Quantitative Aspects of Phase Changes

In this section, we examine the heat absorbed or released in a phase change and the equilibrium nature of the process. 

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Quantitative Aspects of Phase Changes Heat Involved in Phase Changes Equilibrium Nature of Phase Changes Phase Diagrams... 

An Overview of Physical States and Phase Changes Quantitative Aspects of Phase Changes... 

The sixth column of Table 9-1 lists concentration ratios of NH3 to particulate NH4. Most of these are smaller than unity indicating a preference of atmospheric ammonia for the particulate phase. Junge (1963) had previously concluded that the gas phase dominates. The change in outlook can be traced to improvements in sampling techniques and a better discrimination between gaseous and particulate ammonia. The NH3/NH4 ratio should be determined by the rate at which ammonia is tied to aerosol particles following the production of sulfuric and nitric acid, relative to the rates of ammonia supply and its removal from the atmosphere by precipitation. The quantitative aspects of this relation remain to be investigated. 

If for certain values of a parameter A in the differential equation, the qualitative aspect of the solution (i.e., the phase portrait ) of the differential equation remains the same (in other words the changes are only quantitative) such values of A are called ordinary values. If however, for a certain value A = A0 this qualitative aspect changes, such a special value is called a critical or bifurcation value. 

The number of degrees of freedom is represented by/. These are chosen from the list of all quantitatively related aspects of a system that can change. This includes T, P, and the concentrations of c components in each phase, c is the minimum number of components necessary to reproduce the system (ingredients), and p is the number of phases present at equilibrium. A phase is a domain with uniform composition and properties. Examples are a gas, a liquid solution, a solid solution, and solid phases. 

Immunoassays based on phase-modulation spectroscopy have been implemented by two distinctly different approaches. Phase-resolved immunoassays rely on fluorescence intensity measurements, in which the emission of one fluorescent species in a mixture is suppressed, and the remainder is quantitated. Phase fluorescence immunoassays utilize measurements of the phase angle and modulation, which change in response to fluorescence lifetime changes. Common aspects of the theory and instrumentation are discussed in this section, followed by individual discussions of the different approaches. 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

Using empirical potentials, solute-solvent and solvent-solvent interactions can be modeled, allowing many aspects of solvent clusters to be studied semi-quantitatively by classical simulation techniques (Monte Carlo, molecular dynamics). As a specific example we discuss two cluster/sub-strate-type "phase" transitions of the carbazole Ar cluster. This is the smallest system that exhibits all types of two-dimensional transitions observed so far in MC simulations, and, at the same time, allows a direct and intuitive interpretation of the associated structural changes [9-14]. 

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