Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

354 TREATMENT OF EXPERIMENTAL DATA[B],/[A], = r, we measure the time taken for a particular value of a, a*, to bereached, we can writeor8&*)k[A]",:-= constant'rb(ti(a*)- to) = constantwhere Bi(a*) and (ti(a*) - to) represent the reduced time and actual time requiredin the ith experiment for the value of a to reach a*, and [A],, represents the initialconcentration of A used in that experiment. We see immediately that if (a+b) = 1,the time interval throughout the whole series of experiments remains constantindependent of any variations in the initial concentrations. In general, however,the time taken for the reactant concentrations to drop to a particular fraction oftheir initial concentrations depends on the values of these latter quantities asshown in eqn. (22). To obtain the total order of the reaction from the variation ofthis time interval or fractional life with initial concentration, we simply plot log, ,(ti(a*)-t,) against logl,[A],,,; according to eqn. (22), this is a straight line ofslope 1 - (a+b) and hence (a+b) can be obtained immediately. The choice of thevalue of a* is determined primarily by the need to avoid serious errors in the estimationof the time interval ?,(a*)-?,. It follows from eqns. (18) and (22) that, otherthings being equal, the closer a* is to unity, the smaller is the time interval whichmust be measured; this is merely a statement of the obvious fact that, the less thefractional decomposition of the reactants, the less the time required. Clearly therelative error in the time interval increases rapidly as the time interval decreaseswith the result that the uncertainty in the value of the slope of the logarithmicplot of time interval against initial concentration becomes too large to enablea useful estimate of (a+b) to be made. From this point of view, it is desirable tochoose a fairly low value of a* so that the relative error in the smallest time intervalmeasured is small. On the other hand if a* is made very small, say 0.1 or 0.2, thereaction has to be followed to so large an extent that the first method of obtaining(u+b) is far easier in general than the fractional life method under discussion.For these reasons, therefore, values of a* outside the range 0.5 to 0.8 are seldomused. In fact, apart from special situations where the time interval ?,(a*)-to isparticularly easy to measure experimentally, the fractional life method has but oneadvantage over the method based on the superimposition of a/log,, (t-to) anda/log,, 8 curves, viz., that each reaction need not be followed to conversions muchexceeding 50 %.Fractional life method (II):If,for some reason, it is not possible to cover a widerange of initial concentratio-kconditions as is clearly necessary for the applicationof the above method (this could-happen when one of the reactants has a limitedsolubility), essentially the same method can be applied to the data of a singleexperiment. We select a series of values of a (ao, al, a2, . . . aj . . .) such that