Scattered isergonic profiles.
Equations were developed for the geometric analysis of scatter in isergonic profiles as a function of vertical (delta delta H), horizontal (delta delta S), and perpendicular (d) distances from the error line. Sigma (sigma-) correlates with each of these quantities of the test series to produce stastistics (F and r) identical to those for correlation with delta G. The vertical distances, delta delta H, are shown to be identical to the substituent effect, beta delta G; thus, it is concluded that the Hepler approach of implicating a solvation  mechanism on the basis of rho-beta correlations for these distances is not warranted. The uncertainty of points on the isergonic profile for the test series was determined using the joint confidence region ("error contour") approach. Statistical limits of the contour lengths overlapped extremely between members of the test series. To eliminate such overlap for series showing small ranges in substituent effects, extremely precise rate data would be required with N (number of data points) = 4 or 5 for the Arrhenius fits . A comparison between d values and the perpendicular half-widths (w/2) of the error contours resulted in d greater than w/2 for each member of the test series (excluding H), providing that scatter in the profile could not result from experimental error. A further comparison between differences in d, delta d, from suggested beta lines for solvation and w/2 values showed that a beta range as large as 45 degrees could be included in the error contour widths of the test series. Since the average reaction temperature, T, fell within this range solvation effects could not be distinguished statistically from error effects.
D001565:Benzoates; D008962:Models, Theoretical; D013816:Thermodynamics
D001565:Benzoates