Transport processes approximate theories

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic dreoty of gases, and their interrelationship tlu ough k and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests tlrat this should be a dependence on 7 /, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to for the case of molecular inter-diffusion. The Anhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, tlren an activation enthalpy of a few kilojoules is observed. It will thus be found that when tire kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation entlralpy will be a few kilojoules only (less than 50 kJ). 

At present, all these theories are approximate, since all attempts to derive them using molecular mechanics have been largely unsuccessful, because there is a large number of degrees of freedom in describing concentrated polymer solutions. Among these approximate theories, such as those developed by Barrer (IQ), DiBenedetto (ID, and van Krevelen (12), the free-volume theory of diffusion is the only theory sufficiently developed to describe transport processes in concentrated polymer solutions. 

In view of the failure of the rigid sphere model to yield the correct isochoric temperature coefficient of the viscosity, the investigation of other less approximate models of the liquid state becomes desirable. In particular, a study making use of the Lennard-Jones and Devonshire cell theory of liquids28 would be of interest because it makes use of a realistic intermolecular potential function while retaining the essential simplicity of a single particle theory. The main task is to calculate the probability density of the molecule within its cell as perturbed by the steady-state transport process. 

To describe the functioning of the lEMs, theory from the field of charged membranes must be adapted for MCDI to describe the voltage-current relationship and the degree of transport of the colons. This implies that (in contrast to most membrane processes) the theory must be made dynamic (time dependent) because it has to include the fact that across the membrane the salt concentrations on either side of the membrane can be very different, and change in time. This means that approximate, phenomenological approaches based on (constant values for) transport (or transference) numbers or permselectivities are inappropriate, and that instead a microscopic theory must be used. An appropriate theory includes as input parameters the membrane ion diffusion coefficient and a membrane charge density X. 

Transport processes include diffusion, viscous flow, and heat conduction. There are a number of approximate theories of transport processes in liquids, most of which are based on classical statistical mechanics. 

Section 28.6 Approximate Theories of Transport Processes in Liquids... 

The Boltzmann equation is considered valid as long as the density of the gas is sufficiently low and the gas properties are sufficiently uniform in space. Although an exact solution is only achieved for a gas at equilibrium for which the Maxwell velocity distribution is supposed to be valid, one can still obtain approximate solutions for gases near equilibrium states. However, it is evident that the range of densities for which a formal mathematical theory of transport processes can be deduced from Boltzmann s equation is limited to dilute gases, since this relation is reflecting an asymptotic formulation valid in the limit of no coUisional transfer fluxes and restricted to binary collisions only. Hence, this theory cannot without ad hoc modifications be applied to dense gases and liquids. 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

The rate coefficient of a reactive process is a transport coefficient of interest in chemical physics. It has been shown from linear response theory that this coefficient can be obtained from the reactive flux correlation function of the system of interest. This quantity has been computed extensively in the literature for systems such as proton and electron transfer in solvents as well as clusters [29,32,33,56,71-76], where the use of the QCL formalism has allowed one to consider quantum phenomena such as the kinetic isotope effect in proton transfer [31], Here, we will consider the problem of formulating an expression for a reactive rate coefficient in the framework of the QCL theory. Results from a model calculation will be presented including a comparison to the approximate methods described in Sec. 4. 

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