Free diffusional motion

We assume that the mean-free path is much larger than the molecular dimensions see Section 9.2. At very high pressures this assumption of free flight is not valid and the overall reaction rate is controlled by the diffusional motion of the reactants. 

Consider, as an example, molecules at p = 1 atm and T = 298 K. Typically d ( R,.) is 0.3 nm, which implies that A 102 nm. Thus, at moderate—not too high—pressures the mean-free path is several hundred times larger than the molecular dimensions. Reactions in an ideal gas at standard conditions accordingly take place such that free flight prevails between the collisions. Diffusional motion only plays a role at very high pressures. 

Here DR and >, are the diffusion coefficients for the isotropic overall and free internal motions, respectively. Equation 31 assumes a diffusional process for the methyl group. If a jumping process between three equivalent positions separated by 120° is considered,47 the last term becomes C/(6DK + Z), ). Parameters A, B, and C are geometric constants similar to those in Eq. 27, but here the angle is that formed between the methyl C—H vectors and the axis of rotation. Assuming tetrahedral angles, for free internal motion ( >, ) ), 1/7 ,(CH3) is decreased to one-ninth of the value expected for a rigidly attached CH carbon. For slow internal rotation (D, — ) ), l/r1(CH3) becomes one-third of the value of a methine carbon in the same molecule, as predicted by Eq. 16. 

The description of the internal motion of the epoxypropyl ring of 23 is strongly model-dependent. This motion can be satisfactorily approximated either by free rotation about the C-5—C-6 bond or by a jumping process between two stable conformations. Discrimination between these two models from the relaxation data was not possible owing to a fortuitous similarity in the activation energies ( 17 kJ/mol) of the internal and overall diffusional motions.13 Inspection of molecular models indicates, however, that the rotation of the epoxypropyl ring is not sufficiently constrained to justify restricted rotation about the C-5—C-6 bond. 

The activation enthalpy for diffusion obtained at different degrees of coverage is shown in the enclosure. It increases from about 4 to about 6 kcal mole in passing from half to the complete monolayer content and then it decreases progressively toward the value obtained for the free liquid at 0 > 2. This indicates that the diffusional motions are still influenced by the surface for molecules in the third layer. 

Cell dynamics simulations are based on the time dependence of an order parameter, (i) (Eq. 1.23), which varies continuously with coordinate r. For example, this can be the concentration of one species in a binary blend. An equation is written for the time evolution of the order parameter, dir/dt, in terms of the gradient of a free energy that controls, for example, the tendency for local diffusional motions. The corresponding differential equation is solved on a lattice, i.e. the order parameter V (r) is discretized on a lattice, taking a value at lattice point i. This method is useful for modelling long time-scale dynamics such as those associated with phase separation processes. 

Since about 15 years, with the advent of more and more powerfull computers and appropriate softwares, it is possible to develop also atomistic models for the diffusion of small penetrants in polymeric matrices. In principle the development of this computational approach starts from very elementary physico-chemical data - called also first-principles - on the penetrant polymer system. The dimensions of the atoms, the interatomic distances and molecular chain angles, the potential fields acting on the atoms and molecules and other local parameters are used to generate a polymer structure, to insert the penetrant molecules in its free-volumes and then to simulate the motion of these penetrant molecules in the polymer matrix. Determining the size and rate of these motions makes it possible to calculate the diffusion coefficient and characterize the diffusional mechanism. 

Reactions in Solution Molecular motion in liquids is diffusional in place of free flight but the concept of activation energy and stearic requirements survive. Molecules have to jostle their way through the solvent and so the encounter frequency is drastically less than in a gas. Since a molecule migrates only slowly into the region of a possible reaction partner, it also migrates only slowly away from it. 

An(t)/n, the transiently induced birefringence. By this means, deterministic equations of motion, without stochastic terms, can be used via computer simulation to produce spectral features. As we have seen, a stochastic equation such as Eq. (1) is based on assumptions which are supported neither by spectral analysis nor by computer simulation of free molecular diffusion. The field-on simulation allows us the direct use of more realistic fimctions for the description of intermolecular interaction than any diffusional equation which uses stochastically generated intermolecular force fields. 

Figure 1 is a schematic representation of Frenkel s notion an atom or ion can get dislodged from its normal site to form etn interstitial-vacancy pair. He further proposed that they do not always recombine but instead may dissociate and thus contribute to diffusional transport and electrical conduction. They were free to Wcuider about in a "random walk" mcuiner essentially equivalent to that of Brownian motion. . . this meant they should exhibit a net drift in an applied field. 

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