Question

The nucleus of Element X is unstable and decays into Element Y with a probability z. You are given a pure sample of Element X for a physics assignment, but by the time you get around to doing it, there are six times as many atoms of Element Y as Element X. How much time did you wait before observing in terms of z

Answer 1

This can be done very easily using the concept of the radioactive decay probability distribution.

Given XY het N ~ be the initial number of x. n be the number of decayed a nuleuses out of the total N , it is eanal to the number of y. Also given z => as the probability with which X atoms decay to Ý in a unit time interval. Now suppose we take ne radio active atoms and wait for a time t. The probability that natoms have changed or decayed out of N is given by the Binomial distribution. P[n; N:43 = NE Ped C1-pe}" since z is given : the probability that it decays in a time interval Non t is - P(t)=zt From ② eq ♡ becomes P[n; M;t] = N! (24)" (1- 2t - ① (N-n)!n! From the above, the average or expected number of decayed atoms out of Nin time t is N&plt) = N zł

e , the number of Y atoms at a time t = n = NzE so the number og X atoms remaining to decay - N(t) = N-n =N-Nzt Nuct = N{1-2t Now, we are given the ratio og Y atoms at t and X atoms at t to be 6. n M = 6 =) n = 6 NCI-zt = Nzt N[1-2t7 = 6 zt = 6-62€ :: 72 t=6 N - taking in to the base ΟΥ or t = ln (4) is the creauired time.