Theory of corresponding states

The restrictions of the initial version of Flory-lluggins lattice model (P-pLMWL mixtures) were obvious from the very beginning and, first and foremost, to the authors themselves. Indeed, the model did not foresee the chang in volume on mixing and, as a sequence, the additional variation of the configurative entropy of mixing, x, was considered to have a reciprocal dependence on 7 which specifics the existence of the HOST only. 

Since 1960 s, a LCS1 near the critical vapour-liquid temperature of the LMW coiii-poiieiit has been revealed experimentally for many P-f-LMWL systems (tYeemaii and llowlinsoii, 1960 Baker et al., 1962). 

To explain the existence of the LC. ST of such a type, a theory of corresponding states (called the theory of liquid state as well) ha.s received a large d velopment effort by Prigogine et al. (1953), Prigogine (1957), Flory el al. (196dab), Flory (1965, 1970), Patterson and Delmas (1970),, Siow et al. (1972). This approach is baseil on the theory of r-dimcnsional liquids developed by the Brussels school (Ilia Prigogine el al.) 

3c is the number of external (intermolecular) degrees of freedom per segment (mer) with c 1 (Kichinger and Flory, 1968a). 

The molecular characteristics, which depend on the intermolecular interactions only, are called configurational.  

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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. 

The concept of corresponding states was based on kinetic molecular theory, which describes molecules as discrete, rapidly moving particles that together constitute a fluid or soHd. Therefore, the theory of corresponding states was a macroscopic concept based on empirical observations. In 1939, the theory of corresponding states was derived from an inverse sixth power molecular potential model (74). Four basic assumptions were made (/) classical statistical mechanics apply, (2) the molecules must be spherical either by actual shape or by virtue of rapid and free rotation, (3) the intramolecular vibrations are considered identical for molecules in either the gas or Hquid phases, and (4) the potential energy of a coUection of molecules is a function of only the various intermolecular distances. 

The theory of corresponding states only works for certain classes of fluids, thus fluids have been placed into generalized groups based on behavior (75,76). These groupings include the foHowing. 

Simple Fluids. Spherical compounds having Httle molecular interaction, eg, argon, krypton, xenon, and methane, are known as simple fluids and obey the theory of corresponding states. 

Compared to the theory of corresponding states, the reference substance method gives highly accurate results for compounds having sparse experimental data. The corresponding states method gives moderate accuracy for numerous compounds even without actual data. 

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... 

Riedel, L. Chem. Ing. Tech. 26 (1954) 259-264. Liquid density in the saturated state. Extension of the theory of corresponding states. 

Normal Fluids. Asymmetrical compounds having little molecular interaction, eg, carbon monoxide, -butane, and tf-hexane, deviate slighdy from the theory of corresponding states and are considered to be normal fluids. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

We have shown previously (10) that based on Geefs thermodynamic analysis of gas solubility in polymers (12) and the theory of corresponding states (14), the solubility coefficient, a, can be expressed as,... 

In its simplest form, the theory of corresponding states says that if two substances are at the same reduced temperature and reduced pressure, then the other "reduced" properties should be equal. 

Scamehorn et. al. ( ) also developed a reduced adsorption equation to describe the adsorption of mixtures of anionic surfactants, which are members of homologous series. The equations were semi-empirical and were based on ideal solution theory and the theory of corresponding states. To apply these equations, a critical concentration for each pure component in the mixture is chosen, so that when the equilibrium concentrations of the pure component adsorption isotherms are divided by their critical concentrations, the adsorption isotherms would coincide. The advantage of... 

Like the theory of corresponding states on which it is based, Kay s rule provides only approximate values of the quantities it is used to calculate. It works best when used for mixtures of nonpolar compounds whose critical temperatures and pressures are within a factor of two of one another. Reid, Prausnitz, and Poling (see footnote 1) provide more complex but more accurate mixing rules for systems that do not fall into this category. 

Enthalpies and other thermodynamic properties can be estimated by generalized methods based on the theory of corresponding states or additive bond contributions."... 

Krieger and Dougherty applied the theory of corresponding states to obtain the following expression for the viscosity of hard-sphere dispersions ... 

The normal value for the constant is about 20 7, and a large number of different substances approximate fairly closely to this On the other hand, substances such as the alcohols and acids, which are known to be polymerised in the liquid state, give different values for the constant The law therefore is not general1 Although put forward in the first instance as an empirical relation, it has a certain amount of theoretical basis from the standpoint of van der Waals theory of corresponding states The reasoning is as follows —... 

Carbon Monoxide There are fewer experimental data for this fluid than for any of the cryogenic fluids discussed. These data are limited to temperatures above 200°K and a few values at saturation. Property values have been calculated, however, using these existing data and the theory of corresponding states with nitrogen as a model for values down to 70°K with pressures to 300 atm. In view of the low interest in this fluid generally expressed by members of the cryogenic industry, however, these calculated values may be sufficient for some time. 

In the classical empirical theory of corresponding states, the units which are used to reduce the state variables are the critical-point values Pc, Vc, and Tc. There is, however, nothing unique about this choice of units. For example, an alternative formulation of the law of corresponding states has been developed by de Boer and his collaborators. In this formulation, which can also be shown to have an exact theoretical basis, the intermolecular field parameters are used to reduce the state variables. 

CALCULATION OF HEAT CONDUCTANCE OF LIQUIDS BY THE THEORY OF CORRESPONDING STATES. 

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