Isothermal curves

Figure 3.26(a) and (h) gives results for nitrogen on a compact of silica. Curve (a) is a comparison plot in which the adsorption on the compact (ordinates) is plotted against that on the uncompacted powder (abscissae) there is a clear break followed by an increased slope, denoting enhanced adsorption on the compact, at p/p° = 0-15, far below the lower closure point ( 0-42) of the hysteresis loop in the isotherm (curve (b)). A second compact, prepared at 64 ton in" rather than 130 ton in", showed a break, not quite so sharp, at p/p° = 0-35. 

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Equation constants determined from isotherm curve fitting... 

This approach is an alternative to quantitative metallography and in the hands of a master gives even more accurate results than the rival method. A more recent development (Chen and Spaepen 1991) is the analysis of the isothermal curve when a material which may be properly amorphous or else nanocrystalline (e.g., a bismuth film vapour-deposited at low temperature) is annealed. The form of the isotherm allows one to distinguish nucleation and growth of a crystalline phase, from the growth of a preexisting nanocrystalline structure. 

The adsorption/desorption isotherms measured by NMR (equivalent to conventionally measured isotherms), extracted from two different regions of the imaging field of view corresponding to the two ceramics, are shown in Figure 3.5.9. Once these local isotherms are extracted, they are simply the local adsorption for that point in space contained within the material, measured non-invasively and non-destructively. Conventional analysis techniques for adsorption isotherms (such as BET theory) can therefore be applied to the data, to determine the microstructural properties corresponding to that isotherm curve. 

Figure 6.5 Concentratioin esponse plot of inhibition by a slow binding inhibitor that conforms to scheme B of Figure 6.3. The progress curves of Figure 6.4A were fitted to Equation (6.1). The values of vs thus obtained were used together with die velocity of the uninhibited reaction (v0) to calculate the fractional activity (vs/v0) at each inhibitor concentration. The value of Kf9 is then obtained as the midpoint (i.e., die IC50) of die isotherm curve, by fitting die data as described by Equation (6.8).

FIGURE 2.7. Depletion of Inhibitor Stability DSC Curve (A) and Isothermal Curves (B) for an Inhibited Material. 

Typically, sorption isotherms are constructed for a single food ingredient or food system. An alternative approach is to plot the moisture content versus water activity (or relative vapor pressure) values for a variety of as is food ingredients and food systems. The result is a composite food isotherm (Figure 17). The composite isotherm fits the typical shape observed for a sorption isotherm for an individual food system, with a few products falling above or below the isotherm curve (chewing gum, honey, raisins, bread, and colby and cheddar cheeses). Slade and Levine (1991) were the first to construct such a plot using moisture content and aw values from van den... 

Figure 7. Equilibrium water-uptake or isotherm curve at 30 °C. The dashed line signifies the effect of Schroeder s paradox, a change in water uptake at the same chemical potential depending on the phase of water next to the membrane liquid is at A = 22.

Figures—14 Possible region of trajectories for endothermic and for reversible reactions, starting at feed temperature Tj, with heating from the wall at temperature Ti,. Trajectories must be in the shaded region between the adiabatic and isothermal curves and below the equilibrium curve.

Such a family of isothermal curves together with a companion family of isothermal curves of sticking probability, s, vs. 6 would completely describe the adsorption behavior. At a given A and T, 6 will reach that value for which... 

Figure 4. Nitrogen adsorption and desorption isotherm curves and pore size distribution curve (inset) from the adsorption branch of (a) calcined mesoporous silica sphere and (b) calcined mesoporous alumina sphere.

The mercury absorption of HgN03 in 0.1 M NaN03 solutions exhibited a typical Langmuir isotherm curve, as shown in Figure 4a. The maximum loading is 635 mg/g (or 3.2 mmol Hg/g). Similar adsorbing behavior is observed for other mercury species, such as... 

Determine the temperature distribution T = T(x,y,z,t) produced by an initial point source of heat in an infinite graphite crystal. Plot isothermal curves for a fixed temperature as a function of time in ... 

An unusual dependence for adsorption on a uniform surface arises when it is assumed that the rate of change of the surface is considerably slower than the rate of adsorption (see figure). If adsorption and desorption occur rapidly, the state of the surface remains practically unaltered and we then get an adsorption isotherm corresponding to a non-uniform surface with a distribution p(E, 0) of the heat of adsorption (curve 1) [3]. However, when the time interval is considerable, slow adsorption accompanies changes in the properties of the surface, and the amount of gas adsorbed approaches that given by the Langmuir isotherm (curve 2, point B), which describes a state of complete equilibrium (see above). 

Sorption isotherm curves are graphical relationships showing the partitioning between solid and liquid form where mass adsorbed per unit mass of dry solids (S) is plotted against the concentration (C) of the constituent in solution. K is the sorption equilibrium constant N is a constant describing the intensity of sorption. The linear sorption isotherm can be expressed as follows ... 

Isothermal curves derived from this equation are shown in Fig. 2.19. It is clear that this equation fits the experimental data. A comparison of the kinetic equation (2.48) and the Avrami equation shows93 that any experimental data described by the Avrami equation can be approximated by Eq. (2.48) for any arbitrary set of constants. The divergence of the curves does not exceed 1% in the range 0.2 experimental data (in the isothermal case) can be analyzed by both equations with practically the same reliability. Thus the choice of approximating equation depends on the goal of this procedure if we are interested in physi-... 

The capillary condensation theory provides a satisfactory explanation of the phenomenon of adsorption hysteresis, which is frequently observed for porous solids. Adsorption hysteresis is a term which is used when the desorption isotherm curve does not coincide with the adsorption isotherm curve (Figure 5.8). 

Numerous attempts have been made at developing mathematical expressions from postulated adsorption mechanisms to fit the various experimental isotherm curves. The three isotherm equations which are most frequently used are those due to Langmuir, to Freundlich, and to Brunauer, Emmett and Teller (BET). 

Figure 3. Adsorption isotherm curves for xenon on NaY zeolite and NaY zeolite-V205 at 273 and 299.5 K.

As in the cases previously mentioned, the a processes are assigned to the cooperative motions around Tg. Because of these motions the a process can follow a WLF behavior for the shift factor aj of the isothermal curves e". 

Table 3.6 summarizes the characteristic it - A isotherm curves of MA - alt - StM under different pH conditions. 

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