Equations fugacity coefficient

If we adopt as the standard state for gaseous components the state of pure perfect gas at P = 1 bar and T = 298.15 K = f% = 1) and neglect for simphcity the fugacity coefficients, equation 5.304 combined with equation 5.297 gives... 

The generalized methods developed in Sec. 3.6 for the eompressibility faetor Z and in Sec. 6.7 for the residual enthalpy and entropy of pure gases are applied here to tlie fugacity coefficient. Equation (11.34) is put into generalized fonn by substitution of the relations. 

I. Use the partial fugacity coefficient equation derived from Equation 1.21 ... 

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,... 

If we choose to quantify the vapor-phase nonideality using the fugacity coefficient [Equation (7.5)] and the liquid-phase nonideality using the activity coefficient [Equation... 

Here represents the mole fraction in the vapor phase, while Xi represents the liquid mole fraction. Once we have chosen the appropriate liquid reference state (Lewis/ Randall rule or Henry s law), we can solve this problem if we have composition dependencies of the activity coefficient and the fugacity coefficient. Equation (8.2) is actually a set of coupled equations, one for each species i. While Equation (8.2) is completely rigorous (and thus always correct), we have seen that it is not always trivial to calculate these terms. 

Chapter 3 discusses calculation of fugacity coefficient < ). Chapter 4 discusses calculation of adjusted activity coefficient Y fugacity of the pure liquid f9 [Equation (24)], and Henry s constant H. 

The fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure... 

The fugacity coefficient can be found from the equation of state using the thermodynamic relation (Beattie, 1949) ... 

Figures 3 and 4 show fugacity coefficients for two binary systems calculated with Equation (10b). Although the pressure is not large, deviations from ideality and from the Lewis rule are not negligible.

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... 

To use Equation (13), it is first necessary to calculate the true fugacity coefficient (ft. This calculation is achieved by utilizing the Lewis fugacity rule... 

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 ... 

P the other terms provide corrections which at low or moderate pressure are close to unity. To use Equation (2), we require vapor-pressure data and liquid-density data as a function of temperature. We also require fugacity coefficients, as discussed in Chapter 3. 

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. 

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. 

However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

If the data are correlated assuming an ideal vapor, the reference fugacity is just the vapor pressure, P , the Poynting correction is neglected, and fugacity coefficient is assumed to be unity. Equation (2) then becomes... 

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. 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

According to equation 132, is the residual Gibbs energy, G. The dimensionless mXio fjP is a new property called the fugacity coefficient, ( ). ... 

According to equation 184, all fluids having the same value of CO have identical values of Z when compared at the same T and P. This principle of corresponding states is presumed vaHd for all T and P and therefore provides generalized correlations for properties derived from Z, ie, for residual properties and fugacity coefficients, which depend on T and P through Z and its derivatives. 

Effective use of this general equation requires expHcit introduction of the compositions of the phases. This is done either through the activity coefficient, y, or the fugacity coefficient, ( ) Two procedures are in common use. By the gamma—phi approach, activity coefficients for the Hquid phase enter by equation 202 and fugacity coefficients for the vapor phase by equation 164 equation 220 then becomes equation 221 ... 

The second common procedure for VLE calculations is the equation-of-state approach. Here, fugacity coefficients replace the fugacities for both Hquid and vapor phases, and equation 220 becomes equation 226 ... 

Use of equation 247 for actual calculations requires expHcit iatroduction of composition variables. As ia phase-equiUbrium calculations, this is normally done for gas phases through the fugacity coefficient and for Hquid phases through the activity coefficient. Thus, either... 

The fugacity coefficient departure from nonideaHty in the vapor phase can be evaluated from equations of state or, for approximate work, from fugacity/compressibiHty estimation charts. References 11, 14, and 27 provide valuable insights into this matter. 

At system pressures up to several tens of MPa, the fugacity coefficients, ( ) and and the Poynting factor, 7T, are usually near unity. A simplified version of equation 19 can therefore be used for the majority of vapor—Hquid equihbrium problems ... 

Vugacity Coefficients. An exact equation that is widely used for the calculation of fugacity coefficients and fugacities from experimental pressure—volume—temperature (PVT) data is... 

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