Virial equation limitations

When real behaviour is again described by means of the virial equation limited to the second virial coefficient in the form of (6.58), the function Q(n) is changed from the form of (5.58) to... 

This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture. 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

Though limited to pressures where the two-term virial equation in pressure has approximate vahdity, this correlation is applicable to most chemical-processing conditions. As with all generalized correlations, it is least accurate tor polar and associating molecules. 

Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

The virial equation for osmotic pressure must reduce to the van t Hoff equation in the limit of infinite dilution. 

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 mutual dependence of the pressure, volume, and temperature of a substance is described by its equation of state. Many such equations have been proposed for the description of the actual properties of substances (and mixtures) in the gaseous and liquid states. The van der Waals expression is just one of these and of limited applicability. The virial equation of state ... 

Carbon monoxide (CO) PVT behavior is well represented by the Soave-Redlich-Kwong (SRK) equation of state. For quick, approximate calculations over a limited range of conditions, it is proposed to use the virial equation of state of the form... 

The virial equation as a power series of the pressme is to be used in an application for a quick estimation of the pressure-volume relationship of methane at a constant temperature of 200 K. In this application the speciflc volume of methane varies over a range from 2 to 4 mVkmol. Estimate the second and third virial coefficients over this range based on data derived from the Soave equation of state at the range limits. Methane properties Tc = 190.6 K, Pc = 4599 kPa, to = 0.012. 

Recall that the virial equations originate from Taylor expansions about the ideal gas. Alternatives can be obtained by expanding, not about ideal gases, but about hard spheres. Real fluids approach hard-sphere behavior in the isochoric high-temperature limit (4.5.1), so we use the parameter P = 1/RT as the independent variable. Then on truncating the expansion at first-order, we have... 

The main advantage of the virial equation is its theoretical background, as the virial coefficients can be connected to the potential functions of the intermolecular forces. The main disadvantage is that the virial equation is only valid for gases with low or moderate densities. If the equation is truncated after the second term, a rule of thumb says that the density should be p < l/2pc. If it is truncated after the third term, p < 2> Apc should be maintained. This limitation is necessary, because the fourth and all higher virial coefficients are unknown in nearly all practical cases. Even values for the third virial coefficient can hardly be found in the literature. The virial equation can be understood as a Taylor series of the compressibility factor z as a function of the density p at the point po- The reference point po is chosen as the density at zero pressure, where the ideal gas law is valid ... 

The virial equation of state, first proposed on an empirical basis in the nineteenth century and later found to have a solid theoretical basis in statistical mechanics, is sometimes dismissed by geochemists because it only works well at low to moderate densities (see 13.5.1), and there have not been many direct applications in the geochemical literature - Spycher and Reed ( 13.7.3) is an exception. Also it is incapable of representing vapor-liquid equilibria, as do the cubic EoS. However, it is important in any study of the thermodynamics of fluids because references to it are ubiquitous in the literature on equations of state, so an understanding of it and its limitations is fundamental. 

Diamond (2003) use this as a starting point to develop an equation of state for nonelectrolytes, many of which are gaseous under normal conditions. This is of course derived from the virial equation, and has the usual limitation of being valid only at low to moderate densities. Akinfiev and Diamond propose an equation to describe the binary interaction parameter By as a function of P and T. Because By in the virial equation is a function only of T and is independent of P, the development in Akinfiev and Diamond departs from virial theory, using it only as a starting point. For two components, the B terms in (13.47) become AB, called AB, to which they ascribe the temperature dependence... 

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). 

The Virial Equation for an Imperfect Two-dimensional Gas/ temperature limit of equation (60) may be written as ... 

Plyasunov et al. (2000) have proposed a semitheoretical expression for VI based on the fluctuation theory of solution which included the second virial coefficient, Bu, to give a rigorous expression in the low density region. The equation, limited to solutes for which Bu is known or can be estimated, has the form ... 

First, at homogeneous limit the local density becomes a constant, and then DFT naturally gives rise to the EOS in Helmholtz type (Chapman et al., 1989 Lemmon andjacobsen, 2005). Second, EOS can be developed with the knowledge of local structure through Virial equation. Specifically, for the fluid composed of spherically symmetric molecules ofm components, the pressure is determined by... 

The virial equation is limited to gases at low or moderate densities, while many other equations exist that purport to cover the entire fluid region. Nevertheless, the virial equation possess a number of key strengths including ... 

We can see that this expression is identical to the virial equation given by relation [7.20], limited to the second term by identification, the second coefficient of the virial is therefore ... 

At the limit of low pressures, it can be shown that B = B. The second virial coefficient is typically the largest nonideal term in a virial equation, and many lists of virial coefficients give onlyB or B. ... 

Limited-range equations which also represent the experimental data to within their accuracy are mostly available for the gas phase. Virial equations which have been fitted to a wide temperature range are usually of the form... 

There are very few limited-range virial equations for specific fluids in the literature. Values for the second virial coefficients of several gases at specified temperatures are given in Marsh etal. (1993) and Dymond Smith (1980) if a correlating equation is required, it is recommended that these values are fitted to a polynomial in 1/ J. 

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