The Pitzer Acentric Factor

Pitzer derived a term useful in corresponding state predictions. The acentric factor co is defined in terms of vapor pressure and is designed to account for the nonideal behavior of gases. In essence, it encodes information about the nonspherical shape of molecules. Using published values of to, Kier has shown an excellent correlation with k and 

The equation predicts the acentric factor of 20 hydrocarbons within 3% error. 

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Look up the critical temperature and pressure (Tc and Pc) for the species of interest in Table B.l or elsewhere. Also look up the Pitzer acentric factor, selected compounds, and a more complete list can be found in Reid et al. 

The Pitzer acentric factor is a property of pure fluids and has been widely tabulated (for example, see Ref. [ ]). It can be estimated in several ways, one of w hich [5] is given in the Nomenclature. 

A mixture of ethanol and water vapor is being rectified in an adiabatic distillation column. The alcohol is vaporized and transferred from the liquid to the vapor phase. Water vapor condenses—enough to supply the latent heat of vaporization needed by the alcohol being evaporated—and is transferred from the vapor to the liquid phase. Both components diffuse through a gas film 0.1 mm thick. The temperature is 368 K and the pressure is 1 atm. The mole fraction of ethanol is 0.8 on one side of the film and 0.2 on the other side of the film. Calculate the rate of diffusion of ethanol and of water, in kg/m2-s. The latent heat of vaporization of the alcohol and water at 368 K can be estimated by the Pitzer acentric factor correlation (Reid et al., 1987)... 

For the calculation of thermodynamic properties, Equation 23 was used in an empirical manner. Only data for nonpolar normal paraffin hydrocarbon systems were used in the correlation development so that as an approximation, the Pitzer acentric factor, < >, could be taken as an estimate of the collective strength of molecular anisotropies (i.e., 82 = to). Because the use of the resultant correlation for polar systems was anticipated, the parameter y (y =82), referred to herein as the orientation parameter, was used instead of the acentric factor (y < > for other fluids). The equation of state in Equation 23 then takes the form... 

Thermodynamic properties of nonideal hydrocarbon mixtures can be predicted by a single equation of state if it is valid for both the vapor and liquid phases. Although the Benedict-Webb-Rubin (B-W-R) equation of state has received the most attention, numerous attempts have been made to improve the much simpler R-K equation of state so that it will predict liquid-phase properties with an accuracy comparable to that for the vapor phase. The major difficulty with the original R-K equation is its failure to predict vapor pressure accurately, as was exhibited in Fig. 4.3. Following the success of earlier work by Wilson, Soave added a third parameter, the Pitzer acentric factor, to the R-K equation and obtained almost exact agreement with pure hydrocarbon vapor pressure... 

A quantity often used in calculations on real gases is the Pitzer acentric factor, co. Pitzer defined the factor as a means of characterizing deviation from spherical symmetry for use in corresponding state modeP . The acentric factor is obtained from experimental data, as follows co = og P[) —1.0 in which P is the reduced pressure P/P at the reduced temperature of 0.7°C, P being the critical pressure. This definition is consistent with acentric factor values of zero for rare gases. 

The shape factors, 6 and (f, are weak functions of temperature (T) and density (p) and can be regarded as characteristic of the Pitzer acentric factors of fluids a and o, respectively. The superscript c denotes the critical point value. 

So much for historical matters. The current situation is in practice rather simpler than suggested by the function FFor practical reasons it is easier to use Pc and Tc than a and e, and it turns out that the whole set of parameters at can often be condensed into a single parameter. This parameter is now usually based on the slope of the vapor-pressure curve and is called the Pitzer acentric factor a> (Schreiber Pitzer 1989). Values of Pc, Tc and (o have been tabulated for a large number of substances. 

To accomplish this objective, we introduce a third parameter characteristic of classes of molecules. There are many ways to introduce a parameter for classes of molecules we will explore only one— the Pitzer acentric factor, (o. It characterizes how nonspherical a molecule is, thereby assigning it to a class. The definition of (o is somewhat arbitrary ... 

We can extend the principle of corresponding states to account for different classes of molecules, based on the particular nature of the intermolecular interactions involved. One way to accomplish this objective is by introducing a third parameter— the Pitzer acentric factor, w.We then write the compressibiUty factor in terms of z , which accounts for simple molecules, and a correction factor for the nonsphericity ... 

Figure 13.1a shows reduced vapor pressures and Fig. 13.1b reduced liquid molar densities for the parent isotopomers of the reference compounds. Such data can be fit to acceptable precision with an extended four parameter CS model, for example using a modified Van der Waals equation. In each case the parameters are defined in terms of the three critical properties plus one system specific parameter (e.g. Pitzer acentric factor). Were simple corresponding states theory adequate, the data for all... 

Kay s method is known as a two-parameter rule since only pc and Tc for each component are involved in the calculation of z. If a third parameter such as the Pitzer acentric fector, or Vc is included in the determination of the mean compressibility factor, then we would have a three-parameter rule. All the pseudocritical methods do not provide equal accuracy in predicting p V-T properties, but most suffice for engineering work. Stewart et al. reviewed 21 different methods of determining the pseudoreduced parameters by three-parameter rules (see Table 3.6). Although Kay s method was not the most accurate, it was easy to use and not consid-... 

On the acentric factor, Pitzer states "The third parameter is required because the intermolecular force in complex molecules is a sum of interactions between the various parts of the molecules - not just their centers - hence the name acentric factor is suggested". This subject is further discussed, for graduate students, in Example 8.2. 

Table 7.1 summarizes expressions for fugacity coefficients of pure species and mixtures for three cubic equations of state the van der Waals equation, the Redhch-Kwong equation, and the Peng-Robinson equation. You can develop these expressions using Equation (7.8) for pure species or Equation (7.14) for mixtures (see Problems 7.16, 7.17, 7.35, and 7.36). The parameters a, h, and a can be obtained from the critical temperature, the critical pressure, and Pitzers acentric factor, as discussed in Chapter 4. Eor the mixtures, mixing rules given by Equations (7.15), (7.18), and either (7.16) or (7.17) are typically used however, others have proposed alternative mixing rules. 

The corresponding states approach suggested by Pitzer et al. requires only the critical temperature and acentric factor of the compound. For a close approximation, an analytical representation of this method proposed by Reid et al. " for 0.6 [Pg.394]

Although use of an equation based on the two-parameter theorem of corresponding states provides far better results in general than the ideal-gas equation, significant deviations from experiment still exist for all but the simple fluids argon, krypton, and xenon. Appreciable improvement results from the introduction of a third corresponding-states parameter, characteristic of molecular structure the most popular such parameter is the acentric factor , introduced by K. S. Pitzer and coworkers.t... 

For the calculations presented in this paper, we first elected to use three simple cubic equations of state PR-EOS SRK-EOS and RK-EOS. For the pure components, critical properties (Pq, Tq) and Pitzer s acentric factor (ci>) are needed to obtain a and b . Critical properties have been measured for most of the low molecular weight components and are reported by Reid et al. 

Table 11 Critical Properties and Pitzer s Acentric Factor for the Working Materials...

Characterization of the vapor pressure curve was readily accessible in the form of Pitzer s acentric factor. The definition of the acentric factor is... 

For vapor-liquid equilibrium calculations up to moderate pressures, the B equation is suitable and convenient for the vapor phase for its applicability and simple form. Formulas have been derived from statistical theory for the calculation of virial coefficients, including B, from intermo-lecular potential energy functions, but intermolecular energy functions are hardly known quantitatively for real molecules. B is found for practical calculations by correlating experimental B values. Pitzer [1] correlated B of normal flnids in a generalized form with acentric factor to as the third parameter. 

The subscript 0 denotes simple fluids with acentric factor to = 0, and subscript r denotes a reference fluid with acentric factor to,. = 0.3978 from its origin in n-octane. The Lee-Kesler equation constants for Zq and Zy are presented in Table 4.4. The compressibihty factors in Equation (4.239)— Zq, Zy—are at the same [T,.,pJ. This correlation is a three-parameter generahzed correlation that improves the Pitzer correlation to a wider range of states. The lower temperamre bound of the correlation is extended from = 0.8 to 0.3. 

The approach is predictive, and the only input data required are the critical parameters, molecular weight, Pitzer s acentric factor and, for thermal conductivity, the ideal gas specific heat. 

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