Free 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. 

According to the free-radical theory of molecular transformation, thermal crac "". ... 

If the molecule moves without hindrance in a rigid-walled enclosure (the free enclosure ), as assumed in free volume theories, then rattling back and forth is a free vibration, which could be considered as coherent in such a cell. The transfer time between opposite sides of the cell t0 is roughly the inverse frequency of the vibration. The maximum in the free-path distribution was found theoretically in many cells of different shape [74]. In model distribution (1.121) it appears at a > 2 and shifts to t0 at a - oo (Fig. 1.18). At y — 1 coherent vibration in a cell turns into translational velocity oscillation as well as a molecular libration (Fig. 1.19). 

A useful model should account for a reduction of kt and kp with increase in polymer molecular weight and concentration and decrease in solvent concentration at polymerization temperatures both below and above the Tg of the polymer produced. For a mechanistic model this would involve many complex steps and a large number of adjustable parameters. It appears that the only realistic solution is to develop a semi-empirical model. In this context the free-volume theory appears to be a good starting point. 

In addition to temperature and concentration, diffusion in polymers can be influenced by the penetrant size, polymer molecular weight, and polymer morphology factors such as crystallinity and cross-linking density. These factors render the prediction of the penetrant diffusion coefficient a rather complex task. However, in simpler systems such as non-cross-linked amorphous polymers, theories have been developed to predict the mutual diffusion coefficient with various degrees of success [12-19], Among these, the most notable are the free volume theories [12,17], In the following subsection, these free volume based theories are introduced to illustrate the principles involved. 

Yasuda s free volume theory [57] has been proposed to explain the mechanism of permeation of solutes through hydrated homogeneous polymer membranes. The free volume theory relates the permeability coefficients in water-swollen homogeneous membranes to the degree of hydration and molecular size of the permeant by the following mathematical expression ... 

Equation 39 has the structure proposed for the rate constants on the basis of the free volume theory (1,5,9). From this, it would be expected that the models developed from the free volume theory would be very successful in predicting both, the rate behaviour and the molecular properties at high conversions. The reason why these models have been only partially successful stems from the... 

The basic postulate of elementary molecular theories of rubber elasticity states that the elastic free energy of a network is equal to the sum of the elastic free energies of the individual chains. In this section, the elasticity of the single chain is discussed first, followed by the elementary theory of elasticity of a network. Corrections to the theory coming from intermolecular correlations, which are not accounted for in the elementary theory, are discussed separately. 

Resonance such as (5.28a)-(5.28c) is inherently a quantal phenomenon, with no classical counterpart. In NBO language, each of the resonance interactions (5.28a)-(5.28c) corresponds to a donor-acceptor interaction between a nominally filled (donor Lewis-type) and unfilled (acceptor non-Lewis-type) orbital, the orbital counterpart of G. N. Lewis s general acid-base concept. As mentioned above, Lewis and Werner (among others) had well recognized the presence of such valence-like forces in the dative or coordinative binding of free molecular species. Thus, the advent of quantum mechanics and Pauling s resonance theory served to secure and justify chemical concepts that had previously been established on the basis of compelling chemical evidence. 

It would be an advantage to have a detailed understanding of the glass transition in order to get an idea of the structural and dynamic features that are important for photophysical deactivation pathways or solid-state photochemical reactions in molecular glasses. Unfortunately, the formation of a glass is one of the least understood problems in solid-state science. At least three different theories have been developed for a description of the glass transition that we can sketch only briefly in this context the free volume theory, a thermodynamic approach, and the mode coupling theory. 

Table 2.2 Contributions to the system free energy considered by the molecular theory for poleyelectrolyte modified electrodes, Equation 2.16.

Chapter 2 introduces the band theory of solids. The main approach is via the tight binding model, seen as an extension of the molecular orbital theory familiar to chemists. Physicists more often develop the band model via the free electron theory, which is included here for completeness. This chapter also discusses electronic condnctivity in solids and in particular properties and applications of semiconductors. 

The underlying concept of free-volume theory is that the movement of the molecules is intrinsically conditioned to the amount of free volume in a molecular ensemble the less the unoccupied space, the more the collisions among the molecules, which results in a slow response to a perturbation in an equilibrium state [115]. 

The form of W as a function of the set of must be derived from molecular theory or from experimental measurement. It cannot be deduced from the phenomenologic theory of continua, just as the free energy cannot be deduced from thermodynamics. However, the phenomenologic theory imposes the following restrictions on the form of W if the material is isotropic 18 First, W must be an even power function of X,-(restriction A). Second, W must be invariant for permutations of Xt (restriction B). Third, W must be invariant for the transformation of coordinate axes (restriction C). 

The molecular orbital treatment of a crystalline solid considers the outer electrons as belonging to the crystal as a whole (10,11). Sommer-feld s early free electron theory of metals neglected the field resulting... 

Polyvinylchloride was the host polymer in a study of the diffusion of dimethyl-phthalate, dibutylphthalate, and dioctylphthalate, performed by Maklakov, Smechko, and Maklakov 60) between room temperature and 110 °C. Azancheev and Maklakov 61) extended this work to include polystyrene as host, and to dependences of diffusion on concentration. They concluded that the macromolecules did constrain and trap the phthalate molecules at high polymer concentration, but without inhibiting the mobility of these diluents at lower polymer concentrations, e.g., in the gel. They used a version of the free volume theory to give a semi-quantitative explanation of the temperature and molecular size dependence of phthalate diffusion. 

According to the model of random walk in three dimensions, the diffusion coefficient of a molecule i, can be expressed as one-third of the product of its mean free path A, and its mean three-dimensional velocity u, (Eq. 18-7a). In the framework of the molecular theory of gases, u, is (e.g., Cussler, 1984) ... 

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