Acentric factor equation

Second virial coefficients, B, are a fnncBon of temperature and are available for about 1500 compounds in the DIPPR compilaOond The second virial coefficient can be regressed from experimental PX T data or can be reasonably and accurately predicted. Tsonoponlos proposed a predicOon method for nonpolar compounds that requires the criOcal temperature, critical pressure, and acentric factor Equations (2-68) through (2-70) describe the method. 

The acentric factor is calculated using Edmister s equation (1948) ... 

The volumetric properties of fluids are represented not only by equations of state but also by generalized correlations. The most popular generalized correlations are based on a three-parameter theorem of corresponding states which asserts that the compressibiHty factor is a universal function of reduced temperature, reduced pressure, and a parameter CO, called the acentric factor ... 

Three Parameter Models. Most fluids deviate from the predicted corresponding states values. Thus the acentric factor, CO, was introduced to account for asymmetry in molecular stmcture (79). The acentric factor is defined as the deviation of reduced vapor pressure from 0.1, measured at a reduced temperature of 0.7. In equation form this becomes ... 

The acentric factor, CO, was the third parameter used (20) in an equation based on the second virial coefficient. This equation was further modified and is suitable for reduced temperatures above 0.5. 

To correlate to the acentric factor a quadratic Taylor series in terms of the compressibility factor was formulated. This equation is represented as... 

The parameters in the equation are calculated for the simple fluid and the heavy reference fluid with an acentric factor of 0.3978. The parameters in the equation are calculated for the simple fluid and the heavy reference fluid from Eq. (2-79)... 

An afternate method with approximately the same accuracy as the Rackett method is the COSTALD metnod of Hanldnson and Thomson.The critical temperature, a characteristic volume near the critical volume, and an acentric factor optimized for vapor pressure prediction by the Soave equation of state are required input parameters. The method is detailed in the Technical Data Book ... 

Equations of state that are cubic in volume are often employed, since they, at least qualitatively, reproduce the dependence of the compressibility factor on p and T. Four commonly used cubic equations of state are the van der Waals, Redlich-Kwong, Soave, and Peng-Robinson. All four can be expressed in a reduced form that eliminates the constants a and b. However, the reduced equations for the last two still include the acentric factor u> that is specific for the substance. In writing the reduced equations, coefficients can be combined to simplify the expression. For example, the reduced form of the Redlich-Kwong equation is... 

Figure A3.3 compares the experimental (corresponding states) results with the predictions from the van der Waals. modified Berthelot, Dieterici, and Redlich-Kwong equations of state.b The comparison is not so direct for the Soave and Peng-Robinson equations of state, since the reduced equation still includes to, the acentric factor. Figure A3.4 compares the corresponding states line, with the prediction from the Soave equation, using four different values of to. The acentric factors chosen are those for H (o> = —0.218), CH4 (to = 0.011),...

Figure A3.4 Comparison of the experimental r (dashed lines) with the r values calculated from the Soave equation of state (solid lines). Values for the acentric factor are (a) oj = -0.218 (the value for EC), (b) a, = 0.011 (the value for CH4), (c) lU = 0,250 (the value for NEC), and (d) = 0.344 (the value for ECO).

NH3 (a = 0.250), and H20 (u> = 0.344). Thus, results for a wide range of acentric factors are compared. In Figure A3.5, we make the same comparisons with the Peng-Robinson equation. 

For both the Soave and Peng-Robinson equations, the fit is best for uj — 0. The Soave equation, which essentially reduces to the Redlich-Kwong equation when ui — 0, does a better job of predicting than does the Peng-Robinson equation. The acentric factors become important when phase changes occur, and it is likely that the Soave and Peng-Robinson equations would prove to be more useful when 77 < 1. 

In the above equations, co is the acentric factor, and Tc, Pc are the critical temperature and pressure respectively. These quantities are readily available for most components. 

The acentric factor is obtained experimentally. It accounts for differences in molecular shape, increasing with nonsphericity and polarity, and tabulated values are available3. Equation 4.6 can be rearranged to give a cubic equation of the form ... 

Figure 8 depicts how the three popular equation-of-state methods cited previously perform on pure steam. From a theoretical viewpoint, none of the methods has the foundation to handle mixtures of polar/non-polar components. Although the agreement with experimental data is not very satisfactory for any of the methods, the Lee-Kesler equation-of-state does best. It was also found that by slightly adjusting the acentric factor of water, improvement in the representation of the enthalpy of steam can be obtained by this method at 598 K, the conditions of the experimental mixture data, and at other temperatures as well. 

In Figure 10 are shown comparisons of the equation of state methods with the experimental data. The Lee-Kesler methods represent the data the best. Again, if the water acentric factor determined to best represent the pure steam enthalpy data is applied to the mixtures, further improvement is noted for the predictions by the Lee-Kesler method. Use of interaction constants within the Lee-Kesler, or other models, would undoubtedly provide even better representation of the data. 

Commonly encountered cubic equations of state are classical, and, of themselves, cannot rationalize IE s on PVT properties. Even so, the physical properties of iso-topomers are nearly the same, and it is likely in some sense they are in corresponding state when their reduced thermodynamic variables are the same that is the point explored in this chapter. By assuming that isotopomers are described by EOS s of identical form, the calculation of PVT isotope effects (i.e. the contribution of quantization) is reduced to a knowledge of critical property IE s (or for an extended EOS, to critical property IE s plus the acentric factor IE). One finds molar density IE s to be well described in terms of the critical property IE s alone (even though proper description of the parent molar densities themselves is impossible without the use of the acentric factor or equivalent), but rationalization of VPIE s requires the introduction of an IE on the acentric factor. 

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... 

To design a supercritical fluid extraction process for the separation of bioactive substances from natural products, a quantitative knowledge of phase equilibria between target biosolutes and solvent is necessary. The solubility of bioactive coumarin and its various derivatives (i.e., hydroxy-, methyl-, and methoxy-derivatives) in SCCO2 were measured at 308.15-328.15 K and 10-30 MPa. Also, the pure physical properties such as normal boiling point, critical constants, acentric factor, molar volume, and standard vapor pressure for coumarin and its derivatives were estimated. By this estimated information, the measured solubilities were quantitatively correlated by an approximate lattice equation of state (Yoo et al., 1997). 

Values of a-pj and bj for each component of the mixture are obtained with Equations 15-9 through 15-12 from a knowledge of the critical properties and acentric factors of the pure components. 

Extension of Generalized Charts. In 1975, the usefiilness of generalized charts was extended upon the publication of extensive tables of residual enthalpy, entropy, and heat capacity (82). This tabular data has also been converted into graphical form (3). The corresponding equations incorporate the acentric factor Residual enthalpy. 

Partial molar volumes and the isothermal compressibility can be calculated from an equation of state. Unfortunately, these equations require properties of the components, such as critical temperature, critical pressure and the acentric factor. These properties are not known for the benzophenone triplet and the transition state. However, they can be estimated very roughly using standard techniques such as Joback s modification of Lyderson s method for Tc and Pc and the standard method for the acentric factor (Reid et al., 1987). We calculated the values for the benzophenone triplet assuming a structure similar to ground state benzophenone. The transition state was considered to be a benzophenone/isopropanol complex. The values used are shown in Table 1. 

Having a TC, PC, and a volumetric boiling point for each pseudocomponent, the respective acentric factor of each component is next calculated using the well-known equation from Edmister, shown in code line 4720 [18]. 

The carbon di oxi de/lemon oil P-x behavior shown in Figures 4, 5, and 6 is typical of binary carbon dioxide hydrocarbon systems, such as those containing heptane (Im and Kurata, VO, decane (Kulkarni et al., 1 2), or benzene (Gupta et al., 1 3). Our lemon oil samples contained in excess of 64 mole % limonene so we modeled our data as a reduced binary of limonene and carbon dioxide. The Peng-Robinson (6) equation was used, with critical temperatures, critical pressures, and acentric factors obtained from Daubert and Danner (J 4), and Reid et al. (J 5). For carbon dioxide, u> - 0.225 for limonene, u - 0.327, Tc = 656.4 K, Pc = 2.75 MPa. It was necessary to vary the interaction parameter with temperature in order to correlate the data satisfactorily. The values of d 1 2 are 0.1135 at 303 K, 0.1129 at 308 K, and 0.1013 at 313 K. Comparisons of calculated and experimental results are given in Figures 4, 5, and 6. 

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... 

Because of its relative simplicity, the original Redlich/Kwong equation was used in Example 14.S to illustrate the calculation of fugadty coefficients. However, this equation in its original form is rarely satisfactory for VLE calculations, and many modifications have been proposed to make it more suitable. In particular, Soave introduced the acentric factor into the Redlich/Kwong equation by setting 8 equal to a function not only of temperature but also of the acentric factor Soave/Redlich/Kwong (SRK) equation is written ... 

Since the acentric factor is here so small, the two- and three-parameter compressibility-factor correlations are little different. Both the Redlich/Kwong equation and the generalized compressibility-factor correlation give answers very close to the experimental value of 185(atm). The ideal-gas equation yields a result that is high by 14.6 percent. 

Coefficients for the vapor pressure equation are provided near the end of the book. Critical properties and acentric factor are given in the last section of the book. The tabulated values are especially arranged for quick usage with hand calculator or computer. Computer programs, containing data for all compounds, are available for a nominal fee. The programs are in ASCII which can be accessed by other software. Contact Carl L. Yaws, P.O. Box 10053, Lamar University, Beaumont, TX 77710, phone/FAX 409-880-8787. 

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