In a radioactive decay process, the activity is defined as A = - d N d t , where N ( t ) is the number of radioactive nuclei at time t . Two radioactive sources, S 1 and S 2 have same activity at time t = 0 . At a later time, the activities of S 1 and S 2 are A 1 and A 2 , respectively. When S 1 and S 2 have just completed their 3 rd and 7 th half-lives, respectively, the ratio A 1 A 2 is _____.

Step-by-step solution

Activity for both the cases can be written as, A 1 = A 0 e - λ 1 t 1 also A 2 = A 0 e - λ 2 t 2 . Half-life for the reaction can be written as, t 1 2 1 = ln 2 λ 1 and hence at t 1 = 3 ln 2 λ 1 . A 1 = A 0 e - λ 1 3 ln 2 λ 1 ⇒ A 1 = A 0 e - 3 ln 2                                 . . .   i Similarly, for the second case, we can write t 2 = 7 ln 2 λ 2 and therefore, A 2 = A 0 e - λ 2 7 ln