Conditions of stability

We have established the conditions that must be satisfied at equilibrium, but we have not discussed the conditions that determine whether a single-phase system is stable, metastable, or unstable. In order to do so, we consider the incremental variation of the energy of a system, AE, rather than the differential variation of the energy, SE, for continuous virtual variations of the system. Higher-order terms must then be included. The condition of stability is that 

We consider an isolated, homogenous system, and imagine that a part of the single phase is separated from the rest of the phase by a diathermal, nonrigid, permeable wall. By this device we can consider variations of the entropy, volume, and mole numbers of the two parts of the system subject to the conditions of constant entropy, volume, and mole numbers of the 

Identical equations can be written for the double-primed part. The combination of Equation (5.103) and the similar equation for the double-primed part with Equation (5.96) results in the condition that 

To satisfy this condition the first term in the series that is not zero must be greater than zero. The conditions of equilibrium require that (SE + 5E ) must be taken as zero. Therefore, we must examine the conditions based on Equation (5.105) that make 

The use of the conditions expressed by Equations (5.97 -(5.102) and the fact that each second partial derivative in Equation (5.105) is identical to the same derivative for the double-prime part shows that 82E = d2E , and therefore Equation (5.108) can be written as 

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Wilhelm Ostwald was the first to recognise this state of affairs clearly. Indeed, he went further, and made an important distinction. In the second edition of his Lehrbuch der Allgenieinen Chemie, published in 1893, he introduced the concept of metastability, which he himself named. The simplest situation is just instability, which Ostwald likened to an inverted pyramid standing on its point. Once it begins to topple, it becomes ever more unstable until it has fallen on one of its sides, the new condition of stability. If, now, the tip is shaved off the pyramid, leaving a small flat... 

The values of u and A0/AZ are based on assumed conditions of stability class F and stack height wind speed of 2.5 m/s for the stable layer above the inversion. The value of hj incorporates the effect of buoyancy induced dispersion on a/, however, elevated terrain effects are igndred. The equation above is solved by iteration, starting from an initial guess of x , = 5,000 m. The maximum ground-level concentration due to inversion break-up fumigation, Xf, is calculated from ... 

These definitions concern open trajectories (or motions). One can also formulate conditions of stability for closed trajectories (or orbits). 

L. Mandelstam and N. Papalexi were first to establish the theory of the subharmonic resonance based directly on the theory of Poinear6 (Section 6.18). The derivation of this theory, together with the details of the electronic circuits, is given in25, or in an abridged version in6 (pages 464-473). The difficulty of the problem is due to the fact that this case is nonautonomous so that conditions of stability are determined in terms of the characteristic exponents, which always leads to rather long calculations. 

It is necessary to show that the subharmonic resonance whose existence we have just ascertained is stable. Here the condition of stability is very simple, since in the stroboscopic method we deal with the stability of the singular point (and not of the stationary motion). 

From (82) we see that the fractions in (31) are all positive, and since the conditions of stability at points 1 and 2 require that ... 

Observe here that the factorized operator B also is self-adjoint. By means of these operators a sufficient condition of stability becomes... 

The main conditions of stability of these structures formulated based on modeling the compositions of over thirty carbon clusters are given [9] ... 

In this report, this distinction between certified and information values will be drawn on a number of occasions, usually to emphasize the cost of establishing the detailed uncertainty of analytical techniques used to certify materials relative to the perceived benefit of certification. In many situations, a particular analyte—though important scientifically—is studied by a limited number of researchers, in which case it is most practical to establish an information value for that particular reference material by consensus (provided the necessary conditions of stability and homogeneity are met). 

CONDITIONS OF STABILITY IN SUSPENSIONS AND EMULSIONS The following experimental values were obtained by Porter ... 

In other papers by the same group, the effects of sulfur adsorbed or segregated on the Ni surface on corrosion or passivation were described, including the sulfur-induced enhancement of dissolution and the blocking of passivation. It was shown how the conditions of stability of adsorbed sulfur monolayers could be predicted on thermodynamical grounds and this was illustrated by a potential-pH diagram for adsorbed sulfur on nickel in water at 25 °C. (See Refs. 22, 25-29 and papers cited therein.)... 

The calculations shown in Figure 6.4 represent an upper bound, but not an unreasonable one. First, although the void was assumed to remain at constant volume, compaction of the laminate is certainly occurring, and unless resin flow into the bleeder is significant, not much resin volume decrease will occur. Furthermore, if volume were not kept constant, then the voids would grow and likely coalesce, which is a condition of stabilization that is just as bad, if not worse, than that considered in the calculation. Second, it is assumed that Raoult s Law... 

It is not obvious how AF varies with the size of a cluster, because vv depends on the size, but an indirect scheme is available for determining the desired information. For one particular size, R0, there is assumed to be a value of AF for a condition of stability. This means that for a superheated liquid at a stated temperature and pressure, one and only one cluster size is capable of existence for long. This cluster is called a nucleus. A stable cluster is really in a metastable state, as discussed later. However, for any degree of equilibrium, AF must be unaffected by infinitesimal changes in the cluster size. So d(AF)/dR = 0. If vv is defined as the volume occupied by one vapor molecule, then ny = 47r.fi o8/(3ty)- These two manipulations produce a solution for the quantity r v — vl in Eq. (40)... 

Equilibrium models are powerful tools for describing the conditions of stability of redox components in natural water systems. More extended quantitative inferences must be made with great caution because the systems are generally dynamic rather than equilibrated. 

Thus, the conditions to remain in this cyclic trajectory are mnp. However, one can take advantage of the fact that variables m, , and p are not independent to simplify the expression of the conditions of stability of... 

BaAg2(N02)4.H20, from a soln. of the component salts. The formula is here due to M. Oswald, who examined the three nitrites of the alkaline earths, and found only barium forms a double salt. The conditions of stability are represented by AB, Fig. 85. The corresponding diagrams for strontium and calcium nitrites do not show any evidence of the formation of silver strontium nitrite, or of a silver calcium nitrite nor is any strontium barium nitrite formed, for the two salts give a series of mixed crystals, (Ba,Sr)(N02)2.H20. 

In certain instances of poisoning, especially in the case of base metal catalysts, the deactivation can be simply explained by the formation of new bulk solid phases between the base metal and the poison. Examples are the formation of lead vanadates (14) in vanadia catalysts, or of sulfates in copper-chromite and other base metal catalysts (81). These catalyti-cally inactive phases are identifiable by X-ray diffraction. Often, the conditions under which deactivation occurs coincide with the conditions of stability of these inert phases. Thus, a base metal catalyst, deactivated as a sulfate, can be reactivated by bringing it to conditions where the sulfate becomes thermodynamically unstable (45). In noble metal catalysts the interaction is assumed to be, in general, confined to the surface, although bulk interactions have also been postulated. 

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