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The Virial Equation

The virial equation of state was originally developed to describe the P — y T behavior of non-ideal gases. Like the Margules equations, it appeared in the literature around the turn of the century in forms such as that given by Kammerlingh-Onnes in 1901  [Pg.384]

At first sight this looks like nothing more than a polynomial expansion of the ideal gas law. However, it turns out to have real physical significance, and the form of the equation follows directly from statistical mechanics. The details can be found in most textbooks on statistical mechanics (see, for example, Mayer and Mayer, 1940 Hill, 1960, Chapter 15). We will outline the underlying theory very briefly here because virial equations (or similar approaches) appear several times in this book—see for example the discussion of the Pitzer equations for the non-ideal properties of salt solutions in Chapter 17 and Chapter 16 on gas mixtures. [Pg.384]

Remember that ideal gases should have no inter-particle interactions. This gives the first virial coefficient, R, and the ideal gas law. Unfortunately, the atoms or molecules of real gases do interact. The energy E of one mole of a real gas is the sum of the kinetic and intermolecular potential energies of all its molecules, E = K + v. [Pg.384]

In theory, if we knew the functional form of each kind of molecular interaction (such as coulombic interactions oc dipole interactions oc r , interaction between neutral particles oc overlap repulsion at very close distances oc [Pg.384]

The kinetic term here can be evaluated from temperature and particle mass, and it is the second summation that causes the problems. Because we are working with an [Pg.384]

We will discuss first the virial equation, because of its theoretical basis, and then consider the empirical approaches Pitzer s CSP and cubic EoS. [Pg.350]

Because of its theoretical basis, i.e. its development from Statistical Mechanics, the virial equation (Section 8.9) represents the only EoS where rigorous mixing rules for the mixture coefficients are available. Thus  [Pg.350]

Obviously B = Bjf and, as with the case of a pure compound, B is a ftinction of temperature only. The sununations, finally, are over all mixture components. Thus, for a binary mixture  [Pg.351]

Similar expressions exist for the third and higher coefficients. We will restrict, however, our discussion to the virial equation truncated after the second term since higher coefficients, as we have seen, are rarely available. [Pg.351]

Values for the cross coefficient B are determined from experimental data, such as PVT ones, for the binary system i-j. But such experimental Bfj values are not often available and, consequently, we must resort to some estimation technique, just as we did with B for pure compounds. [Pg.351]

A statistical mechanical treatment shows that the equation of state of a real gas can be expressed as a power series in /Vm as given by equation (A3.3) [Pg.627]

From a molecular point of view, this equation implies that the internal energy of the gas does not depend upon the separation of the gaseous molecules, potential energy due to attractions and repulsions between the molecules is not present, and the internal energy is a function only of the temperature. [Pg.627]

In 1901 Kammerlingh-Onnes first used a power series to describe the PVT properties of a gas. This is now known as a virial equation of state, and can be written in various related ways  [Pg.378]

Often it is more convenient to use P and T as variables, in which case the equation becomes [Pg.378]

Because it is an equation of state, it (at least in principle) allows calculation of aU thermodynamic parameters. For example, chemical potentials in terms of virial parameters are (starting with (13.25)) [Pg.378]

Other examples will be cited later on. For a complete summary, see Pitzer (1995, p. 133). [Pg.379]


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. [Pg.26]

The virial equation of state is a power series in the reciprocal molar volume or in the pressure ... [Pg.27]

For a pure vapor the virial coefficients are functions only of temperature for a mixture they are also functions of composition. An important advantage of the virial equation is that there are theoretically valid relations between the virial coefficients of a mixture and its composition. These relations are ... [Pg.28]

At moderate densities. Equation (3-lOb) provides a very good approximation. This approximation should be used only for densities less than (about) one half the critical density. As a rough rule, the virial equation truncated after the second term is valid for the present range... [Pg.29]

The virial equation is appropriate for describing deviations from ideality in those systems where moderate attractive forces yield fugacity coefficients not far removed from unity. The systems shown in Figures 2, 3, and 4 are of this type. However, in systems containing carboxylic acids, there prevails an entirely different physical situation since two acid molecules tend to form a pair of stable hydrogen bonds, large negative... [Pg.31]

The Virial Equation of State, Pergamon Press, Oxford (1969)... [Pg.38]

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. [Pg.59]

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. [Pg.82]

As discussed in Chapter 3, the virial equation is suitable for describing vapor-phase nonidealities of nonassociating (or weakly associating) fluids at moderate densities. Equation (1) gives the second virial coefficient which is used directly in Equation (3-lOb) to calculate the fugacity coefficients. [Pg.133]

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 ... [Pg.354]

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. [Pg.423]

The virial equation for the pressure is also modified by the tliree-body and higher-order temrs, and is given in general by... [Pg.475]

The virial equations are unsuitable forhquids and dense gases. The simplest expressions appropriate (in principle) for such fluids are equations cubic in molar volume. These equations, inspired by the van der Waals equation of state, may be represented by the following general formula, where parameters b, 9 5, S, and Tj each can depend on temperature and composition ... [Pg.485]

SemiempiricalRelationships. Exact thermodynamic relationships can be approximated, and the unknown parameters then adjusted or estimated empirically. The virial equation of state, tmncated after the second term, is an example of such a correlation (3). [Pg.232]

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... [Pg.233]

Correlation Methods Vapor densities are not correlated as functions of temperature alone, as pressure and temperature are both important. At high temperatures and very low pressures, the ideal gas law can be applied whde at moderate temperature and low pressure, vapor density is usually correlated by the virial equation. Both methods will be discussed later. [Pg.399]

Virial Equations of State The virial equation in density is an infinite-series representation of the compressiDility factor Z in powers of molar density p (or reciprocal molar volume V" ) about the real-gas state at zero density (zero pressure) ... [Pg.529]

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... [Pg.529]

An alternative form of the virial equation expresses Z as an expansion in powers of pressure about the real-gas state at zero pressure (zero density) ... [Pg.529]

Equation (4-187) is the virial equation in pressure, and B, C, D, . . . , are the pressure-series virial coefficients. Like the density-series coefficients, they depend on temperature and composition only. Moreover, the two sets of coefficients are related ... [Pg.529]

Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, ATChP J., 25, pp. 931-938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B given by Eq. (4-188), is usually preferred ... [Pg.529]

The MS closure results from s = 2. The HNC closure results from s = 1. In the latter two expressions, additional adjustable parameters occur, namely ( for the RY closure and for the BPGG version of the MS approximation. However, even when adjustable, these parameters cannot be chosen at will, as they should be chosen such that they eliminate the so-called thermodynamic inconsistency that plagues many approximate integral equations. We recall that a manifestation of this inconsistency is that there is a difference between the pressure as computed from the virial equation (10) and as computed from the compressibility equation (20). Note that these equations have been applied to a very asymmetric mixture of hard spheres [53,54]. Some results of the MS closure are plotted in Fig. 4. The MS result for y d) = g d) is about the same as the MV result. However, the MS result for y(0) is rather poor. Using a value between 1 and 2 improves y(0) but makes y d) worse. Overall, we believe the MS/BPGG is less satisfactory than the MV closure. [Pg.149]

Finally, in this part of the work we would like to discuss to some extent practical tools to obtain thermodynamic properties of adsorbed fluids. We have mentioned above that the compressibility equation is the only simple recipe, for the moment, to obtain the thermodynamics of partly quenched simple fluids. The reason is that the virial equation is difficult to implement it has not been tested for partly quenched systems. Nevertheless, for the sake of completeness, we present the virial equation in the form [22,25]... [Pg.303]

A. Milchev, K. Binder. Osmotic pressure, atomic pressure and the virial equation of state of polymer solutions Monte Carlo simulations of a bead-spring model. Macromol Theory Simul 5 915-929, 1994. [Pg.630]

In their classic review on Continuous Distributions of the Solvent , Tomasi and Persico (1994) identify four groups of approaches to dealing with the solvent. First, there are methods based on the elaboration of physical functions this includes approaches based on the virial equation of state and methods based on perturbation theory with particularly simple reference systems. For many years... [Pg.254]

Equations of state, such as the virial equation, demonstrate that o remains finite as p— 0. See Appendix 3 for a discussion of the virial and other equations of state. [Pg.252]


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