The half-life of a radioactive substance is 30 minutes. The time (in minute) taken between 40% decay and 85% d

Question

The half-life of a radioactive substance is 30 minutes. The time (in minute) taken between 40% decay and 85% decay of the same radioactive substance is:

Accepted Answer

1. **Understand Decay vs. Remaining:** Radioactive decay calculations are based on the amount of substance *remaining* ($N$), not the amount decayed.
   - At 40% decay, the amount remaining $N_1 = (100 - 40)\% = 60\%$ of the initial amount $N_0$.
   - At 85% decay, the amount remaining $N_2 = (100 - 85)\% = 15\%$ of the initial amount $N_0$.
2. **Analyze the Change:** Compare the remaining amounts $N_1$ and $N_2$.
   $$ \frac{N_1}{N_2} = \frac{60\%}{15\%} = 4 $$
   - Since $4 = 2^2$, the substance has reduced by a factor of 4, which corresponds to exactly **2 half-lives**.
   - Sequence: $60\% \xrightarrow{T_{1/2}} 30\% \xrightarrow{T_{1/2}} 15\%$.
3. **Calculate Time:**
   - Given half-life ($T_{1/2}$) = 30 minutes.
   - Time taken = $2 	imes T_{1/2} = 2 	imes 30 	ext{ min} = 60 	ext{ min}$.

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