0/ phase boundary

The pressure of a vapor that is in equilibrium with its condensed phase is called the vapor pressure of the substance. Vapor pressure increcises with temperature because, as the temperature is raised, more molecules have sufficient energy to leave their neighbors in the liquid. To determine the vapor pressure, a small amount of liquid can be introduced into the near-vacuum at the top of a mercury barometer and the depression of the column measured (Fig. 3.6). To ensure that the pressure exerted by the vapor is truly the vapor pressure, enough Hquid must be added for some to remain after the vapor forms, for only then are the liquid and vapor phases in equilibrium. The temperature can be changed to determine another point on the curve, and so on (Fig. 3.7). 

The plot of the vapor pressure against temperature is also the liquid-vapor boundary in a phase diagram. To appreciate that interpretation, suppose we have a liquid in a cylinder fitted with a piston. If at some temperature we apply a pressure greater than the vapor pressure of the liquid, the vapor is eliminated, the piston rests on the surface of the liquid, and the system moves to one of the points in the liquid region of the phase diagram. If instead we reduce the pressure on the system to a value below the vapor pressure at that temperature, the system moves to one of the points in the vapor region of the diagram. At the vapor pressure itself, vapor and liquid are in equihbrium, and the state of the system is represented by a point on the phase boundary. 

The same approach can be used to plot the solid-vapor boimdcuy, which is a graph of the vapor pressure of the solid against temperature. The sublunatioii vapor pressure of a soHd, the pressure of the vapor in equiUbrium with a sohd at a particular temperature, is usually much lower than that of a hquid because the molecules are more strongly bound together in the solid thcui in the hquid. 

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The discussion focuses on two broad aspects of electrical phenomena at interfaces in the first we determine the consequences of the presence of electrical charges at an interface with an electrolyte solution, and in the second we explore the nature of the potential occurring at phase boundaries. Even within these areas, frequent reference will be made to various specialized treatises dealing with such subjects rather than attempting to cover the general literature. One important application, namely, to the treatment of long-range forces between surfaces, is developed in the next chapter. 

It is this potential difference that is discussed in Chapter IV in connection with monomolecular films. Since it is developed in the space between the phases, none of the uncertainties of phase boundary potentials is involved. 

The work function across a phase boundary, discussed in Sections V-9B and VIII-2C, is strongly affected by the presence of adsorbed species. Conversely,... 

Figure A2.4.12 shows the two possibilities that can exist, m which the Galvani potential of the solution, (jig, lies between ( )(I) and ( )(n) and in which it lies below (or, equivalently, above) the Galvani potentials of the metals. It should be emphasized that figure A2.4.12 is highly schematic in reality the potential near the phase boundary in the solution changes initially linearly and then exponentially with distance away from the electrode surface, as we saw above. The other point is that we have assumed that (jig is a constant in the region between the two electrodes. This will only be true provided the two electrodes are iimnersed in the same solution and that no current is passing.

In order to describe any electrochemical cell a convention is required for writing down the cells, such as the concentration cell described above. This convention should establish clearly where the boundaries between the different phases exist and, also, what the overall cell reaction is. It is now standard to use vertical lines to delineate phase boundaries, such as those between a solid and a liquid or between two innniscible liquids. The junction between two miscible liquids, which might be maintained by the use of a porous glass frit, is represented by a single vertical dashed line, j, and two dashed lines, jj, are used to indicate two liquid phases... 

In fact, some care is needed with regard to this type of concentration cell, since the assumption implicit in the derivation of A2.4.126 that the potential in the solution is constant between the two electrodes, caimot be entirely correct. At the phase boundary between the two solutions, which is here a semi-pemieable membrane pemiitting the passage of water molecules but not ions between the two solutions, there will be a potential jump. This so-called liquid-junction potential will increase or decrease the measured EMF of the cell depending on its sign. Potential jumps at liquid-liquid junctions are in general rather small compared to nomial cell voltages, and can be minimized fiirther by suitable experimental modifications to the cell. 

At low currents, the rate of change of die electrode potential with current is associated with the limiting rate of electron transfer across the phase boundary between the electronically conducting electrode and the ionically conducting solution, and is temied the electron transfer overpotential. The electron transfer rate at a given overpotential has been found to depend on the nature of the species participating in the reaction, and the properties of the electrolyte and the electrode itself (such as, for example, the chemical nature of the metal). 

As the temperature of the liquid phase is increased, the system ultimately reaches a phase boundary, the bubble point at which the gas phase (vapour) begins to appear, with the composition shown at the left end of the horizontal two-phase tie-line . As the temperature rises more gas appears and the relative amounts of the two phases are detemiined by applying a lever-ami principle to the tie-line the ratio of the fractionof molecules in the gas phase to that hn the liquid phase is given by the inverse of the ratio of the distances from the phase boundary to the position of the overall mole fraction Xq of the system. 

With a further increase in the temperature the gas composition moves to the right until it reaches v = 1/2 at the phase boundary, at which point all the liquid is gone. (This is called the dew point because, when the gas is cooled, this is the first point at which drops of liquid appear.) An unportant feature of this behaviour is that the transition from liquid to gas occurs gradually over a nonzero range of temperature, unlike the situation shown for a one-component system in figure A2.5.1. Thus the two-phase region is bounded by a dew-point curve and a bubble-point curve. 

According to the Porod law [28], the intensity in the tail of a scattering curve from an isotropic two-phase structure havmg sharp phase boundaries can be given by eqnation (B 1.9.81). In fact, this equation can also be derived from the deneral xpression of scattering (61.9.56). The derivation is as follows. If we assume qr= u and use the Taylor expansion at large q, we can rewrite (61.9.56) as... 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

In practice, colloidal systems do not always reach tlie predicted equilibrium state, which is observed here for tlie case of narrow attractions. On increasing tlie polymer concentration, a fluid-crystal phase separation may be induced, but at higher concentration crystallization is arrested and amorjihous gels have been found to fonn instead [101, 102]. Close to the phase boundary, transient gels were observed, in which phase separation proceeded after a lag time. 

Shorthand Notation for Electrochemical Cells Although Figure 11.5 provides a useful picture of an electrochemical cell, it does not provide a convenient representation. A more useful representation is a shorthand, or schematic, notation that uses symbols to indicate the different phases present in the electrochemical cell, as well as the composition of each phase. A vertical slash ( ) indicates a phase boundary where a potential develops, and a comma (,) separates species in the same phase, or two phases where no potential develops. Shorthand cell notations begin with the anode and continue to the cathode. The electrochemical cell in Figure 11.5, for example, is described in shorthand notation as... 

The geometry of Fig. 10.3 leads to a result known as Snell s law, which relates the refractive index of the medium to the angles formed by two wave fronts with the interface. Defining 6q and 6, respectively, as the angles between the phase boundary and the wave front under vacuum and in the medium of refractive index n, show that Snell s law requires n = sin Oo/sind. 

In Figure 1, the force balance in Cartesian coordinates for a body not intersected by phase boundaries is... 

Because the reaction takes place in the Hquid, the amount of Hquid held in the contacting vessel is important, as are the Hquid physical properties such as viscosity, density, and surface tension. These properties affect gas bubble size and therefore phase boundary area and diffusion properties for rate considerations. Chemically, the oxidation rate is also dependent on the concentration of the anthrahydroquinone, the actual oxygen concentration in the Hquid, and the system temperature (64). The oxidation reaction is also exothermic, releasing the remaining 45% of the heat of formation from the elements. Temperature can be controUed by the various options described under hydrogenation. Added heat release can result from decomposition of hydrogen peroxide or direct reaction of H2O2 and hydroquinone (HQ) at a catalytic site (eq. 19). 

Al—Li [12042-37-4] 5. The nature of the phase relationships involving 5 has been the subject of much discussion. Portions of the metastable phase boundaries have not yet been agreed upon. 

Fig. 1. Sodium nitrite solubiUty in water where (---) represents soHd-phase boundaries (1,2,4,5).

Glassification of Phase Boundaries for Binary Systems. Six classes of binary diagrams have been identified. These are shown schematically in Figure 6. Classifications are typically based on pressure—temperature (P T) projections of mixture critical curves and three-phase equiHbria lines (1,5,22,23). Experimental data are usually obtained by a simple synthetic method in which the pressure and temperature of a homogeneous solution of known concentration are manipulated to precipitate a visually observed phase. 

Fig. 6. Qualitative piessuie—tempeiatuie diagiams depicting ctitical curves for the six types of phase behaviors for binary systems, where C or Cp corresponds to pure component critical point G, vapor 1, Hquid U, upper critical end point and U, lower critical end point. Dashed curves are critical lines or phase boundaries (5). (a) Class I, the Ar—Kr system (b) Class 11, the CO2—CgH g system (c) Class 111, where the dashed lines A, B, C, and D correspond to the H2—CO, CH —H2S, He—H2, and He—CH system, respectively (d) Class IV, the CH —C H system (e) Class V, the C2H -C2H OH...

Samples can be removed for analysis, phase volumes can be measured to determine mixture composition and molar volumes (70), and phase boundaries can be measured. Many different configurations of view cells have been proposed. Some are capable of pressures ia excess of 100 MPa (14,500 psi). The cell coateats may be viewed safely through the sapphire wiadow by use of a mirror, video camera, or borescope. 

Fig. 2. PT diagram for a pure substance that expands on melting (not to scale). For a substance that contracts on melting, eg, water, the fusion curve. A, has a negative slope point / is a triple state point c is the gas—Hquid critical state (—) are phase boundaries representing states of two-phase equiUbrium ...

Under equiUbrium or near-equiUbrium conditions, the distribution of volatile species between gas and water phases can be described in terms of Henry s law. The rate of transfer of a compound across the water-gas phase boundary can be characterized by a mass-transfer coefficient and the activity gradient at the air—water interface. In addition, these substance-specific coefficients depend on the turbulence, interfacial area, and other conditions of the aquatic systems. They may be related to the exchange constant of oxygen as a reference substance for a system-independent parameter reaeration coefficients are often known for individual rivers and lakes. 

Figure 1 shows the mechanistic picture developed by C. M. Starks (1,2) for Hquid—Hquid PTC in a graphical form. The catalyst cation extracts the more hpholilic anion Y from the aqueous to the nonpolar organic phase where it is present in the form of a poorly solvated ion pair Y ]. This then reacts rapidly with RX, and the newly formed ion pair X ] returns to the aqueous phase for another exchange process X — Y . In practice most catalyst cations used are rather lipophilic and do not extract strongly into the aqueous phase so that the anions are exchanged at the phase boundary. 

Fig. 5. Metastable Fe—Ni—Cr "temary"-pliase diagram where C content is 0.1 wt % and for alloys cooled rapidly from 1000°C showing the locations of austenitic, duplex, ferritic, and martensitic stainless steels with respect to the metastable-phase boundaries. For carbon contents higher than 0.1 wt %, martensite lines occur at lower ahoy contents (43). A is duplex stainless steel, eg. Type 329, 327 B, ferritic stainless steels, eg. Type 446 C, 5 ferrite + martensite D, martensitic stainless steels, eg. Type 410 E, ferrite + martensite F, ferrite + pearlite G, high nickel ahoys, eg, ahoy 800 H,...

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