Solid-liquid-vapor system, equilibrium condition

The pressure-temperature phase diagrams also serve to highlight the fact that the polymorphic transition temperature varies with pressure, which is an important consideration in the supercritical fluid processing of materials in which crystallization occurs invariably at elevated pressures. Qualitative prediction of various phase changes (liquid/vapor, solid/vapor, solid/liquid, solid/liquid/vapor) at equilibrium under supercritical fluid conditions can be made by reference to the well-known Le Chatelier s principle. Accordingly, an increase in pressure will result in a decrease in the volume of the system. For most materials (with water being the most notable exception), the specific volume of the liquid and gas phase is less than that of the solid phase, so that... 

For ascertaining the process conditions of RESS and PGSS, it is essential to have knowledge of the equilibrium solubility of the solute in dense gas (SCF phase) and vice versa, and also the P-T trace for the solid-liquid-vapor (S-L-V) phase transition of the drug substance. If all three phases coexist, there is only a single degree of freedom for a binary system, and a P-T trace of the S-L-V equilibrium is sufficient to determine the phase equilibrium compositions. 

The equilibrium condition for a solid-liquid-vapor (SLV) system at a specified temperature and pressure may be written in terms of the fugacities of the solid-forming component as... 

The development of SCF processes involves a consideration of the phase behavior of the system under supercritical conditions. The influence of pressure and temperature on phase behavior in such systems is complex. For example, it is possible to have multiple phases, such as liquid-liquid-vapor or solid-liquid-vapor equilibria, present in the system. In many cases, the operation of an SCF process under multiphase conditions may be undesirable and so phase behavior should first be investigated. The limiting case of equilibrium between two components (binary systems) provides a convenient starting point in the understanding of multicomponent phase behavior. 

Independent of the CVD system, certain constants must be adhered to. The precursor is one of the most important components of the CVD system and is often referred to as the source. The first step in the CVD process is vaporization of the precursor, if it does not already exist as a gas at ambient conditions. The precursor should have sufficient vapor pressure, at least 100 mtorr at delivery temperature, to achieve reasonable deposition rates (16). Ambient temperature liquids are preferred to solids, since it is easier to maintain a constant flux of the precursor in the vapor phase. This is due to the fact that liquids rapidly reestablish equilibrium upon removal of vapors,... 

When we consider a one-component, two-phase system, of constant mass, we find similar relations. Such two-phase systems are those in which a solid-solid, solid-liquid, solid-vapor, or liquid-vapor equilibrium exists. These systems are all univariant. Thus, the temperature is a function of the pressure, or the pressure is a function of the temperature. As a specific example, consider a vapor-liquid equilibrium at some fixed temperature and in a state in which most of the material is in the liquid state and only an insignificant amount in the vapor state. The pressure is fixed, and thus the volume is fixed from a knowledge of an equation of state. If we now add heat to the system under the condition that the temperature (and hence the pressure) is kept constant, the liquid will evaporate but the volume must increase as the number of moles in the vapor phase increases. Similarly, if the volume is increased, heat must be added to the system in order to keep the temperature constant. The change of state that takes place is simply a transfer of matter from one phase to another under conditions of constant temperature and pressure. We also see that only one extensive variable—the entropy, the energy, or the volume—is necessary to define completely the state of the system. 

Next consider the triple point of the single-component system at which the solid, liquid, and vapor phases are at equilibrium. The description of the surfaces and tangent planes at this point are applicable to any triple point of the system. At the triple point we have three surfaces, one for each phase. For each surface there is a plane tangent to the surface at the point where the entire system exists in that phase but at the temperature and pressure of the triple point. There would thus seem to be three tangent planes. The principal slopes of these planes are identical, because the temperatures of the three phases and the pressures of the three phases must be the same at equilibrium. The three planes are then parallel. The last condition of equilibrium requires that the chemical potential of the component must be the same in all three phases. At each point of tangency all of the component must be in that phase. Consequently, the condition... 

Case I a temperature at which the vapor pressure of the solid is greater than that of the liquid. At this temperature the solid requires a higher pressure of vapor than the liquid to be in equilibrium with the vapor. Thus, as vapor is released from the solid to try to achieve equilibrium, the liquid absorbs vapor in an attempt to reduce the vapor pressure to its equilibrium value. The net effect is a conversion from solid to liquid through the vapor phase. In fact, no solid can exist under these conditions. The amount of solid steadily decreases and the volume of liquid increases. Finally, there is only liquid in the right compartment, which comes to equilibrium with the water vapor, and no further changes occur in the system. The temperature for Case 1 must be above the melting point of ice, since only the liquid state can exist. 

Figure 1 Isotopic change under open- and closed-system Rayleigh conditions for evaporation with a fractionation factor a =1.01 for an initial liquid composition of = 0. The of the remaining water (solid line A), the instantaneous vapor being removed (solid line B), and the accumulated vapor being removed (solid line C) all increase during singlephase, open-system, evaporation under equilibrium conditions. The of water (dashed line D) and...

As in liquid-liquid or vapor-liquid equilibria, when a liquid or vapor is in contact with a sorbent, equilibrium is established at the solid surface between the compositions of a solute in the two phases. This is expressed in terms of the concentration of the solute in the sorbent as a function of its concentration in the fluid phase. Whereas phase equilibrium in vapor-liquid or liquid-liquid systems can be estimated based on the thermodynamic condition of equality of component fugacities in the phases, no valid theory exists for predicting solid-fluid systems. Equilibrium concentrations for these systems must be based on experimental data. 

Alternatively, thermodynamic phase equilibrium in a model system can be evaluated by beginning the simulation with two (or more) phases in the same simulation volume, in direct physical contact (i.e., with a solid-fluid interface). This approach has succeeded [79], but its application can be problematic. Some of the issues have been reviewed by Frenkel and McTague [80]. Certainly the system must be large (recent studies [79,81,82] have employed from 1000 up to 65,000 particles) to permit the bulk nature of both phases to be represented. This is not as difficult for solid-liquid equilibrium as it is for vapor-liquid, because the solid and liquid densities are much more alike (it is a weaker first-order transition) and the interfacial free energy is smaller. However, the weakness of the transition also implies that a system out of equilibrium experiences a smaller driving force to the equilibrium condition. Consequently, equilibration of the system, particularly at the interface, may be slow. 

In a situation compatible with the lubrication approximation, perturbations due to the proximity of a solid surface are weak. In this case, the translational invariance of an unbounded two-phase system is weakly broken, and both the shift of the equilibrium chemical potential due to interactions with the solid surface and the deviation from the zero-order density profile are small. Since molecular interactions have a power decay with a nanoscopic characteristic length, this should be certainly true in layers exceeding several molecular diameters. A necessary condition for the perturbation to remain weak even as the liquid-vapor and liquid-solid interfaces are drawn together still closer, as it should happen in the vicinity of a contact line, is smallness of the dimensionless Hamaker constant % = asps/p — 1- Even under these conditions, the perturbation, however, ceases to be weak when the density in the layer adjacent to the solid deviates considerably from p+. This means that low densities near the solid surface are strongly discouraged thermodynamically, and a... 

With aqueous solutions of electrolytes we have two types of equilibrium to consider phase equilibrium and chemical or ionic reaction equilibrium. Phase equilibrium of interest are primarily vapor-liquid and liquid-solid, though vapor-llquid-solid is often of great importance as, for example, in carbonate systems. The necessary condition of phase equilibrium is that the chemical potential of any species i in phase a is equal to the chemical potential of that same species 1 in phase b or... 

Important to any measurement of citrus juice volatile flavor components is the presence of (i-limonene, since this compound is naturally present as the most concentrated component in all of the natural citrus oils. Also, the solubility of d-limonene in aqueous media must be considered, since after liquid phase saturation, the headspace concentration remains constant. It has long been established for d-limonene and similar nonpolar flavor compounds over water that meaningful headspace measurement techniques [e.g., solid-phase microextraction (SPME)] require equilibrium of the vapor and liquid phase concentrations. Equilibrium may take a number of hours for static (unstirred) experiments and less than 1 hr for stirred systems. These conditions have been discussed elsewhere, and solubility and activity coefficients of d-limonene in water and sucrose solutions have been determined [1,2]. More recently, the chemical and physical properties as well as citrus industry applications of d-limonene and other citrus essential oils have been compiled [3]. Although not specific to d-limonene, important relationships affecting behavior of flavor release and partitioning between the headspace and the liquid phase of a number of food systems have also been discussed [4]. 

The quantity ffsv represents the surface energy associated with equilibrium between the solid surface and the vapor of the liquid (wet systems). If vacuum conditions prevail (dry systems), the surface energy of the solid/vacuum interface is represented by the term infinitely dilute vapor. A solid surface in the presence of vapor differs from a solid surface in vacuum by the degree of adsorption of vapor molecules on the solid surface—a process that can occur spontaneously and thus can result in a lower surface... 

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