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NEET PHYSICSMedium

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:

A

30 days

B

50 days

C

60 days

D

15 days

Step-by-Step Solution

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 $N_2 = 1 - 2/3 = 1/3$ .

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 $t_1$ .

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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