Unstable equilibrium

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are... 

Recall that a Hopf bifurcation is termed supercritical if its bifurcation diagram is as shown schematically in Fig. 6.2.2a. Correspondingly, in this case a stable limit cycle is born around the equilibrium, unstable hereon, only at a critical (bifurcation) value of the control parameter A = Ac. In contrast, in the subcritical case (Fig. 6.2.2b), the equilibrium is surrounded by limit cycles already for A < Ac, with an unstable limit cycle separating the stable one from the still stable equilibrium. At the bifurcation A = Ac the unstable limit cycle dies out with the equilibrium, unstable hereon, surrounded by a stable limit cycle. Thus the main feature of the subcritical case (as opposed to the supercritical one) is that a stable equilibrium and a stable limit cycle coexist in a certain parameter range, with a possibility to reach the limit cycle through a sufficiently strong perturbation of the equilibrium. 

Monomer-dimer equilibrium. Unstable at r.t. Reacts N, - CpjTiNtTiCpi as dark-blue complex. 

In eq. (IV.3) the first term is positive and increases proportionally to r2 with increasing r. The second term may be negative (in the case of supersaturation - Ap>0), with absolute value increasing proportionally to r3. Thus, in the case of supersaturation, the W(r) curve must pass through a maximum. This maximum is characteristic to some critical particle size, rcr, corresponding to the critical nucleus of a new phase. The critical nucleus of size rzx exists in equilibrium (unstable equilibrium) with the mother medium, i.e. the p = ir condition (p,. is the chemical potential of the substance in the nucleus) stays valid. For such a nucleus, one can write in agreement with eq. (1.13) that... 

Relatively stable Relatively stable Secondary growth Secondary growth Equilibrium Unstable... 

Solvent mixing, less relevant commercially, is widely used in scientific studies to determine the natures of blends. By using dilute solutions of the components the polymers, miscible or immiscible in bulk, can be combined homogeneously. Slow removal of solvent from inherently immiscible polymer mixtures allows Hquid-liquid phase separation to proceed and the polymers to segregate. However, rapid solvent removal or co-precipitation into a large volume of non-solvent can result in intimate mixtures of even immiscible polymers results may depend on the solvent used. Thus, non-equilibrium, unstable mixtures of inherently immiscible polymers can be produced. Such mixtures may segregate when heated above the TgS of the samples when molecular mobility permits. This situation is encoimtered many times in studies of PCL blends. 

The concepts introduced are illustrated in Fig. 1 with the use of a simple mechanical model. This model is analogous to the well known illustration of the Lagrange-Dirichlet theorem concerning stability of potential mechanical systems (a heavy ball on a smooth curved surface). In the present case, a heavy body, say, a cylinder, is placed on a rigid, "geared" cylindrical surface. The body is restricted from the left-hand-side motion with a constraint, a kind of racheting mechanism. The state of the system in Case 1 is subequilibrium, consequently, stable. That in Case 2 is in equilibrium and stable. Case 3 corresponds to an equilibrium, neutral state, and Case 4 to an equilibrium, unstable state. State 5 is nonequilibrium and, consequently, unstable. 

Systems involving an interface are often metastable, that is, essentially in equilibrium in some aspects although in principle evolving slowly to a final state of global equilibrium. The solid-vapor interface is a good example of this. We can have adsorption equilibrium and calculate various thermodynamic quantities for the adsorption process yet the particles of a solid are unstable toward a drift to the final equilibrium condition of a single, perfect crystal. Much of Chapters IX and XVII are thus thermodynamic in content. 

The preceding conclusion is easily verified experimentally by arranging two bubbles with a common air connection, as illustrated in Fig. II-2. The arrangement is unstable, and the smaller of the two bubbles will shrink while the other enlarges. Note, however, that the smaller bubble does not shrink indefinitely once its radius equals that of the tube, its radius of curvature will increase as it continues to shrink until the final stage, where mechanical equilibrium is satisfied, and the two radii of curvature are equal as shown by the dotted lines. 

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

Dichlorine monoxide is the anhydride of hypochlorous acid the two nonpolar compounds are readily interconvertible in the gas or aqueous phases via the equilibrium CI2 O + H2 0 2H0Cl. Like other chlorine oxides, CI2O has an endothermic heat of formation and is thus thermodynamically unstable with respect to decomposition into chlorine and oxygen. Dichlorine monoxide typifies the chlorine oxides as a highly reactive and explosive compound with strong oxidhing properties. Nevertheless, it can be handled safely with proper precautions. 

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

The iron-carbon solid alloy which results from the solidification of non blastfurnace metal is saturated with carbon at the metal-slag temperature of about 2000 K, which is subsequendy refined by the oxidation of carbon to produce steel containing less than 1 wt% carbon, die level depending on the application. The first solid phases to separate from liquid steel at the eutectic temperature, 1408 K, are the (f.c.c) y-phase Austenite together with cementite, Fe3C, which has an orthorhombic sttiicture, and not die dieniiodynamically stable carbon phase which is to be expected from die equilibrium diagram. Cementite is thermodynamically unstable with respect to decomposition to h on and carbon from room temperature up to 1130 K... 

With the microfocus instrument it is possible to combine the weak Raman scattering of liquid water with a water-immersion lens on the microscope and to determine spectra on precipitates in equilibrium with the mother liquor. Unique among characterization tools, Raman spectroscopy will give structural information on solids that are otherwise unstable when removed from their solutions. 

The three branches of the equilibrium configurations after point T are labeled T1, T2, and T3 in Figure 6-26. Branch T2 is a continuation of the saddle shape of solution ST, but this branch is unstable, so the other branches are the real, physical solution because they are stable. Branch T1 has a larger than Ky. If L is about 50% bigger than the... 

Such a stationary value of V can be a relative maximum, a relative minimum, a neutral point, or an inflection point as shown in Figure B-1. There, Equation (B.1) is satisfied at points 1, 2, 3, 4, and 5. By inspection, the function V(x) has a relative minimum at points 1 and 4, a relative maximum at point 3, and an inflection point at point 2. Also shown in Figure B-1 at position 5 is a succession of neutral points for which all derivatives of V(x) vanish. A simple physical example of such stationary values is a bead on a wire shaped as in Figure B-1. That is, a minimum of V(x) (the total potential energy of the bead) corresponds to stable equilibrium, a maximum or inflection point to unstable equilibrium, and a neutral point to neutral equilibrium. 

The plate buckling equations inherently cannot be derived from the equilibrium of a differential element. Instead, the buckling problem represents the departure from the equilibrium state when that state becomes unstable because the in-plane load is too high. The departure from the equilibrium state is accompanied by waves or buckles in the surface of the plate. That is, the plate cannot remain flat when the... 

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