The half-life of a radioactive nucleus is 50 days. The time interval $(t_2 - t_1)$ between the time $t_2$ when

Question

The half-life of a radioactive nucleus is 50 days. The time interval $(t_2 - t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it has decayed and the time $t_1$ when $\frac{1}{3}$ of it had decayed is:

Accepted Answer

1. **Analyze Remaining Amounts:** Radioactive decay calculations use the amount of substance *remaining* (undecayed), not the amount decayed.
   - At time $t_1$: Fraction decayed = $1/3$. Fraction remaining $N_1 = 1 - 1/3 = 2/3$.
   - At time $t_2$: Fraction decayed = $2/3$. Fraction remaining $N_2 = 1 - 2/3 = 1/3$.
2. **Compare the States:** Observe the relationship between the remaining amounts at $t_1$ and $t_2$.
   $$ \frac{N_2}{N_1} = \frac{1/3}{2/3} = \frac{1}{2} $$
   - This shows that the amount of substance at $t_2$ is exactly half of the amount at $t_1$.
3. **Apply Half-Life Definition:** The time required for a radioactive substance to reduce to half of its initial value is defined as one half-life ($T_{1/2}$) .
   - Therefore, the time interval $\Delta t = t_2 - t_1$ is exactly equal to one half-life.
   - Given $T_{1/2} = 50$ days.
   - $\Delta t = 50$ days.

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