Virial expansion model

Generally speaking, the virial expansion model, asymptotically exact for very large aspect ratios, is applicable only when the volume fraction of rods is very small (second virial approximation), whereas the lattice model may be more reliable at rod concentrations in typical lyotropic phases. The implications of both models are still being explored today. Inclusion of a distribution of rod... 

We start with the basic relationships ( Ansatz ) of collision-induced spectra (Section 5.1). Next we consider spectral moments and their virial expansions (Section 5.2) two- and three-body moments of low order will be discussed in some detail. An analogous virial expansion of the line shape follows (Section 5.3). Quantum and classical computations of binary line shapes are presented in Sections 5.4 and 5.5, which are followed by a discussion of the symmetry of the spectral profiles (Section 5.6). Many-body effects on line shape are discussed in Sections 5.7 and 5.8, particularly the intercollisional dip. We conclude this Chapter with a brief discussion of model line shapes (Section 5.10). 

It has been argued that, in the low-density limit, intercollisional interference results from correlations of the dipole moments induced in subsequent collisions (van Kranendonk 1980 Lewis 1980). Consequently, intercollisional interference takes place in times of the order of the mean time between collisions, x. According to what was just stated, intercollisional interference cannot be described in terms of a virial expansion. Nevertheless, in the low-density limit, one may argue that intercollisional interference may be modeled as a sequence of two two-body collisions in this approximation, any irreducible three-body contribution vanishes. 

The LCM is a semi-theoretical model with a minimum number of adjustable parameters and is based on the Non-Random Two Liquid (NRTL) model for nonelectrolytes (20). The LCM does not have the inherent drawbacks of virial-expansion type equations as the modified Pitzer, and it proved to be more accurate than the Bromley method. Some advantages of the LCM are that the binary parameters are well defined, have weak temperature dependence, and can be regressed from various thermodynamic data sources. Additionally, the LCM does not require ion-pair equilibria to correct for activity coefficient prediction at higher ionic strengths. Thus, the LCM avoids defining, and ultimately solving, ion-pair activity coefficients and equilibrium expressions necessary in the Davies technique. Overall, the LCM appears to be the most suitable activity coefficient technique for aqueous solutions used in FGD hence, a data base and methods to use the LCM were developed. 

To seek a reasonable accurate analytical approximation for the available area, as a function of 6S = Nsnr /A and 6y = Nynr /A one should have accurate values for a reasonable number of coefficients in the low-density expansion of the binaiy RSA model, which is not a trivial task. Even for binaiy mixtures of disks at equilibrium, a problem that received much more attention than RSA, analytical expressions are known only for the first three terms of the virial expansion [21], The values of the fourth and fifth terms, obtained using laborious numerical calculations, were reported only for a few values of y and molar fractions of the two types of disks [22], In the non-equilibrium RSA of binaiy particles, one should take into account, when calculating the higher terms of the series, not only various y and molar fractions, but also the order of deposition of particles. Furthermore, as already noted, it is not clear whether the involved calculations needed to obtain the next unknown terms of the low-density expansion would improve much the accuracy of estimating the jamming coverage. 

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. 

The prescription for determining the functions is that an m-decomposable model should be correct if the system consisted only of m molecules. Formulae such as Eqs. (6.2) and (6.3) involve the Mayer / function that, for a pair decomposable case, is / j) = exp [—/3m( )(0, y)] — 1. A natural accommodation of nonpair-decomposable interactions in this case takes the goal of insuring that successive terms in a virial expansion are ordered by the density. This is the historical approach (Ursell, 1927), and is called an Ursell expansion. In this language, fa (j) is an Ursell function (Stell, 1964 Munster, 1969). Again the idea is to require that the desired m-body Ursell function makes the product of Eq. (6.2) correct if just m molecules are involved. Thus for the case that only two molecules are involved... 

It is helpful to contrast the view we adopt in this book with the perspective of Hill (1986). In that case, the normative example is some separable system such as the polyatomic ideal gas. Evaluation of a partition function for a small system is then the essential task of application of the model theory. Series expansions, such as a virial expansion, are exploited to evaluate corrections when necessary. Examples of that type fill out the concepts. In the present book, we establish and then exploit the potential distribution theorem. Evaluation of the same partition functions will still be required. But we won t stop with an assumption of separability. On the basis of the potential distribution theorem, we then formulate additional simplified low-dimensional partition function models to describe many-body effects. Quasi-chemical treatments are prototypes for those subsequent approximate models. Though the design of the subsequent calculation is often heuristic, the more basic development here focuses on theories for discovery of those model partition functions. These deeper theoretical tools are known in more esoteric settings, but haven t been used to fill out the picture we present here. 

There are two major theoretical approaches to the understanding of orientational ordering of extended chain polymers one is a model wherein rod-shaped particles are confined to a lattice under the restriction that segments may not overlap with one another the other utilizes a virial expansion that accounts for mutual orientational correlations and interactions of pairs, triplets, etc. of elongated particles at varied concentrations. 

Computer simulations, Monte Carlo or molecular dynamics, in fact appear to be the actual most effective way of introducing statistical averages (if one decides not to pass to continuous distributions), in spite of their computational cost. Some concepts, such as the quasi-structure model introduced by Yomosa (1978), have not evolved into algorithms of practical use. The numerous versions of methods based on virial expansion, on integral equation description of correlation functions, on the application of perturbation theory to simple reference systems (the basic aspects of these... 

When the molecular Interaction Is strong, as Is the case of ionomers in DMF, a virial expansion is not sufficient to describe the interference effect. We use a simple effective potential model with an effective diameter. This model, which treats the macroions (lonomers) as if they were neutral but have an effective size, was originally used by Doty and Steiner (231 to analyze light scattering data from protein solutions. The equation obtained is... 

To obtain a manageable equation for the adsorption isotherm in this model, without loosing the generality of a continuous model, we make use of a virial expansion for the two-dimensional spreading pressure (f) of the adsorbed phase [24] ... 

Our purpose in this section is to derive a set of useful expressions for the chemical potentials starting with the principles of statistical mechanics. The expressions we shall obtain take the form of virial expansions similar to those of the Edmond and Ogston (6) but having a very different theoretical basis. Our model parameters are isobaric-isothermal virial coefficients which are about an order of magnitude smaller than the osmotic virial coefficients in the Edmond and Ogston model. We shall develop the theory neglecting the effect of polydispersity because we empirically did not find this to be very important at the level of accuracy commonly attainable in experimental phase diagrams for these systems. 

Also, the equation of state, and the virial expansion can easily be written for this model ... 

Plasma. model The analyticity theorem implies that the pressure p and particle density n as functions of hj and z (9) are analytic for z i 0. However, infrared singularities show up for vanishing fugacity, i.e. in the dilute limit, implying that there exists no virial expansion. In fact, one obtains a quite interesting and unusual equation of state near Oj = 2 which will be... 

For systems ranging from dilute to moderate concentrations, the most commonly used model is based on a virial expansion developed by Pitzer (1). For a system with one solute, MX, having Vj cations of charge Zj and Vx anions of charge Zx, the excess Gibbs energy can be written as... 

In Pitzer s model the Gibbs excess free energy of a mixed electrolyte solution and the derived properties, osmotic and mean activity coefficients, are represented by a virial expansion of terms in concentration. A number of summaries of the model are available (i,4, ). The equations for the osmotic coefficient (( )), and activity coefficients (y) of cation (M), anion (X) and neutral species (N) are given below ... 

Equation 9.5-8 is modeled after the virial expansion for gaseous mixtures, and, in fact, the constants in the expansion (2, 3,3,4, 4, 6, etc.) are those that arise in that equation. 

The Pitzer model adds a virial expansion to a simplified version of the D-H equation and begins by describing the total excess free energy of an electrolyte solution as... 

Although Eq. (2.28) has been derived in the context of the lattice-gas (Ising) model for a binary system with a critical composition of 1/2, it can be reinterpreted quite generally for a liquid-gas system. Derive this form from the virial expansion for an imperfect gas and show that the difference in the grand potential per unit volume, g(n), between a system with a local density of n and the grand potential of the bulk gas or liquid system, gh, at coexistence (where the liquid and gas have equal grand potentials), has the form ... 

Many approximations used in modeling thermodynamic properties are based on the Taylor series. Examples are the virial expansions for the equation of state and the Redlich-Kister expansion of the excess Gibbs energy. Let/(x) and all its derivatives be continuous and single-valued on [a, b]. Then the Taylor series provides an approximation to f b) if we know/at a nearby point x = a and if we can evaluate derivatives of/ atx = a,... 

We might just as easily have made use of some other statistical theory of dense gases and liquids to describe G(0) and /(0). However, the Carnahan - Starling model seems preferable since it does not involve unwieldy numerical calculation and leads to simple analytical expressions. In expressing these functions for low-concentration systems, it is also possible to use the standard technique of virial expansions. 

Simultaneously, conditions of thermal (equality of temperatures), mechanical (equality of pressures) and phase equilibrium (equality of chemical potentials of both components in both phases) must be satisfied. The constraints are defined by a set of equations nonlinear both in model parameters and in fliermodynamic quantities. When correlating vapor-liquid equilibrium data at low and moderate pressures, liquid phase is described in terms of a G -model and vapor phase is either considered as an ideal gas or its nonideality is expressed by an equation of state, e. g. the virial expansion limited to the second virial coefficient (Chap. 1.6). 

---