Statistical-Mechanical Methods

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

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]. 

Many other approaches for finding a correct structural model are possible. A short description of ab-initio, density functional, and semiempirical methods are included here. This information has been summarized from the paperback book Chemistry with Computation An Introduction to Spartan. The Spartan program is described in the Computer Software section below.65 Another description of computational chemistry including more mathematical treatments of quantum mechanical, molecular mechanical, and statistical mechanical methods is found in the Oxford Chemistry Primers volume Computational Chemistry,52... 

What was the distinction between quantum chemistry and chemical physics After the Journal of Chemical Physics was established, it was easy to say that chemical physics was anything found in the new journal. This included molecular spectroscopy and molecular structures, the quantum mechanical treatment of electronic structure of molecules and crystals and the problem of chemical binding, the kinetics of chemical reactions from the standpoint of basic physical principles, the thermodynamic properties of substances and calculation by statistical mechanical methods, the structure of crystals, and surface phenomena. 

While Debye and HUckel recognized the short-range repulsive forces between ions by assuming a hard-core model, the statistical mechanical methods then available did not allow a full treatment of the effects of this hard core. Only the effect on the electrostatic energy was included—not the direct effect of the hard core on thermodynamic properties. 

Stephen J. Paddison received a B.Sc.(Hon.) in Chemical Physics and a Ph.D. (1996) in Physical/Theoretical Chemistry from the University of Calgary, Canada. He was, subsequently, a postdoctoral fellow and staff member in the Materials Science Division at Los Alamos National Laboratory, where he conducted both experimental and theoretical investigations of sulfonic acid polymer electrolyte membranes. This work was continued while he was part of Motorola s Computational Materials Group in Los Alamos. He is currently an Assistant Professor in the Chemistry and Materials Science Departments at the University of Alabama in Huntsville, AL. Research interests continue to be in the development and application of first-principles and statistical mechanical methods in understanding the molecular mechanisms of proton transport in fuel-cell materials. 

Spectroscopic measurements may, in certain cases, yield direct information on these interactions. On the other hand, thermodynamic values, obtained by measuring certain bulk properties of the system, require the aid of statistical mechanical methods to be related to specific interactions between the solute and the solvent. However, the thermodynamic aspects of the solute-solvent interactions reflect the preference of the solute for one solvent over another and, thereby, determine distribution of the solute in a solvent extraction system. 

Smaller single-component, single-atom adsorbates, such as Ar, in zeolite A have also been investigated (20, 28, 29, 130). Kono and Takasaka (130) calculated the sorption characteristics using classical statistical mechanics methods. The particular MC method that they used is applicable at sorbate concentrations higher than infinite dilution. They found that a London... 

The virial coefficients B(T), C(T), D(T),... are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B T) to be explicitly evaluated by statistical mechanical methods as... 

The absolute entropies of small molecules can be calculated by statistical mechanical methods. Table 2.1 shows the results of such calculations for liquid propane. The largest contributions to the entropy come from the translational and rotational freedom of the molecule, and much smaller contributions from vibrations electronic terms are insignificant. Although exact calculations of this type become intractable for large biological molecules, the relative sizes of the contributions from different types of motions are similar to those in small molecules. Thus entropy is associated primarily with translation and rotation. This relationship is very different from enthalpy, in which electronic terms are dominant and translational and rotational energies are comparatively small. 

This chapter s statistical mechanics method was used to generate phase diagrams as illustrations of multicomponent hydrate equilibria concepts at one isotherm, 277.6 K, the most common temperature in a pipeline on the ocean floor at water depths beyond 600 m. Section 5.2.1 shows the fit if the method to single (simple) hydrates, before the extension to binary hydrate guests in Section 5.2.2. Section 5.2.3 shows the final extension to ternary mixtures of CH4 + C2H6 + C3H8 and indicates an industrial application. Most of the discussion in this section was extracted from the thesis of Ballard (2002) and the paper by Ballard and Sloan (2001). 

In some other cases, more elaborate statistical mechanics methods are needed to calculate the free energies of the reactants and the transition state. This occurs whenever the range of geometries sampled by the system goes well beyond the vicinity of the relevant stationary point, that is, the reactant minimum or the saddle point. Some examples of this type of behavior will be described below. Also, in some cases, atomic motion is not well described by classical mechanics, and although TST incorporates some quantum mechanical aspects, it does not typically include others, and more advanced methods are needed to describe reactions in such cases. Again, some examples will be given below. 

Symbol Gibbs free energy is denoted eponymously by G, after Josiah Willard Gibbs, ca. 1873, who single-handedly created much of chemical thermodynamics. In the older literature F was sometimes used. Equation since G = H TS, the free energy of a molecule can be calculated from its enthalpy (above) and entropy at temperature 7 the entropy is calculated by standard statistical mechanics methods [130]. 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

No being Avogadro s number, the number of molecules in a mole, given in Eq. (3.10) of Chap. IV. We observe that Eq. (3.14) is exactly the same as (1.21), determined by thermodynamics, except that now we have found the quantities Uo, the arbitrary constant in the energy, and i, the chemical constant, in terms of atomic constants. Similarly, we can show that all the other formulas of Sec. 1 follow from our statistical mechanical methods, using Eqs. (3.15) and (3.16) for the constants which could not be evaluated from thermodynamics. 

Figure 2. Examples of macroscopic and statistical mechanical methods for studying...

The combined quantum chemical statistical mechanical method QMSTAT was originally published in 1996 for Hartree-Fock (HF) quantum chemistry [20], This formulation has been applied in a number of studies, to which we will return in a later section. In 2006, an extension of QMSTAT was published with another quantum chemical method which enabled studies of excited states and multiconfigurational systems [21]. The two formulations have a lot in common and the discussion below applies to both formulations except when stated otherwise. 

Theoretically it is possible to calculate thermodynamical data by means of statistical mechanical methods. Elowever, these are laborious and moreover the (empirical ) spectroscopic data required to this end are usually lacking. 

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