Equilibrium curve pressure dependence

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],... 

The Yl/A isotherms of the racemic and enantiomeric forms of DPPC are identical within experimental error under every condition of temperature, humidity, and rate of compression that we have tested. For example, the temperature dependence of the compression/expansion curves for DPPC monolayers spread on pure water are identical for both the racemic mixture and the d- and L-isomers (Fig. 13). Furthermore, the equilibrium spreading pressures of this surfactant are independent of stereochemistry in the same broad temperature range, indicating that both enantiomeric and racemic films of DPPC are at the same energetic state when in equilibrium with their bulk crystals. 

For a binary mixture under constant pressure conditions the vapour-liquid equilibrium curve for either component is unique so that, if the concentration of either component is known in the liquid phase, the compositions of the liquid and of the vapour are fixed. It is on the basis of this single equilibrium curve that the McCabe-Thiele method was developed for the rapid determination of the number of theoretical plates required for a given separation. With a ternary system the conditions of equilibrium are more complex, for at constant pressure the mole fraction of two of the components in the liquid phase must be given before the composition of the vapour in equilibrium can be determined, even for an ideal system. Thus, the mole fraction yA in the vapour depends not only on X/ in the liquid, but also on the relative proportions of the other two components. 

An interesting example of a one-component systems is SiOa, which can exist in five different crystalline forms or as a liquid or a vapor. As C = 1, the maximum number of phases that can coexist at equilibrium is three. Each phase occupies an area on the T P diagram the two-phase equilibria are represented by curves and the three-phase equilibria by points. Figure 13.1 (2, p. 123), which displays the equUi-brium relationships among the sohd forms of Si02, was obtained from calculations of the temperature and pressure dependence of AG (as described in Section 7.3) and from experimental determination of equUibrium temperature as a function of equilibrium pressure. 

The value of sj v is almost constant (6-7 kcal mol" ) in the measured temperature range and the positive value means that the vacancy-vacancy interaction is repulsive. On the other hand, the value of (/iNis + ) changes sign from minus to plus with increasing temperature. Upon substituting eqns (1.145) for (jUnjs + fi ) and, from eqns (1.146) and (1.147), eqn (1.145) can be rewritten as the relation between y, Uj, and T, as shown in Fig. 1.36. The curves for phase boundaries (thicker curves), i.e. the upper curve for coexistent condensed phases (Ni. S phase + adjacent sulfur rich phase) and the lower curve for coexistent condensed phases (Ni. S phase + adjacent sulfur poor phase), were taken from Refs 26 and 27, in which the temperature dependence of Ps. for coexistent samples was investigated in detail. (As mentioned in Section 1.2, the relationship between the equilibrium sulfur pressure for coexistent condensed phases and temperature must show one to one correspondence. Rau calculated <5 in Nij S for the coexistent phases by substitution of the data from refs 26 and 27 for Os and T into eqn (1.145).)... 

In this chapter we get to know the second essential equation of surface science — the Kelvin5 equation. Like the Young-Laplace equation it is based on thermodynamic principles and does not refer to a special material or special conditions. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors which are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid. 

Figure 5.3 shows how the extent of gas adsorption on to a solid surface might vary with temperature at a given pressure. Curve (a) represents physical adsorption equilibrium and curve (b) represents chemisorption equilibrium. The extent of adsorption at temperatures at which the rate of chemisorption is slow, but not negligible, is represented by a non-equilibrium curve, such as (c), the location of which depends on the time allowed for equilibrium. 

This behavior can be explained if we resort to the imaginary desorption PCT curve at 275°C in Fig. 1.33. The equilibrium plateau pressure at 275°C is higher than 0.1 MPa at which desorption is carried out and at this temperature MgH2 can desorb at atmospheric pressure. However, the kinetics of desorption will depend on the driving force (as shown in Fig. 1.26). Since a larger mass of hydride will produce... 

They are not exactly equilibrium curves because their position and shape depend on feed gas composition and equilibrium pressure. Their value is that they show the maximum extent to which SOz can be oxidized in a catalyst bed. They provide a visual picture of how catalytic S02 oxidation can be optimized. 

Fig. 11. Pressure dependence of the IR spectra of H2 adsorbed at 20 K on high-surface-area (230m g ), sintered (dOm g ). and smoke (10m g ) MgO samples, parts (a), (b), and (c), respectively. The upper curve of each series of spectra has been collected after an elapsed time allowing the surface species to reach the equilibrium conditions and corresponds to the maximum H2 coverage (Fjj, = 1010 kPa), while the bottom spectrum has been recorded after prolonged outgassing at 20 K (Ph, <10 Pa). All spectra have been vertically shifted for the sake of clarity. Note that the ordinate scale is progressively expanding when passing from part (a) to part (c), to account for the loss of band intensity with the decrease of the MgO surface area. (Adapted with permission from Gribov et al. (,124).)...

For the small concentrations of interest in flashing furfural residues (5 % by weight of furfural in water corresponds to a mere 0.977 % by mole), this ratio can be well approximated by the initial slope of the vapor/liquid equilibrium curve. Referred to mass fractions, this slope is known as the amplification factor k . Its dependence on pressure is illustrated in Figure 124. As can be seen, k increases strongly with decreasing pressure. 

Due to the total pressure dependency of the chemical potential, the equilibrium curves with their azeotrope points can be shifted by applying different pressures. Azeotrope rectification can be performed either in batch or in continuous mode. On a technical... 

For individual components, a two-parameter approximation [7-9] of pressure vs. temperature dependence along the phase equilibrium curve is used ... 

At the lowest pressure in the figure, P = 0.133 bar, the vapor-liquid equilibrium curve intersects the liquid-liquid equilibrium curve. Consequently, at this pressure, depending on the temperature and composition, we may have only a liquid, two liquids, two liquids and a vapor, a vapor and a liquid, or only a vapor in equilibrium. The equilibrium state that does exist can be found by first determining whether the composition of the liquid is such that one or two liquid phases exist at the temperature chosen. Next, the bubble point temperature of the one or either of the two liquids present is determined (for example, from experimental data or from known vapor pressures and an activity coefficient model calculation). If the liquid-phase bubble point temperature is higher than the temperature of interest, then only a liquid or two liquids are present. If the bubble point temperature is lower, then depending on the composition, either a vapor, or. a vapor and a liquid are present. However, if the temperature of interest is equal to the bubble point temperature and the composition is in the range in which two liquids are present, then a vapor and two coexisting liquids will be in equilibrium. 

Gas chromatography allows the determination of the adsorption isotherm by integration of the retention curve that shows the dependence of the retention volume on the equilibrium vapor pressure of the adsorbate [131] ... 

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