Kinetic theory also Chap

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

P. G. Tait, Trans. Roy. Soc. Edinburgh, 33 (1887). See also E. H. Kennard, Kinetic Theory of Gases, chaps. 3 and 4, McGraw-Hill Book Company, Inc., New York, 1938. 

Enskog s dense gas theory for rigid spheres is also used as basis developing granular flow models. The modifications suggested extending the dense gas kinetic theory to particulate flows are discussed in chap 4. 

It is also interesting to note that although the local, two-body equilibrium solution is indeed a particular solution to Eq. (6.77), substitution of the equilibrium solution into Eq. (6.79) leads to nonsensical results. Therefore, the equilibrium solution is not self-consistent to leading order. There are numerous extensive treatments on the development, modifications, and treatment of the Boltzmann transport equation in gas kinetic theory (see the Further Reading section at the end of Chap. 3). More modern approaches, originally due to Bogolubov, consider the time scales of the expansion procedure more carefully. These so-called multiple time-scales methods are powerful procedures... 

It is important to note that this distribution function (2.244), defined so that it resembles (2.243) but with the constant values of n, v and T in (2.243) replaced by the corresponding functions of r and t, remains a solution to (2.242). This distribution function, which is called the local Maxwellian, makes the kinetic theory much more general and practically relevant. Both the absolute- and local Maxwellians are termed equilibrium distributions. This result relates to the local and instantaneous equilibrium assumption in continuum mechanics as discussed in Chap. 1, showing that the assumption has a probabilistic fundament. It also follows directly from the local equilibrium assumption that the pressure tensor is related to the thermodynamic pressure, as mentioned in Sect. 2.5. 

Chapter 2 contains a summary of the basic concepts of kinetic theory of dilute and dense gases. This theory serves as basis for the development of the continuum scale conservation equations by averaging the governing equations determining the discrete molecular scale phenomena. This method is an alternative to, or rather both a verification and an extension of, the continuum approach described in Chap. 1. These kinetic theory concepts also determine the basis for a group of models used describing granular flows, further outlined in Chap. 4. 

In Chap. 11 the laws governing the behavior of gases were presented. The fact that gases exert pressure was stated, but no reasons why gases should exhibit such behavior were given. The kinetic molecular theory explains all the gas laws that we have studied and some additional ones also. It describes gases in terms of the behavior of the molecules that make them up. 

The basic, macroscopic theories of matter are equilibrium thermodynamics, irreversible thermodynamics, and kinetics. Of these, kinetics provides an easy link to the microscopic description via its molecular models. The thermodynamic theories are also connected to a microscopic interpretation through statistical thermodynamics or direct molecular dynamics simulation. Statistical thermodynamics is also outlined in this section when discussing heat capacities, and molecular dynamics simulations are introduced in Sect 1.3.8 and applied to thermal analysis in Sect. 2.1.6. The basics, discussed in this chapter are designed to form the foundation for the later chapters. After the introductory Sect. 2.1, equilibrium thermodynamics is discussed in Sect. 2.2, followed in Sect. 2.3 by a detailed treatment of the most fundamental thermodynamic function, the heat capacity. Section 2.4 contains an introduction into irreversible thermodynamics, and Sect. 2.5 closes this chapter with an initial description of the different phases. The kinetics is closely link to the synthesis of macromolecules, crystal nucleation and growth, as well as melting. These topics are described in the separate Chap. 3. 

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