Equations virial expansion equation

As for the case of membrane osmometry, non-ideality is accounted for by a virial expansion (Equation (28)). 

Generally speaking, many one-component state equations have been proposed (Read et al., 1977), but the van der Waals equation 34 and the virial expansion equation... 

Extension of the virial expansion (Equation 5.4 41) up to the third term has the form (Duplantier, 1986b)... 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibiHty factor Z, defined as Z = PV/RT having either molar density, p[ = V ), or pressure, P, as the independent variable of expansion ... 

The PirialExpansion. Many equations of state have been proposed for gases, but the virial equation is the only one having a firm basis in theory (1,3). The pressure-expHcit form of the virial expansion is... 

The equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the Hquid region. The compressibihty factor can be used to accurately determine gas volumes when used in conjunction with a virial expansion or an equation such as equation 53 (77). However, the prediction of saturated Hquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibihty factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

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... 

If the gas of charges (plasma) is sufficiently dilute, we could hope a priori that its equation of state would be described by the virial expansion ... 

Finally, we study the structure of the generalized Boltzmann operator. It can be expressed in terms of the transport operator, which allows one to obtain the virial expansion of the generalized Boltzmann equation. The remarkable point here is that the generalized Boltzmann operator can be expressed in terms of non-connected contributions to the transport operator. This happens for the correction proportional to c3 (c = concentration) and for the following terms in the virial expansion of the generalized Boltzmann operator. 

This is a simple and important result. It equates VPIE to the isotopic difference of standard state free energies on phase change, plus a small correction for vapor phase nonideality, here approximated through the second virial coefficient. Therefore Equation 5.8 is limited to relatively low pressure. As T and P increase third and higher virial corrections may be needed, and at even higher pressures the virial expansion must be abandoned for a more accurate equation of state. 

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]. 

A classical equation of state is normally composed of a truncated Taylor series in the independent variables, normalized to the critical point conditions (e.g., van der Waals, virial expansion, etc.). All these sorts of equations yield similar (so-called classical ) asymptotic behavior in their derivative properties at the critical point. 

Virial expansion coefficients B, C, D and B , C, D in equations 9.23 and 9.24 are simple functions of T and are specific to each gaseous species. Virial expansion equations such as equations 9.23 and 9.24 do not reproduce the behavior of real gases in the vicinity of the critical point and under high P and T conditions with satisfactory precision. 

The difference between equation 9.38 and the corresponding virial expansion of equation 9.23 lies in term A, which substitutes 1 and which is a function of T, as are the remaining coefficients. Equation 9.38 also finds application in a corresponding states notation ... 

A quite different approach to the detonation product state has been to treat it as solidlike. Jones and Miller6 performed equilibrium calculations on TNT with this idea in mind. They used an equation in which the volume was a virial expansion in the pressure. Other solidlike equations are cited in Ref. 2, but these have mostly been used for computing the state parameters with an assumed product state. The modified Kistia-kowsky-Wilson equation of interest to us liere appears to be one of several possible compromises between the hard-sphere molecule approach and the solid state approach. 

Equation (95) is obtained from the virial expansion of the equation of state for rigid spheres for higher densities the rigid-sphere equation of state obtained from the radial distribution function by Kirkwood, Maun, and Alder has to be used (K10, Hll, p. 649). When Eq. (95) is substituted in Eqs. (92), (93), and (94) one then obtains the rigorous expressions for the coefficients of viscosity, thermal conductivity, and selfdiffusion of a gas composed of rigid spheres. 

Equation (2.1) is written in a virial expansion form by introducing the density p = 1/V and the reduced variables for density, temperature, and pressure... 

This is a virial expansion form of the osmotic pressure analogous to the van der Waals fluid. Dusek and Patterson examined this equation and predicted the presence of two phases, i.e. collapsed and swollen phases. % is temperature dependent and is given by,... 

In 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed as... 

The virial expansion (2.30) can also be used to expand empirical equations of state in power series form, allowing term-by-term comparisons of one equation with another. To illustrate this procedure, let us obtain the virial expansion for the Van der Waals equation (2.26), which can be rewritten as... 

In these equations ns is the solvent refractive index, dn/dc the refractive index increment, c the polymer concentration in g/ml, T the temperature in K, R the gas constant, NA Avogadro s number, and n the osmotic pressure. Equation (B.8) follows from Eq. (B.7) by using the familiar virial expansion of the osmotic pressure... 

Equations 10 and 11 are known as virial expansions, and the coefficients B C, D. .. and B, C, D,... are called vitial coefficients. For a given substance, these coefficients are functions of temperature only (1—3). 

The virial expansion has a far more solid theoretical justification than does the van der Waals equation. It can be shown quite generally that ... 

Finally, we should note that the a and b coefficients in the van der Waals equation are related to B(T) in the virial expansion. In the limit that b <some algebraic manipulation (Problem 7-7) that... 

Many of the other viscosity equations are simply empirical virial expansions, and typically have the form [37,215] ... 

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. 

Although the virial expansions might seem very complicated, because they contain an infinite number of terms, their power lies in the fact that usually only a few terms must be considered. As pressure is reduced and molar volume gets very large, the higher terms in the expansion become negligible and only the first two terms need be considered. Equation (26) then becomes... 

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