Corresponding states, theory

Generalized Correla.tions. A simple and rehable method for the prediction of vapor—Hquid behavior has been sought for many years to avoid experimentally measuring the thermodynamic and physical properties of every substance involved in a process. Whereas the complexity of fluids makes universal behavior prediction an elusive task, methods based on the theory of corresponding states have proven extremely useful and accurate while still retaining computational simplicity. Methods derived from corresponding states theory are commonly used in process and equipment design. 

Numerous other methods have been used to predict properties of gases and Hquids. These include group contribution, reference substance, approaches, and many others. However, corresponding states theory has been one of the most thoroughly investigated methods and has become an important basis for the development of correlation and property estimation techniques. The methods derived from the corresponding states theory for Hquid and gas property estimation have proved invaluable for work such as process and equipment design. 

Generalized Correlations for Viscosity. Gas viscosity has also been predicted by corresponding states theory (90) using... 

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... 

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. 

It is often of industrial interest to be able to predict the equilibrium sorption of a gas in a molten polymer (e.g., for devolatilization of polyolefins). Unfortunately, the Prigogine-Flory corresponding-states theory is limited to applications involving relatively dense fluids 3,8). An empirical rule of thumb for the range of applicability is that the solvent should be at a temperature less than 0.85 Tp, where Tp is the absolute temperature reduced by the pure solvent critical temperature. 

Unfortunately, relatively little work has been done on the solution thermodynamics of concentrated polymer solutions with "gathering". The definitive work on the subject Is the article of Yamamoto and White (17). The corresponding-states theory of Flory (11) does not account for gathering. We therefore restrict our consideration here to multicomponent solutions where the solvents and polymer are nonpolar. For such solutions, gathering Is unlikely to occur. 

There Is a large body of experimental literature relating to polymer fractionation In liquid-liquid equilibria. In addition, numerous authors have analyzed polymer fractionation using Flory-Huggins theory. We have considered use of the corresponding states theory to model polymer fractionation for the ethylene/ polyethylene system at reactor conditions (18). Results of the... 

The corresponding-states theory of polymer solution thermodynamics, developed principally by Prigogine and Flory, has provided a reliable predictive tool requiring only minimal information. We have seen here several examples of the use of the corresponding-states theory. We have also seen that the corresponding-states theory is a considerable improvement over the older Flory-Huggins theory. 

Corresponding States Expressions. In corresponding states theory (l2) the basic parameters characterizing a liquid are the reduced... 

On account of its underlying assumptions this model has alternatively been called the Average Potential Model (APM)11 or the Corresponding States Theory of Mixtures.12 Various slightly different versions of it have been developed, depending on the precise definition of the average potential acting on a molecule j11 12 extensions and comments on this model have been published by several authors.13-16... 

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... 

Appendix A Effect of Temperature and Pressure on Surface Tension of Liquids (Corresponding States Theory)... 

It is well known that the corresponding states theory can provide much useful information about the thermodynamics and transport properties of fluids. For example, the most useful two-parameter empirical expression, which relates the surface tension, y, to the critical temperature, Tc, is given as... 

The Hamaker constants of nonpolar fluids and polymeric liquids can be obtained using an expression similar to Equation (67) in combination with the corresponding state theory of thermodynamics and an expression for interfacial energy based on statistical thermodynamics (Croucher 1981). This leads to a simple, but reasonably accurate and useful, relation for Hamaker constants for nonpolar fluids and polymeric liquids. We present in this section the basic details and an illustration of the use of the equation derived by Croucher. 

Corresponding-states theory (Prigogine, 1957 Flory, 1970) incorporates features of the pure component properties and liquid structure in the mixture equation-of-state, producing nonzero values of AFm and contributing enthalpic and entropic terms beyond those in Flory-Huggins theory. The theory assumes that all pure components and mixtures obey the same universal equation-of-state, e.g., (Flory, 1970)... 

Nakajima, A. Hamada, F. Yasue, K. Fujisawa, K., "Thermodynamic Studies Based on Corresponding States Theory of Solutions of Polystyrene in Ethyl Methyl Ketone," Die Makromol. Chem., 178, 197 (1974). 

These data clearly show that the magnitude of y is proportional to the chain length of the alkanes. This is to be expected based on the previous relation given by Stefan on the dependence of the magnitude of y on the heat of evaporation. The data of surface tension vs. temperature can be analyzed as follows. It is well known that the corresponding states theory can provide much useful information about the thermodynamics and transport properties of fluids. For example, the most useful two-parameter empirical expression that relates the surface tension, y, to the critical temperature is given as ... 

The main advantage of using reduced variables is that these variables make the fundamental similarities and correspondences between different materials much more obvious than is possible otherwise. For example, many (but certainly not all) existing equations-of-state are corresponding states theories, where all materials have the same reduced density p at a given... 

The Sanchez-Lacombe equation-of-state provides a good example to help clarify the rather abstract discussion given above. It will now be discussed further. It is given by Equation 3.26 for a pure molecular liquid or gas. The variable r is defined by Equation 3.27, where M is the molecular weight and R is the gas constant. If T, p and p are known, Equation 3.26 can be solved iteratively to estimate the density as a function of temperature and pressure. Since the reduced density p depends on M through the variable r defined by Equation 3.27, it is not equal for all molecules at the same combination of T and p values. Consequently, for ordinary molecules, the Sanchez-Lacombe equation-of-state is not a corresponding states theory. 

Figure 3.7. Reduced density as a function of reduced temperature and reduced pressure for polymers, calculated by using the Sanchez-Lacombe equation-of-state in the limit of infinite molecular weight where it becomes a corresponding states theory. Each curve is labeled by the value of the reduced pressure that was used in its calculation.

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