A nuclear reactor starts producing a radionuclide of half - life T at a constant rate R starting at time t = 0 The activity of the radionuclide at t = T is found to the A. Then `(R )/(4)` is

Options

A. `2:1`
B. `3:2`
C. `4:1`
D. `2:3`

Answer

To solve the problem, we need to understand the behavior of the radionuclide being produced in the nuclear reactor. The key points to consider are the half-life of the radionuclide, the rate of production, and the relationship between these quantities.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a radionuclide with a half-life \( T \).
- It is produced at a constant rate \( R \) starting from time \( t = 0 \).
- We need to find the relationship between \( R \) and the activity \( A \) at time \( t = T \).

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

3. **Set Up the Differential Equation**:
- The change in the number of radionuclide atoms \( n \) over time can be described by the equation:
\[ \frac{dn}{dt} = R - \lambda n \]
- Here, \( R \) is the rate of production, and \( \lambda n \) is the rate of decay.

4. **Integrate the Equation**:
- Rearranging gives:
\[ \frac{dn}{R - \lambda n} = dt \]
- Integrate both sides from \( 0 \) to \( n \) and \( 0 \) to \( t \):
\[ \int_0^n \frac{dn}{R - \lambda n} = \int_0^t dt \]

5. **Perform the Integration**:
- The left-hand side integrates to:
\[ -\frac{1}{\lambda} \ln |R - \lambda n| \Big|_0^n = -\frac{1}{\lambda} [\ln |R - \lambda n| - \ln R] = -\frac{1}{\lambda} \ln \left(\frac{R - \lambda n}{R}\right) \]
- The right-hand side integrates to \( t \).

6. **Evaluate at \( t = T \)**:
- Substitute \( t = T \) into the equation:
\[ -\frac{1}{\lambda} \ln \left(\frac{R - \lambda n}{R}\right) = T \]

7. **Relate Activity to Number of Nuclei**:
- The activity \( A \) is given by:
\[ A = \lambda n \]
- Rearranging gives:
\[ n = \frac{A}{\lambda} \]

8. **Substitute \( n \) back into the equation**:
- Substitute \( n \) into the integrated equation to find the relationship between \( R \) and \( A \).

9. **Final Calculation**:
- After substituting and simplifying, we find that:
\[ R = 2A \]

10. **Find \( \frac{R}{4} \)**:
- Therefore, we can conclude:
\[ \frac{R}{4} = \frac{2A}{4} = \frac{A}{2} \]

### Final Answer:
\[ \frac{R}{4} = \frac{A}{2} \]