Fugacity coefficient from virial equation

Example 1.16 Estimation of fugacity coefficients from virial equation Derive a relation to estimate the fugacity coefficients by the virial equation... 

Fugacity Coefficients from the Virial Equation of State... 

For areal gas at temperature T and pressure />, Eq. 7.8.16 or 7.8.18 allows us to evaluate the fugacity coefficient from an experimental equation of state or a second virial coefficient. We can then find the fugacity from / = (pp. 

The fugacity coefficients are determined from equations of state. The equations available for the vapor phase fugacity coefficients are listed in Figure 11. The two B-W-R equations and the Soave equation are also used for liquid phase fugacity coefficients. The Soave equation and the Hayden-0 Connell virial equation are very recent additions to the system. Therefore, if they are eliminated from consideration, the Prausnitz-Chueh modification of the Redllch-Kwong equation and the BWR equations have received the most usage. 

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 ... 

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. 

IF BINARY SYSTEM CONTAINS NO ORGANIC ACIDS. THE SECOND VIRTAL coefficients ARE USED IN A VOLUME EXPLICIT EQUATION OF STATE TO CALCULATE THE FUGACITY COEFFICIENTS. FOR ORGANIC ACIDS FUGACITY COEFFICIENTS ARE PREDICTED FROM THE CHEMICAL THEORY FOR NQN-IOEALITY WITH EQUILIBRIUM CONSTANTS OBTAINED from METASTABLE. BOUND. ANO CHEMICAL CONTRIBUTIONS TO THE SECOND VIRIAL COEFFICIENTS. 

When i = J, all equations reduce to the appropriate values for a pure species. When i j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of By and dBjj/dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4-183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of and for the mixture are then given by Eqs. (4-193) and (4-194), and values of In i for the component fugacity coefficients are given by Eq. (4-196). 

Gamma/Phi Approach For many XT E systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactoiy for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written... 

Restriction to relatively low pressures allows calculation of the fuga coefficients in Eq. (12.2) from the simplest form of the virial equation of s< the two-term expansion in P (Eq. (3.31)]. In this case the expression for k, fugacity coefficient for species k in solution, follows from Eq. (11.48) ... 

If we chose our standard state for the activity coefficient to be the pure liquid solvent at pressure P and temperature T, then the above equation becomes the expression for the standard state fugacity. The fugacity coefficient at the saturation pressure can be calculated from the second virial coefficient... 

Due to lack of space, the few results presented here are primarily intended to demonstrate the validity of the proposed method. The pore space of the adsorbent is assumed to consist of slit-shaped pores of width 15 A, with parameters chosen to model activated carbon. The porosity values are fixed at q = 0.45 and qp = 0.6. The feed stream is atemary gas mixture of H2/CH4/C2H6. The vtqx>r-phase fugacities were computed from the virial equation to second order, using coefficients taken from Reid et al ... 

In the gas law for real gases, the molar volume can be expressed with one or two virial coefficients according to the equations from Redlich-Kwong or Prausnitz [18, 19], With low pressures, the dependency of the fugacity coefficient can be neglected. 

Fugacity coefficients are important in considering the boiling of hydrothermal fluids, and have been approached from the virial equation as well as from numerous modifications of the van der Waals equation, the best known of these being the (modified) Redlich-Kwong equation. However, they are of minor importance in most environmental situations, and are routinely assumed to be 1.0, so that the activity of gaseous solution components is equal to the partial pressure (Equation (3.14)). 

Fig. 16.4. Solubility of liquid decane in nitrogen gas at 50°C, calculated from Henry s Law coefficient and decane fugacities based on ideal gas and Lewis Fugacity Rule approximations and the virial equation. Experimental data are solid dots. After Prausnitz (1969)...

That is, at low pressures we ignore the pressure dependence of all activity coefficients and all standard-state fugacities. In the 3 phase, values for the activity coefficients depend on the choice made for the standard-state fugacity for example, if the Lewis-Randall standard state is chosen for all components (5.1.5), then the y,- would be obtained from a model for the excess Gibbs energy. Common choices for the standard state are discussed in 10.2. In the a phase, values for the fugacity coefficients are obtained from a volumetric equation of state now, either pressure-explicit or volume-explicit models may be chosen. Fortunately at low pressures, either the ideal-gas law or a virial equation may be sufficiently accurate. 

Figure 12.12 Supercritical enhancement of the solubility of solid methane(l) in fluid hydro-gen(2) at 76 K. Points are experimental data of Hiza and Herring [8]. Solid line is from the ideal-gas law dashed line is the ideal-gas result corrected by a Poynting factor dash-dot line is the approximation (12.2.14) with the fugacity coefficient computed from the simple virial equation via (12.2.16). Figure after Chueh and Prausnitz [7].

From the historical point of view and also from the number of applications in the literature, the common method is to use activity coefficients for the liquid phase, i.e., the polymer solution, and a separate equation-of-state for the solvent vapor phase, in many cases the truncated virial equation of state as for the data reduction of experimental measurements explained above. To this group of theories and models also free-volume models and lattice-fluid models will be added in this paper because they are usually applied within this approach. The approach where fugacity coefficients are calculated from one equation of state for both phases was applied to polymer solutions more recently, but it is the more promising method if one has to extrapolate over larger temperature and pressure ranges. 

The knowledge of equations of state for gas phases permits the calculation of activity coefficients via fugacity coefficients. Equations of state for general practical use such as the virial equation (and others) are not known for condensed phases (liquids and solids). However, as shown by Planck and Schottky, the passage from the gaseous to the liquid or solid state does not change the structure of Eq. (87) and leads to the general formulation for the chemical potentials,... 

Another commonly used equation of state is the virial equation. We discuss the calculation of fugacity from virial coefficients in Chapter 13, 13.5.1. 

Figure 13.2 The fugacity of water at 400as a function of pressure, calculated from virial coefficients and Equation (13.30). The dashed line represents fugacity from the NIST program steam.

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