Formulation of the Kinetic Theory

Andersen, H. C. Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. III. Cluster Analysis of the Renormalized Interactions and a Second Diagrammatic Representation of the Correlation Functions. J. Phys. Chem. B 2003, 107, 10234-10242. 

The connection between the classical and quantum formulations of the transport coefficients has been studied by applying the WKB method to the quantum formulation of the kinetic theory (B16, B17). In this way it was shown that at high temperatures the quantum formulas for the transport coefficients may be written as a power series in Planck s constant h. When the classical limit is taken (h approaches zero), then the classical formulas of Chapman and Enskog are obtained. 

I. Formulation of the Kinetic Theory 2. Dumbbell Suspension Models... 

Freedom in the choice of these assertions seems to be restricted essentially by only one requirement the scheme has to be self-consistent. This tendency to axiomatize is an important factor throughout the new development of the kinetic theory. It first attracted the general attention of mathematicians174 after the appearance of the program formulated by W. Gibbs in the preface of his Elementary Principles of Statistical Mechanics (1901). 

The thermodynamics of irreversible processes should be set up from the scratch as a continuum theory, treating the state parameters of the theory as field variables [32]. This is also the way in which classical fluid mechanic theory is formulated. Therefore, in the computational fluid dynamics literature, the transport phenomena and the extensions of the classical thermodynamic relations are both interpreted as closures of the fluid dynamic theory. The validity of the thermodynamic relations for fluid dynamic systems has been approached from the viewpoint of the kinetic theory of gases [13]. However, any Arm distinction between irreversible thermodynamics and fluid mechanics... 

The dependence type found corresponds well with the ideas about initiation of crystalline materials by impact or shock [101,103,104] (see also Refs. [26,47] and quotations herein) when a molecular crystal receives shock or impact, lattice vibrations (phonons) are excited at first. The phonon energy must then be converted into bond stretching frequencies (vibrons) with subsequent spontaneous localisation of vibrational energy in the nitro (explosophore) groupings [105,106] and then with consequential bond breaking. Conclusions of this type also correspond to an older simplified idea formulated by Bernard [107,108] on the basis of the kinetic theory of detonation the only explosophore groups should be compressed ahead of the shock wave as a result of the activation of explosive molecules. 

The condensed-phase atomic recombination process has been extensively studied from both experimental and theoretical points of view. This apparently simple reaction is actually rather complex, and our knowledge of the process is still very far from being complete. We have already referred to this type of reaction several times to illustrate certain features of the kinetic theory formulation. We now give a more detailed and coherent discussion. 

A fully microscopic treatment of this problem is a very difficult task. It is usually the motion of some internal coordinate of a complex molecule that is important for the description of the isomerization reaction (cf. Sections III and IV). A microscopic theory at the same level as that for the bimolecular processes described in the previous sections would entail a full description (or model) of the internal structure of the molecule and its interactions with the surrounding solvent. The collision dynamics for such a process are necessarily complex, but a theory at this detailed level is not out of the question for some models of small molecule isomerization reactions. However, it is probably premature to embark on such a program, since the implications of the kinetic theory for the reactions for which it is more easily formulated have not yet been fully explored. 

Since its original formulation, the basis of the kinetic theory has been the subject... 

The thermodynamic formulation of reaction rates is also particularly useful in discussing rates in ideal solutions. Indeed, the concept of collision between molecules and the derivations of the kinetic theory of gases seem to be useless in the condensed state. Yet, the results of transition-state theory are not limited to the treatment of ideal gas mixtures. In particular, these results can also be couched in the language of the colU on theory. This may appear surprising since the concept of collision in condensed phases is not a fruitful one. Yet it is found that normal reactions in solution exhibit a rate constant described by (2.5.3) with a probability factor P close to unity. 

We will use classical mechanics throughout, since most of the topics we deal with are more easily presented in terms of classical, rather than quantum, mechanics. Those topics for which a quantum formulation is essential, such as inelastic collisions of polyatomic molecules, are not discussed here, but appropriate literature references will be given. In addition, we assume that some of the elementary concepts of the kinetic theory of gases, such as the mean free path, are already familiar to the reader. 

The molecular model approach to the quantum mechanical formulation of a kinetic theory of velocity of reaction rates at surfaces is a little-developed subject, and the first difficulties are as follows ... 

The early study of catalysis by acids and bases was concerned chiefly with the use of catalysed reactions for investigating general problems of physical chemistry. For example, the first correct formulation of the kinetic laws of a first-order reaction was made by Wilhelmy in 1850 in connection with his measurements of the catalytic inversion of cane sugar by acids. Catalytic reactions also played an important part in the foundation of the classical theory of electrolytic dissociation towards the end of the nineteenth century, and kinetic measurements (notably on the... 

As is well known, early studies of coordination compounds primarily involved thermodynamically stable and kinetically inert complexes. These studies, mainly by Werner, Blomstrand, and Jorgensen, made possible the formulation of the coordination theory by Werner and led to the discovery of the spatial structure of complexes. The resolution of optical isomers, obviously the most spectacular evidence for the octahedral configuration of certain complexes, could be achieved only by the study of inert complexes. Consequently, the study of complex equilibria at this time was of secondary Importance. 

These particular forms of the kinetic theory results are somewhat cumbersome to write down, although less so than some other formulations (McCourt et al. 1990 Monchick et al. 1965). Their particular virtues are their complete generality, that they are particularly suitable for machine calculations and that they show explicitly that once the effective cross sections are available the transport properties of an arbitrary mixture of components are easily evaluated. 

Elementary collision theory was formulated after the development of the kinetic theory of gases although it has been superceded by transition state theory, it is frequently used because its limitations can be explained and even anticipated. 

These theories proceeded from the kinetic theory of gases through modified equations of state, such as the van der Waals equation, in an attempt to correct for the effects of the interatomic or intermole-cular forces resulting from the close proximity of the components in a dense gas. Born and Green/ following an approach initiated by Kirkwood, developed a set of molecular distribution functions adequate for the formulation of a kinetic theory of liquids. The formulation is mathematically complex, however, and has not yielded much in the way of practical results. 

The thermodynamic formulation of the transition state theory is useful in considerations of reactions in solution when one is examining a particular class of reactions and wants to extrapolate kinetic data obtained for one reactant system to a second system in which the same function groups are thought to participate (see Section 7.4). For further discussion of the predictive applications of this approach and its limitations, consult the books by Benson (59) and Laidler (60). Laidler s kinetics text (61) and the classic by Glasstone, Laidler, and Eyring (54) contain additional useful background material. 

Another defect problem to which the ion-pair theory of electrolyte solutions has been applied is that of interactions to acceptor and donor impurities in solid solution in germanium and silicon. Reiss73>74 pointed out certain difficulties in the Fuoss formulation. His kinetic approach to the problem gave results numerically very similar to that of the Fuoss theory. A novel aspect of this method was that the negative ions were treated as randomly distributed but immobile while the positive ions could move freely. 

Since the discovery of the deuterium isotope in 1931 [44], chemists have long recognized that kinetic deuterium isotope effects could be employed as an indicator for reaction mechanism. However, the development of a mechanism is predicated upon analysis of the kinetic isotope effect within the context of a theoretical model. Thus, it was in 1946 that Bigeleisen advanced a theory for the relative reaction velocities of isotopic molecules that was based on the theory of absolute rate —that is, transition state theory as formulated by Eyring as well as Evans and Polanyi in 1935 [44,45]. The rate expression for reaction is given by... 

The relativistic formulation of Thomas-Fermi theory started at the same time as the original non-relativistic one, the first work being of Vallarta and Rosen [9] in 1932. The result they arrived at can be found by replacing the kinetic energy fimctional by the result of the integration of the relativistic kinetic energy in terms of the momentum p times the number of electrons with a given momentum p from /i = 0 to the Fermi momentum p = Pp. ... 

---