A radioactive isotopes is being produced at a constant rate X. Half life of substance is Y .
After some time the no of radioactive nuclei become constant. The value of constant is

Options

A. `(XY)/("in"(2))`
B. XY
C. (In 2) XY
D. `X/Y`

Answer

To solve the problem, we need to analyze the situation where a radioactive isotope is produced at a constant rate \( X \) and decays over time with a half-life \( Y \). We want to find the constant number of radioactive nuclei \( n \) when the production and decay rates balance each other.

### Step-by-Step Solution:

1. **Understand the Decay Equation**:
The decay of radioactive isotopes can be described by the equation:
\[ \frac{dn}{dt} = -\lambda n \]
where \( \lambda \) is the decay constant.

2. **Incorporate the Production Rate**:
Since the isotopes are being produced at a constant rate \( X \), we modify the equation to include this production:
\[ \frac{dn}{dt} = X - \lambda n \]

3. **Set the Rate of Change to Zero**:
When the number of radioactive nuclei becomes constant, the rate of change \( \frac{dn}{dt} \) becomes zero:
\[ 0 = X - \lambda n \]

4. **Rearrange the Equation**:
Rearranging the equation gives us:
\[ \lambda n = X \]
or
\[ n = \frac{X}{\lambda} \]

5. **Relate the Decay Constant to Half-Life**:
The decay constant \( \lambda \) is related to the half-life \( Y \) by the formula:
\[ \lambda = \frac{\ln 2}{Y} \]

6. **Substitute \( \lambda \) into the Equation for \( n \)**:
Substituting \( \lambda \) into the equation for \( n \):
\[ n = \frac{X}{\frac{\ln 2}{Y}} = \frac{X \cdot Y}{\ln 2} \]

7. **Final Result**:
Therefore, the constant number of radioactive nuclei \( n \) is given by:
\[ n = \frac{XY}{\ln 2} \]