The half-life of a radioactive nucleus is `50` days. The time interval `(t_2 -t_1)` between the time `t_2` when `(2)/(3)` of it has decayed and the time `t_1` when `(1)/(3)` of it had decayed is

Options

A. 50 days
B. 60 days
C. 15 days
D. 30 days

Explanation

To solve the problem, we need to find the time interval \( (t_2 - t_1) \) between the times when \( \frac{2}{3} \) of the radioactive nuclei have decayed and when \( \frac{1}{3} \) of them have decayed. The half-life of the radioactive nucleus is given as 50 days.

### Step-by-step Solution:

1. **Understanding Decay**:
- When \( \frac{1}{3} \) of the nuclei have decayed, \( \frac{2}{3} \) of the original nuclei remain undecayed.
- When \( \frac{2}{3} \) of the nuclei have decayed, \( \frac{1}{3} \) of the original nuclei remain undecayed.

2. **Setting Up the Equations**:
- Let \( N_0 \) be the initial number of undecayed nuclei.
- At time \( t_1 \) (when \( \frac{1}{3} \) has decayed):
\[ N(t_1) = N_0 \left( \frac{2}{3} \right) = N_0 \left( \frac{1}{2} \right)^{\frac{t_1}{T}} \]
Here, \( T \) is the half-life (50 days).

- At time \( t_2 \) (when \( \frac{2}{3} \) has decayed):
\[ N(t_2) = N_0 \left( \frac{1}{3} \right) = N_0 \left( \frac{1}{2} \right)^{\frac{t_2}{T}} \]

3. **Forming the Equations**:
- From the equations above, we can write:
\[ \frac{2}{3} = \left( \frac{1}{2} \right)^{\frac{t_1}{T}} \quad \text{(1)} \]
\[ \frac{1}{3} = \left( \frac{1}{2} \right)^{\frac{t_2}{T}} \quad \text{(2)} \]

4. **Dividing the Equations**:
- Dividing equation (1) by equation (2):
\[ \frac{\frac{2}{3}}{\frac{1}{3}} = \frac{\left( \frac{1}{2} \right)^{\frac{t_1}{T}}}{\left( \frac{1}{2} \right)^{\frac{t_2}{T}}} \]
- This simplifies to:
\[ 2 = \left( \frac{1}{2} \right)^{\frac{t_1 - t_2}{T}} \]

5. **Equating Exponents**:
- Since the bases are the same, we can equate the exponents:
\[ 1 = \frac{t_2 - t_1}{T} \]
- Rearranging gives:
\[ t_2 - t_1 = T \]

6. **Substituting the Half-Life**:
- We know \( T = 50 \) days, so:
\[ t_2 - t_1 = 50 \text{ days} \]

### Final Answer:
The time interval \( (t_2 - t_1) \) is **50 days**.