An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

Atoms and Nuclei PYQ — JEE Advanced 2016 · Shift 2 · Q43 with Solution

View written solutionFree

Correct answer: C

Let the initial radiation level be $R_0$ .

The laboratory is safe when radiation falls to the permissible level $R_p$ .

Given: $R_0 = 64R_p$
Radioactive decay follows: $R = R_0\left(\frac{1}{2}\right)^{t/T_{1/2}}$ where half-life $T_{1/2} = 18$ days.
For the laboratory to become safe: $R = R_p$

So, $R_p = 64R_p\left(\frac{1}{2}\right)^{t/18}$
Cancel $R_p$ from both sides: $1 = 64\left(\frac{1}{2}\right)^{t/18}$

Since $64 = 2^6$ , $1 = 2^6 \cdot 2^{-t/18}$ $2^{6 - t/18} = 1$
Therefore, $6 - \frac{t}{18} = 0$ $\frac{t}{18} = 6$ $t = 108\text{ days}$
So the minimum time required is: $\boxed{108\text{ days}}$
Checking options:
- A: $64$ days $\rightarrow$ incorrect
- B: $90$ days $\rightarrow$ incorrect
- C: $108$ days $\rightarrow$ correct
- D: $120$ days $\rightarrow$ incorrect

Therefore, the correct answer is Option C.