Stability of a system

The relative stability of a system is inversely related to its relative potential energy. 

B, APPLICATION OF THEOREM TO CLOSEDLOOP STABILITY. To check the stability of a system, we are interested in the roots or zeros of the characteristic equation. If any of them lie in the right half of the s plane, the system is unstable. For a closedloop system, the characteristic equation is... 

Numerous other applications could be listed in which electrokinetic characterization provides a convenient experimental way of judging the relative stability of a system to coagulation. Paints, printing inks, drilling muds, and soils are examples of additional systems with properties that are extensively studied and controlled by means of the f potential. 

Setting out the requirements for homoaromaticity and homoantiaromaticity in the manner above, in principle, makes it easy to identify a system as homoaromatic. Clearly, an appropriate geometry or structure of the species in question is required. This pertains not only to the appropriate placement of the AOs at the homoconjugative centres, but also to the structural changes associated with the cyclic delocalization of (4q + 2) 71-electrons. The cyclic delocalization should also be reflected in the stability of a system and its spectro-... 

It is possible and probably very likely that both types of electronic effects are occurring in the acetal function. In other words, 2 could be more stable than 2 because 2 is stabilized relative to 2 by a partial electron transfer and 2 is destabilized relative to 2 by electronic repulsions. There is presently no experimental technique to differentiate between the two effects. At the present time, many chemists, including myself, prefer to consider the anomeric effect as a stabilizing rather than a destabilizing effect. The main reason is that the concept of stabilization of a system through electronic delocalization is a well established principle in organic chemistry. The resonance theory is indeed based on this principle. I believe that this concept rather than the dipole - dipole or electron pair - electron pair repulsions allows the organic chemist to rationalize his results better. 

To describe the chemical reactivity in the context of DFT, there are several global and local quantities useful to understand the charge transfer in a chemical reaction, the attack sites in a molecule, the chemical stability of a system, etc. In particular, there are processes where the spin number changes with a fixed number of electrons such processes demand the SP-DFT version [27,32]. In this approach, some natural variables are the number of electrons, N, and the spin number, Ns. The total energy changes, estimated by a Taylor series to the first order, are... 

Let us illustrate the sense of the "rite by investigating the stability of a system with two variables. Let us take the system... 

The conditions of stability developed in Section 5.15 suggest a boundary between stable and unstable systems. This boundary is determined by the conditions that one of the quantities that determine the stability of a system becomes zero at the boundary at one side of the boundary the appropriate derivative has a value greater than zero, whereas on the other side its value is less than zero. The derivative is a function of the independent... 

The stability of a system may be discussed in terms of Figures 5.9 and 5.10. According to Equation (5.53) the Gibbs energy of a system at equilibrium must have the smallest possible value at a given temperature and pressure. Therefore, the lower curves on either side of the point of intersection represent the stable states of a one-phase system, and the upper curves represent metastable states. For any essentially unstable state the surface would have to be concave upward. The cut of this surface and the plane perpendicular to the appropriate axis is shown by the broken line in Figures 5.9 and 5.10. These curves are obtained when it is assumed that the equation of state and its derivatives are continuous as shown in Figure 5.1. 

A breach in stability of a system may result from internal causes (the aging of its elements) or external causes associated with the unfavorable influence of the environment (an ill-intentioned enemy, in particular). The survivability of biological systems is determined by environmental conditions where human interference with nature is an important factor. In connection with this and the prospects of constructing artificial biological systems, optimization problems also arise an increase of productivity of a biological system being the main criterion of optimality. 

Not all foams are wanted though. Foams, other than flotation froths, are generally not wanted in the process industries where they tend to interfere with process unit operations and may cause upsets. Some agents will act to reduce the foam stability of a system (termed foam-breakers or defoamers) while others can prevent foam formation in the first place (foam preventatives, foam inhibitors). There are many such agents and Kerner [327] describes several hundred different formulations for foam... 

A CNDO study on benzo[l,2-c 4,5-c ]dithiophene predicted that it would be more polar than thieno[3,4-c]thiophene, and that excess electron density would be distributed at the bridgehead carbon atoms. A singlet ground state was also predicted <75ACR139>. The heats of atomization, charge densities, and resonance energies of a series of benzodithiophenes were calculated <80T27il>. A [c] fusion led to less resonance stabilization of a system than a [b] fusion. 

Example 12.3 Stability under both dissipative and convective effects In some cases, both dissipative as well as convective effects determine the stability of a system. Some examples of such stability are the onset of free convection in a layer of fluid at rest, leading to Benard convection cells, and the transition from laminar to turbulent flow. For stability considerations, two limiting cases exist (i) in the case of ideal fluids, dissipative processes are neglected, and (ii) in purely dissipative systems, no convection effects occur. 

The term chelate effect refers to the enhanced stability of a complex system containing chelate rings as compared to the stability of a system that is as similar as possible but contains none or fewer rings. As an example, consider the following equilibrium constants ... 

The positive curvature is not realized in actual systems from the thermodynamic requirement for the stability of a system, that is, the second derivative of the excess Gibbs energy, AG = y, with respect to Aq 0 variable be negative, at T and P constant ... 

Obviously, the inspection of stability of a system comprising several elementary chemical reactions needs analysis of the sign of the parameter... 

The Relationship between Entropy and Dynamic Stability of a System 301... 

THE RELATIONSHIP BETWEEN ENTROPY AND DYNAMIC STABILITY OF A SYSTEM... 

Legendre Transforms and Stability of a System 1.22.1 Generalized Legendre Transforms... 

This is the classical argument introduced by van Heerden in 1953 for the adiabatic stirred tank. It is a most important one to grasp firmly for it can be used in more complicated situations to get some insight into the stability of a system. However, its limitations must be also thoroughly understood. In particular, it can be used to establish instability, but it does not count conclusively for stability because of several reasons. First, we should be suspicious of a single condition for a system in which there are two variables. Second, the diagram for the heat generation was drawn in a rather special way, for the steady state-mass balance equation, f 7), was first... 

Stability The stability of a system is determined by its response to inputs. A stable system remains stable unless it is excited by an external source, and it should return to its original state once the perturbation is removed and the system cannot supply power to the output irrespective of the input. The system is stable if its response to the impulse excitation approaches 0 at long times or when every bounded input produces a bounded output. Mathematically this means that the function does not have any singularities that caimot be avoided. The impedance Z(s) must satisfy the following conditions Z s) is real when s is real (that is, when 0) and Re[Z(5)] > 0 when v > 0 [ = v -i- ja>, see Section... 

What is the major advantage of the Routh-Hurwitz criterion for examining the stability of a system ... 

Therefore, if s lies to the left of the imaginary axis, then a <0 and z < 1, which is the rule for stability of discrete-time systems. Figure 29.12 shows the corresponding regions for stability of a system in continuous form (Figure 29.12a) and discrete-time form (Figure 29.12b). 

The reactivity and stability of a system can be studied applying the concept of chemical hardness. Incorporation of the concepts of hardness and softness into DFT has led to the mathematical identification of t] as the second derivative of the total energy with respect to the number of electrons N [41,42] ... 

One way to see the effect of the reduction in dimensionality that occurs as one goes from time series to phase space portrait to Poincare section is to consider the stability of a system as it is reflected in the stability of points in the Poincare section. The single point, which corresponds to the Poincare section of a simple limit cycle, can be treated in the same way an equilibrium point (or steady state point) is treated, even though, here, we are considering the stability of the periodic state, that is, the limit cycle. A small perturbation added to this point in the cross section will be found to decay back toward the point itself, if the limit cycle is stable, or to evolve away from the point, if the cycle is unstable. The stability properties of the point in the Poincare section are the same as the stability properties of the limit cycle to which it corresponds. Hence, a stable point in the section means that the limit cycle is stable, and an unstable one means that the limit cycle is unstable. And, furthermore, any bifurcations which occur for the point in the cross section also correspond to bifurcations which the limit cycle undergoes as a parameter is varied. 

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