Statistical mechanical principles

T. L. Hill, Statistical Mechanics Principles and Selected Applications. McGraw-Hill, New York, 1956. 

The important statistical mechanical principle of microscopic reversibility asserts that the mechanism of any chemical reaction considered in the reverse direction must be exactly the inverse of the mechanism of the forward reaction. A consequence of this principle is that if the mechanism of a reaction is known, that of the reverse reaction is also known. Furthermore, it follows that the forward and reverse reactions catalyzed by an enzyme must occur at the same active site on the enzyme and the transition state must be the same in both directions. The principle of microscopic reversibility is often useful when the likelihood of a given mechanism is being considered. If a mechanism is proposed for a reversible reaction in one direction the principle of microscopic reversibility will give an unambiguous mechanism for the reverse reaction. Sometimes this reverse mechanism will be chemically untenable and, recognizing this, the enzymologist can search for a better one. 

STATISTICAL MECHANICS Principles and Applications, Terrell L. Hill. Standard text covers fundamentals of statistical mechanics, applications to fluctuation theory, imperfect gases, distribution functions, more. 448pp. 5X 8X. 

Statistical mechanical principles suggest that the conformational entropy Sc of I mol of conformers occupying domains is given by... 

In this chapter we have described some recent applications of various computational methods to understanding some basic principles of complex catalytic and electrocatalytic processes. These methods rely on either quantum-mechanical or statistical-mechanical principles, or a combination of both, and obviously the level of detail and the kind of insight into a certain catalytic problem will depend on the chosen method. 

A t) ical Anneal-Flex run on a molecule such as the vitamin D3 ketone 1 consists of 20 runs of 1000 steps per temperature at 30 temperatures. Since the acceptance rate is usually around 30%, there are about 180,000 accepted steps or 9,000 lines of data for each 20-run file. In classical statistical mechanics, one Anneal-Flex run can be considered as one member of an ensemble [30]. The collection of twenty runs is the ensemble. In this type of formulation, the numerical value of the quantity of interest is obtained by calculating averages over this ensemble. While the quantities that we are interested in are too complicated to be represented by a single number, the same statistical mechanical principles can be used to create the distribution functions which accurately represent dihedral space. 

Hill TL. Statistical Mechanics Principles and Selected Applications. Original copyright New York McGraw-Hill 1956. Unabridged and unaltered republication New York Dover 1987. Chapters 1 and 2 especially Sects. 4, 5,10,11, and 13. 

Before outlining Onsager s theory, we collect and quickly summarize some statistical mechanical principles needed to formulate the theory. 

It is possible to calculate from statistical mechanical principles the approximate conformations of the adsorbed caseins, by assuming that they are flexible, and composed of chains of hydrophilic and hydrophobic amino acids (91). The calculations of these model systems show many of the features of the actual measured properties, especially the tendency of the adsorbed 6-casein to protrude further from the inter face than the Ogj-casein (92). These calculations have in turn been used to explain the differing stability of the two different types of emulsions (93). These calculations have considerable success in explaining both the structure and stability of casein-coated emulsions, but are less adaptable to explain the behavior of more rigid protein surfactants. However, the same principles have been used to explain the apparently anomalous adsorption of phosvitin (16). 

Hill, T.L. (1956), Statistical Mechanics, Principles and Selected Applications. 

The DA equation (4.2-5) is obtained by assuming the temperature invariance of the adsorption potential at constant loading and a choice of the Weibull s distribution to describe the filling of micropore over the differential molar work of adsorption. It can be shown to be a special case of an isotherm equation derived from the statistical mechanical principles when the loading is appreciable (Chen and Yang, 1994). They derived the following isotherm... 

Hill TL (1956) Statistical Mechanics Principles and Selected Apphcalions. McGraw-Hill, New York Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136 864-871 Holt AC, Hoover WG, Gray SG, Shortle DR (1970) Comparison of the lattice-dyrrarrrics and cell-model approximations with Monte-Carlo thermodynamic properties. Physica 49 61-76 Hoover WG (1983) Non-equihbrittm molecttlar-dynamics. Arm Rev Phys Chem 34 103-127 Horiuchi H, Ito E, Weidner DJ (1987) Perovskite-type MgSiOs single-crystal x-ray-drffiaction study. Am Min 72 357-360... 

Equation 4.65 reveals that the root-mean-squared speed increases as the temperature increases and decreases as the molar mass of the gas-phase species increases. Although it is beyond the scope of this textbook, the kinetic theory of gases can also be used (through the application of statistical mechanics principles) to derive the complete distribution of velocities for a gas-phase species ... 

We have devoted a greater part of this book to the experimental features of ion sorption in complex systems as soils are and, as introduced in Chapter 1, particularly most of this third part is dedicated to its physicochemical modeling. The interest, as of this writing, is mostly oriented to models capable of predicting the state of chemical species with reasonable accuracy, either natural or pollutants, in a given soil under determinate conditions. Thus, to date most models are phenomenological, primarily based in experimental correlations even when theories based on fundamental statistical mechanics principles may be desirable, this appears to be not feasible for the time being. 

Prominent among the methods for exploring the atomic scale dynamics of a system, including relaxation and rare events, are temperature-accelerated dynamics (TAD) [39], hyperdynamics [40] and parallel replica [41], all developed by Voter and coworkers. These techniques build on statistical mechanics principles for infrequent event systems, and as such do not make any prior assumptions regarding the atomistic mechanisms. They are designed to simply allow the system to evolve more quickly from state to state than they would in normal MD, provided that the barriers are relatively high compared to kT. 

Boltzmann order principle As the entropy of a system is decreased, the probability that the system will become more ordered is increased. This statistical-mechanical principle is basic to the understanding of equilibrium structures. 

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