Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

378 TREATMENT OF EXPERIMENTAL DATAcorresponding to determinations at different initial concentrations, there being rivalues in the first group, r2 values in the second group, . . . r,, values in the uthgroup, . . . and r, values in the uth and final group. Clearlyo=ur=Crll11= 1For example, the six first-order rate coefficients listed in Appendix 2 can be dividednaturally into 3 groups according to the initial concentration of isoprene.TABLE 3Run [Z~oprene]~ 10' k 10' s(k)-mole.litre-l SeC-1 SeC-11231.6907.097 0.0236.944 0.0157.275 0.0104 7.155 0.0363.22057.050 0.0336 0.815 7.091 0.012In other words, the runs within a group are replicate experiments. The problemwhich we have to solve is whether or not there is any significant difference betweenthe rate coefficients of the various groups. If there is no significant difference betweenthe rate coefficients so that the values are not dependent on the initialconcentrations of the reactants, we confirm that the rate expression under investigationrepresents the data adequately. In such a case where the values from a seriesof sub-sets do not differ significantly from one another, we can pool the wholeset to obtain a grand average-a set of this type is called an homogeneous set. Wecan also obtain an estimate of the standard error of the grand average as will beshown shortly.We should note in passing that we do not really need a statistical analysis of thedata presented in Table 3 to convince us that the six rate coefficients constitutean homogeneous set. The sub-set of runs 1,2 and 3, which were carried out, as faras was possible, under identical conditions, shows a scatter of values overlappingthose obtained under different initial concentrations (runs 4, 5 and 6).We approach the general .problem in the following way. Firstly, we relabel ourrate coefficients, kvi, where u specifies the group and i specifies the particular kwithin the group; thus, k,, specifies the third value of the rate coefficient in thefirst group. Secondly, we evaluate the average of the rate coefficients,(a) for each group, Ell