The half-life of a radioactive sample is \(T\). The fraction of the initial mass of the sample that decays…

The half-life of a radioactive sample is \(T\). The fraction of the initial mass of the sample that decays in an interval \(T / 2\) is
  1. \(\frac{1}{\sqrt{2}}\)
  2. \(\sqrt{2}\)
  3. \(\frac{(\sqrt{2}-1)}{\sqrt{2}}\)
  4. \(\frac{(\sqrt{2}+1)}{\sqrt{2}}\)

Solution

Fraction remains after \(n\) half-lives is given as \(\frac{N}{N_0}=\left(\frac{1}{2}\right)^n=\left(\frac{1}{2}\right)^{t / T} \quad\) [where, \(T\) is half-life] \(\Rightarrow \quad \frac{N}{N_0}=\left(\frac{1}{2}\right)^{\frac{t}{T}}\) Given, \(\quad t=\frac{T}{2}\) \(\therefore \quad \frac{N}{N_0}=\left(\frac{1}{2}\right)^{\frac{T / 2}{T}}=\left(\frac{1}{2}\right)^{\frac{1}{2}}=\frac{1}{\sqrt{2}}\)

Asked in: AP EAMCET 2020 (21 Sep Shift 2)

Practice more Structure of Atoms and Nuclei questions on Aicharya