Fugacity coefficient corresponding states

The standard state fugacity coefficient is calculated from a Pitzer-type corresponding states correlation ... 

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. 

The fugacity coefficient ratio J can be estimated by assuming that the Lewis and Randall rule11 applies, at least approximately, for the mixture, so that each component has the same fugacity coefficient that it would have if it were a pure gas at the same total pressure. The Principle of Corresponding States can then be used to compare the fugacity coefficients of the three components. At p = 60 atm (61 bar) and in the temperature range from 900 to 1600 K, the reduced temperatures and pressures for the components of the equilibrium... 

The PR eos has been modified by Stryjek and Vera to extend to polar substances that do not follow the three-parameter principle of corresponding states. The modified eos is fitted to the vapor pressure of polar substances with additional substance-specific parameters. The PRSV equation has been described in Equation (4.163) et seq. The free-energy-matched mixture eos parameters are given in Equations (4.436) and (4.438) the fugacity coefficients are given in Equation (4.439). PRSV eos using the UNIEAC activity coefficient predicts the vie data for both ethanol/water mixtures at 423-623°K and acetone/water mixtures at 373-523°K from low to high pressure. 

An alternative is to use the activity coefficient approach. We do this here using regular solution theory and corresponding states for the fugacity coefficients. The starting point is the equilibrium condition... 

The Chao-Seader method12 is an example of the use of multiple equations of state for the calculation of K values. The Redlich-Kwong equation of state is used to compute the vapor-phase fugacity coefficient the Hildebrand equation for the calculation of the liquid-phase activity coefficient y/% and an extension of Pitzer s modified form of the principle of corresponding states for the calculation of the liquid-phase fugacity coefficient 4> ... 

Consequently, the fugacity coefficient can be tabulated in the corresponding-states manner. The corresponding-states correlation for the fugacity coefficient of nonpolar gasesjmd liquids given in Fig. 7.4-1 was obtained using Eq. 7.4-15b and the compressibility correlation (Fig. 6.6-3). 

In the case of the subcooled liquid, which involves an extrapolation into the solid region, the vapor pressure is usually so low that the fugacity coefficient is. close to unity, and the fugacity of this hypothetical liquid is equal to the extrapolated vapor pressure. For the supercritical liquid, however, the extrapolation is above the critical temperature of the liquid and yields very high vapor pressures, so that the fugacity of this hypothetical liquid is equal to the product of the extrapolated vapor pressure and the fugacity coefficient (which is taken from the corresponding-states plot of Fig. 

Gas-liquid system (GLC) The solute concentrations in both phases will be again expressed in mole fractions, the standard concentration and standard physical state of the solute in the stationary (liquid) phase will be defined in the same way as with the liquid-liquid system, and a hypothetical pure solute in a state of ideal gas at a unit pressure and at the temperature of the system will be chosen as a standard state for the solute in the mobile (gaseous) phase. Thus, and may be written as /is Yis is is 3 d /iiy = J iM/ - CiM where is the fugacity coefficient (mean value) of the solute in the mixture with carrier gas, and p is the mean pressure in the column. The corresponding standard fugacities (jc s = 1 and Yis = ffw = T = 1, and = 1) are f° = and = 1, so that, according to equation 43,... 

Lee and Kesler (reference cited) found an accurate representation for compressibility of both gases and liquids by combining BWR-EOS with corresponding states law. They generated departure functions for enthalpy, entropy, fugacity coefficient and heat capacity. Tables are given in Reid et al. (1987), whereas illustrative graphs are presented in Perry (1997). The method is similar to that developed for compressibility. As an example, the enthalpy departure function may be calculated with the relation ... 

In the case of hydrocarbon mixtures the Chao-Seader (1961) method is often used. The fugacity coefficients may be calculated by a corresponding states formulation, as the sum of two contributions, for spherical molecule and deviation from sphericity ... 

Despite widespread use of the ideal K-value concept in industrial calculations, particularly during years prior to digital computers, a sound thermodynamic basis does not exist for calculation of the fugacity coefficients for pure species as required by (4-85). Mehra, Brown, and Thodos discuss the fact that, for vapor-liquid equilibrium at given system temperature and pressure, at least one component of the mixture cannot exist as a pure vapor and at least one other component cannot exist as a pure liquid. For example, in Fig. 4.3, at a reduced pressure of 0.5 and a reduced temperature of 0.9, methane can exist only as a vapor and toluene can exist only as a liquid. It is possible to compute vl or f v for each species but not both, unless vl = vy, which corresponds to saturation conditions. An even more serious problem is posed by species whose critical temperatures are below the system temperature. Attempts to overcome these difficulties via development of pure species fugacity correlations for hypothetical states by extrapolation procedures are discussed by Prausnitz. ... 

The fugacity coefficient v" of a pure species at temperature T and pressure P can be determined directly from an equation of state by means of (4-51). If Pvapor fugacity coefficient. For P>P , v° is the fugacity coefficient of the liquid. Saturation pressure corresponds to the condition vi = vy. Integration of (4-51) with the R-K equation of state gives... 

The fugacity coefficients (f/P) for the various species may be determined from a corresponding states chart if one knows the reduced temperature and pressure corresponding to the species in question. Therefore ... 

Determine the equilibrium composition that is achieved at 300 bar and 700 K when the initial mole ratio of hydrogen to carbon monoxide is 2. You may use standard enthalpy and Gibbs free energy of formation data. For purposes of this problem you should not neglect the variation of the standard heat of reaction with temperature. You may assume ideal solution behavior but not ideal gas behavior. You may also use a generalized fugacity coefficient chart based on the principle of corresponding states as well as the heat capacity data listed below. 

Equation (11) also assumes that the reaction is ideal, i.e. that the fugacity coefficients are equal to 1. The equilibrium constant K contains a partition function corresponding to the vibrational frequency vt. Transition-state theory assumes that the vibration has a very low frequency, so that the corresponding partition function can be factorized in the following form ... 

This equation can be used to calculate the fugacity coefficient with the help of a pressure-explicit equation of state. As RTln (p is by definition equivalent to the residual Gibbs energy (g — g )r.p, the above equation is equivalent to the corresponding equation in Table 2.2. 

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