Statistical thermodynamics phase transitions

The temperature Tp in equations 21 and 22 is the temperature at the maximum fuel generation rate during the course of the linear temperature history. Defining a characteristic heating rate p = A ATp, where ATp = BT IEg is the half-width of the pyrolysis temperature interval, equation 22 takes the form of a (statistical) thermodynamic phase transition temperature... 

Thermodynamic, statistical This discipline tries to compute macroscopic properties of materials from more basic structures of matter. These properties are not necessarily static properties as in conventional mechanics. The problems in statistical thermodynamics fall into two categories. First it involves the study of the structure of phenomenological frameworks and the interrelations among observable macroscopic quantities. The secondary category involves the calculations of the actual values of phenomenology parameters such as viscosity or phase transition temperatures from more microscopic parameters. With this technique, understanding general relations requires only a model specified by fairly broad and abstract conditions. Realistically detailed models are not needed to un-... 

Ben-Shaul, A. and Gelbart, W. M. (1994). Statistical Thermodynamics of Amphi-phile Self-assembly Structure and Phase Transitions in Micellar Solutions. Chapter 1. Springer, Berlin. 

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

We shall now discuss the phase transition from the viewpoint of statistical thermodynamics. " The total free energy G can be expressed as a function of N (total number of cation sites = total number of anion sites), (total number of anions), (number of cations on the A sites), Ag, A,-, and Aq as G = G(A,Ax,Aa,Ab,Ac,Ad) (1.234)... 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

Below the statistical-thermodynamical calculation of configurational heat capacity of ordering twocomponent fullerite from fullerenes = C(, , [Pg.219]

A thermodynamic treatment, similar to that used for microemulsions, as well as an approximate statistical mechanical one, are developed to explain the phase transition in monolayers of insoluble surfactants [3.8], A similar thermodynamic approach is applied to multilamellar liquid crystals, and it is shown that, for a given set of interactions and bending moduli, only narrow ranges of the thicknesses of the water and oil layers are allowed [3.9]. 

Using an interparticle potential, the characterization of the equilibrium state is possible by thermodynamic analysis. Van Megen and Snook [10,11] have adopted the statistical approach to predict the disorder—order phase transitions in concentrated dispersions that are stabilized electrostatically. Using the perturbation theory for the disordered phase and the cell model for the ordered phase, they have estimated the particle concentrations in the two coexisting phases when an electrostatically stabilized dispersion undergoes phase separation. Recently, Cast et al. [12] have used a similar approach to construct phase diagrams for colloidal dispersions that have free polymer molecules in solution. Using the interaction potential of Asakura... 

A general thermodynamic theory and a statistical thermodynamic approach are presented, which describe the phase transitions in insoluble monolayers, particularly the inclined transition from a liquid-expanded to a liquid-condensed phase. 

The first goal of this article is to determine the conditions under which the transition from a liquid-expanded (LE) to a liquid-condensed (LC) phase is horizontal or inclined, A simple criterion involving the ranges of attractive and repulsive interactions is suggested, which can explain qualitatively the nature of the transition. Then, a thermodynamic approach for. systems that exhibit inclined transitions is presented, followed by the derivation on the basis of statistical thermodynamics of a two-dimensional equation of state for the surface pressure against the surface area per molecule. 

In Section A.l, the general laws of thermodynamics are stated. The results of statistical mechanics of ideal gases are summarized in Section A.2. Chemical equilibrium conditions for phase transitions and for reactions in gases (real and ideal) and in condensed phases (real and ideal) are derived in Section A.3, where methods for computing equilibrium compositions are indicated. In Section A.4 heats of reaction are defined, methods for obtaining heats of reaction are outlined, and adiabatic flame-temperature calculations are discussed. In the final section (Section A.5), which is concerned with condensed phases, the phase rule is derived, dependences of the vapor pressure and of the boiling point on composition in binary mixtures are analyzed, and properties related to osmotic pressure are discussed. 

Extending the current state of knowledge could involve measurements at high temperature with shock experiments to access transition pressures closer to the thermodynamic limit. Progress is currently being made to study the transition in individual nanocrystal particles, to both eliminate the ensemble statistics and allow for individual transitions to be observed on -femtosecond time scales. The study of nanocrystal solid-solid phase transitions to oxide nanosystems should also prove to be useful in understanding the microscopic process of solid-solid transitions relevant to geophysically important systems. 

David W. Oxtoby is a physical chemist who studies the statistical mechanics of liquids, including nucleation, phase transitions, and liquid-state reaction and relaxation. He received his B.A. (Chemistry and Physics) from Harvard University and his Ph.D. (Chemistry) from the University of California at Berkeley. After a postdoctoral position at the University of Paris, he joined the faculty at The University of Chicago, where he taught general chemistry, thermodynamics, and statistical mechanics and served as Dean of Physical Sciences. Since 2003 he has been President and Professor of Chemistry at Pomona College in Claremont, California. 

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