Statistical thermodynamics crystals

K (66.46 e.u.) with the spectroscopic value calculated from experimental data (66.41 0.009 e.u.) (295, 289) indicates that the crystal is an ordered form at 0°K. Thermodynamic functions of thiazole were also determined by statistical thermodynamics from vibrational spectra (297, 298). 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

The distinct properties of liquid-crystalline polymer solutions arise mainly from extended conformations of the polymers. Thus it is reasonable to start theoretical considerations of liquid-crystalline polymers from those of straight rods. Long ago, Onsager [2] and Flory [3] worked out statistical thermodynamic theories for rodlike polymer solutions, which aimed at explaining the isotropic-liquid crystal phase behavior of liquid-crystalline polymer solutions. Dynamical properties of these systems have often been discussed by using the tube model theory for rodlike polymer solutions due originally to Doi and Edwards [4], This theory, the counterpart of Doi and Edward s tube model theory for flexible polymers, can intuitively explain the dynamic difference between rodlike and flexible polymers in concentrated systems [4]. 

Figure 1 also shows that plasticized polyvinyl chloride begins to flow at a lower temperature. This is to be expected in view of the fact that equilibrium melting temperature of polymer crystals is depressed by monomeric diluents. A statistical thermodynamic treatment by Flory (13), showed that this effect depends on the nature of the polymer, concentration of the diluent, and the degree of polymer-diluent interaction in the following manner ... 

Monovacancy. Statistical thermodynamics requires that if a vacancy is formed by removing an atom from the crystal and depositing it on the surface, then the free energy of the crystal must decrease as the number of created vacancies increases until a minimum in this free energy is reached. Because a minimum in the free energy exists for a certain vacancy concentration in the crystal, the vacancy is a stable point defect. The following facts about vacancies have been obtained experimentally (12). 

The statistical thermodynamic method discussed here provides a bridge between the molecular crystal structures of Chapter 2 and the macroscopic thermodynamic properties of Chapter 4. It also affords a comprehensive means of correlation and prediction of all of the hydrate equilibrium regions of the phase diagram, without separate prediction schemes for two-, three-, and four-phase regions, inhibition, and so forth as in Chapter 4. However, for a qualitative understanding of trends and an approximation (or a check) of prediction schemes in this chapter, the previous chapter is a valuable tool. 

The structural interpretation of dielectric relaxation is a difficult problem in statistical thermodynamics. It can for many materials be approached by considering dipoles of molecular size whose orientation or magnitude fluctuates spontaneously, in thermal motion. The dielectric constant of the material as a whole is arrived at by way of these fluctuations but the theory is very difficult because of the electrostatic interaction between dipoles. In some ionic crystals the analysis in terms of dipoles is less fruitful than an analysis in terms of thermal vibrations. This also is a theoretically difficult task forming part of lattice dynamics. In still other materials relaxation is due to electrical conduction over paths of limited length. Here dielectric relaxation borders on semiconductor physics. 

Yet more important was the publication by Schottky and Wagner (1930) of their classical paper on the statistical thermodynamics of real crystals (41). This clarified the role of intrinsic lattice disorder as the equilibrium state of the stoichiometric crystal above 0° K. and led logically to the deduction that equilibrium between the crystal of an ordered mixed phase—i.e., a binary compound of ionic, covalent, or metallic type—and its components was statistical, not unique and determinate as is that of a molecular compound. As the consequence of a statistical thermodynamic theorem this proposition should be generally valid. The stoichiometrically ideal crystal has no special status, but the extent to which different substances may display a detectable variability of composition must depend on the energetics of each case—in particular, on the energetics of lattice disorder and of valence change. This point is taken up below, for it is fundamental to the problems that have to be considered. 

First approaches to the quantitative understanding of defects in stoichiometric crystals were published in the early years of the last century by Frenkel in Russia and Schottky in Germany. These workers described the statistical thermodynamics of solids in terms of the atomic occupancies of the various crystallographic sites available in the stmcture. Two noninteracting defect types were envisaged. Interstitial defects consisted of atoms that had been displaced from their correct positions into normally unoccupied positions, namely, interstitial sites. Vacancies were positions that should have been occupied but were not. 

If macroscopic thermodynamics are applied to materials containing a popnlation of defects, particnlarly nonstoichio-metric compounds, the defects themselves do not enter into the thermodynamic expressions in an exphcit way. However, it is possible to construct a statistical thermodynamic formahsm that will predict the shape of the free energy-temperatnre-composition curve for any phase containing defects. The simplest approach is to assnme that the point defects are noninteracting species, distributed at random in the crystal, and that the defect energies are constant and not a ftmction either of concentration or of temperatnre. In this case, reaction eqnations similar to those described above, eqnations (6) and (7), can be used within a normal thermodynamic framework to deduce the way in which defect populations respond to changes in external variables. 

Statistical thermodynamic treatments of defect populations have lead to an explanation of existence of grossly nonstoichio-metric crystals in terms of microdomains of ordered structure. The model considers that the nonstoichiometric matrix is made up of a mosaic of small regions of ordered defect-free structures, the microdomains. To account for stoichiometric variation, one can postulate that at least two microdomains with different compositions occur. However, compositional change might simply arise at the surface of the domain. For example, if there are compositionally identical microdomains, one of which is bounded by an anion surface and one by a cation surface, variation in the two populations can give rise to compositional variation. In a strict sense, as each microdomain is ordered, the concept of a defect is redundant, except for... 

The idea of a fixed crystal structure in which single cages contained at most one guest proved irresistible to statistical thermodynamicists. After an initial effort by Powell, Royal Dutch Shell workers van der Waals and Platteeuw generated a method that still stands today as a principal, regular industrial use of statistical thermodynamics. However, the model was not suitable for manual calculations (as were the methods of Katz in item 3 above), but required access to then-scarce computers, which limited its application to large companies or major universities. Widespread adoption of the model awaited the proliferation of personal computers. 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

Boehm, R. E., Martire, D. E., and Madhusudana, N. V., A statistical thermodynamic theory of thermotropic linear main chain polymeric liquid crystals, Macwmolecules, 19, 2329-2341 (1986). 

Simha, R., Jain, R. K., Statistical thermodynamics of polymer crystal and melt. Journal of Polymer Science, Polymer Physics Edition, 16(8), pp. 1471-1489 (1978). 

The principles of polymer fractionation by solubility or crystallization in solution have been extensively reviewed on the basis of Hory-Huggins statistical thermodynamic treatment [58,59], which accounts for melting point depression by the presence of solvents. For random copolymers the classical Flory equation [60] applies ... 

The basic, macroscopic theories of matter are equilibrium thermodynamics, irreversible thermodynamics, and kinetics. Of these, kinetics provides an easy link to the microscopic description via its molecular models. The thermodynamic theories are also connected to a microscopic interpretation through statistical thermodynamics or direct molecular dynamics simulation. Statistical thermodynamics is also outlined in this section when discussing heat capacities, and molecular dynamics simulations are introduced in Sect 1.3.8 and applied to thermal analysis in Sect. 2.1.6. The basics, discussed in this chapter are designed to form the foundation for the later chapters. After the introductory Sect. 2.1, equilibrium thermodynamics is discussed in Sect. 2.2, followed in Sect. 2.3 by a detailed treatment of the most fundamental thermodynamic function, the heat capacity. Section 2.4 contains an introduction into irreversible thermodynamics, and Sect. 2.5 closes this chapter with an initial description of the different phases. The kinetics is closely link to the synthesis of macromolecules, crystal nucleation and growth, as well as melting. These topics are described in the separate Chap. 3. 

Flory PJ (1954) Theory of crystallization in copolymers. Trans Faraday Soc 51 848-857 Flory PJ (1956) Statistical thermodynamics of semi-fiexible chain molecules. Proc R Soc London A234 60-73... 

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