Three-phase equilibria

Consider a closed system containing Ny...N/ moles of k nonreacting 

In order, now, that Eq. 12.6.4 be satisfied for arbitrary changes in all independent variables  

We conclude that when a three-phase closed system reaches equilibrium at constant T and P, the chemical potential and fugacity of component i must be the same in all three phases. 

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There is one other three-phase equilibrium involving clathrates which is of considerable practical importance, namely that between a solution of Q, the clathrate, and gaseous A. For this equilibrium the previous formulas and many of the following conclusions also hold when replacing fiQa by fiQL, the chemical potential of Q in the liquid phase. But a complication then arises since yqL and the difference are not only... 

Let us first consider the three-phase equilibrium ( -clathrate-gas, for which the values of P and x = 3/( +3) were determined at 25°C. When the temperature is raised the argon content in the clathrate diminishes according to Eq. 27, while the pressure can be calculated from Eq. 38 by taking yA values following from Eq. 27 and the same force constants as used in the calculation of Table III. It is seen that the experimental results at 60°C and 120°C fall on the line so calculated. At a certain temperature and pressure, solid Qa will also be able to coexist with a solution of argon in liquid hydroquinone at this point (R) the three-phase line -clathrate-gas is intersected by the three-phase line -liquid-gas. At the quadruple point R solid a-hydroquinone (Qa), a hydroquinone-rich liquid (L), the clathrate (C), and a gas phase are in equilibrium the composition of the latter lies outside the part of the F-x projection drawn in Fig. 3. The slope of the three-phase line AR must be very steep, because of the low solubility of argon in liquid hydroquinone. 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

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... 

Two- and Three-Phase Equilibrium Calculations for Coal Gasification and Related Processes... 

A constant interaction parameter was capable of representing the mole fraction of water in the vapor phase within experimental uncertainty over the temperature range from 100°F to 460°F. As with the methane - water system, the temperature - dependent interaction parameter is also a monotonically increasing function of temperature. However, at each specified temperature, the interaction parameter for this system is numerically greater than that for the methane - water system. Although it is possible for this binary to form a three-phase equilibrium locus, no experimental data on this effect have been reported. 

Nevertheless, a calculated locus is included for completeness and to indicate the possible region of three-phase equilibrium. 

The pressure-temperature-composition diagram presented by Morey is shown in Fig. 8. The vapor pressure of pure water (on the P-T projection) terminates at the critical point (647 K, 220 bar). The continuous curve represents saturated solutions of NaCl in water, i.e., there is a three-phase equilibrium of gas-solution-solid NaCl. The gas-phase pressure maximizes over 400 bar at around 950 K. Olander and Liander s data for a 25 wt. % NaCl solution are shown, and T-X and P X projections given. At the pressure maximum, the solution phase contains almost 80% NaCl. 

Therefore, the physical meaning of the solubility curve of a surfactant is different from that of ordinary substances. Above the critical micelle concentration the thermodynamic functions, for example, the partial molar free energy, the activity, the enthalpy, remain more or less constant. For that reason, micelle formation can be considered as the formation of a new phase. Therefore, the Krafft Point depends on a complicated three phase equilibrium. 

The KTTS depends upon an absolute zero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equilibrium, together with specification of an interpolation instrument and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

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. 

D. Two-box sediment/water model combined with three-phase equilibrium (Table 23.7) ... 

According to this equation the maximum number of phases that can be in equilibrium in a binary system is = 4 (F= 0) and maximum number of degrees of freedom needed to describe the system = 3 (n=l). This means that all phase equilibria can be represented in a three-dimensional P,T,x-space. At equilibrium every phase participating in a phase equilibrium has the same P and T, but in principle a different composition x. This means that a four-phase-equilibrium (F=0) is given by four points in P, 7, x-space, a three-phase equilibrium (P=l) by three curves, a two-phase equilibrium (F=2) by two planes and a one phase state (F= 3) by a region. The critical state and the azeotropic state are represented by one curve. 

In a binary system more than two fluid phases are possible. For instance a mixture of pentanol and water can split into two liquid phases with a different composition a water-rich liquid phase and a pentanol-rich liquid-phase. If these two liquid phases are in equilibrium with a vapour phase we have a three-phase equilibrium. The existence of two pure solid phases is an often occuring case, but it is also possible that solid solutions or mixed crystals are formed and that solids exists in more than one crystal structure. 

Often the essentials of phase diagrams in P,7,x-space are represented in a P,7-projection. In this type of diagrams only non-variant (F=0) and monovariant (F=l) equilibria can be represented. Since pressure and temperature of phases in equilibrium are equal, a four-phase equilibrium is now represented by one point and a three-phase equilibrium with one curve. Also the critical curve and the azeotropic curve are projected as a curve on the P, 7-plane. A four-phase point is the point of intersection of four three-phase curves. The point of intersection of a three-phase curve and a critical curve is a so-called critical endpoint. In this intersection point both the three-phase curve and the critical curve terminate. 

Figure 2.2-2 Possible locations of one- and two-phase equilibria around a three-phase equilibrium in a Pjc- section.

In a PyX- or 7 -section a three-phase equilibrium is represented by three points, which give the compositions of the three coexisting phases. All mixtures with composition on the line through these points will split into three phases. Figure 2.2-2 gives a schematic example of a three-phase equilibrium aPy in a P c-section. Around the three-phase equilibrium three two-... 

In a thorough review of calorimetric studies of clathrates and inclusion compounds, Parsonage and Staveley (1984) presented no direct calorimetric methods used for natural gas hydrate measurements. Instead, the heat of dissociation has been indirectly determined via the Clapeyron equation by differentiation of three-phase equilibrium pressure-temperature data. This technique is presented in detail in Section 4.6.1. 

Treybal, in his book Liquid Extraction [1], works equilibrium material balances with triangular coordinates. The most unique and simple way to show three-phase equilibrium is a triangular diagram (Fig. 7.1), which is used for extraction unit operation in cumene synthesis plants [2], In this process benzene liquid is used as the solvent to extract acetic acid (the solute) from the liquid water phase (the feed-raffinate). The curve D,S,P,F,M is the equilibrium curve. Note that every point inside the triangle has some amount of each of the three components. Points A,... 

Experimental results are presented for high pressure phase equilibria in the binary systems carbon dioxide - acetone and carbon dioxide - ethanol and the ternary system carbon dioxide - acetone - water at 313 and 333 K and pressures between 20 and 150 bar. A high pressure optical cell with external recirculation and sampling of all phases was used for the experimental measurements. The ternary system exhibits an extensive three-phase equilibrium region with an upper and lower critical solution pressure at both temperatures. A modified cubic equation of a state with a non-quadratic mixing rule was successfully used to model the experimental data. The phase equilibrium behavior of the system is favorable for extraction of acetone from dilute aqueous solutions using supercritical carbon dioxide. 

Figure 4. Phase equilibrium behavior for the system water (1)-acetone (2) - carbon dioxide (3) at 333 K experimental phase compositions ( ) and tie-lines (—) predicted tie-lines (—) predicted three-phase equilibrium compositions (A).

When the surface energy is forced on the interface of two adherents, the surface energy can be also studied by the adsorption isotherm as a function of the amount adsorbed at finite dilution (or concentration). The theoretical and applied studies on adsorption isotherm of the solid surfaces have been widely carried out with the finite concentration, since Thomas Young described the three-phase equilibrium in 1805 [71]. 

For temperature T and the three-phase equilibrium pressure P, Eq. (11. for low-pressure VLE has a double application ... 

The three-phase equilibrium pressure P as given by Eq. (13.58) is therefore ... 

Figure 13.15 is drawn for a single constant pressure equilibrium phase compositions, and hence the locations of the lines, change with pressure, but the general nature of the diagram is the same over a range of pressures. For the majority of systems the species become more soluble in one another as the temperature increases, as indicated by lines CG and DH of Fig. 13.15. If this diagram is drawn for successively higher pressures, the corresponding three-phase equilibrium temperatures increase, and lines CG and DH extend further and further until they meet at the liquid/liquid critical point Af, as shown by Fig. 13.16. The temperature at which this occurs is known as the upper critical solution temperature, and at this temperature the two liquid phases become identical and merge into a single phase.

Solution The three-phase equilibrium temperature t is determined from Eq. (A), here written as ... 

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