecular interactions, the values of... 

The virial coefficients B, B, B , and C describe the thermodynamic properties of a pure electrolyte. The second virial coefficients B, B and B arise from binary interactions and have an ionic strength dependence similar to that suggested by statistical mechanics ... 

The great appeal of the virial equations derives from their interpretations in terms of molecular theory. Virial coefficients can he calculated from potential fonctions describing interactions among moleculas. More importantly, statistical mechanics provides rigorous expressions for the composition dependeace of ihe virial coefficients. Thus, the nth virial coefficient of a mixture is nth order in the mole fractions ... 

This expansion is completely analogous to the virial expansion of the pressure, where Ps is the number density of the solute and is the th virial coefficient of the osmotic pressure. There is a complete analogy between virial expansion of the pressure in the density in the gaseous phase and the expansion of the osmotic pressure in the solute density in solution (McMillan and Mayer, 1945). The statistical mechanical expression for obtained from the McMillan-Mayer... 

Theoretically, the second and the third virial coefficient can be calculated from models for the potential functions and with the help of the statistical mechanics. However, this procedure is rarely used in practical applications and tends to be extremely complex and even approximate for nonspherical molecules. 

Excluded volume n. The volume surrounding and including a given object, which is excluded to another object. This terminology comes from the statistical mechanics of gases, where this function arises in the leading order concentration expression (virial coefficient) for the pressure in the case of gas particles that repeal each other with a hard-core volume exclusion. 

Suzuki and Schnepp (1971) have shown that the specific heat of solid a-N2 can be calculated in good agreement with experiment from the results of Schnepp and Ron (1969). These authors also have shown that the potential model of Kuan and others gives good results for second virial coefficients of N2 gas. They used the statistical mechanics results for the diatomic model given by Sweet and Steele (1967). 

Wong and Sandler followed a different approach by matching the Helmholtz function at infinite pressure from the cubic equation of state and from an activity coefficient model. This approach ensures consistency with statistical mechanics requirements that the second virial coefficient of a mixture has a quadratic dependence on composition. For the case of the Peng-Robinson equation of state, the Wong-Sandler mixing rules are ... 

The criterion used to choose the topics covered in this book was their usefulness in application to problems in chemistry and biochemistry. Thus cluster expansion methods for a real gas, although very useful for the development of the theory of real gases per se, was judged not useful except for the second virial coefficient. Similarly, the statistical mechanical extensions of the theory of ionic solutions beyond the Debye-Huckel limiting law were judged not useful in actual applications. Some important topics may have been missed either because of my lack of familiarity with them or because I failed to appreciate their potential usefulness. I would be grateful to receive comments or criticism from readers on this matter or on any other aspect of this book. 

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