Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

2 CONSISTENCY WITH EQNS. OF TYPE -d[A]/dt = k[A]"[BIb 365averageofthetwotimest,-, andt,,,;i.e.,tj - t (t,-i+t,+,),andsothecontributionsof all values of f(a) intermediate between the first and last are seriouslyunderweighted. Indeed, in the special case when the time intervals between successiveconcentration determinations are equal so that t, = 3 (tj-, + ti+ ,) the weightingfactor becomes zero for all values ofj from 1 to (n- 1) inclusive. In this situation,none of the intermediate concentration terms contribute to the mean valueof k and we obtainf(ao)- - 1 f(an))h-tn-1As far as this method of obtaining an average value of k is concerned, the intermediateconcentration determinations at equally spaced time intervals might aswell not have been made!The general conclusion to be drawn from this analysis is that there is littlepoint in performing the laborious operation of calculating individual k values fromsuccessive pairs of observations in order to obtain a mean value since this procedureplaces undue weight on the initial and final observations which in practice areoften the least accurate.Method 2. Calculation of the rate coefficient using a graphical method.As we have already pointed out, each of the integrated forms of the rate equationmay be writtenf(ao)-f(a) = k'(t-to)where k' is proportional to the rate coefficient k. Clearly a plot of f(a) against(t- to) should be linear with an intercept equal to f(ao) at (t - to) = 0 and a slope,-k'. From the slope and the value of m, the rate coefficient follows at once. Weshall not consider at this point the arguments for and against constraining the lineto pass through f(ao) at (t - to) = 0. For the moment, we shall adopt this proceduresince the object of this section is merely to present some of the problems whicharise in the treatment of data; the recommended method is always to make useof the data of a number of experiments as described in the next section.In view of our previous discussion, it is clearly of importance to evaluate thecontribution of each experimental observation to the average value of k so obtained.This depends on the criterion according to which the straight line through thepoints is constructed. If we assume that the line is drawn so that the algebraic sumof the deviations, S, of the experimental points from the line is close to zero, wecan deduce the contribution of each point to the average value of k as follows.References p. 407