Elementary Kinetic Theory of Gases

The science of mechanics constitutes a vast number of sub-disciplines commonly considered beyond the scope of the standard chemical engineering education. However, when dealing with kinetic theory-, granular flow- and population balance modeling in chemical reactor engineering, basic knowledge of the principles of mechanics is required. Hence, a very brief but essential overview of the disciplines of mechanics and the necessary prescience on the historical development of kinetic theory are given before the more detailed and mathematical principles of kinetic theory are presented. 

Mechanics is a branch of physics concerned with the motions of physical bodies and the interacting forces. The science of mechanics is commonly divided into two major disciplines of study called classical mechanics and quantum mechanics. Classical mechanics, which can be seen as the prime discipline of mechanics, describes the behavior of bodies with forces acting upon them. Quantum mechanics is a relatively modern held of science invented at about 1900. The term classical mechanics normally refers to the motion of relatively large objects, whereas the study of motion at the level of the atom or smaller is the domain of quantum mechanics. Classical mechanics 

Jakobsen, Chemical Reactor Modeling, doi 10.1007/978-3-540-68622-4 2, Springer-Verlag Berlin Heidelberg 2008 

There have apparently been two parallel developments of the statistical mechanics theory, which are typified by the work of Boltzmann [6] and Gibbs [33]. The main difference between the two approaches lies in the choice of statistical unit [15]. In the method of Boltzmann the statistical unit is the molecule and the statistical ensemble is a large number of molecules constituting a system, thus the system properties are said to be calculated as 

Statistical mechanics is normally further divided into two branches, one dealing with equilibrium systems, the other with non-equilibrium systems. Equilibrium statistical mechanics is sometimes called statistical thermodynamics [70]. Kinetic theory of gases is a particular field of non-equilibrium statistical mechanics that focuses on dilute gases which are only slightly removed from equilibrium [28]. 

In a series of impressive publications. Maxwell [95-98] provided most of the fundamental concepts constituting the statistical theory recognizing that the molecular motion has a random character. When the molecular motion is random, the absolute molecular velocity cannot be described deterministically in accordance with a physical law so a probabilistic (stochastic) model is required. Therefore, the conceptual ideas of kinetic theory rely on the assumption that the mean flow, transport and thermodynamic properties of a collection of gas molecules can be obtained from the knowledge of their masses, number density, and a probabilistic velocity distribution function. The gas is thus described in terms of the distribution function which contains information of the spatial distributions of molecules, as well as about the molecular velocity distribution, in the system under consideration. An important introductory result was the Maxwellian velocity distribution function heuristically derived for a gas at equilibrium. It is emphasized that a gas at thermodynamic equilibrium contains no macroscopic gradients, so that the fluid properties like velocity, temperature and density are uniform in space and time. When the gas is out of equilibrium non-uniform spatial distributions of the macroscopic quantities occur, thus additional phenomena arise as a result of the molecular motion. The random movement of molecules from one region to another tend to transport with them the macroscopic properties of the region from which they depart. Therefore, at their destination the molecules find themselves out of equilibrium with the properties of the region in which they arrive. At the continuous macroscopic level the net effect 

The kinetic theory of multi-component non-reactive mixtures was first described by Maxwell [95, 97] and Stefan [142, 143] and later thoroughly described by Hirschfelder et al. [55]. Hirschfelder et al. [55] also considers reactive systems. The latest contributions are reviewed by Curtiss and Bird [26, 27]. 

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. 

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In order to predict the value of the frequency factor, one may assume that all collisions between reactant molecules with sufficient activation energy result in the instantaneous formation of the reaction products. With this simple hypothesis (collision theory), if the activation energy is known, then the problem of computing the reaction rate reduces to the problem of computing the rate of collision between the appropriate reactant molecules in the ideal gas mixture. This last problem is easily solved by the elementary kinetic theory of gases. 

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