The radioactivity of a sample is `R_(1)` at a time `T_(1)` and `R_(2)` at time `T_(2)`. If the half-life of the specimen is T, the number of atoms that have disintegrated in the time `(T_(2) -T_(1))` is proporational to

Options

A. `(R_(1)T_(1) - R_(2)T_(2))`
B. `(R_(1)-R_(2))T`
C. `(R_(1) - R_(2))//T`
D. `(R_(1) - R_(2))(T_(1) - T_(2))`

Explanation

To solve the problem, we need to determine how the number of atoms that have disintegrated in the time interval \( (T_2 - T_1) \) is related to the activities \( R_1 \) and \( R_2 \) at times \( T_1 \) and \( T_2 \), respectively, and the half-life \( T \) of the specimen.

### Step-by-Step Solution:

1. **Understanding Radioactivity**:
The radioactivity \( R \) of a sample is proportional to the number of undecayed nuclei \( N \) present in the sample. This can be expressed as:
\[ R = \lambda N \]
where \( \lambda \) is the decay constant.

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

3. **Calculating Number of Atoms**:
At time \( T_1 \), the number of atoms \( N_1 \) can be expressed as:
\[ N_1 = \frac{R_1}{\lambda} \]
At time \( T_2 \), the number of atoms \( N_2 \) is:
\[ N_2 = \frac{R_2}{\lambda} \]

4. **Finding the Number of Disintegrated Atoms**:
The number of atoms that have disintegrated between times \( T_1 \) and \( T_2 \) is given by:
\[ N_{\text{disintegrated}} = N_1 - N_2 \]
Substituting the expressions for \( N_1 \) and \( N_2 \):
\[ N_{\text{disintegrated}} = \frac{R_1}{\lambda} - \frac{R_2}{\lambda} = \frac{R_1 - R_2}{\lambda} \]

5. **Substituting for \( \lambda \)**:
Now, substituting \( \lambda \) in terms of half-life \( T \):
\[ N_{\text{disintegrated}} = \frac{R_1 - R_2}{\frac{\ln 2}{T}} = \frac{T(R_1 - R_2)}{\ln 2} \]

6. **Conclusion**:
The number of atoms that have disintegrated in the time interval \( (T_2 - T_1) \) is proportional to \( (R_1 - R_2) \).

### Final Answer:
The number of atoms that have disintegrated in the time \( (T_2 - T_1) \) is proportional to \( (R_1 - R_2) \).