Quantitative expression

AS )) the function to be minimized is exp (-AS p/R)/ [36]. A quantitative expression for AS can be found by noting that the A monomers in an unstrained loop (N > 4) have essentially two possible confonnations, pointing either inwards or outwards. For loops smaller than a critical size the inward ones are in an apolar environment, since the enclosed water no longer has bulk properties, and the outward ones are in polar bulk water hence the electrostatic charges on... 

Let us consider tire case of a donor-acceptor pair where tire acceptor, after capturing excitation from tire donor, can emit a photon of fluorescence. If tire excitation light is linearly polarized, tire acceptor emission generally has a different polarization. Common quantitative expressions of tliis effect are tire anisotropy of fluorescence, r, or tire degree of polarization,... 

As already indicated in Section 3.1, the study of mesoporous solids is closely bound up with the concept of capillary condensation and its quantitative expression in the Kelvin equation. This equation is, indeed, the basis of virtually all the various procedures for the calculation of pore size... 

In a good solvent, the end-to-end distance is greater than the 1q value owing to the coil expansion resulting from solvent imbibed into the domain of the polymer. The effect is quantitatively expressed in terms of an expansion factor a defined by the relationship... 

The reciprocal of a radical reactivity ratio is sometimes used to quantitatively express the reactivity of monomer M2 by comparing its rate of addition to radical Mi - relative to the rate of Mi adding to Mi-. 

The objective of the Flory-Krigbaum theory is to find a quantitative expression for the placement probability n(d) of the two coils as a function of their separation d. There are three stages to the derivation ... 

A proposed mechanism for toughening of mbber-modifted epoxies based on the microstmcture and fracture characteristics (310—312) involves mbber cavitation and matrix shear-yielding. A quantitative expression describes the fracture toughness values over a wide range of temperatures and rates. 

A quantitative expression of these observations is shown in equation 1, where is the observed absorption maximum for the unsymmetrical carbocyanine and Xj is the arithmetic mean (isoenergetic wavelength) for the absorption maxima of the related symmetrical dyes. 

In the same way that the first law of thermodynamics cannot be formulated without the prior recognition of internal energy as a property, so also the second law can have no complete and quantitative expression without a prior assertion of the existence of entropy as a property. 

DFT turns out to be well suited to quantitative expression of some of the qualitative concepts introduced in Section 1.2, such as electronegativity, hardness, and softness. The pnnciple of maximum hardness (p. 22) can be denved as a consequence of DFT, as can the concepts of hardness and softness. ... 

This idea can be quantitatively expressed by defining activation hardness as the difference between the LUMO-HOMO gap for the reactant and that for the rr-complex intermedi-... 

Partition coefficient Quantitative expression of the partition equilibrium of... 

Risk characterization is tlie process of estimating tlie incidence of a healtli effect under tlie various conditions of human or animal exposure as described in the exposure assessment. It evolves from both dose exposure assessment and toxicity response assessment. The data are then combined to obtain qualitative and quantitative expression of risk. 

In all cases the reaction of the solution can be quantitatively expressed by the magnitude of the hydrogen ion (or hydroxonium ion) concentration, or, less frequently, of the hydroxide ion concentration, since the following simple relations between [H + ] and [OH-] exist ... 

The stability of complex ions varies within very wide limits. It is quantitatively expressed by means of the stability constant. The more stable the complex, the greater is the stability constant, i.e. the smaller is the tendency of the complex ion to dissociate into its constituent ions. When the complex ion is very stable, e.g. the hexacyanoferrate(II) ion [Fe(CN)6]4", the ordinary ionic reactions of the components are not shown. 

With these goals in mind, several investigators have undertaken to set down quantitative expressions which will predict propellant burning rates in terms of the chemical and physical properties of the individual propellant constituents and the characteristics of the ingredient interactions. As in the case of ignition, the basic approach taken in these studies must consider the different types of propellants currently in use and must make allowances for their differences. In the initial combustion studies, the effort was primarily concerned with the development of combustion models for double-base propellants. With the advent of the heterogeneous composite propellants, these studies were redirected to the consideration of the additional mixing effects. 

The first-formed individual atoms (ions or molecules) of B cannot be regarded as a distinct and separate phase but initially, at least, are expected to tend to conform to the structure of, and retain their former positions with reference to, the reactant phase A. During the continued accumulation of atoms (ions or molecules) of B, the consequent increase in total deformation strain energy will lead to a transformation to the structure characteristic of the stable product, solid B. This is quantitatively expressed [28] as a change in free energy by... 

Some changes are accompanied by a change in volume. Because a larger volume provides a greater range of locations for the molecules, we can expect the positional disorder of a gas and therefore its entropy to increase as the volume it occupies is increased. Once again, we can use Eq. 1 to rum this intuitive idea into a quantitative expression of the entropy change for the isothermal expansion of an ideal gas. 

Many quantitative expressions relating to acids and bases are greatly simplified and more easily remembered when we use logarithms. The quantity pX is a generalization of pH ... 

These simplified relationships offer a clearer insight into the dependence of the equilibrium swelling ratio qm on the quality of the solvent as expressed by Xh on the extent of cross-linking. Because of the nature of the approximations introduced to obtain Eqs. (40) and (40 ), their use as quantitative expressions must be limited to networks of very low degrees of cross-linking in good solvents. 

Finally, it is hoped that this book will facilitate the broadening of perspective in the domain of safety research, for example, to enable research of the correlations existing between quantitative expression of risk and recorded aocidents. 

The present analysis shows that when a thermodynamic gradient is first applied to a system, there is a transient regime in which dynamic order is induced and in which the dynamic order increases over time. The driving force for this is the dissipation of first entropy (i.e., reduction in the gradient), and what opposes it is the cost of the dynamic order. The second entropy provides a quantitative expression for these processes. In the nonlinear regime, the fluxes couple to the static structure, and structural order can be induced as well. The nature of this combined order is to dissipate first entropy, and in the transient regime the rate of dissipation increases with the evolution of the system over time. 

Inasmuch as the interaction energy can be related to the transition moments, Forster has been able to develop a quantitative expression for the rate of energy transfer due to dipole-dipole interactions in terms of experimental parameters<4 aB-30) ... 

Dependence of the measured value (signal intensity), y, from other constituents than the analyte A, present in the measuring sample, quantitatively expressed by the respective partial differential quotient. 

Spiro [27] has derived quantitative expressions for the catalytic effect of electron conducting catalysts on oxidation-reduction reactions in solution in which the catalyst assumes the Emp imposed on it by the interacting redox couples. When both partial reaction polarization curves in the region of Emp exhibit Tafel type kinetics, he determined that the catalytic rate of reaction will be proportional to the concentrations of the two reactants raised to fractional powers in many simple cases, the power is one. On the other hand, if the polarization curve of one of the reactants shows diffusion-controlled kinetics, the catalytic rate of reaction will be proportional to the concentration of that reactant alone. Electroless metal deposition systems, at least those that appear to obey the MPT model, may be considered to be a special case of the general class of heterogeneously catalyzed reactions treated by Spiro. 

The most successful quantitative expression of color is that known as the CIE (Commission Internationale de l Eclairage) system [13]. This methodology assumes that color may be expressed as the summation of selected spectral components (blue, green, and red hues) in a three-dimensional manner. The CIE system is based on the fact that human sight is trichromatic in its color perception, and that two stimuli will produce the same color if each of the three tristimulus values (X, Y, and Z) are equal for the two. Detailed summaries of the CIE and other quantitative systems for color measurement are available [14,15],... 

The advantage of the multidentate ligand over the monodentate ligand is quantitatively expressed, according to Schwarzenbach (1952), by the quantity Chel (chelate effect) defined as (25). 

The concept of mole fraction of a component used in Equation (4.1) is a convenient measure of concentration when dealing with trace quantities and dilute solutions, often experienced in environmental systems. This is especially the case with transport phenomena and equilibrium between phases, where it results in simple quantitative expressions. The phenomena of interest when dealing with the exchange of odorous compounds and oxygen between wastewater and a sewer atmosphere are, in this respect, relevant examples. 

Constraints in optimization arise because a process must describe the physical bounds on the variables, empirical relations, and physical laws that apply to a specific problem, as mentioned in Section 1.4. How to develop models that take into account these constraints is the main focus of this chapter. Mathematical models are employed in all areas of science, engineering, and business to solve problems, design equipment, interpret data, and communicate information. Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. For the purpose of optimization, we shall be concerned with developing quantitative expressions that will enable us to use mathematics and computer calculations to extract useful information. To optimize a process models may need to be developed for the objective function/, equality constraints g, and inequality constraints h. 

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