Condensation radius

In contrast to the clear tendency of the measured condensed fraction to decrease upon dilution, the behavior of the condensation radius Rm appears to be more complicated. There does not seem to exist a simple monotonic convergence of R towards Ru- Rather, for high densities the measured condensation distance is larger than the Manning radius, while for the investigated low densities it is smaller. Unfortunately, a clear-cut statement is difficult, since the localization of the point of inflection in P as a function of ln(r) is only possible with an error estimated to be of the order of 1%. 

FIGURE 23.23 (a) CapiUary condensation radius for typical hydrotreater light oil feed mixed with 50 mole % hydrogen, plotted as a function of relative pressure referred to the dew point. The dew point is in the region of 32 har. (b) Pore size distribution for an alumina hydrotreatment catalyst support. 

This leads to the following expression for the condensation radius ... 

Figure 29 The scaled condensation radius determined from Eq. [269] as a function of the scaled surface charge density cta/cto for a plane (solid line), cylinder (dashed Hnes Kd

Figure 30 The NLDH potential at the condensation radius for the systems of Figure 29.

Unlike the cylindrical case of Eq. [265], this expression is not a function of Koa alone. We have plotted this fraction in Figure 36 for two values of the Debye constant (0.01 and 0.1 and for two different sphere radii (10 and 20 A). For micelles of radius 20 A, there is a noticeable difference between Kd = 0.01 and 0.1 A in the rate at which the limiting fraction plateau is reached. The condensation radius determined from Eq. [269] and the corresponding potential as a function of the scaled charge density are displayed in Figures 29 and 30, respectively, and compared with those for the plane and cylinder. Comments follow those given earlier for a charged cylinder. 

Critical radius R / X c True condensation radius Rj/ X True condensation radius V ... 

Dust and complex organics located at the water condensation radius of M giants and supergiants. 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

In the LS analysis, an assembly of drops is considered. Growth proceeds by evaporation from drops withi < R and condensation onto drops R > R. The supersaturation e changes in time, so that e (x) becomes a sort of mean field due to all the other droplets and also implies a time-dependent critical radius. R (x) = a/[/"(l)e(x)]. One of the starting equations in the LS analysis is equation (A3.3.87) withi (x). 

If the adsorbent contains mesopores, capillary condensation will occur in each pore when the relative pressure reaches a value which is related to the radius of the pore by the Kelvin equation, and a Type IV isotherm will... 

We consider first a cylinder closed at one end, B (Fig. 3.11(a)). Capillary condensation commences at that end to form a hemispherical meniscus r, and are equal to one another and therefore to r , which in turn is equal to r, the radius of the core (cf. Equation (3.7) and Fig. 3.7). Thus capillary condensation, to fill the whole pore, takes place at the relative pressure... 

Thus, as pointed out by Cohan who first suggested this model, condensation and evaporation occur at difi erent relative pressures and there is hysteresis. The value of r calculated by the standard Kelvin equation (3.20) for a given uptake, will be equal to the core radius r,. if the desorption branch of the hysteresis loop is used, but equal to twice the core radius if the adsorption branch is used. The two values of should, of course, be the same in practice this is rarely found to be so. 

The variant of the cylindrical model which has played a prominent part in the development of the subject is the ink-bottle , composed of a cylindrical pore closed one end and with a narrow neck at the other (Fig. 3.12(a)). The course of events is different according as the core radius r of the body is greater or less than twice the core radius r of the neck. Nucleation to give a hemispherical meniscus, can occur at the base B at the relative pressure p/p°)i = exp( —2K/r ) but a meniscus originating in the neck is necessarily cylindrical so that its formation would need the pressure (P/P°)n = exp(-K/r ). If now r /r, < 2, (p/p ), is lower than p/p°)n, so that condensation will commence at the base B and will All the whole pore, neck as well as body, at the relative pressure exp( —2K/r ). Evaporation from the full pore will commence from the hemispherical meniscus in the neck at the relative pressure p/p°) = cxp(-2K/r ) and will continue till the core of the body is also empty, since the pressure is already lower than the equilibrium value (p/p°)i) for evaporation from the body. Thus the adsorption branch of the loop leads to values of the core radius of the body, and the desorption branch to values of the core radius of the neck. 

An essential feature is the involvement of 6A, the additional area of multilayer exposed during the particular step as the group of pores loses its capillary condensate. 5A is calculated from the volume and radius of the group, using the geometry of the cylinder (column 15). The total area of multilayer which is thinned down during any step is obtained by summing the SA contributions in all the lines above the line of the step itself (column 16). 

Consider stage i in the desorption process where the thickness of the adsorbed film is and the pores of radius r, have just lost their capillary condensate. The volume of multilayer lining the pores of any radius r, where r > ri, will then be (since the pores are cylindrical) ... 

Fig. 3J0 Plot of cumulative pore volume against logarithm of r the effective pore radius, (o) For charcoal AY4 A by mercury intrusion O by capillary condensation of benzene, (b) For zinc chloride carbon AYS A by mercury intrusion O by capillary condensation of benzene x by capillary condensation of benzene, after mercury intrusion followed by distillation of mercury under vacuum at temperature rising to 350°C. (Courtesy...

Perhaps the best known explanation of reproducible hysteresis in mercury porosimetry is based on the ink bottle model already discussed in connection with capillary condensation (p. 128). The pressure required to force mercury with a pore having a narrow (cylindrical) neck of radius r, will be... 

Figure 1.9 The balance of endothermic surface energy and the exothermic formation of the stable condensed phase during nucleation from the vapour phase. The critical radius, above which the nuclei become stable, is where the resultant Gibbs energy change has zero slope...

The hydrogen atom has a high ionization energy (1312kJmol ) and in this it resembles the halogens rather than the alkali metals. Removal of the Is electron leaves a bare proton which, having a radius of only about 1.5 x 10 pm, is not a stable chemical entity in the condensed phase. However, when bonded to other species it is well known in solution and in... 

L tj) or, factor from Table 10-46 or, mean radius of bend, in. or, rcQux raBo, mol condensate returned/mol product withdrawn or, volume fraction of phase, dimensionless. 

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