Stable states

Northrup S H and Hynes J T 1980 The stable states picture of chemical reactions. I. Formulation for rate constants and initial condition effects J. Chem. Phys. 73 2700-14... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Figure C3.6.7(a) shows tire u= 0 and i )= 0 nullclines of tliis system along witli trajectories corresponding to sub-and super-tlireshold excitations. The trajectory arising from tire sub-tlireshold perturbation quickly relaxes back to tire stable fixed point. Three stages can be identified in tire trajectory resulting from tire super-tlireshold perturbation an excited stage where tire phase point quickly evolves far from tire fixed point, a refractory stage where tire system relaxes back to tire stable state and is not susceptible to additional perturbation and tire resting state where tire system again resides at tire stable fixed point.

This is the most common and stable state of chromium in aqueous solution. The Cr ion, with 2d electrons, forms mainly octahedral complexes [CrX ], which are usually coloured, and are kweticallv inert, i.e. the rate of substitution of X by another hgand is very slow consequently a large number of such complexes have been isolated (see below, under chromium(III) chloride). 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

FIGURE 2 8 sp Hybridization (a) Electron configuration of carbon in its most stable state (b) Mixing the s orbital with the three p orbitals generates four sp hybrid orbitals The four sp hybrid orbitals are of equal energy therefore the four valence electrons are distributed evenly among them The axes of the four sp orbitals are directed toward the corners of a tetrahedron... 

Absorption of electromagnetic radiation causes a molecule to be excited from Its most stable state (the ground state) to a higher energy state (an excited state)... 

Energy of activation (Section 3 2) Minimum energy that a re acting system must possess above its most stable state in or der to undergo a chemical or structural change... 

A radioisotope is an atom the nucleus of which is not stable and which decays to a more stable state by the emission of various radiations. Radioactive isotopes, also called nucHdes or radionucHdes, are important to many areas of scientific research, as well as ia medical and iadustrial appHcations (see... 

As-polymerized PVDC is not in its most stable state annealing and recrystaUization can raise the temperature at which it dissolves (78). Low crystallinity polymers dissolve at a lower temperature, forming metastable solutions. However, on standing at the dissolving temperature, they gel or become turbid, indicating recrystaUization into a more stable form. 

Ores are mined and are then refined in an energy intensive process to produce pure metals, which in turn are combined to make alloys (see Metallurgy Mineral RECOVERY and processing). Corrosion occurs because of the tendency of these refined materials to return to a more thermodynamically stable state (1—4). The key reaction in corrosion is the oxidation or anodic dissolution of the metal to produce metal ions and electrons... 

Removable partial dentures are partial dentures that restore function, maintain surrounding dentition in a stable state, and are removable. They require no (or minimal) dental surgery on neighboring teeth, as opposed to the removal of sound, healthy enamel for the insertion of fixed partial dentures... 

The test must be conducted at the rated current, at a frequency with a tolerance of -t-2% and -5% of the rated frequency and the voltage in a sinusoidal waveform, as much as practicable. See also Section 11.3.2. The test is earned out until the temperature reaches almost a stable state, i.e. when the variation does not exceed 1°C per hour. To shorten the test duration, the current may be enhanced during the initial period to reach a fast, stable state. 

Fig. 5.6. Plot of the Gibbs functions for ice and water os functions of temperature. Below the melting point Tn, > Gjjj and ice is the stable state of HjO above T , G...

A closed system moving slowly through a series of stable states is. said to undergo a reversible process if that process can be completely reversed in all thermodynamic respects, i.e. if the original. state of the system itself can be recovered (internal reversibility) and its surroundings can be restored (external irreversibility). An irreversible process is one that cannot be reversed in this way. 

Consider first the steady flow of fluid through a control volume CV between prescribed stable states X and Y (Fig. 2.1) in the presence of an environment at ambient temperature Tj, (i.e. with reversible heat transfer to that environment only). The maximum work which is obtained in reversible flow between X and Y is given by... 

The actual work output in a real irreversible process between stable states X and Y is therefore... 

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