Virial formulation

The virial methods differ conceptually from other techniques in that they take little or no explicit account of the distribution of species in solution. In their simplest form, the equations recognize only free ions, as though each salt has fully dissociated in solution. The molality m/ of the Na+ ion, then, is taken to be the analytical concentration of sodium. All of the calcium in solution is represented by Ca++, the chlorine by Cl-, the sulfate by SO4-, and so on. In many chemical systems, however, it is desirable to include some complex species in the virial formulation. Species that protonate and deprotonate with pH, such as those in the series COg -HCOJ-C02(aq) and A1+++-A10H++-A1(0H), typically need to be included, and incorporating strong ion pairs such as CaSO aq) may improve the model s accuracy at high temperatures. Weare (1987, pp. 148-153) discusses the criteria for selecting complex species to include in a virial formulation. 

The sum is over all pairs of molecules the angle bracket notation as usual denotes an ensemble average of the enclosed quantity, and < )(r y) is the potential of interaction between molecules i and /. The p term is the energy associated with induced moments (polarization). It is treated in many ways, including setting it to zero the nontrivial treatments are discussed in later sections. The pressure, p, is obtained using the molecular virial formulation ... 

However, if one focuses on the adsorption of a fluid in heterogenous matrices [32,33] and/or on the fluctuations in an adsorbed fluid, it is inevitable to perform developments similar to those above in the grand canonical ensemble. Moreover, this derivation is of importance for the formulation of the virial route to thermodynamics of partially quenched systems. For this purpose, we include only some basic relations of this approach. 

The potential energy function prohibits double occupancy of any site on the 2nnd lattice. In the initial formulation, which was designed for the simulation of infinitely dilute chains in a structureless medium that behaves as a solvent, the remaining part of the potential energy function contains a finite repulsion for sites that are one lattice unit apart, and a finite attraction for sites that are two lattice units apart [153]. The finite interaction energies for these two types of sites were obtained by generalizing the lattice formulation of the second virial coefficient, B2, described by Post and Zimm as [159] ... 

In the virial methods, therefore, the activity coefficients account implicitly for the reduction in the free ion s activity due to the formation of whatever ion pairs and complex species are not included in the formulation. As such, they describe not only the factors traditionally accounted for by activity coefficient models, such as the effects of electrostatic interaction and ion hydration, but also the distribution of species in solution. There is no provision in the method for separating the traditional part of the coefficients from the portion attributable to speciation. For this reason, the coefficients differ (even in the absence of error) in meaning and value from activity coefficients given by other methods. It might be more accurate and less confusing to refer to the virial methods as activity models rather than as activity coefficient models. 

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Most liquid-crystalline polymer solutions have a large second virial coefficient ( > 10 4 cm 3mol/g2) [41], which means that it is rather difficult to find poor or theta solvents for these polymers and that liquid-crystalline polymers in solution interact repulsively. This fact is essential in formulating their static solution properties (osmotic pressure, phase separation, etc.). 

In addition to the above effects, the intermolecular interaction may affect polymer dynamics through the thermodynamic force. This force makes chains align parallel with each other, and retards the chain rotational diffusion. This slowing down in the isotropic solution is referred to as the pretransition effect. The thermodynamic force also governs the unique rheological behavior of liquid-crystalline solutions as will be explained in Sect. 9. For rodlike polymer solutions, Doi [100] treated the thermodynamic force effects by adding a self-consistent mean field or a molecular field Vscf (a) to the external field potential h in Eq. (40b). Using the second virial approximation (cf. Sect. 2), he formulated Vscf(a), as follows [4] ... 

In particular, it is well known that, if the macromolecule is supercooled below the 0 temperature, the phase transition isotropic coil-isotropic globule occurs. We emphasize that for the semiflexible macromolecule this is the peculiar phase transition between two metastable states. It should be recalled that the theory of the transition isotropic coil-isotropic globule for the model of beads is formulated in terms of the second and third virial coefficients of the interactions of beads , B and C24). This transition takes place slightly below the 0 point and its type depends on the value of the ratio C1/2/a3 if Cw/a3 I, the coil-globule transition is the first order phase transition with the bound of the macromolecular dimensions, and if C1/2/a3 1, it is a smooth second order phase transition (see24, 25)). 

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. 

Pitzer (1973) re-examined the statistical mechanics of aqueous electrolytes in water and derived a different but semi-empirical method for activity coefficients, commonly termed the Pitzer specific-ion-interaction model. He fitted a slightly different function for behavior at low concentrations and used a virial coefficient formulation for high concentrations. The results have proved extremely fruitful for modeling activity coefficients over a very large range of molality. The general equation is... 

While a great deal of progress has proved possible for the case of the hydrogen atom by direct solution of the Schrodinger wave equation, some of which will be summarized below, at the time of writing the treatment of many-electron atoms necessitates a simpler approach. This is afforded by the semi-classical Thomas-Fermi theory [4-6], the first explicit form of what today is termed density functional theory [7,8]. We shall summarize below the work of Hill et al. [9], who solved the Thomas-Fermi (TF) equation for heavy positive ions in the limit of extremely strong magnetic fields. This will lead naturally into the formulation of relativistic Thomas-Fermi (TF) theory [10] and to a discussion of the role of the virial in this approximate theory [11]. 

Using eqs. (l)-(9), along with empirical pure-electrolyte parameters 3 ), 3 > 3 and and binary mixture parameters 0, one can reproduce experimental activity-coefficient data typically to a few percent and in all cases to + 20%. Of course, as noted above, the most accurate work on complex, concentrated mixtures requires that one include further mixing parameters and also for calculations at temperatures other than 25°C, include the temperature dependencies of the parameters. However, for FGD applications, a more important point is that Pitzer1s formulation appears to be a convergent series. The third virial coefficients... 

Pitzer s formulation offers a satisfactory and desirable way to model strong electrolyte activity coefficients in concentrated and complex mixtures. When sufficient experimental data are available, one can make calculations which are considerably more accurate than those presented in this paper. Attaining high accuracy requires not only experimentally-based parameters but also that one employ third virial coefficients and additional mixing terms and include explicit temperature dependencies for the various parameters. 

The standard form of the Virial EOS formulates the compressibility as an infinite series of the inverse molar volume or pressure, as shown by the equations (5.3) and (5.4). In the low-pressure region, up to 15 bar, Virial EOS is the most accurate. The formulation known as second virial coefficient is sufficient for technical computations. Virial EOS is able to handle a variety of chemical classes, including polar species. Hayden and O Connell have proposed one of the best correlations for the second virial parameter. The method is predictive, because considers only physical data, as dipole moment, critical temperature, critical pressure, and the degree of association between the interacting components. Usually Virial EOS it is an option to describe the vapour phase... 

The original Virial EOS was applicable only to the gas phase. This limitation incited the development of extended forms, as the Benedict-Webb-Rubin (BWR) correlation (equation 5.5). This equation may contain sophisticated terms with a large number of parameters, mostly between 10 and 20, and need substantial experimental data for tuning. The extended Virial-type EOS s have lost much of their interest after the arrival of various cubic EOS in the last decades. Some formulations are still used for special applications, notably in gas processing and liquefaction, as BWR-Lee-Starling (BWR-LS) equation, one of the most accurate for hydrogen rich hydrocarbon mixtures. Note that extended Virial EOS may calculate not only volumetric properties, but also VLE. 

To obtain a virial form of state equation suitable for a real gas, engineers used numerous experimental data. They have been used to formulate empirical equations of state. In these equations pressure is presented as polynomial of substance molar density p = jo with coefficients dependent on temperature. Out of many suggested equations of state, the most widely accepted is the Benedict-Weber-Rubin equation (BWR) and its modifications. One of successful (from the viewpoint of computational accuracy) modifications of empirical equations of state is the Starling-Khan equation... 

Equally, the following theorems of the virial theorem type can be formulated ... 

Equations (1.5-12)-(1.5-15) together constitute the most common formulation for predicting or correlating subcritical VLE at low to moderate pressures. When using the formulation for VLE predictions, one requites data or correlations for pure-component vapor pressures (e.g., Antoine equations), for the activity coefRcients (e.g., the UNIQUAC equation or the UNIFAC correlation), for the second virial coefficients (e.g., one of the correlations referent in Section 1.3-2), and for the molar volumes of the saturated liquid (e.g., the Rackett equation - for v ). The actual VLE calculations are iterative and require the use of a computer, details are given in the monograph by Ptausnitz et al. ... 

The knowledge of equations of state for gas phases permits the calculation of activity coefficients via fugacity coefficients. Equations of state for general practical use such as the virial equation (and others) are not known for condensed phases (liquids and solids). However, as shown by Planck and Schottky, the passage from the gaseous to the liquid or solid state does not change the structure of Eq. (87) and leads to the general formulation for the chemical potentials,... 

There have been significant advances on both the theoretical and experimental fronts in elucidating the behavior of nuclear material at very low densities, densities less than 0.1 of saturation. On the theoretical front, a virial (density expansion) EoS has been formulated by... 

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