Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

2 CONSISTENCY WITH EQNS. OF TYPE -d[A]/dt = k[A]"[BIb 3716, is termed a residual. Eqns. (58) and (59) give the values of k; and Ii which makeEi a minimum in accordance with the criterion given earlier.s(k;) being an estimate of the standard error in k; .In writing these expressions, it is implicitly assumed that the observations ui,are subject to random errors and that the reaction times (tij- ti,,) are free fromrandom errors. The assumption that the random errors occur solely in u is reasonablesince, in the majority of kinetic studies, time can be measured with muchgreater precision than concentration. When the random errors in the determinationof time are commensurate with the random errors in the determination ofconcentration, the above equations are no longer valid; we shall not discuss thissituation in view of the difficulties which arise but would suggest that, if possible,every effort should be made to avoid it by redesigning the experimental procedure.(i) The weights wijThe real problem in utilizing expressions (58) to (63) is the assignment of theweights, wij. To proceed further, it is necessary to digress into a discussion of somebasic statistical ideas.Let us return to our basic parameter, u. Suppose that it were possible to make avery large number of determinations of u at some particular value of the reactiontime, t- to, by performing a large number of identical experiments. Supposefurther that all systematic errors had been eliminated and that the observed valuesof u were distributed normally about a mean value of A; i.e., the frequency withwhich a value of u lying between u and u + du is found is given by1The quantity a'(.) is termedthe variance of the population of u or more simply thevariance of u and is the average of the squares of the deviations of the u valuesfrom the population mean, A; i.e.a'(.)1 i=R= - C(ui-A)*'R i=1t This equation must not be confused with the equation giving an estimate of the variance of thepopulation, sa(a), from a limited sample of r a-values, viz.1 i=rsa(a) = - Cr-1 i=lwhere Cr is the sample mean.References p. 407