Frequently a species can react in different ways to give a variety of products For example toluene can be nitrated at the ortho meta or para positions we shall consider the simplest case, that of two competing irreversible first- order reactions: A overset(k_(1)) rarr C and A overset (k_(2)) rarr D where the stoichimetric coefficients are taken as unity for simplicity. THe rate law is ((d[A])/(dt))=-k_(1)[A] -k_(2)[A] =-(k_(1)+k_(2))[A] Rightarrow [A] = [A]_(0)e^-(k_(1)+k_(2))t For C, we have ((d[C])/(dt))=k_(1)[A] =k_(1)[A]_(0)e^-(k_(1)+k_(2))t Multiplication by dt and integration from time 0(where [C]_(0)=0 to an arbitary time t gives [C] =(k_(1)[A]_(0))/(k_(1)+k_(2))(1-e^-(k_(1)+k_(2))t) Similarly integration of ((d[D])/(dt)) gives [D] = (k_(2)[A]_(0))/(k_(1)+k_(2))(1-e^-(k_(1)+k_(2))t) The sum of the rate constants k_(1)+k_(2) appears in the exponentials for both [C] and [D] At any time, we also have, [C]/[D] = k_(1)/k_(2) At high temperature, acetic acid decomposition into CO_(2) and CH_(4) and simultaneously into CH_(2)CO (ketene) and H_(2)O (i) CH_(3)COOH overset (k_(1)=3s^(-1)rarr CH_(4)+CO_(2) (ii) CH_(3)COOH overset(k_(2)=4s^(-1)rarr CH_(2)CO+H_(2)O What is the fraction of acetic acid is reacting as per reaction(i)?