Dual parameters

The DSP treatment is also instructive for the Na,Np-rearrangement. The rate constants for solvolysis in TFE ( s) yield the dual-parameters equation given in Scheme 8-11, and those for the corresponding Na,Np-rearrangement ( r) yield the equation of Scheme 8-12 (Ravenscroft and Zollinger, 1988). 

In principle it should be possible to predict quantitatively the reactivity of such species containing nucleophilic homolytic leaving groups towards diazonium ions, by using a dual parameter equation. One parameter serves as a measure of the donor property of the particle the other parameter is the redox potential. However, the complex nature of kinetics of homolytic dediazoniations is likely to be a great obstacle in attempts to calculate rate constants referring only to the radical-generation step. 

Titus, J.A., Haugland, R.P., Sharrow, D.M., and Segal, J. (1982) Texas Red, a hydrophilic, red-emitting fluorophore for use with fluorescein in dual parameter flow microfluorometric and fluorescence microscopic studies./. Immunol. Meth. 50, 193-204. 

Relative activation enthalpies (Aif) in Table 2 were converted to o% kx k ) at 298 K, and were plotted against Hammett a constants. Here, we used enthalpies, because the size of the entropy and hence the free energy depend much on low frequencies, which are less reliable than higher frequencies, especially for compounds with weak interactions such as TS (8). The use of free energy (AG ) gave similar correlations with more scattered points. As for the Hammett o constant, we used dual-parameter o constants in the form of the Yukawa-Tsuno equation (LArSR equation) (9) as defined in eq 3. Here, the apparent a constant (aapp) has a variable resonance contribution parameter (r), which varies depending on the nature of the reaction examined for t-cumyl... 

The pKa values of amidines containing two substituted phenyl rings (one at the imino nitrogen and the other at the amidino carbon atom) do not obey a simple dual parameter equation, but fit the following equation ... 

Thermal isomerization of ketenimines to nitriles " are known to take place via recombination of the radical pair. Rate constants for the isomerization of diphenyl N-(substituted benzyl) ketenimines at several temperatures were measured. The Hammett dual parameters equation (log ky/kg = pa + p a ) " shows a better correlation than either of the Hammett single correlations (log ky/kg = pa and log ky/kg = p a"). This... 

Yukawa and Tsuno have still another approach, also involving dual parameters Yukawa Tsuno Bull. Chem. Soc. Jpn. 1959, 32. 971. For a review and critique of this method see Shorter, in Chapman Shorter. Ref. 18. pp. 119-173, pp. 126-144. This article also discusses the Swain-Lupton and Taft oh cr approaches. For yet other approaches,... 

A DSP analysis, in which contributions due to inductive and resonance effects of substituents are considered separately, has been applied in order to obtain a deeper level of insight into the electronic factors influencing pyridinium ion acidities. It has been said that a ... single parameter treatment has the merit of simplicity but where this suggests correlations which are unrealistic in terms of chemical experience, then a dual parameter approach is necessary. 53 The correlation has the form of Eq. (8). While only one set of inductive-effect... 

Correlations of p/sTa values with the Hammett or dual-parameter equations have been reported for 5-substituted 4-amino-2-... 

Fig. 4.2. The four histograms and six dual-parameter plots derived from the data from a four-parameter cytometer. Each of the six dual-parameter plots could be drawn with the x and y axes reversed.

Fig. 4.5. Quadrants are cursors or markers for delineating the intensity (according to channel number) of cells of interest in dual-parameter analysis.

Fig. 4.6. Identical single-color histograms derived from quite different cell samples. The dual-parameter dot plots allow us to distinguish these two populations of cells.

Dual-parameter correlations constitute the standard procedure for analysis of much flow cytometric data. Most cytometers provide us with four or five or more parameters of information. Because humans, in general, are used to thinking in two dimensions, actually correlating three or more parameters with each other can seem rather difficult— in terms of both computer software and our ability to keep track of the strategy (Fig. 4.8). For purposes of data analysis, the most com-... 

Quadrant For dual-parameter analysis, a two-dimensional plot of particles according to the correlated fluorescence intensities of two colors is, traditionally, divided into four quadrants. The division is based on the background fluorescence of the unstained control sample. The quadrants, from this division, will contain (1) cells that have stained with the first fluorochrome only (2) cells that have stained with both fluorochromes (3) cells that remain unstained and (4) cells that have stained with the second fluorochrome only. In other words, the quadrants are defined so that they delineate the two types of single positive cells (1 and 4), double-negative cells (3), and double-positive cells (2). 

In a similar competition study, Jiang et al. observed a value of -0.99 for the addition of CF3- to p-substituted phenyl acetylenes. They found that applying a dual parameter equation (o+ ct- ) led to a much better correlation [126]. 

M. Misono, E. Ochiai, Y. Saito, and Y. Yoneda, A new dual parameter scale for the strength of Lewis acids and bases with evaluation of their softness, J. Inorg. Nucl. Chem. 29 2685 (1967). 

Earlier, for a more diverse group of one-hundred molecules, containing heteroatoms and most typical functional groups, e.g. OH, NH2, OCH3, CN, N02, F, Cl etc., we have shown that their normal boiling points and critical temperatures (and other critical constants) can be related in dual-parameter relationships to some of our computed molecular surface properties66,72,73. Our best relationships for TbP and Tc are of the forms given in equations 18 and 19 the linear correlation coefficients are 0.949 and 0.914, respectively ... 

For the more limited variety of molecules in Table 4, we have found that surface area and vrelationships between molecular properties and and Tc. These are given in equations 20 and 21 ... 

For correlation analysis, the Y-T equation can be applied directly by least-squares analysis of the dual parameter relationship. However, improvement of the correlation is not of much interest since the increased number of parameters in the Y-T equation should inevitably improve the precision. Particularly in the investigation of reaction mechanisms, insufficient improvement could be more important for indicating the involvement of mechanistic complexity. The correlation will be frequently modified by various extraneous factors. We therefore illustrate the behaviour of substituent effects in several typical benzylic solvolyses in terms of the Y-T relationship. 

Swain-Lupton equation A dual-parameter approach to the correlation analysis of substituent effects, which involves a field constant (F) and a resonance constant (R). The original treatment was modified later. The procedure has been considerably applied, but also much criticized. 

The nucleophilicities are found to be dependent on electronic, steric, and symbiotic effects, and limited series obeyed a constant selectivity , a reactivity-selectivity or a dual-parameter linear free-energy relationship. The conclusion made was that because of different blends of the effects, the construction of a substrate-independent nucleophilicity scale was impossible at present, but an approximate scale was presented. In nucleophilic reactions on relatively long lived vinyl cations, the steric effects predominate, but at constant steric effects, reactivity-selectivity relationships were found for very short series of substrates. Additional data are required for constructing more reliable nucleophilicity scales toward neutral and positively charged vinylic carbons. 

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