Statistical mechanics second virial coefficient

Schenter, G.K. The development of effective classical potentials and the quantum statistical mechanical second virial coefficient of watta-. J. Chetn. Phys. 117, 6573 (2002)... 

Pople J 1954 Statistical mechanics of assemblies of axially symmetric molecules II. Second virial coefficients Proc. R. Soc. A 221 508... 

Its precise basis in statistical mechanics makes the virial equation of state a powerful tool for prediction and correlation of thermodynamic properties involving fluids and fluid mixtures. Within the study of mixtures, the interaction second virial coefficient occupies an important position because of its relationship to the interaction potential between unlike molecules. On a more practical basis, this coefficient is useful in developing predictive correlations for mixture properties. 

The virial coefficients B(T), C(T), D(T),... are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B T) to be explicitly evaluated by statistical mechanical methods as... 

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. 

The mixture second virial coefficient 5 is a function of temperature and composition. Its exact composition dependence is given by statistical mechanics, and this makes tire virial equation preeminent among equations of state where it is applicable, i.e., to gases at low to moderate pressures. The equation giving this composition dependence is ... 

In the virial equation as given by Eq. (3.12), the first term on the right is unity, and by itself provides the ideal-gas value for Z. The remaining terms provide corrections to the ideal-gas value, and of these the term B/ V is the most important. As the two-body-interaction term, it is evidently related to the pair-potential function discussed in the preceding section. For spherically symmetric intermolecularforce fields, statistical mechanics provides an exact expression relating the second virial coefficient B to the pair-potentialfunctionW() ) ... 

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 equation, which may be derived on the basis of statistical mechanics, between the second virial coefficient and the interaction energy has the simple form... 

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 ... 

Note that the Virial coefficients B, C, or B, C , are function only of temperature. Virial EOS has a theoretical basis in statistical mechanics. For example, the B/V term accounts for interactions between pairs of molecules C/V describes three molecule interactions, etc. The form Z=l+B/V called the second virial coefficient has found wide applications, but it is restricted to gas phase calculations at moderate pressures up to 20 bar. 

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 second experimental quantity relevant to the problem of H(f)0 is the second virial coefficient in the density expansion of the osmotic pressure. Consider the statistical mechanical expression for given in (4.1.2), which for simplicity of notation is written as... 

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... 

The same equations are used for mixtures of gases. In this case the coefficients are combinations of the coefficients of the pure components. They are combined according to mixing rules, which are also specified by statistical mechanics for any number of components. The second virial coefficient is, for a mixture of m components,... 

Theta solvent n. A solvent, at a particular temperature, in which the polymer is at the edge of solubility and exists in the form of a statistical coil. Long-range forces between polymer molecular segments are balanced by polymer solvent interactions. At these conditions the second virial coefficient becomes zero and entropy is at its minimum. Kamide K, Dobashi T (2000) Physical chemistry of polymer solutions. Elsevier, New York. Flory PJ (1969) Statistical mechanics of chain molecules. Interscience Publishers Inc., New York. Flory PJ (1953) Principles of polymer science. The Cornell University Press, Ithaca, NY. 

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). 

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. 

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 ... 

We round out this introduction to the virial equation of state by reference to its theoretical foundation. Thus statistical mechanics permits deduction of an expression for pVin terms of either the grand partition function or the radial distribution function. The leading term in the expansion of the latter function corresponds to pairwise interaction between molecules, and indicates the following relation between the second virial coefficient and the potential energy (r) of the interacting pair, when this depends only on the distance r between molecular centres ... 

Al. Statistical-mechanical computations of the second virial coefficient and pair-correlation-coefficients of liquid N2, O2,... 

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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