Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

390 TREATMENT OF EXPERIMENTAL DATAIt is obvious from the preceding discussion that it is almost essential to determinethe magnitude of the physical property, 4, at equally spaced time intervals. If theobservations are irregularly spaced in time, only a limitednumber of pairs of experimentalobservations can be obtained which are separated by the same interval; anincreased number of paired Cpr,,(br,+5 values could be obtained by interpolationbut this is less satisfactory than using experimental values.It should also be clear that the same procedure can be applied to every run in aseries of experiments at different initial concentrations. Consequently we obtaina series of values for the rate coefficients: k,, kz . . . ki . . . etc., in just the sameway as we would obtain had it been possible to treat the data by the more conventionalmethod described earlier. The method of averaging these rate coefficientsfollows the lines already described.Mangelsdorf's method.2 Mangelsdorf has presented an alternative method ofprocessing the data, again utilizing paired values of the physical property $separated by a constant time interval, t. As before, we write4t,-$m = (40-4m)e-k(t,-'o)= (40-4m)e4rj+r-$m-k(tj+ r -to)Instead of subtracting these two equations to eliminate on the left-hand sideas suggested by Guggenheim, we divide one by the other to eliminate the factor(40-$m)e-k(fj-to) and obtainEqn. (83) shows that a plot of $,+, against 4t, is a straight line of slope e-kr andintercept at &, = 0 of 4m(l -e-kr). Instead of using the intercept, the value of 4acan be found in a rather simpler way from the point of intersection of the abovestraight line with the line $,,+, = $t,; from eqn. (83), the co-ordinates of thispoint are clearly (q5m, 4m). Strictly speaking, the least squares formulae previouslygiven [eqns. (58) to (63)] cannot be used to obtain the best values of the slope,intercept, and their associated standard errors: The reason is that both &r, and$t,+r are subject to random errors whereas eqns. (58) to (63) pertain to the situationwhere only the dependent variableis subject to error. To take account of thepresence of random errors in both variables in a proper fashion requires us toexamine the statistics of the straight line in considerable detail. This would be outof place in an article of this type. However, since the two variables are characterizedby the same standard error (assuming this to be independent oft,), it is reasonableto proceed with the computation of the slope of the line as if only the observed$t,+r values were subject to error.