Quadratic relationship

Electrostrictive materials are materials that exhibit a quadratic relationship between mechanical stress and the square of the electric polari2ation (14,15). Electrostriction can occur in any material. Whenever an electric field is appHed, the induced charges attract each other, thus, causing a compressive force. This attraction is independent of the sign of the electric field and can be approximated by... 

Quantitative controllable variables are ftequentiy related to the response (or performance) variable by some assumed statistical relationship or model. The minimum number of conditions or levels per variable is determined by the form of the assumed model. For example, if a straight-line relationship can be assumed, two levels (or conditions) may be sufficient for a quadratic relationship a minimum of three levels is required. However, it is often desirable to include some added points, above the minimum needed, so as to allow assessment of the adequacy of the assumed model. 

Fig. 4.4. The piezoelectric charge produced by elastic strain in x-cut quartz and z-cut lithium niobate is well represented by a quadratic relationship without a need for fourth-order contributions.

Baczek, T., Markuszewski, M., Kaliszan, R. Linear and quadratic relationships between retention and organic modifier content in eluent in reversed phase high-performance liquid chromatography a systematic comparative statistical study. 

This quadratic relationship results from the fact that the number of oxygen atoms is proportional to the cumulative number N of supernovas and that the number of high-energy... 

In the course of the investigation (146) it was also ex ined whether Eq. (13) should be replaced by a quadratic relationship between In k and do. but no marked improvement was found in the fit of retention factor composition data. 

See Table 3.2). One additional remark should be made with respect to Fig. 3.2. In this figure parameter p is calculated with the aid of the experimental viscosity-shear rate relation. As a consequence, the deviation from the quadratic relationship is less pronounced than in Fig. 3.1. 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

In contrast, however, a quadratic relationship between SH and e2qQlh was used in a recent report [71 ] based on the earlier correlation of Berglund and Vaughan [75]. 

As was true for statistically significant linear relationship between (Tmeta and x was found. Quadratic relationships for o-meta similar in form to the relationships of Equations 4a and 4b are presented in Equations 5a and 5b for the meta-substituted TFMS derivatives. Both X2 and x terms are again necessary for optimum correlation although the final correlation relationship in Equation 5b for o-meta is not nearly as statistically significant (r2 = 0.44) as the final relationship for [Pg.263]

The dependence of k (retention factor) on the volume percentage of the modifier is a subject of great controversy. One school of thought claims a linear dependence [36,37], whereas another advocates a quadratic relationship [38, 39] and indicates that deviation from linearity will be more pronounced at high concentrations of the modifier. 

Another possibility is to obtain physico-chemical models. Consider for instance the optimization of pH and solvent strength. The subject was studied among others by Marques and Schoenmakers [66], Schoenmakers et al. [67,68] and Bourguignon et al. [62] for the optimization of the separation of a mixture of chlorophenols. Marques and Schoenmakers proposed the following model based on a quadratic relationship between log k and the solvent composition () and on dissociation equilibria ... 

All these coefficients are unknown and are estimated by fitting experimental results to the model. The question is then how to construct the model. The authors reasoned that the quadratic relationship between log k and O requires 3 levels and that the sigmoid relationship between log A and pH could be described by four or five points. They... 

The curvature often observed in plots of log A- versus tp led [161 to a quadratic relationship ... 

The calculations yield S and AS° values of appreciable magnitude, +18 eu (cal moff and +10 eu, respectively, and in contrast to the free energies, calculated S and H quantities depart substantially from the quadratic relationship given by Eq. 112. In the case of small a and also small 7.,s-, one expects, from Eq. 112, the value of S /AS° to be approximately 0.5, whereas the calculations yield a ratio of approximately 2 (the distinction is pronounced even when the sizable estimated statistical uncertainties ( 5 eu) in the calculated entropies is taken account of). For this result to be compatible with Eq. 112, it would require a sizable positive value of /..S, but in fact the simulation results indicated a.s 0. Thus, for reaction 107, as represented by the simulation and model molecular Hamiltonian [36], we infer that near room temperature the separate entropy and enthalpy quantities are not well accounted for by a harmonic model, whereas, due to compensating effects, harmonic behavior is recovered when they are combined in the free-energy quantities. 

However, in the case of protonated amines, quadratic relationships were obtained with MeOH. l general, for polar solutes, linearity holds within a limited range using mobile phases rich in aqueous component. In contrast, for... 

This simple, linear relationship is the most frequently used. However. Bratsch (sec Footnote 28 and Table 5.6) has presented evidence for a somewhat better-fitting quadratic relationship. 

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