Unstable solution

Sodium forms unstable solutions in Hquid ammonia, where a slow reaction takes place to form sodamide and hydrogen, as foUows ... 

When the loop is closed, will the system in (iv) be stable or unstable Solution... 

Penicillin was discovered before the war, but could only be prepared in highly dilute, impure, and unstable solutions. Up to 1943, when chemical engineers first became involved with the project, industrial manufacturers used a batch purification process that destroyed or inactivated about two-thirds of the penicillin produced. Within 7 months of their involvement, chemical engineers at an oil company (Shell Development Company) had applied their... 

The above two examples were chosen so as to point out the similarity between a physical experiment and a simple numerical experiment (Initial Value Problem). In both cases, after the initial transients die out, we can only observe attractors (i.e. stable solutions). In both of the above examples however, a simple observation of the attractors does not provide information about the nature of the instabilities involved, or even about the nature of the observed solution. In both of these examples it is necessary to compute unstable solutions and their stable and/or unstable manifolds in order to track and analyze the hidden structure, and its implications for the observable system dynamics. 

The enthalpy of decomposition of this peracid is relatively high (AHcj = -1.83 kJ/g - average risk according to CHETAH criterion C,). Its aqueous solutions are unstable. Solutions that contain 80% of peracid detonate when they are stirred (even at -10°C). The usual way of preparing this peracid involves the effect of hydrogen peroxide when metaboric acid is present however, although this operating method was followed, serious accidents have occurred. 

A solution in which the concentration of the solute is equal to the solubility is called a saturated solution. If the concentration is lower, the solution is said to be unsaturated. It is also possible to prepare a supersaturated solution, an unstable solution containing a greater concentration of solute than is present in a saturated solution. Such a solution deposits the excess solute if a crystal of the solute is added to it. It is prepared by dissolving solute at one temperature and carefully changing the temperature to a point where the solution is unstable. 

MRH Ammonia, 5.86/25, aniline 6.44/17, dimethylhydrazine 6.69/19 Ammonia dissolved in 99.6% peroxide gave an unstable solution which exploded violently [1]. In the absence of catalysts, cone, peroxide does not react immediately with hydrazine hydrate. This induction period has caused a number of explosions and accidents owing to sudden reaction of accumulated materials [2], 1,1-Dimethylhydrazine is hypergolic with high-test peroxide [3],... 

Supersaturated A supersaturated solution is one in which more solute is dissolved than is necessary to make a saturated solution. A supersaturated solution is unstable solute molecules may crash out of solution given the slightest perturbation. The situation is like that of Wile E. Coyote, who runs off a cliff and remains suspended in the air until he looks down — at which point he inevitably falls. Supersaturated solutions cire most easily created by heating the solution to the point where it can accommodate more solute than it could normally handle. 

Fig. 16. Bifurcation diagram of temporal dissipative structures, c (maximal amplitude of the oscillation minus the homogeneous steady-state value) is sketched versus B for a two-dimensional system with zero flux boundary conditions. The first bifurcation occurs at B = Bn and corresponds to a stable homogeneous oscillation. At B, two space-dependent unstable solutions bifurcate simultaneously. They become stable at B a and Bfb. Notice that as it is generally the case Bfa Bfb.

There are ammoniates of PtCl2, of halides of other platinum metals and of cobalt and nickel, too, some of which have been mentioned before in, Section 50. The cobalt complexes clearly show the importance of the completed d shells for the stability of the complex. Non complex compounds of trivalent cobalt are very unstable. Solutions of divalent cobalt in ammonia, however, are readily oxidized by air, because the NH3 complex of trivalent cobalt Co(NH3)6 3+ClT has eighteen electrons used in bond formation, whereas the ion Co(NH3) + would have nineteen electrons. 

Whenever this cohdition is satisfied, the resulting region of unstable solutions lies strictly between the crossing point and the maximum in the stationary-state loci. 

Fig. 5.1. (a) A typical parameter plane showing a locus of Hopf bifurcation points. For any given value of the parameter k on the ordinate we may construct a horizontal (broken line) the Hopf bifurcation points, pi and pi, are then located as shown. The corresponding stationary-state loci, shown in (b) and (c), have unstable solutions between pi and pi. ... 

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

Amylose, although water soluble, gives an unstable solution which irreversibly precipitates. It is mainly responsible for the deep blue coloration given by starch and iodine. Solutions of amylopectin are relatively stable. The iodine-binding capacity, on the other hand, is very low. A small amount of covalently bound phosphate normally appears with starch but its exact location within the molecule is not known. 

Fig. 15.3 Illustration of a stable and unstable solution to the model problem (Eq. 15.5) by the forward (explicit) Euler method.

Injection into open tubular columns split routine for introducing small sample volume into open tubular column splitless best for trace levels of high-boiling solutes in low-boiling solvents on-column best for thermally unstable solutes and high-boiling solvents best for quantitative analysis... 

Exercise. Show that (VI.9.7) has one unstable solution. All other solutions are locally stable but not globally. 

This statement could be proved in the manner similar to that used in Section 8.2. It is important to note that the correlation dynamics of the Lotka and Lotka-Volterra model do not differ qualitatively. A stationary solution exists for d = 3 only. Depending on the parameter k, different regimes are observed. For k kq the correlation functions are changing monotonously (a stable solution) but as k < o> the spatial oscillations of the correlation functions (unstable solution) are observed. In the latter case a solution of non-steady-state equations of the correlation dynamics has a form of the non-linear standing waves. In one- and two-dimensional cases there are no stationary solutions of the Lotka model. 

Since A and B are not necessarily zero, this difference equation will be stable only if b < 1 or equivalently only if At <, 2( ilK)1/2. For CO this implies At has to be less than or equal 4.89 x 10 15 s. Verlet s method gave stable solutions to the equations of motion for At = 2.5 x 10-15 s but unstable solutions for At = 5 x 10"15 s. This is in remarkable agreement with the above results. 

Combining all four possible kinds of features of the curves a, 6, c, d in various ways and taking into account must always have either one solution or three, we obtain the possible types of kinetic curves illustrated in Figs. 2-6. The dotted line on all of the curves indicates the region of unstable solutions. 

By analogy we conclude that, of the two steady solutions under the limit (see Fig. 1), only the lower one corresponds to a stable regime (At in Fig. 2) the upper curve corresponds to an unstable solution from which a small deviation leads either to explosion or to the lower stable regime (the point A2 in Fig. 2). 

Powerful oxidizing agent. Unstable solutions of ozone explode on warming.1... 

Dotted curves represent unstable solutions determined by means of numerical techniques... 

Fig. 3.7. One-parameter continuation of steady states with continuation parameter / solid lines = stable solutions dashed lines = unstable solutions yA = yc = 7, (a) y = 0.5,...

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