Definition of a Phase

Strictly speaking, a phase is defined as a region in a material in which the structure and composition are homogeneous. This assumes that the system is in chemical equilibrium, a situation which would eventually occur because of diffusion. However, for reasons that will become clear shortly, such a situation rarely occurs in a multicomponent solid, even if the system is isomorphous (aU components completely soluble) because of the slow inter-diffusion of components within a solid. Therefore, practically speaking, we generally define a phase as a region in which the crystal structure is the same and the components are the same but allow for the fact that chemical equilibrium may not yet have been reached. 

For example, if we dope a semiconductor with an impurity by diffusion, there will be a higher concentration of dopant atoms near the surface, but we do not consider this a separate phase unless we exceed the solid solubility with the concentration of dopant atoms causing them to form a precipitate. The precipitate would be considered a second phase. Similarly, if we solidify a solid solution (isomorphous) system, the resulting polycrystalline alloy is considered a single phase because all of the grains have the same crystal structure, even though the first-to-freeze grains may have a different composition than the last-to-freeze. On the other hand, a liquid or gas may have the same composition as a solid, but clearly have different structures and therefore must be considered as separate phases. 

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The strict definition of a phase is any homogeneous and physically distinct region that is separated from another such region by a distinct boundary . For example a glass of water with some ice in it contains one component (the water) exhibiting three phases liquid, solid, and gaseous (the water vapour). The most relevant phases in the oil industry are liquids (water and oil), gases (or vapours), and to a lesser extent, solids. 

It must however be pointed out that a sound experimental definition of a phase diagram can be obtained from the results of a number of concerted investigations such as thermal analysis, thermodynamic analysis, micrographic examination and phase analysis and identification by means of techniques such as X-ray diffraction measurements, microprobe analysis, etc. 

Considering now a few common methods and techniques for the experimental definition of a phase diagram, it may be useful to refer to their usual classification into two groups polythermal and isothermal methods. 

We do have to be careful in the way we apply the definition of a phase to the n-butylammonium vermiculite system. According to Gibbs [13], a phase is any homogeneous and physically distinct part of a system that is separated from other parts of the system by definite boundary surfaces. Because the gel can be lifted out of the supernatant fluid on a spatula, it clearly justifies description as a phase in the latter sense, but it is inhomogeneous on the nanometer-to-micron (colloidal) length scale. It can only be defined as homogeneous on the macroscopic length scale. The same considerations apply to the tactoid phase. 

A strict definition of a phase is a domain with a homogenous chemical composition and physical state (Atkins 1998). Typically, phase behavior is associated with equilibrium mixtures of hquid, vapor and solid phases in a mixture of two or more substances as a fimctiorr of temperature, pressure and compositiorr. hr fats, when discussing phases, it usually refers to their polymorphic form, sohd/hquid states, or compositionally differerrt ffactiorrs within a complex mixtrrre. Therefore, when phases of fats are discussed it is important to carefully define what is being used as the definition for... 

Fig. 12. Definition of a phase angle with respect to an imaginary wave scattered at the unit-cell origin.

Discussion of phases and miscibility requires a proper definition. The general thermodynamic definition of a phase states that a phase is a region of constant or continuously changing physical and chemical properties, the state of which can be described by thermodynamic relations. 

Being aware of this, an appropriate definition of a phase could be ... 

The term interface implies a two-dimensional structure. It is clear in nearly all practical cases in polymer science that the regions between phases are three-dimensional in nature. These regions are also often likely not to be isotropic, but of a compositionally graded nature which means they do not meet the strict definition of a phase. In this chapter, the terms interface and interphase will be used essentially interchangeably. 

Phase equilibrium deals with phases, so we need a working definition of a phase. A phase is a mass of matter, not necessarily continuous, in which there are no sharp discontinuities of any physical properties over short distances. An equilibrium phase is one that (in the absence of significant gravitational, electrostatic, or magnetic effects) has a completely uniform composition throughout. In this book we will deal almost exclusively with equilibrium phases. 

With foams, one is dealing with a gaseous state or phase of matter in a highly dispersed condition. There is a definite relationship between the practical application of foams and colloidal chemistry. Bancroft (4) states that adopting the very flexible definition that a phase is colloidal when it is sufficiently finely divided, colloid chemistry is the chemistry of bubbles, drops, grains, filaments, and films, because in each of these cases at least one dimension of the phase is very small. This is not a truly scientific classification because a bubble has a film round it, and a film may be considered as made up of coalescing drops or grains. ... 

The outer electrical potential of a phase is the electrostatic potential given by the excess charge of the phase. Thus, if a unit electric charge is brought infinitely slowly from infinity to the surface of the conductor to a distance that is negligible compared with the dimensions of the conductor considered (for a conductor with dimensions of the order of centimetres, this distance equals about 10 4cm), work is done that, by definition, equals the outer electric potential ip. 

FIGURE 20. Top Definition of the phase relationship of the three localized basis 7r-orbitals of a norbornadiene molecule with an exocyclic double bond in position 7. Bottom Correlation diagram of the experimental orbital energies j = —/ of norbornadiene 75, 7-methylidenenorbornadiene 196 and 7-isopropylidenenorbomadiene 206, with those of the corresponding monoenes... 

Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ij)a of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By just outside we mean a position very close to the surface, but so fax away that the image interaction with the phase can be ignored in practice, that means a distance of about 10 5 — 10 3 cm from the surface. Obviously, the outer potential i/ a U a measurable quantity. 

The characterization, the full identification, of a phase requires a complete and detailed description of its structure. As examples, consider the data, obtained from X-ray diffraction experiments, reported in Table 3.2 for stoichiometric and variable composition phases. An explanation of the various symbols used will be given in the following paragraphs. For a general reference to symbols, definitions and... 

A review about the Zintl phases has been published by Sevov (2002) from the introduction of this publication we quote a few remarks. It was preliminary observed that the number of Zintl phases has increased many-fold since Zintl s time and that the definition of a Zintl phase has never been very exact often compounds that include non-metals have been considered in this family. The paper by Sevov is mainly dedicated to clearly intermetallic Zintl phases (that is phases containing main group metals, semi-metals or semiconductors only). Attention has therefore been dedicated to compounds of alkali metals with the elements of the 13th, 14th and 15th groups (without B, Al, C, N and P). To this end the following definitions and statements have been considered. 

The QET is not the only theory in the field indeed, several apparently competitive statistical theories to describe the rate constant of a unimolecular reaction have been formulated. [10,14] Unfortunately, none of these theories has been able to quantitatively describe all reactions of a given ion. Nonetheless, QET is well established and even the simplified form allows sufficient insight into the behavior of isolated ions. Thus, we start out the chapter from the basic assumptions of QET. Following this trail will lead us from the neutral molecule to ions, and over transition states and reaction rates to fragmentation products and thus, through the basic concepts and definitions of gas phase ion chemistry. 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

The first satisfactory definition of entropy, which is quite recent, is that of Kittel (1989) entropy is the natural logarithm of the quantum states accessible to a system. As we will see, this definition is easily understood in light of Boltzmann s relation between configurational entropy and permutability. The definition is clearly nonoperative (because the number of quantum states accessible to a system cannot be calculated). Nevertheless, the entropy of a phase may be experimentally measured with good precision (with a calorimeter, for instance), and we do not need any operative definition. Kittel s definition has the merit to having put an end to all sorts of nebulous definitions that confused causes with effects. The fundamental P-V-T relation between state functions in a closed system is represented by the exact differential (cf appendix 2)... 

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