From a radioactive substance `n_1` nuclei decay per second at an instant when total number of nuclei are `n_2`. Find half-life of the radioactive substance.

Explanation

To find the half-life of a radioactive substance given the decay rate and the total number of nuclei, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have `n_1` nuclei decaying per second.
- The total number of nuclei present at that instant is `n_2`.

2. **Write the Rate of Decay**:
- The rate of radioactive decay is given by the equation:
\[ -\frac{dn}{dt} = \lambda n \]
- Here, \( \lambda \) is the decay constant, and \( n \) is the total number of nuclei.

3. **Substitute the Known Values**:
- We know that the decay rate \( -\frac{dn}{dt} \) is equal to \( n_1 \) and the total number of nuclei \( n \) is equal to \( n_2 \).
- Therefore, we can write:
\[ n_1 = \lambda n_2 \]

4. **Solve for the Decay Constant \( \lambda \)**:
- Rearranging the equation gives us:
\[ \lambda = \frac{n_1}{n_2} \]

5. **Use the Half-Life Formula**:
- The half-life \( t_{1/2} \) of a radioactive substance is given by the formula:
\[ t_{1/2} = \frac{\ln 2}{\lambda} \]

6. **Substitute \( \lambda \) into the Half-Life Formula**:
- Now substituting the expression for \( \lambda \) into the half-life formula:
\[ t_{1/2} = \frac{\ln 2}{\frac{n_1}{n_2}} = \frac{n_2 \ln 2}{n_1} \]

7. **Final Result**:
- Therefore, the half-life of the radioactive substance is:
\[ t_{1/2} = \frac{n_2 \ln 2}{n_1} \]