Statistical mechanics postulates

An alternative way of deriving the BET equation is to express the problem in statistical-mechanical rather than kinetic terms. Adsorption is explicitly assumed to be localized the surface is regarded as an array of identical adsorption sites, and each of these sites is assumed to form the base of a stack of sites extending out from the surface each stack is treated as a separate system, i.e. the occupancy of any site is independent of the occupancy of sites in neighbouring stacks—a condition which corresponds to the neglect of lateral interactions in the BET model. The further postulate that in any stack the site in the ith layer can be occupied only if all the underlying sites are already occupied, corresponds to the BET picture in which condensation of molecules to form the ith layer can only take place on to molecules which are present in the (i — l)th layer. 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

In conclusion, under the hypothesis that the reaction has no barrier in excess of its endoergicity, Att/°j(0) = 0, the enthalpy of reaction 3.10 is given by the Arrhenius activation energy for the forward reaction minus a heat capacity term. This term can be estimated by using statistical mechanics, provided that a structure for the activated complex is available. It is often found that T A Cjj > is fairly small, ca. — 1 kJmol-1 at 298.15 K [60], and therefore, the alternative assumption of a,i Ar//" is commonly accepted if T is not too high. Finally, note that either 3,1 Ar//." or Atf/°j(0) = 0 are not equivalent (see equation 3.22) to another current (but probably less reliable) postulate, Ea- = 0. 

The acid-catalyzed ester hydrolysis provides a good target for MM treatments. DeTar first used hydrocarbon models in which an ester was approximated by an isoalkane (74) and the intermediate (75) by a neoalkane (76). He assumed that if the rate of reaction truly is not influenced by polar effects but is governed only by steric effects of R, as has been generally postulated, the rate must be proportional to the energy difference (AAH ) between 74 and 76. The AAH f is mainly determined by the van der Waals strain in these branched alkanes. Nonsteric group increment terms were carefully adjusted, and statistical mechanical corrections for conformer populations... 

If we accept the first three postulates, we can lift each of these approximations using statistical mechanics and the companion techniques of computer simulation. But to do so we must consider a material for which complete thermodynamic and the necessary structural information is available. Ve, therefore, consider the Lennard-Jones fluid in most of the following discussion. 

The basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

The second postulate is well established by statistical mechanics. 

The simplest, self-consistent model of the diffuse-ion swarm near a planar, charged surface like that of a smectite is modified Gouy-Chapman (MGQ theory [23,24]. The basic tenets of this and other electrical double layer models have been reviewed exhaustively by Carnie and Torrie [25] and Attard [26], who also have made detailed comparisons of model results with those of direct Monte Carlo simulations based in statistical mechanics. The postulates of MGC theory will only be summarized in the present chapter [23] ... 

Two important objectives of statistical mechanics are (1) to verify the laws of thermodynamics from a molecular viewpoint and (2) to make possible the calculation of thermodynamic properties from the molecular structure of the material composing the system. Since a thorough discussion of the foundations, postulates, and formal development of statistical mechanics is beyond the scope of this summary, we shall dispose of objective (1) by merely stating that for all cases in which statistical mechanics has successfully been developed, the laws quoted in the preceding section have been found to be valid. Furthermore, in discussing objective (2), we shall merely quote results the reader is referred to the literature [3-7] for amplification. 

Note the qualitative — not merely quantitative — distinction between the thermodynamic (Boltzmann-distribution) probability discussed in Sect. 3.2. as opposed to the purely dynamic (quantum-mechanical) probability Pg discussed in this Sect. 3.3. Even if thermodynamically, exact attainment of 0 K and perfect verification [22] that precisely 0 K has been attained could be achieved for Subsystem B, the pure dynamics of quantum mechanics, specifically the energy-time uncertainty principle, seems to impose the requirement that infinite time must elapse first. [This distinction between thermodynamic probabilities as opposed to purely dynamic (quantum-mechanical) probabilities should not be confused with the distinction between the derivation of the thermodynamic Boltzmann distribution per se in classical as opposed to quantum statistical mechanics. The latter distinction, which we do not consider in this chapter, obtains largely owing to the postulate of random phases being required in quantum but not classical statistical mechanics [42,43].]... 

The structure of the chapter is as follows. In Section 2, we review the basic postulates of the equilibrium thermodynamics. The equilibrium statistical mechanics based on generalized entropy is formulated in a general form in Section 3. In Section 4, we describe the Tsallis statistics and analyze its possible connection with the equilibrium thermodynamics. The main conclusions are summarized in the final section. 

As thermodynamics required postulates or laws, so does statistical mechanics. Gibbs postulates which define statistical mechanics are (1) Thermodynamic quantities can be mapped onto averages over all possible microstates consistent with the few macrosopic parameters required to specify the state of the system (here, NVE). (2) We construct the averages using an ensemble . An ensemble is a collection of systems identical on the macroscopic level but different on the microscopic level. (3) The ensemble members obey the principle of equal a priori probability . That is, no one ensemble member is more important or probable than another. 

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