Three phase solid-liquid-vapor equilibrium

Calculation of Three-Phase Solid-Liquid-Vapor Equilibrium Using an Equation of State... 

If a mixture of ice and water at 1 atm pressure and 0°C is placed in an insulated container and all of the air is pumped away and the container sealed, what will happen As was shown in the previous section, at pressures lower than 1 atm, the melting point of ice is above 0°C. Water will, therefore, be solid at 0°C and reduced pressure. However, when some liquid water freezes, its latent heat is released and the temperature of the system is slightly increased. Equilibrium is reestablished at the higher temperature and reduced pressure. The pressure in the system is the vapor pressure of both liquid and solid water slightly above 0 K. The new equilibrium point of the system is called the triple point of water and is at 0.0098°C and 611 Pa. Three phases—solid, liquid, and gas—coexist at the triple point, and the chemical potential of water in each of the phases must be equal ... 

The phase diagram (Figure 1-18) indicates the existence of three phases solid, liquid, and gas. The conditions under which they exist are separated by three equilibrium lines the vapor pressure line TA, the melting pressure line TC, and the sublimation pressure line BT. The three lines meet at point T,... 

Lyophilization (Freeze Drying) Lyophilization is most frequently used for heat-labile dosage forms that are unstable in aqueous formulation. The principle of lyophilization can be seen by reference to the phase equilibrium diagram for water (Fig. 15). Water at atmospheric pressure and ambient temperatures is stable in its liquid phase at lOO C the liquid phase attains an equilibrium with its vapor phase. Above 100 C water is stable in its vapor phase. At atmospheric pressures and 0 C the solid (ice) and liquid phases of water are in equilibrium with each other. At vacuum pressures a temperature (the eutectic point) can be reached where the three phases, solid, liquid, and vapor are all in equilibrium with each other. At even lower temperatures and pressures the solid phase comes into equilibrium with the liquid phase. The significance of this is that an aqueous solution can be concentrated by evaporation (sublimation) at low pressures without any necessity for significant heat input. 

Let us consider a somewhat unusual application of this type of instrument. In the manufacture of titanium it is important to keep liquid TiC at a particular high temperature and pressure in a vessel. TiC has a triple point (i.e. the pressure and temperature conditions at which all three phases - solid, liquid, and vapor - of a substance are at equilibrium) in the neighborhood of the particular technical conditions. By using a y-density gauge it is possible to detect when the triple point is exceeded because of the disappearance of the vapor-liquid interface. This allows a simple method of control of the process conditions in the vessel. 

The triple point (where all three phases, solid, liquid, and gas, are in equilibrium) is achieved, by recent estimates, at a temperature of4200 K and a pressure of 100 atm, as shown in the vapor-pressure curve of Fig. 3.7.I M A great deal of uncertainty still remains regarding these values of pressure and temperature, reflecting in part the difficulty of conducting experiments under such extreme conditions. 

Systems at the triple point contain three phases, solid, liquid, and vapor, all in equilibrium with one another. Consequently,... 

A triple point is a point where three phase boundaries meet on a phase diagram. For water, the triple point for the solid, liquid, and vapor phases lies at 4.6 Torr and 0.01°C (see Fig. 8.6). At this triple point, all three phases (ice, liquid, and vapor) coexist in mutual dynamic equilibrium solid is in equilibrium with liquid, liquid with vapor, and vapor with solid. The location of a triple point of a substance is a fixed property of that substance and cannot be changed by changing the conditions. The triple point of water is used to define the size of the kelvin by definition, there are exactly 273.16 kelvins between absolute zero and the triple point of water. Because the normal freezing point of water is found to lie 0.01 K below the triple point, 0°C corresponds to 273.15 K. 

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. 

When one component is present in three phases at equilibrium, the phase rule states that the system is invariant and possesses no degrees of freedom. This implies that such a system at equilibrium can only exist at one definite temperature and one definite pressure, which is termed the triple point. For instance, the solid/liquid/vapor triple point of water is found at a temperature of 273.16K and a pressure of 4.58 torr. 

We start by extending the Gibbs phase rule to multiple-component systems, in its most general form. We will confine our development of multiple-component systems to relatively simple ones, having two or three components at most. However, the ideas we will develop are generally applicable, so there will be little need to consider more complicated systems here. One example of a simple two-component system is a mixture of two liquids. We will consider that, as well as the characteristics of the vapor phase in equilibrium with the liquid. This will lead into a more detailed study of solutions, where different phases (solid, liquid, and gas) will act as either the solute or solvent. 

Point A on a phase diagram is the only one at which all three phases, liquid, solid, and vapor, are in equilibrium with each other. It is called the triple point. For water, the triplepoint temperature is 0.01°C. At this temperature, liquid water and ice have the same vapor pressure, 4.56 mm Hg. 

In the three areas of the phase diagram labeled solid, liquid, and vapor, only one phase is present. To understand this, consider what happens to an equilibrium mixture of two phases when the pressure or temperature is changed. Suppose we start at the point on AB... 

For a pure substance, having three phases in equilibrium results in a triple point that is invariant. When pure solid, liquid, and gaseous water are in equilibrium, the temperature is fixed at a value of 273.16 K, and the pressure of the gas is fixed at the vapor pressure value (0.6105 kPa). 

In Fig. 8.8, we see that sulfur can exist in any of four phases two solid phases (rhombic and monoclinic sulfur), one liquid phase, and one vapor phase. There are three triple points in the diagram, where various combinations of these phases, such as monoclinic solid, liquid, and vapor or monoclinic solid, rhombic solid, and liquid, coexist. However, four phases in mutual equilibrium (such as the vapor, liquid, and rhombic and monoclinic solid forms of sulfur, all in mutual equilibrium) in a one-component system has never been observed, and thermodynamics can be used to prove that such a quadruple point cannot exist. 

The phase equilibrium for pure components is illustrated in Figure 4.1. At low temperatures, the component forms a solid phase. At high temperatures and low pressures, the component forms a vapor phase. At high pressures and high temperatures, the component forms a liquid phase. The phase equilibrium boundaries between each of the phases are illustrated in Figure 4.1. The point where the three phase equilibrium boundaries meet is the triple point, where solid, liquid and vapor coexist. The phase equilibrium boundary between liquid and vapor terminates at the critical point. Above the critical temperature, no liquid forms, no matter how high the pressure. The phase equilibrium boundary between liquid and vapor connects the triple point and the... 

The three interfacial surface energies, as shown at the three-phase junction in Figure 2.29, can be used to perform a simple force balance. The liquid-solid interfacial energy plus the component of the liquid-vapor interfacial energy that lies in the same direction must exactly balance the solid-vapor interfacial energy at equilibrium ... 

To explore Young s equation still further, suppose we distinguish between ysv and ySo, where the former describes the surface of a solid in equilibrium with the vapor of a liquid and the latter a solid in equilibrium with its own vapor. Since Young s equation describes the three-phase equilibrium, it is proper to use ysv in Equation (44). The question arises, however, what difference, if any, exists between these two y s. In order to account for the difference between the two, we must introduce the notion of adsorption. In the present context adsorption describes the attachment of molecules from the vapor phase onto the solid surface. All of Chapter 9 is devoted to this topic, so it is unnecessary to go into much detail at this point. The extent of this attachment depends on the nature of the molecules in the vapor phase, the nature of the solid, and the temperature and the pressure. 

TRIPLE POINT. The temperature and pressure at which the solid, liquid, and vapor of a substance are in equilibrium with one another. Also applied to similar equilibrium between any three phases, Le., two solids and a liquid, etc. The triple point of water is +0.072 C at 4.6 mmHg it is of special importance because it is the fixed point for the absolute scale of temperature. 

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... 

The phase rule(s) can be used to distinguish different types of equilibria based on the number of degrees of freedom. For example, in a unary system, an invariant equilibrium (/ = 0) exists between the liquid, solid, and vapor phases at the triple point, where there can be no changes to temperature or pressure without reducing the number of phases in equilibrium. Because / must equal zero or a positive integer, the condensed phase rule (/ = c — p + 1) limits the possible number of phases that can coexist in equilibrium within one-component condensed systems to one or two, which means that other than melting, only allotropic phase transformations are possible. Similarly, in two-component condensed systems, the condensed phase rule restricts the maximum number of phases that can coexist to three, which also corresponds to an invariant equilibrium. However, several invariant reactions are possible, each of which maintains the number of equilibrium phases at three and keeps / equal to zero (L represents a liquid and S, a solid) ... 

Contact angle — The contact angle is the angle of contact between a droplet of liquid and a flat rigid solid, measured within the liquid and perpendicular to the contact line where three phases (liquid, solid, vapor) meet. The simplest theoretical model of contact angle assumes thermodynamic equilibrium between three pure phases at constant temperature and pressure [i, ii]. Also, the droplet is assumed to be so small that the force of gravity does not distort its shape. If we denote the - interfacial tension of the solid-vapor interface as ysv. the interfacial tension of the solid-liquid interface as ySL and the interfacial tension of the liquid-vapor interface as yLV, then by a horizontal balance of mechanical forces (9 < 90°)... 

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