Pure virial equation

Spycher and Reed (1988) use a more or less pure virial equation approach to obtaining fugacities of components in gas mixtures. Their results are therefore... 

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 ... 

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. 

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 ... 

Purely phenomenological as well as physically based equations of state are used to represent real gases. The deviation from perfect gas behaviour is often small, and the perfect gas law is a natural choice for the first term in a serial expression of the properties of real gases. The most common representation is the virial equation of state ... 

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The virial equation of state discussed in Section 7.2 is applicable to gas mixtures with the condition that n represents the total moles of the gas mixture that is, n = f= l n,. The constants and coefficients then become functions of the mole fractions. These functions can be determined experimentally, and actually the pressure-volume-temperature properties of some binary mixtures and a few ternary mixtures have been studied. However, sometimes it is necessary to estimate the properties of gas mixtures from those of the pure gases. This is accomplished through the combination of constants. 

Two further crude approximations have been used for the virial equation of state. The first is that the virial coefficients combine linearly. This combination of constants results in an equation of state that is additive in the properties of the pure components. In such a mixture Dalton s and Amagat s laws still hold, and the mixture may be called an ideal mixture of real gases. The assumption is probably the crudest that can be used and is... 

This ratio of / to p for a non-ideal gas of a pure substance may be calculated from the equation of state for real gases such as the virial equation and the van der Wools equation. 

Although we have omitted an identifying subscript in the preceding equations, their application so far has been to the development of generated correlations for pure gases only. In the remainder of this section we show how the virial equation may be generalized to allow calculation of fugacity coefficients < , of species in gas mixtures. 

As discussed in Chap. 3, equations of state provide concise descriptions of the PVT behavior for pure fluids. The only equation of state that we have used extensively is the two-term virial equation,... 

It was van t Hoff, winner of the very first Nobel prize in chemistry, who perceived an analogy between the properties of dilute solutions and the gas laws. We will see that many physical properties of dilute solutions, such as the amount of light scattered or the viscosity, can be written as a virial equation in the number of molecules (moles), N, or concentration of solute, c. We have written a general form of a virial equation in Equation 12-4, using the quantity P to represent some measured property of the solution and P0to represent the property of the pure solvent. 

The first application of such equations to dilute solutions actually came from van t Hoff s measurements of the osmotic pressure of 1% solutions of cane sugar in water (relative to pure water), where the analogy to the virial equation of a gas expressed as a power series in the pressure is more direct. Accordingly, we will start our discussion of molecular weight measurements by considering osmotic pressure. 

Equation (2-62) is the key to the application of colligative properties to polymer molecular weights. We started with Eq. (2-53), which defined an ideal solution in terms of the mole fractions of the components. Equation (2-62), which followed by simple arithmetic, expresses the difference in chemical potential of the solvent in the solution and in the pure state in terms of the mass concentrations of the solute. This difference in chemical potential is seen to be a power series in the solute concentration. Such equations are called virial equations and more is said about them on page 65. 

Values of (p,- for species i in solution are readily found from equations of state. The simplest form of the virial equation provides a useful example. Written for a gas mixture it is exactly the same as for a pure species ... 

Whether the adsorption isotherm has been determined experimentally or theoretically from molecular simulation, the data points must be fitted with analytical equations for interpolation, extrapolation, and for the calculation of thermodynamic properties by numerical integration or differentiation. The adsorption isotherm for a pure gas is the relation between the specific amount adsorbed n (moles of gas per kilogram of solid) and P, the external pressure in the gas phase. For now, the discussion is restricted to adsorption of a pure gas mixtures will be discussed later. A typical set of adsorption isotherms is shown in Figure 1. Most supercritical isotherms, including these, may be fit accurately by a modified virial equation. ... 

Equation 7-14 is used to calculate the reference state fugacity of liquids. Any equation of state can be used to evaluate ([) . For low to moderate pressures, the virial equation is the simplest to use. The fugacities of pure gases and gas mixtures are needed for estimating many thermodynamic properties, such as entropy, enthalpy, and fluid phase equilibria. For pure gases, the fugacity is... 

Data are readily available for pure component and binary interaction second virial coefficients for a large number of components and binaries. Binary interaction coefficients are required for extending the equation to mixtures. The simplicity of the equation, the availability of coefficient data, and its ability to represent mixtures are some of the reasons the virial equation of state is a viable option for representing gases at densities up to about 70% of the critical density. It may be used for calculating vapor phase properties at these conditions but is not applicable to dense gases or liquids. 

The pure-component fugacity coefficient is obtained by combining Equation 1.21 with the truncated virial equation ... 

Equation (P3.7.7) shows the difference in chemical potential of the solvent in the solution and in the pure state to be a power series in solute concentration. Such equations are called virial equations. 

Compute the fugacities of pure ethane and pure butane at 373.15 K and 1, 10, and 15 bar, assuming the virial equation of state can describe these gases at these conditions. 

The virial equation of state in Table 4.2 provides a sound theoretical basis for computing P-v-T relationships of polar as well as nonpolar pure species and mixtures in the vapor phase. Virial coefficients B, C, and higher can, in principle, be determined from statistical mechanics. However, the present state of development is such that most often (4-34) is truncated at B, the second virial coefficient, which is estimated from a generalized correlation. - In this form, the virial equation is accurate to densities as high as approximately one half of the critical. Application of the virial equation of state to phase equilibria is discussed and developed in detail by Prausnitz et al. and is not considered further here. 

Thus, pressure-explicit equations of state for pure substance 1 (for the first integral) and for the gas mixture (the second integral) are required. Five different equations of state have been used in the analysis of this system (1) the five-constant Beattie-Bridgeman equation (2) the eight-constant Benedict-Webb-Rubin equation (3) the twelve-constant modified Martin-Hou equation and (4) and (5), the virial equation using two sets of virial coefficients. The first of these uses pure-substance second and third virial coefficients calculated from the Lennard-Jones 6-12 potential with interaction coefficients determined by the method of Ewald [ ]. The second set differs only in the second virial coefficients and interaction coefficient, these being found using the Kihara potential Solutions of the theoretical equa-... 

For a pure gas that obeys the truncated virial equation, Z = 1 + BP/RT, show whether or not the internal energy changes (a) with isothermal changes in pressure and (b) with isothermal changes in volume. 

Consider a gas that obeys the simple virial equation Z = 1 + BP IRT. Determine whether this substance can become mechanically unstable. Is your conclusion affected by whether the gas is pure or a mixture ... 

In Eq. (77) p° is the standard pressure, p° = l atm quantity p

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