Limits of Stability

It is noted in Sections XVII-10 and 11 that phase transformations may occur, especially in the case of simple gases on uniform surfaces. Such transformations show up in q plots, as illustrated in Fig. XVU-22 for Kr adsorbed on a graphitized carbon black. The two plots are obtained from data just below and just above the limit of stability of a solid phase that is in registry with the graphite lattice [131]. 

Fig. 4.41 Potential-pH diagram for tin. The full lines enclose the zones where the stability of various compounds or ions makes the indicated ion possible. The broken lines indicate the limits of stability of water at a pressure of 1 atm (after Reference 4 and Cebelcor)...

The Flade potential, which is the negative potential limit of stability of the oxide film. At potentials more negative than the Flade potential the oxide film is unstable with respect to its reduction or dissolution, or both, since the rates of these two processes exceed that of film formation. 

The reason is that the backspillover ions desorb to the gas phase directly from the three-phase-boundaries or react directly at the three-phase-boundaries (electrocatalysis, A=l) before they can migrate on the gas-exposed electrode surface and promote the catalytic reaction. The limits of NEMCA are set by the limits of stability of the effective double layer at the metal/gas interface. 

These calculated decreases are valid only in case the change in structure is not compensated by deformation of the polyhedra. Some compensating deformation will always occur the rules governing deformation (Section 9) show that it will be small in case the radius ratio approaches the lower limit of stability for the polyhedron, and will increase with the radius ratio. 

The most stable core configuration for an inner core of one spheron is KM, with an outer core of nine spherons. For icosahedral packing the outer core contains 12 spherons. These structures, which we may take as defining the limits of stability for a one-spheron inner core, have been discussed in the preceding section. 

Compared with rigid microgels, the intrinsic viscosity of microgels prepared from the comonomer mixture mentioned before is higher, but the slope of the curve in Fig. 56 is still low because the composition of these microgels was close to the limit of stability. 

A proportional controller is used to control a process which may be represented as two non-interacting first-order lags each having a time constant of 600 s (10 min). The only other lag in the closed loop is the measuring unit which can be approximated by a distance/velocity lag equal to 60 s (1 min). Show that, when the gain of a proportional controller is set such that the loop is on the limit of stability, the frequency of the oscillation is given by ... 

As discussed in Section 7.10.4, Volume 3, the Bode stability criterion states that the total open loop phase shift is —n radians at the limit of stability of the closed loop system. 

Blue fluorescent emitters based on fused polyaromatic ring systems have long been known and systematic work has steadily improved the efficiencies and colors, while pushing the limits of stability in an operational device. A sky blue based on styrylamine doped 2-methyl-9,10-di(2-naphthyl)anthracene OLED was reported to provide the highest efficiency device (Scheme 3.99) [365],... 

A critical point must satisfy two conditions. The first defines the limit of stability. In the procedure of Heidemann and Khalil, this condition takes the form that the determinant of matrix Q is zero, where the elements of Q are... 

The other type of stability limit is associated with the mixture flow and its relationship to the laminar flame itself. This stability limit, which includes the phenomena of flashback, blowoff, and the onset of turbulence, describes the limitations of stabilizing a laminar flame in a real experimental situation. 

This situation is extremely important. We have found the limit of stability of a second-order system. The roots of the characteristic equation are... 

The direct-substitution method is a simple and useful method for finding the values of parameters m the characteristic equation that put the system just at the limit of stability. 

With continuous systems we made root locus plots in the s plane. Controller gain was varied from zero to infinity, and the roots of the closedloop characteristic equation were plotted. Time constants, damping coefficients, and stability could be easily determined from the positions of the roots in the s plane. The limit of stability was the imaginary axis. Lines of constant closedloop damping coefficient were radial straight lines from the origin. The closedloop time constant was the reciprocal of the distance from the origin. 

The limit of stability becomes the imaginary axis in the log-z plane and the region of stability is the left half of the log-z plane. This is analogous to the situation in the continuous s plane. 

Recent experiments by Akamatsu et al. (1993) show that pressure has a significant elfect on Fe-Mg intracrystalline distribution (dK /dP = 2 Mbar ). At the upper pressure limit of stability of olivine (P 0.12 Mbar, corresponding to a depth of about 400 km p-wave seismic discontinuity, cf section 5.2.3), the increase in with respect to the P = 1 bar condition, Tand composition being equal, is about 0.24. For a Fogo olivine, this variation is similar to that induced by a T increase of about 250 °C (cf figure 5.11). 

Two master variables, pH and pe, can be used to define the limits of stability of the UO2 spent fuel matrix under repository conditions. The Pourbaix diagram depicted in Fig. 2 clearly indicates the stability space of UO2 and consequently the desired chemical conditions that ensure the correct performance of the waste matrix. 

The presence of shared edges, and particularly shared faces, in a coordinated structure tends to decrease its stability this effect is large for cations with large valence and small coordination number, and is especially large in case the radius ratio approaches the lower limit of stability of the polyhedron. 

Figure 5.1 A diagram showing the theoretical and practical limits of stability of ions in solutions of pH between 0 and 14...

Couples with E values outside the practical limits of stability do not necessarily cause the destruction of the solvent. Some reactions, although they may be thermodynamically feasible, are kinetically very slow. The Co3J"/Co2+ couple has an E value of + 1.92 V, and Co3+ should not exist in aqueous solution. However, its oxidation of water to oxygen is very slow, and solutions containing [Co(H20)6]3+ evolve dioxygen slowly. The value for the A13 + /A1 couple is -1.66 V, and indicates that aluminium should dissolve in acidic aqueous solutions. A stable protective oxide layer on the metal s surface normally retards the reaction, a circumstance gratefully acknowledged by aeroplane manufacturers. 

The oxidation potentials show that, whereas unprotonated DBMA and unpro-tonated HF-phase were oxidised on Pt under these conditions (at 0.94 V and 1.81 V vs Hg/Hg2Cl2 sat., respectively), the protonated species, i.e., the form in which they would exist in ECF, were not, at least up to the limits of stability of the electrolyte. The authors cite this as evidence against the ECbECN mechanism, where the electrochemical properties of the organic compound to be fluorina-ted, particularly its ability to be oxidised, are stated to be decisive. 

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