Phase equilibrium surfaces

From the viscosity, as well as the phase equilibrium, surface tension, and density measurements it is evident that the system KF-K2M0O4-B2O3 is very complex. Beside the chemical reactions, the polymerization tendency of the melts, especially in the region of higher contents of boron oxide, makes this system difficult to study. 

Because of the close relationship between the MNM transition and the vapor-liquid transition, it is to be expected that immiscibility in the mercury-helium system reaches up to the critical point, or even into the supercritical region. This expectation is confirmed by measurements of the phase diagram at very low helium concentrations and at pressures close to the critical pressure of pure mercury. The experiments extend up to 1610 °C and to pressures up to 3325 bar (Marceca et al., 1996). The p — T — X phase equilibrium surface obtained is qualitatively like the one shown schematically in Fig. 6.4 for a binary fiuid-fluid system of the first kind. The critical line starts at the critical point of pure mercury (Tc(l) = 1478 °C, Pc(l) = 1673 bar) and runs to higher temperatures and pressures as the helium composition X2 increases. 

The foregoing is an equilibrium analysis, yet some transient effects are probably important to film resilience. Rayleigh [182] noted that surface freshly formed by some insult to the film would have a greater than equilibrium surface tension (note Fig. 11-15). A recent analysis [222] of the effect of surface elasticity on foam stability relates the nonequilibrium surfactant surface coverage to the foam retention time or time for a bubble to pass through a wet foam. The adsorption process is important in a new means of obtaining a foam by supplying vapor phase surfactants [223]. 

The strong dependence of the PES on molecular orientation also leads to strong coupling between rotational states, and hence rotational excitation/de-excitation in the scattering. This has been observed experimentally for H2 scattering from Cu surfaces. Recent work has shown that for H2 the changes m rotational state occur almost exclusively when the molecular bond is extended, that is, longer than the gas-phase equilibrium value [ ]. 

This database provides thermophysical property data (phase equilibrium data, critical data, transport properties, surface tensions, electrolyte data) for about 21 000 pure compounds and 101 000 mixtures. DETHERM, with its 4.2 million data sets, is produced by Dechema, FIZ Chcmic (Berlin, Germany) and DDBST GmhH (Oldenburg. Germany). Definitions of the more than SOO properties available in the database can be found in NUMERIGUIDE (sec Section 5.18). 

In situations where the intermediates are bound very weakly or the temperature is high enough for the equilibrium to be shifted sufficiently towards the gas phase, the surface is mostly empty, and we may use the approximation ... 

CO oxidation is often quoted as a structure-insensitive reaction, implying that the turnover frequency on a certain metal is the same for every type of site, or for every crystallographic surface plane. Figure 10.7 shows that the rates on Rh(lll) and Rh(llO) are indeed similar on the low-temperature side of the maximum, but that they differ at higher temperatures. This is because on the low-temperature side the surface is mainly covered by CO. Hence the rate at which the reaction produces CO2 becomes determined by the probability that CO desorbs to release sites for the oxygen. As the heats of adsorption of CO on the two surfaces are very similar, the resulting rates for CO oxidation are very similar for the two surfaces. However, at temperatures where the CO adsorption-desorption equilibrium lies more towards the gas phase, the surface reaction between O and CO determines the rate, and here the two rhodium surfaces show a difference (Fig. 10.7). The apparent structure insensitivity of the CO oxidation appears to be a coincidence that is not necessarily caused by equality of sites or ensembles thereof on the different surfaces. 

Cox MP, Ertl G, Imbihl R, Riistig J. 1983. Non-equilibrium surface phase transitions during the catalytic oxidation of CO on Pt(lOO). Surf Sci 134 L517. 

The objective here is to construct the equilibrium surface in the T-P-x space from a set of available experimental VLE data. In general, this can be accomplished by using a suitable three-dimensional interpolation method. However, if a sufficient number of well distributed data is not available, this interpolation should be avoided as it may misrepresent the real phase behavior of the system. 

In porous media, liquid-gas phase equilibrium depends upon the nature of the adsorbate and adsorbent, gas pressure and temperature [24]. Overlapping attractive potentials of the pore walls readily overcome the translational energy of the adsorbate, leading to enhanced adsorption of gas molecules at low pressures. In addition, condensation of gas in very small pores may occur at a lower pressure than that normally required on a plane surface, as expressed by the Kelvin equation, which relates the radius of a curved surface to the equilibrium vapor pressure [25],... 

Where applications to industrial combustion systems involve a relatively limited set of fuels, fire seeks anything that can bum. With the exception of industrial incineration, the fuels for fire are nearly boundless. Let us first consider fire as combustion in the gas phase, excluding surface oxidation in the following. For liquids, we must first require evaporation to the gas phase and for solids we must have a similar phase transition. In the former, pure evaporation is the change of phase of the substance without changing its composition. Evaporation follows local thermodynamics equilibrium between the gas... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

The mechanism of transfer of solute from one phase to the second is one of molecular and eddy diffusion and the concepts of phase equilibrium, interfacial area, and surface renewal are all similar in principle to those met in distillation and absorption, even though, in liquid-liquid extraction, dispersion is effected by mechanical means including pumping and agitation, except in standard packed columns. 

J.C. Shelton, H.R. Patil, J.M. Blakely, Equilibrium segregation of carbon to a nickel (111) surface A surface phase transition, Surface Science, 43 (1974) 493-520. 

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. 

Standard state free energies (AG°,js) and entropies (ASacjs) may also be determined from GSC retention data if ideal conditions are assumed. For the adsorbate behaving as an ideal gas in the mobile phase, the standard state is defined as a partial pressure of 1 atm. The adsorbed standard state is defined as a two-dimensional perfect gas at 1 atm where the mean distance between adsorbed molecules is the same as in the three dimensional gas phase standard state. Thus, the sorbate equilibrium surface concentration Cg becomes 4.07 x 10 9/T (moles/cm2) and the gas phase sorbate concentration becomes 4.07 x 10- /TK ,. 

An adsorption isotherm for a single gaseous adsorptive on a solid is the function which relates at constant temperature the amount of substance adsorbed at equilibrium to the pressure (or concentration) of the adsorptive in the gas phase. The surface excess amount rather than the amount adsorbed is the quantity accessible to experimental measurement, but, at lower pressures, the difference between the two quantities becomes negligible (see Appendix II, Part I, 1.1.11). 

Figure 9 gives some results obtained with vitreous silica. Three runs were made with 5, 15, and 45 sq. meters of total surface area in suspension. Apparently, the data indicate again the approach toward the two-phase equilibrium for polymeric silicic acid—i.e., independent of the total amount of solid material, the equilibrium concentration of about 110 //grams Si02/ml., as indicated in Figure 1, is approached in all three cases. Only the rate of dissolution is different, being determined by the size of the exposed surface area. 

Solid-Phase Chemical Equilibrium. For the growth of multicomponent films, the solid film composition must be predicted from the gas-phase composition. In general, this prediction requires detailed information about transport rates and surface incorporation rates of individual species, but the necessary kinetics data are rarely available. On the other hand, the equilibrium analysis only requires thermodynamic data (e.g., phase equilibrium data), which often are available from liquid-phase-epitaxy studies, as discussed by Anderson in Chapter 3. 

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