Corresponding states molecular theory

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. 

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. 

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

We therefore believe one of the most pressing problems in the field of free-volume theory and glass-transition theory to be the development of new concepts and obtaining of new parameters for the corresponding states. Miller37 as early as 1968 introduced the idea that the glass transition corresponds to the iso-relaxation state, which at molecular weights of polymers below the critical ones may be replaced by an iso-viscous state. 

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. 

The emphasis of the theoretical discussion is (1) derivation and interpretation of the sum on states perturbation theory for charge polarization (2) development of physical models for the hyperpolarizability to assist molecular design (e.g., reduction of molecular orbital representations to the corresponding anharmonic oscillator description for hyperpolarizability). 

Eotvos deduced his equation theoretically from considerations of corresponding states of liquids of similar molecular constitution, which are rather difficult to follow. The central point of the theory is, however, that surfaces should be compared on the basis of the number of molecules per unit area, which is, if the molecules are similar in shape and symmetrically packed, proportional to (Jfv)1. 

The perturbation theories [2, 3] go a step beyond corresponding states the properties (e.g., Ac) of some substance with potential U are related to those for a simpler reference substance with potential Uq by a perturbation expansion (Ac = Aq + A + Aj + ). The properties of the simple reference fluid can be obtained from experimental data (or from simulation data for model fluids such as hard spheres) or corresponding states correlations, while the perturbation corrections are calculated from the statistical mechanical expressions, which involve only reference fluid properties and the perturbing potential. Cluster expansions involve a series in molecular clusters and are closely related to the perturbation theories they have proved particularly useful for moderately dense gases, dilute solutions, hydrogen-bonded liquids, and ionic solutions. 

Liquids are neither characterised by the random chaotic motion of molecules, which one find in gases, nor by the perfect order of moleculars arrangement in solids. They occupy an intermediary position where molecules are more disorderly than those of a solid, but much less disorderly than those of gases. Because of this fact the enthalpy change when a crystal melts is always positive and the corresponding entropy change is also positive. This implies that there is less of order when a crystal melts. The liquid is thus intermediate between the complete order of the crystalline state and the complete disorder of the gaseous state. Because of this fact, the development of a molecular theory for liquids has posed formidable difficulties. 

The equation demands that the molecular volumes of all substances at the same reduced temperature and reduced pressure should be the same fraction of their critical volumes. As the equation is intended to apply to aU liquid and sohd substances, it postulates a very great similarity in the physical behaviour of substances. It is not to be wondered at, therefore, that the theory only gives a very rough picture of the facts. Apart from some marked exceptions, it has been found that the physical properties of various substances may be compared best at equal reduced pressures and temperatures, i.e. when the substances are in corresponding states. 

Physical properly estimation methods may be classified into six general areas (1) theory and empirical extension of theory, (2) corresponding states, (3) group contributions, (4) computational chemistry, (5) empirical and quantitative structure property relations (QSPR) correlations, and (6) molecular simulation. A quick overview of each class is given below to provide context for the methods and to define the general assumptions, accuracies, and limitations inherent in each. 

Numerous attempts have been made to develop fluid models on the basis of molecular thermodynamics, taking into account the intermolecular forces. It is beyond the scope of this book to review these theories, and, in any case, the theoretical models are not necessarily the ones that are most widely used. The success of a model rests on its ability to represent real fluids. The principle of corresponding states is another approach that provides the foundation for some of these models. A number of models, or equations of state, that have proven their practical usefulness for phase equilibrium and enthalpy departure calculations are presented in this section. 

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.

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. 

Crystal states, where one molecule is ionized, and the conduction band contains one electron, can be used as those intermediate states, as has been shown in (31) (the role of such intermediate states in theory of photoconductivity of molecular crystals has been discussed by Lyons (32)). The use of intermediate states becomes indispensable when the second-order perturbation theory is applied in the case of a degenerate term. According to (33), correct linear combinations of crystal states, containing one molecule in a triplet state and all remaining in the ground state, can be found by perturbation theory when in the corresponding secular equation the following effective Hamiltonian is used... 

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 Theory. All are able to model nonpolar systems fairly successfully, but most are increasingly challenged 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... 

Mass transfer, an important phenomenon in science and engineering, refers to the motion of molecules driven by some form of potential. In a majority of industrial applications, an activity or concentration gradient serves to drive the mass transfer between two phases across an interface. This is of particular importance in most separation processes and phase transfer catalyzed reactions. The flux equations are analogous to Ohm s law and the ratio of the chemical potential to the flux represents a resistance. Based on the stagnant-film model. Whitman and Lewis [25,26] first proposed the two-film theory, which stated that the overall resistance was the sum of the two individual resistances on the two sides. It was assumed in this theory that there was no resistance to transport at the actual interface, i.e., within the distance corresponding to molecular mean free paths in the two phases on either side of the interface. This argument was equivalent to assuming that two phases were in equilibrium at the actual points of contact at the interface. Two individual mass transfer coefficients (Ld and L(-n) and an overall mass transfer coefficient (k. ) could be defined by the steady-state flux equations ... 

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