question_answer

(i) What is meant by half-life of a radioactive element?

(ii) The half-life of a radioactive substance is 50 s. Calculate

(a) the decay constant and

(b) time taken for the sample to decay by \[\frac{3}{4}th\] of the initial value.

Answer:

(ii) (a) \[0.01386\,{{s}^{-1}}\]    (b) 100 s

(i) Half-life of a radioactive element is defined as the time during which half the number of atoms present initially in the sample of the radioactive element will decay or it is the time during which number of atoms left undecayed in the sample is half the total number of atoms present in the sample initially. It is represented by \[{{T}_{1/2}}.\]

(ii) Here, \[{{T}_{1/2}}=50\,\,s\]

(a) Decay constant, \[\lambda =\frac{0.693}{{{t}_{1/2}}}\Rightarrow \lambda =\frac{0.693}{50}=\frac{0.693}{50}=0.01386\,{{s}^{-1}}\] (b) \[N={{N}_{0}}{{\left( \frac{1}{2} \right)}^{n}}\]                              ? (i) where,   n = number of half-lives, N = number nuclei left after n half-life And      \[{{N}_{0}}=\] original number of nuclei. Here,     \[N={{N}_{0}}-\frac{3}{4}{{N}_{0}}\Rightarrow N=\frac{1}{4}{{N}_{0}}\] \[\Rightarrow \]   \[\frac{{{N}_{0}}}{4}={{N}_{0}}{{\left( \frac{1}{2} \right)}^{n}}\]                 [From Eq. (i)] \[\Rightarrow \]   \[{{\left( \frac{1}{2} \right)}^{2}}={{\left( \frac{1}{2} \right)}^{n}}\] \[\Rightarrow \]   \[n=2\] Since,    \[n=\frac{T}{{{t}_{1/2}}}\]             \[2=\frac{\text{Total}\,\,\text{time}\,\,\text{taken}}{50\,s}\] \[\Rightarrow \]   Total time taken = 100 s