Statistical mechanical theory

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

It seems appropriate to assume the applicability of equation (A2.1.63) to sufficiently dilute solutions of nonvolatile solutes and, indeed, to electrolyte species. This assumption can be validated by other experimental methods (e.g. by electrochemical measurements) and by statistical mechanical theory. 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

Kirkwood J G and Buff F P 1951 Statistical mechanical theory of solutions I J. Chem. Phys. 19 774... 

Stell G, Patey G N and H0ye J S 1981 Dielectric constant of fluid models statistical mechanical theory and its quantitative implementation Adv. Chem. Phys. 48 183... 

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

Kubo R, Yokota M and Nakajima S 1957 Statistical-mechanical theory of irreversible processes. Response to thermal disturbance J. Phys. Soc. Japan 12 1203... 

Yamamoto T 1960 Quantum statistical mechanical theory of the rate of exchange chemical reactions in the gas phase J. Chem. Phys. 33 281... 

Salto N, Takahashi K and Yunoli Y 1967 The statistical mechanical theory of stiff chains J Phys. See. Japan 22 219... 

J. G. Kirkwood and J. Boss, The Statistical Mechanical Basis of the Boltzmann Equation, in I. Frigogine, ed., Transport Processes in Statistical Mechanics, pp. 1-7, Interscience Publishers, Inc., New York, 1958. Also, J. G. Kirkwood, The Statistical Mechanical Theory of Transport Processes I. General Theory, J, Chem, Phys, 14, 180 (1946) II. Transport in Gases, J, Chem. Phys, 15, 72 (1947). 

He is the author of two other books. Nonequilibrium Thermodynamics (1962) and Vector Analysis in Chemistry (1974), and has published research articles on the theory of optical rotation, statistical mechanical theory of transport processes, nonequilibrium thermodynamics, molecular quantum mechanics, theory of liquids, intermolecular forces, and surface phenomena. 

The Universal Quasichemical (UNIQUAC) is a two-parameter (t12, x21) model based on statistical mechanical theory. Activity coefficients are obtained by... 

M. Tokuyama and I. Oppenheim, Statistical-mechanical theory of Brownian motion— translational motion in an equilibrium fluid, Physica A 94, 501 (1978). 

Using the statistical mechanical theory, Longuet-Higgins and Pople obtained a correlation function for the velocities in a liquid consisting of... 

Suspension Model of Interaction of Asphaltene and Oil This model is based upon the concept that asphaltenes exist as particles suspended in oil. Their suspension is assisted by resins (heavy and mostly aromatic molecules) adsorbed to the surface of asphaltenes and keeping them afloat because of the repulsive forces between resin molecules in the solution and the adsorbed resins on the asphaltene surface (see Figure 4). Stability of such a suspension is considered to be a function of the concentration of resins in solution, the fraction of asphaltene surface sites occupied by resin molecules, and the equilibrium conditions between the resins in solution and on the asphaltene surface. Utilization of this model requires the following (12) 1. Resin chemical potential calculation based on the statistical mechanical theory of polymer solutions. 2. Studies regarding resin adsorption on asphaltene particle surface and... 

One major question of interest is how much asphaltene will flocculate out under certain conditions. Since the system under study consist generally of a mixture of oil, aromatics, resins, and asphaltenes it may be possible to consider each of the constituents of this system as a continuous or discrete mixture (depending on the number of its components) interacting with each other as pseudo-pure-components. The theory of continuous mixtures (24), and the statistical mechanical theory of monomer/polymer solutions, and the theory of colloidal aggregations and solutions are utilized in our laboratories to analyze and predict the phase behavior and other properties of this system. 

Quantum mechanical calculations are appropriate for the electrons in a metal, and, for the electrolyte, modern statistical mechanical theories may be used instead of the traditional Gouy-Chapman plus orienting dipoles description. The potential and electric field at any point in the interface can then be calculated, and all measurable electrical properties can be evaluated for comparison with experiment. 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

Spectroscopic Methods, [Biological] Applications of Spectroscopy, EPR, Recent Advances in (Smaller). Spectroscopy, Infrared, Use in Biology (Lecomte). Spectroscopy of Transition-Group Complexes (Jorgensen) Statistical-Mechanical Theory of Transport Processes. X. The Heat of Transport in Binary Liquid Systems (Bearman, Kirkwood, Fixman). ... 

Systems (Friedmann). . . . . . Transport Process, Statistical-Mechanical Theory of. X 4 225... 

Extension of the Peturbed Hard Chain Correlation (Statistical Mechanical Theory of Fluids)" (2, 5). Extend the PHC program under development to include additional compounds including water. This work is an attempt to combine good correlations for phase equilibrium, enthalpy, entropy, and density into a single model. 

Subsequently major theoretical advances were made, principally by Mayer, in creating an adequate statistical mechanical theory in which both long-range electrostatic forces and short-range forces of whatever origin were properly considered. 

Clearly, any measurement that differentiates between the properties of high and low temperature forms of H20(as), and/or delineates the relationship between H20(as) and liquid H20, can be used to test the hypotheses advanced vis a vis their structures. These and the experimental tests suggested, together with the construction of continuous random network models more sophisticated than that for Ge(as), the increased use of computer simulation, and exploitation of the available experimental information to guide the choice of appproximations in a statistical mechanical theory should increase our understanding of H20(as) and, uitimately, liquid H20. 

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

The microscopic structure of water at the solution/metal interface has been the focus of a large body of literature, and excellent reviews have been published summarizing the extensive knowledge gained from experiments, statistical mechanical theories of varied sophistication, and Monte Carlo and molecular dynamics computer simulations. To keep this chapter to a reasonable size, we limit ourselves to a brief summary of the main results and to a sample of the type of information that can be gained from computer simulations. 

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