The half-life of a radioactive sample is `T`. If the activities of the sample at time `t_(1)` and `t_(2)` `(t_(1) lt t_(2))` and `R_(1)` and `R_(2)` respectively, then the number of atoms disintergrated in time `t_(2)-t_(1)` is proportional to

Options

A. `(R_1-R_2)T`
B. `(R_1+R_2)T`
C. `(R_1R_2)/(R_1+R_2)T`
D. `(R_1+R_2)/T`

Answer

To solve the problem, we need to determine the number of atoms disintegrated in the time interval \( t_2 - t_1 \) in terms of the activities \( R_1 \) and \( R_2 \) at times \( t_1 \) and \( t_2 \) respectively.

### Step-by-Step Solution:

1. **Understanding Activity**:
The activity \( R \) of a radioactive sample is defined as the rate of disintegration of atoms, which can be expressed as:
\[ R = -\frac{dN}{dt} = \lambda N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive atoms present.

2. **Establishing Relationships**:
At time \( t_1 \):
\[ R_1 = \lambda N_1 \]
At time \( t_2 \):
\[ R_2 = \lambda N_2 \]
Here, \( N_1 \) and \( N_2 \) are the number of atoms at times \( t_1 \) and \( t_2 \) respectively.

3. **Finding the Number of Atoms**:
From the equations above, we can express \( N_1 \) and \( N_2 \) in terms of the activities:
\[ N_1 = \frac{R_1}{\lambda} \]
\[ N_2 = \frac{R_2}{\lambda} \]

4. **Calculating Atoms Disintegrated**:
The number of atoms disintegrated between \( t_1 \) and \( t_2 \) is given by:
\[ \text{Atoms Disintegrated} = N_1 - N_2 = \frac{R_1}{\lambda} - \frac{R_2}{\lambda} = \frac{R_1 - R_2}{\lambda} \]

5. **Substituting for Decay Constant**:
The decay constant \( \lambda \) can be related to the half-life \( T \) of the sample using:
\[ \lambda = \frac{\ln 2}{T} \]
Substituting this into the equation for atoms disintegrated:
\[ \text{Atoms Disintegrated} = \frac{R_1 - R_2}{\frac{\ln 2}{T}} = \frac{(R_1 - R_2) T}{\ln 2} \]

6. **Proportionality**:
From the final expression, we can see that the number of atoms disintegrated is proportional to:
\[ (R_1 - R_2) T \]

### Conclusion:
Thus, the number of atoms disintegrated in the time interval \( t_2 - t_1 \) is proportional to \( (R_1 - R_2) T \).