Equations virial equation

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 ... 

Fugacity coefficients and hence activity coefficients can be calculated with the help of appropriate equations of state (see Section IV). This is possible, however, only for the gas phase (van der Waals equation, Redlich-Kwong equation, virial equation) for condensed phases no useful general equations of state are available. Experimental determination of activity coefficients in condensed phases is based on the study of equilibria. There are numerous methods, but only typical examples will be given. 

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. 

Numerous empirical equations of state have been proposed but the theoretically based virial equation (Mason and Spurling, 1969) is most useful for our purposes. We use this equation for systems which do not contain carboxylic acids. 

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

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 ... 

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... 

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... 

A component in a vapor mixture exhibits nonideal behavior as a result of molecular interactions only when these interactions are very wea)c or very infrequent is ideal behavior approached. The fugacity coefficient (fi is a measure of nonideality and a departure of < ) from unity is a measure of the extent to which a molecule i interacts with its neighbors. The fugacity coefficient depends on pressure, temperature, and vapor composition this dependence, in the moderate pressure region covered by the truncated virial equation, is usually as follows ... 

This chapter uses an equation of state which is applicable only at low or moderate pressures. Serious error may result when the truncated virial equation is used at high pressures. 

The Virial Equation of State, Pergamon Press, Oxford (1969)... 

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. 

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. 

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. 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

We conclude with the matter of adsorbate-adsorbate interactions these give rise to deviations from Henry s law behavior. These may be expressed in the form of a virial equation, much as is done for imperfect gases. Following Steele [8], one can write... 

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 ... 

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

Virial equation represents the experimental compressibility of a gas by an empirical equation of state ... 

The Langmuir model is discussed in reference 19 the Volmer in reference 20 and the van der Waals and virial equations in reference 8. 

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 ... 

Cubic equations, although simple and able to provide semiquantitative descriptions of real fluid behavior, are not generally useful for accurate representation of volumetric data over wide ranges of T and P. For such appHcations, more comprehensive expressions with large numbers of adjustable parameters are needed. 7h.e simplest of these are the extended virial equations, exemplified by the eight-constant Benedict-Webb-Rubin (BWR) equation of state (13) ... 

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 ... 

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). 

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... 

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. 

At slightly higher pressures up to a reduced pressure of about 0.4, the truncated virial equation, Eq. (2-67), is commonly used for all types of organic fluids. 

Since reduced pressure is below 0.4, use virial equation (2-67 ). Calculate B by tbe Tsonopoulos method, Eq. (2-68). 

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