Corresponding-states calculations

An overriding conclusion from all of these studies is that, while extended corresponding-states calculations are very accurate for systems of similar molecules, the predictions tend to decrease in accuracy as the system of interest deviates in size and shape from methane. More generally, as the components in a mixture become more dissimilar both in size and polarity, there is a marked decrease in the accuracy of the predictions. With this situation in mind, a research program was initiated by Ely and co-workers in the mid-1980 s to... 

Eq.7.6.4provides the theoretical basis for the so-called geometric mean rule approximation used in corresponding states calculations involving nonpolar mixtures. 

Liquid densities can be calculated according two types of methods, both based on the principle of corresponding states. 

The COSTALD (Corresponding states liquid density) method was originally developed for calculating the densities of liquefied gases its use has become generally widespread. 

The other method is to employ the principle of corresponding states and calculate the Cp/ of the mixture in the liquid phase starting from the mixture in the ideal gas state and applying an appropriate correction ... 

The specific heat of gases at constant pressure is calculated using the principle of corresponding states. The for a mixture in the gaseous state is equal to the sum of the C g of the ideal gas and a pressure correction term ... 

Calculation of thermophysical properties of gases relies on the principle of corresponding states. Viscosity and conductivity are expressed as the sum of the ideal gas property and a function of the reduced density ... 

A single calculation of the discrete path integral with a fixed length of time t can be employed to compute the state conditional probability at many other times. It is possible to use segments of the path of time length At, 2At,..., NAt sampled in trajectories of total length of NAt and to compute the corresponding state conditional probabilities. The result of the calculations will make it possible to explore the exponential relaxation of P Ao B,t) for times between 0 and t. 

Corresponding states have been used in other equations. For example, the Peng-Robinson equation is a modified RedHch-Kwong equation formulated to better correlate vapor—Hquid equiHbrium (VLE) vapor pressure data. This equation, however, is not useful in reduced form because it is specifically designed to calculate accurate pressure data. Reduced equations generally presuppose knowledge of the pressure. 

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. 

Critical Temperature The critical temperature of a compound is the temperature above which a hquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. 

The method of Lee and KesleF is the preferred method if the critical temperature and the critical pressure of the hydrocarbon is known or can be reasonably predicted by the methods of the first section. The corresponding states method is shown in equation (2-31) with the simple fluid and correction terms to be calculated from equations (2-32) and (2-33), respectively, for any Tr-... 

For prediction of vapor density of pm e hydi ocaihon and nonpolar gases, tbe corresponding states method of Pitzer et al. is tbe most accurate method, witb errors of less than 1 percent except in tbe critical region where errors of up to 30 percent can occur. Tbe method correlates tbe compressibibty factor by Eq. (2-75), after which tbe density can be calculated by Eq. (2-75) ... 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

Fig. 10. The mole fraction of carbon dioxide in saturated solutions in air at — 110°C (above the lower critical end point). The full line is the experimental curve of Webster and the dashed curves are 1, an ideal gas mixture 2, an ideal gas mixture with Poynting s correction and 3, the solubility calculated from Eq. 8 and the principle of corresponding states.

Madame Ivirstine Meyer (1900) has shown that the discrepancies are not to be explained by errors in the critical data the law of corresponding states can be tested without making use of these constants, and differences between the observed and calculated magnitudes are still apparent. D. Berthelot (Journ. de Phys., 1903) has deduced some new equations. 

While virial coefficients can be calculated from statistical-mechanical formulas, for practical work it is usually more convenient to employ semi-empirical correlations. Most of these correlations are based on the principle of corresponding states and as a result their applicability is limited to normal... 

Several authors, notably Leland and co-workers (L2), have discussed vapor-liquid equilibrium calculations based on corresponding-states correlations. As mentioned in Section II, such calculations rest not only on the general assumptions of corresponding-states theory, but also on the additional assumption that the characterizing parameters for a mixture do not depend on temperature or density but are functions of composition only. Further, it is necessary clearly to specify these functions (commonly known as mixing rules), and experience has shown that if good results are to be obtained, these... 

A theoretical analysis of the helium-xenon system was reported by Zandbergen and Beenakker (Zl), who based their calculations on the Prigogine-Scott theory of corresponding states for mixtures (PI 5, S2). We cannot here go into the details of their analysis, but will merely indicate the essential elements. Zandbergen and Beenakker use the three-liquid theory to obtain an expression for the volumes of helium-xenon mixtures as a function of temperature, pressure, and composition. This expression is... 

For helium, a = 2.56 A and e/k = 10.22°K, where k is Boltzmann s constant. For xenon, e/k = 221 °K and a = 4.10. Because of symmetry, the subscript 1 may refer to either helium or xenon from the assumption of corresponding-states behavior, v°2 should be independent of the component chosen for subscript 1. Equation (108) is an equation of state for the binary mixture, and from it the phase behavior can be calculated without further assumptions. 

Mostinski, I.L. Brit. Chem. Eng. 8 (1963) 580. Calculation of boiling heat transfer coefficients, based on the law of corresponding states. 

To illustrate the application of corresponding-states theory to polymer solution calculations, we consider two cases of sol-vent/polymer vapor-liquid equilibria. The first case we consider is that of the chloroform/polystyrene solution. The second is that of benzene/polyethylene oxide. 

Our best estimation for the vertical excitation energies for states of A, symmetry are reported in Table 12. They correspond to a ground state calculated at CI( 6) level using orbitals optimized for the neutral molecule with the MCSCF/SD expansion, and excited Rydberg states calculated at the level using orbitals optimized for the positive ion... 

Sterbacek, Z., Biskup, B. and Tausk, P. (1979) Calculation of Properties Using Corresponding-state Methods (Elsevier). 

Plocker, U., Knapp, H. and Prausnitz, J. (1978) Ind. Eng. Chem. Proc. Des. and Dev. 17, 243. Calculation of high-pressure vapour-liquid equilibria from a corresponding-states correlation with emphasis... 

Mostinski IL (1963) Calculation of Boiling Heat Transfer Coefficients, Based on the Law of Corresponding States, Br Chem Eng, 8 580. 

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