A radioactive element $X$ converts into another stable element $Y$. Half life of $X$ is 2 hours. Initially…

A radioactive element $X$ converts into another stable element $Y$. Half life of $X$ is 2 hours. Initially only $X$ is present. After a time $t$, if the ratio of atoms of $X$ to $Y$ is $1: 4$, then the value of $t$ is
  1. 2 hours
  2. 4 hours
  3. between 4 hours and 6 hours
  4. 6 hours

Solution


Initially at $t=0$, amount of $X$ in sample $=5 X$ After time ' $t$ ' amount of $X$ remained in sample $=X$ From, $\frac{N}{N_0}=\left(\frac{1}{2}\right)^{\frac{t}{T}}$ We have, $\frac{X}{5 X}=\left(\frac{1}{2}\right)^{\frac{t}{2}} \Rightarrow\left(\frac{1}{2}\right)^{\frac{t}{2}}=\frac{1}{5}$ $\therefore T=2 \mathrm{~h}$ As, $\frac{1}{2^2} < \frac{1}{5} < \frac{1}{2^3} \Rightarrow \frac{1}{2^2} < \frac{1}{2^{t / 2}} < \frac{1}{2^3}$ $2 < t / 2 < 3 \Rightarrow 4 < t < 6$

Asked in: AP EAMCET 2018 (24 Apr Shift 1)

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