Statistical mechanics virial equations

Although PVT equations of state are based on data for pure fluids, they are frequently appHed to mixtures. 7h.e virial equations are unique in that rigorous expressions are known for the composition dependence of the virial coefficients. Statistical mechanics provide exact mixing rules which show that the nxh. virial coefficient of a mixture is nxh. degree in the mole fractions ... 

Although the virial equation itself is easily rationalized on empirical grounds, the mixing rules of Eqs. (4-183) and (4-184) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by... 

The equilibrium between a pure solid and a gaseous mixture is one of very few classes of solution for which an exact treatment can be made by the methods of statistical mechanics. The earliest work on the theory of such solutions was based on empirical equations, such as those of van der Waals,45 of Keyes,44 and of Beattie and Bridgemann.3 However, the only equation of state of a gas mixture that can be derived rigorously is the virial expansion,46 66... 

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. 

Even though the van der Waals equation is not as accurate for describing the properties of real gases as empirical models such as the virial equation, it has been and still is a fundamental and important model in statistical mechanics and chemical thermodynamics. In this book, the van der Waals equation of state will be used further to discuss the stability of fluid phases in Chapter 5. 

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. 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

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. 

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 statistical mechanical verification of the adsorption Equation 11 proceeds most conveniently with use of the expression for y given by Equation 5. An identical starting formula is obtained via the virial theorem or by differentiation of the grand partition function (3). We simplify the presentation, without loss of generality, by restricting ourselves to multicomponent classical systems possessing a potential of intermolecular forces of the form... 

Virial treatment provides a general method of analysing the low-coverage region of an adsorption isotherm and its application is not restricted to particular mechanisms or systems. If the structure of the adsorbent surface is well defined, virial treatment also provides a sound basis for the statistical mechanical interpretation of the adsorption data (Pierotti and Thomas, 1971 Steele, 1974). As indicated above, Kl in Equation (4.5) is directly related to kH and therefore, under favourable conditions, to the gas-solid interaction. 

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. 

Milner s contribution (1912) was direct. He attempted to find out the virial equation for a mixture of ions. However. Milner s statistical mechanical approach lacked the mathematical simplicity of the ionic-cloud model of Debye and Hiickel and proved too unwieldy to yield a general solution testable by experiment. Nevertheless, his contribution was a seminal one in that for the first time the behavior of an ionic solution had been linked mathematically to the interionic forces. 

Rm is the molar refraction or Lorentz-Lorenz function, and As, Ba, Cb, are the refractivity virial coefficients. The rigorous derivation of the statistical mechanical equations for / m> Ab, and Bb, corresponding to equations (5)—(11), is complicated by the variation of the electric field... 

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... 

Constraint dynamics is just what it appears to be the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to mclude a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rj,r2...rN and velocities V=Vi,V2,...vn of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as l dV/dt -i- s = 0 where I and s are functions of R and V only. Gauss principle states that the constrained equations of motion can be written as ... 

Mairy otlrer equations of state Irave been proposed for gases, but the virial equations are tire oirly ones having a finrr basis in theory. Tire methods of statistical mechanics allow derivation of tire virial equations and provide physical sigirificance to the virial coefficients. Thus, for the expansionin 1/ F, the term B/V arises on account of interactions between pairs of molecules (Sec. 7d.2) the C / term, on account of tlrree-body interactions etc. Since two-... 

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... 

Statistical mechanics gives relationships between the distribution functions and the bulk properties of fluids. The total internal energy of a fluid is given by the energy equation, the pressure is given by the virial equation, and the isothermal compressibility is given by the compressibility equation, see e. g.. Ref. 11. Through the Kirkwood-Buff formulas (0,... 

One of the exact results we do have from statistical mechanics is the virial equation of state... 

Most other approaches to the calculation of activity coefficients for solution components, including solid and gaseous as well as liquid solutions, have used some form of a virial equation as a starting point. A virial equation is simply an equation for the ideal state (e.g., the ideal gas equation) followed by an ascending polynomial in one of the state variables. It seems to work well as a basis for activity coefficients because the form of the equation has a basis in 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 ... 

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