Second-Order Analysis

Not fully resolved. [Pg.141]

Experiments were then designed in which absorbance readings were acquired over the full time course of the reaction. These experiments employed a low concentration of PyO as the limiting reagent and a large excess of phosphine. The data fit a precise pseudo-second-order analysis, and gave kcat = 1.5xl04Lmol 1 s-1 in benzene at 298 K a precision of 5% was estimated. 

There are two competing and equivalent nomenclature systems encountered in the chemical literature. The description of data in terms of ways is derived from the statistical literature. Here a way is constituted by each independent, nontrivial factor that is manipulated with the data collection system. To continue with the example of excitation-emission matrix fluorescence spectra, the three-way data is constructed by manipulating the excitation-way, emission-way, and the sample-way for multiple samples. Implicit in this definition is a fully blocked experimental design where the collected data forms a cube with no missing values. Equivalently, hyphenated data is often referred to in terms of orders as derived from the mathematical literature. In tensor notation, a scalar is a zeroth-order tensor, a vector is first order, a matrix is second order, a cube is third order, etc. Hence, the collection of excitation-emission data discussed previously would form a third-order tensor. However, it should be mentioned that the way-based and order-based nomenclature are not directly interchangeable. By convention, order notation is based on the structure of the data collected from each sample. Analysis of collected excitation-emission fluorescence, forming a second-order tensor of data per sample, is referred to as second-order analysis, as compared with the three-way analysis just described. In this chapter, the way-based notation will be arbitrarily adopted to be consistent with previous work. 

The resulting Gibbs-energy functions agree well with those obtained (9 ) by second-order analysis. 

Second-Order Analysis of Conical Intersections Applications to Photochemistry and Photophysics of Organic Molecules... 

Second-Order Analysis at an Intersection Intersection Space of Fulvene... 

The kot, values obtained from such a second order analysis are shown as a function of relative sulfur concentration in Table I. The observed dependence of koj, on sulfur concentration indicates that the rate expression shown in Equation 1 is inadequate and that a concentration term in sulfur is required. Assuming an unknown order of reaction b with respect to sulfur, an expanded rate expression is obtained. 

Substituent EfFects. The eflFect on the rate of a variety of electron-donating and -withdrawing para substituents on thiophenol were examined. Pseudo-second order analysis similar to those shown in Figure 1 were carried out but for only one thiol sulfur concentration ratio. The rate data obtained under specified conditions are shown in Table II. The thiols have been listed in order of increasing Hammett a value of the substituents—i.e., most eflFective electron donor to most eflFective withdrawer. Examination of these data shows that there is a marked substituent eflFect on the reaction. The electron-donating para substituents, methoxy and methyl, increased the rate while the electron-withdrawing groups, fiuoro, chloro, and bromo, decreased the rate. Two substituents, p-amino and p-nitro, are anomalous in that the reaction rates do not obey the Hammett... 

The kinetics of this reaction have been investigated using simple mass spectrometric analysis. Low concentrations of reagents ( 5 x 10 mol cm ) were used with second-order analysis for measurements of the rate constant for the extremely rapid primary reaction... 

It was shown that delamination failure could be induced by shear stresses caused by second-order bending, which resulted from second-order lateral deformations [23]. The measured deformed shapes along the specimen height at delamination initiation are shown in Figure 7.40 for temperatures up to 180 °C. On the basis of second-order analysis, they were described according to Eq. (7.16) ... 

A buckling load of columns can be used as a parameter in the second order analysis of braced piers assinning that the second order bending moment has a sine-shaped distribution EN 1992-1 2004). For braced piers (Figure 1 b), the second order eccentricity of the appMed compressive force Ng is defined as ... 

The analysis methods used in the design process make use of hand computations, three dimensional structural analysis software, STAAD Pro (2009) and spreadsheets. The structural analysis software is required to be capable of conducting static linear elastic analysis, non-linear second order analysis, buckling/stability, and modal (frequency) analysis. The steps and procedures used for preliminary design, detailed analysis, and member optimization, are described in the following sections. 

Simplified models for buckling, second order analysis and modal analysis, using STAAD Pro, were compared with textbook and hand computation solutions to verify their accuracy. 

P(l) Depending on the specific nature of the structure, the limit state being considered and the specific conditions of design or execution, for the ultimate limit states a linear elastic or a second-order analysis method shall be used. 

P(3) In this section, the term non-linear analysis relates to analyses which take account of non-linearity of the material behaviour or of specially formulated connections. Analyses which take account of non Unear behaviour resulting from the deflection of elements are termed second order analyses (thus a non-linear second order analysis takes account of both effects). 

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