Phase one-component

Moreover, using the generalized Gibbs-Duhem equations (A2.1.27) for each of the two one-component phases,... 

While the Gibbs phase rule provides for a qualitative explanation, we can apply the Clapeyron equation, derived earlier [equation (5.71)], in conjunction with studying the temperature and pressure dependences of the chemical potential, to explain quantitatively some of the features of the one-component phase diagram. 

Stoichiometric saturation defines equilibrium between an aqueous solution and homogeneous multi-component solid of fixed composition (10). At stoichiometric saturation the composition of the solid remains fixed even though the mineral is part of a continuous compositional series. Since, in this case, the composition of the solid is invariant, the solid may be treated as a one-component phase and Equation 6 is the only equilibrium criteria applicable. Equations 1 and 2 no longer apply at stoichiometric saturation because, owing to kinetic restrictions, the solid and saturated solution compositions are not free to change in establishing an equivalence of individual component chemical potentials between solid and aqueous solution. The equilibrium constant, K(x), is defined identically for both equilibrium and stoichiometric saturation. 

For all one-component phase diagrams, choose the correct statement from the following list. 

Shown below is a one-component phase diagram. There exist 4 different phases, namely, a, P, y and 5. 

Thorstenson and Plummer (1977), in an elegant theoretical discussion (see section on The Fundamental Problems), discussed the equilibrium criteria applicable to a system composed of a two-component solid that is a member of a binary solid solution and an aqueous phase, depending on whether the solid reacts with fixed or variable composition. Because of kinetic restrictions, a solid may react with a fixed composition, even though it is a member of a continuous solid solution. Thorstenson and Plummer refer to equilibrium between such a solid and an aqueous phase as stoichiometric saturation. Because the solid reacts with fixed composition (reacts congruently), the chemical potentials of individual components cannot be equated between phases the solid reacts thermodynamically as a one-component phase. The variance of the system is reduced from two to one and, according to Thorstenson and Plummer, the only equilibrium constraint is IAP g. calcite = Keq(x>- where Keq(x) is the equilibrium constant for the solid, a function of... 

Thus, for each equilibrium state, the chemical potential of the adsorbate is equal to that of the adsorptive in the gas phase. In the case of a single adsorptive, the adsorbed state may be regarded as a one-component phase, which has lost one degree of freedom. Equation (2.18) indicates that it is sufficient to specify two of the variables... 

Again, the primary phase particles of the required substance modifica tion (material precursors) are usually very small. When seeds of the synthe sized phase are used, these primary particles are identical in size to the seeds. In the homogeneous liquid solutions or gas mixtures, the size ofpri mary particles is determined by the nucleation processes. The small size of the primary phase particles can influence considerably the chemical poten tial of the phase to be formed. For example, in the case of spherical parti cles, the chemical potential is determined by equation (1.5). Hence, the equilibrium partial pressure, p, of the saturated vapor or concentration, c of the saturated solution of the substance—for example, of the synthe sized one component phase—is determined by the Kelvin Thomson equation... 

When solids or gases are present in a reaction either as products or reactants, mole fractions and partial pressures (respectively) are normally used as a measure of their concentration. If the solid is a pure, one component phase, its mole fraction is used and its value is unity. Likewise, the solvent mole fraction (which is practically unity in very dilute solutions) is used as a measure of its concentration. 

The stoichiometric saturation concept assumes that a solid-solution can under certain circumstances behave as if it were a pure one-component phase. In such a situation, the dissolution of a solid-solution Bi xCxA can be expressed as ... 

In order to demonstrate the performance of the Doppler-burst envelope integral value method for the estimation of the instantaneous particle velocity vector and the particle mass flux or concentration, measurements were performed in a liquid spray issuing from a hollow cone pressure atomizer (cone angle 60°) and a swirling flow which exhibits complex particle trajectories (Sommerfeld and Qiu 1993). All the measurements were conducted using the one-component phase-Doppler anemometer. The integration of the mass flux profiles provided the dispersed phase mass flow rate which agreed to 10 % with independent measurements of the mass flow rate (Sommerfeld and Qiu 1995). 

The previous section showed how the van der Waals equation was extended to binary mixtures. However, much of the early theoretical treatment of binary mixtures ignored equation-of-state effects (i.e. the contributions of the expansion beyond the volume of a close-packed liquid) and implicitly avoided the distinction between constant pressure and constant volume by putting the molecules, assumed to be equal in size, into a kind of pseudo-lattice. Figure A2.514 shows schematically an equimolar mixture of A and B, at a high temperature where the distribution is essentially random, and at a low temperature where the mixture has separated into two virtually one-component phases. 

The description of phase equilibria makes use of the partial molar free enthalpies, i, called also chemical potentials. For one-component phase equilibria the same formalism is used, just that the enthalpies, G, can be used directly. The first case treated is the freezing point lowering of component 1 (solvent) due to the presence of a component 2 (solute). It is assumed that there is complete solubility in the liquid phase (solution, s) and no solubility in the crystalline phase (c). The chemical potentials of the solvent in solution, crystals, and in the pure liquid (o) are shown in Fig. 2.26. At equilibrium, ft of component 1 must be equal in both phases as shown by Eq. (1). A similar set of equations can be written for component 2. By subtracting j,i° from both sides of Eq. (1), the more easily discussed mixing (left-hand side, LHS) and crystallization (right-hand side, RHS) are equated as Eq. (2). 

A and denoting the free energy densities of the one component phases. Being in close contact, neighboring blocks, interact with each other across the interfaces and we have to inquire about the related interfacial energy. We know that it must vanish for equal concentrations and increase with the concentration difference, independent of the direction of change. The simplest expression with such properties is the quadratic term... 

There are a number of applications of one-component phase diagrams in ceramics. One such application is the development of commercial production of synthefic diamonds from graphite. For this, high temperatures and high pressures are necessary. This is evident from the phase diagram of carbon shown in Figure 4.1 [1]. 

Perhaps the simplest and easiest type of phase diagram to understand is that for a one-component system, in which composition is held constant (i.e., the phase diagram is for a pure substance) this means that pressure and temperature are the variables. This one-component phase diagram (or unary phase diagram, sometimes also called a pressure-temperature [or P-T diagram) is represented as a two-dimensional plot of... 

For one-component phase diagrams, the logarithm of the pressure is plotted versus the temperature solid-, liquid-, and vapor-phase regions are found on this type of diagram. 

Prototypical pure substances occur in at least one solid phase, a liquid phase, and a vapour (gas) phase and can be defined/identified in terms of their one-component-phase diagram, triple point(s) connecting the phases. Intermediate substances can be identified by their location in phase diagrams. Substances that seem to occur in only one phase or which cannot be put into bottles for one reason or another can be situated relative to protot3q>ical substances. Different research contexts may lead to different classifications of non-standard cases, but often a distinction of component, substance, and species will resolve some ambiguities. 

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