Isoparametrical relationship

The utmost generalization of the isokinetic relationship is the so called isoparametrical relationship (171), considering the dependence of a given... 

Analysis of the structure-reactivity cross-correlations shows the existence of isoparametric relationships in the reactions of Y-substituted benzyl bromides with X-substituted anilines in dioxane and in its mixtures with DMSO at 40 The kinetics of hydrolysis of 1-arylethyl ethers of salicyclic acid, catalysed intramolecularly by 0-CO2H, have been studied. Analysis of substituent effects in both arylethyl and leaving groups provides the most detailed available mechanistic insight into a reaction involving efficient intramolecular proton-transfer catalysis. The mechanism is very different from classical general acid-base catalysis. ... 

It follows that for a special value of one parameter, the observed value of y is independent of the second parameter. This happens at Ii= a2/ai2 or I2 = -ai/ai2 any of these values determines y= a -aia2/ai2, the so called isoparametrical point. The argument can evidently be extended to more than two independently variable parameters. Experimental evidence is scarce. In the field of extrathermodynamic relationships, i.e., when j and 2 are kinds of a constants, eq. (84) was derived by Miller (237) and the isoparametrical point was called the isokinetic point (170). Most of the available examples originate from this area (9), but it is difficult to attribute to the isoparametrical point a definite value and even to obtain a significant proof that a is different from zero (9, 170). It can happen—probably still more frequently than with the isokinetic temperature—that it is merely a product of extrapolation without any immediate physical meaning. 

A 3D isoparametric, finite-element analysis has been carried out by Bangash. Figure 9.25 shows the finite-element mesh scheme for a dynamic model for a nuclear shelter. Figure 9.26 gives the relationships between pressure and time. The results are given in Fig. 9.27. 

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