Dominant first-order response

It ean be seen in Figure 5.17 that the pole at the origin and the zero at. v = —1 dominate the response. With the eomplex loei, ( = 0.7 gives K a value of 15. ITowever, this value of K oeeurs at —0.74 on the dominant real loeus. The time response shown in Figure 5.20 shows the dominant first-order response with the oseillatory seeond-order response superimposed. The settling time is 3.9 seeonds, whieh is outside of the speeifieation. 

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

From (218) it is clear that the reduced quantity 5 has a very simple structure = — yBjBy. The reduced Hessian E is constructed in two steps The first and time-consuming step, dominating the calculation of the first order response, is the construction of the <7 vector... 

The preference for the yn-isomer can be described via the combination of primary and secondary stereo-electronic effects in the o/jt-model, or via two antiperiplanar lone pair interactions using the x-model. Using the more familiar o/ t-model, the jyn-conformation benefits from hyperconjugative n a stabilization analogous to the anomeric effect, in which in-plane lone pairs can donate into the orbital of the carbonyl moiety (Figure 6.93). This is a second-level interaction that is weaker than the dominant first-order n tt c=o interaction between the p-type lone pair of oxygen and the carbonyl it-system. However, since the first order interaction does not change upon the rotation, it is the second order effect that is responsible for the difference in stability. 

Not all instruments behave as a first-order system. If several capacities exist in a series connection and none dominates, i.e., is much larger than the others, then the response differs from a first-order system. The step response of a higher-()rder sys-... 

The key to obtaining pore size information from the NMR response is to have the response dominated by the surface relaxation rate [19-26]. Two steps are involved in surface relaxation. The first is the relaxation of the spin while in the proximity of the pore wall and the other is the diffusional exchange of molecules between the pore wall and the interior of the pore. These two processes are in series and when the latter dominates, the kinetics of the relaxation process is analogous to that of a stirred-tank reactor with first-order surface and bulk reactions. This condition is called the fast-diffusion limit [19] and the kinetics of relaxation are described by Eq. (3.6.3) ... 

Simple models are used to Identify the dominant fate or transport path of a material near the terrestrial-atmospheric Interface. The models are based on partitioning and fugacity concepts as well as first-order transformation kinetics and second-order transport kinetics. Along with a consideration of the chemical and biological transformations, this approach determines if the material is likely to volatilize rapidly, leach downward, or move up and down in the soil profile in response to precipitation and evapotranspiration. This determination can be useful for preliminary risk assessments or for choosing the appropriate more complete terrestrial and atmospheric models for a study of environmental fate. The models are illustrated using a set of pesticides with widely different behavior patterns. 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

Other more unusual pictures that have been proposed include the libron model [45], where librations (molecular rotations) are responsible for ph T). It was suggested that the T2 behavior arose from the dominance of two-libron processes, and that the first-order, single-libron contribution is small by symmetry. The T dependence of the spin susceptibility was discussed within the same framework, in terms of a band-narrowing effect [46]. However, this approach has been criticized [44], and furthermore, no evidence has been found that librations play any role in metal-semiconductor transitions [9]. 

Kinetic transitions between well-defined micromorphologies are usually dominated not by second order but by first order thermodynamics. Recent ideas have shown how propagating fronts of the new nucleating phase may be responsible for the limiting rate. ... 

The popularity of the SOS methods in calculations of non-linear optical properties of molecules is due to the so-called few-states approximations. The sum-over-states formalism defines the response of a system in terms of the spectroscopic parameters, like excitations energies and transition moments between various excited states. Depending on the level of approximation, those states may be electronic or vibronic or electronic-vibrational-rotational ones. Under the assumption that there are few states which contribute more than others, the summation over the whole spectrum of the Hamiltonian can be reduced to those states. In a very special case, one may include only one excited state which is assumed to dominate the molecular response through the given order in perturbation expansion. The first applications of two-level model to calculations of j3 date from late 1970s [93, 94]. The two-states model for first-order hyperpolarizability with only one excited state included can be written as ... 

In the present contribution we will discuss the direction of the changes of the NLO response and the solvatochromic behavior as a function of solvent polarity of the D-tt-A chromophores. The best starting point for these considerations seems to be the simple two-state model for the first-order hyperpolarizability (/ ) [8]. To avoid the extreme complexity of the sum-over-states (SOS) expression [101], Oudar and Chemla proposed the relation between the dominant component of 13 along the molecular axis (let it be the x-axis) and the spectroscopic parameters of the low-lying CT transition [8]. The use of the two-level approximation in the static case (ru = 0.0) has lead to the following expression for the static component of the first-order hyperpolarizability tensor ... 

In Chapter 2, it was stated that if the process is greater than first order but without time delay, a reasonable choice for the scaling factor p can be based on the dominant time constant of the process. In this case, we can let ar = i to cancel this dominant pole in G(s), which gives t = allowing us to dioose a to bring about the desired closed-loop response speed. 

Ascorbic add. The ascorbic acid loss during storage showed an initial period of rapid loss followed by a period of slow loss. This behaviour can be described by a two simultaneous first order kinetic model. This means that two simultaneous reactions with different rates are responsible for the ascorbic acid degradation. It has been referred in literature that ascorbic acid is degraded by two simultaneous reactions one aerobic and other one anaerobic. The aerobic reaction dominates first and it is fairly rapid, while the anaerobic reaction dominates later and it is quite slow (6,7). Equation (1) was used to fit the experimental data. 

Figure 7.6 includes two limiting cases T2/T1 = 0, where the system becomes first-order, and T2/T1 = 1, the critically damped case. The larger of the two time constants, Ti, is called the dominant time constant. The S-shaped response becomes more pronounced as the ratio of T2/T1 becomes closer to one. 

The time constant r characterizes the response of the first-order system and is discussed in greater detail in the next section. All higher-order systems can be broken down into sets of first-order systems, and the time constants of these LDEs can be used to ascertain the relative importance of each from a dynamic response perspective. That is, the dominant, or largest, time constant will determine the speed of the response. The commonly used rule of thumb is that any subsystem with a time constant an order of magnitude (10 times) less than the dominant time constant can be described by steady-state or algebraic equations. 

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