Corresponding-states extended

The principle of corresponding states, extended as above to mixtures of acentric molecules, has been applied to the calculation of many of the properties needed for the design of separation equipment. The examples reviewed briefly here are taken from our own work on cryogenic fluids, liquified natural gas (LNG), mixtures of hydrocarbons, and mixtures of carbon dioxide with hydrocarbons. In all this work methane was used as the reference substance. 

The principle of corresponding states, extended on the basis of kinetic theory and some knowledge of intermolecular forces, allows the correlation of the transport coefficients of low-density gases and their mixtures over a very wide temperature range. The ability to operate in a predictive mode is especially valuable for mixtures of any number of components, for which accurate experimental data are extremely scanty. 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from. 32-term MBWR equation of state transport properties are from extended corresponding states model, t = triple point c = critical point. 

Corresponding-states correlations for pure components rest upon several fundamental assumptions which are discussed elsewhere (G3, H4, H6). To extend such correlations to mixtures, it is necessary to make an additional fundamental assumption, viz, that the characterizing parameters chosen (TcMr FCm, and (oM) are independent of temperature and density and are... 

Lee and Kesler (1975) extended the Benidict- Webb-Rubin equation to a wider variety of substances, using the principle of corresponding states. The method was modified further by Plocker et al. (1978). 

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. 

Fig. 13.1 Reduced vapor pressure and molar density vs. reciprocal reduced temperature for HoO, CH4, H2, and 4He. In each case, were simple corresponding states theory adequate, all data would lie on a single master curve. Using extended CS the curves are fit to acceptable precision, (a) (top) = reduced vapor pressures, (b) (bottom) = reduced liquid molar densities... 

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... 

In solid-state NMR [1,51-64], the magnetic coupling between the fullerene anions has to be taken into accoimt. In the case of metal intercalated fullerides that have metallic properties a contribution from the conduction electrons must be added, a phenomenon called the Knight shift . Even if this additional shift affects the C-chemical resonance, the correspondence between extended and discrete systems of comparable Cjq oxidation state is quite close [1]. 

The Law of Corresponding States has been extended to cover mixtures of gases which are closely related. As was brought out in Chapter 2, obtaining the critical point for multicomponent mixtures is somewhat difficult therefore, pseudocritical temperature and pseudocritical pressure have been invented. 

Thompson. William H.. "A Molecular Association Factor for Use in the Extended Theorem of Corresponding States." Ph D. dissertation, Pennsylvania State University. University Park. 1966. 

For higher gas densities, the Lee-Kesler method described below provides excellent predictions for nonpolar and slightly polar fluids. Extended four-parameter corresponding-states methods are available for polar and slightly associating compounds. 

The primary key to Soave s early success was the recognition of the need to extend from two-parameter to three-parameter corresponding states. With two parameters, it is possible to fit the critical temperature and critical pressure of any compound, but a general description of the vapor pressure curve is unattainable. 

At low and moderate pressures, the viscosity of a gas is nearly independent of pressure and can be correlated for engineering purposes as a function of temperatnre only. Eqnations have been proposed based on kinetic theory and on corresponding-states principles these are reviewed in The Properties of Gases and Liquids [15], which also inclndes methods for extending the calculations to higher pressures. Most methods contain molecular parameters that may be fitted to data where available. If data are not available, the parameters can be estimated from better-known quantities such as the critical parameters, acentric factor, and dipole moment. The predictive accuracy for gas viscosities is typically within 5%, at least for the sorts of small- and medinm-sized, mostly organic, molecules used to develop the correlations. 

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. 

---