Q19 / 120 · JEE Main 2007 · Chemistry

A radioactive element gets spilled over the floor of a room. Its half-life period is $30$ days. If the initial activity is ten times the permissible value, after how many days will it be safe to enter the room?

  • A
    $1000$ days
  • B
    $300$ days
  • C
    $10$ days
  • Correct answer
    D
    $100$ days

Solution

Activity $\left(-\frac{\mathrm{dN}}{\mathrm{dt}}\right) \propto \mathrm{N}$ $\mathrm{N}=\mathrm{N}_{\circ}\left(\frac{1}{2}\right)^{\mathrm{n}}$ $\frac{\mathrm{N}}{\mathrm{N}_{\circ}}=\left(\frac{1}{2}\right)^{\mathrm{n}}$ $\frac{1}{10}=\left(\frac{1}{2}\right)^{\mathrm{n}} \Rightarrow 10=2^{\mathrm{n}}$ $\log 10=\mathrm{nlog} 2$ $\Rightarrow \mathrm{n}=\frac{1}{0.301}=3.32$ $\mathrm{t}=\mathrm{n} \times \mathrm{t}_{112}$ $=3.32 \times 30=99.6 \text { days }$ Hence, (D) is correct.

Hence, the correct answer is option: 'D'.

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