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

Explanation

To solve the problem step-by-step, we will analyze the relationship between radioactivity, the number of atoms, and the time interval given the half-life of the specimen.

### Step 1: Understand the Definitions
- **Radioactivity (R)** is defined as the number of disintegrations per unit time. It is proportional to the number of radioactive nuclei present in the sample.
- The relationship between radioactivity (R) and the number of nuclei (N) is given by:
\[ R = \lambda N \]
where \(\lambda\) is the decay constant.

### Step 2: Express Radioactivity at Two Different Times
At time \(T_1\):
\[ R_1 = \lambda N_1 \]
At time \(T_2\):
\[ R_2 = \lambda N_2 \]

### Step 3: Relate the Decay Constant to Half-Life
The decay constant \(\lambda\) is related to the half-life \(T\) by the formula:
\[ \lambda = \frac{0.693}{T} \]

### Step 4: Calculate the Number of Atoms Disintegrated
The number of atoms that have disintegrated between times \(T_1\) and \(T_2\) can be expressed as:
\[ N_1 - N_2 \]
Using the expressions for \(N_1\) and \(N_2\) from the radioactivity equations:
\[ N_1 = \frac{R_1}{\lambda} \quad \text{and} \quad N_2 = \frac{R_2}{\lambda} \]
Thus:
\[ N_1 - N_2 = \frac{R_1}{\lambda} - \frac{R_2}{\lambda} = \frac{R_1 - R_2}{\lambda} \]

### Step 5: Substitute the Decay Constant
Substituting \(\lambda\) into the equation:
\[ N_1 - N_2 = \frac{R_1 - R_2}{\frac{0.693}{T}} = \frac{(R_1 - R_2) T}{0.693} \]

### Step 6: Determine Proportionality
The number of atoms that have disintegrated, \(N_1 - N_2\), is proportional to:
\[ R_1 - R_2 \]
Thus, we can conclude that:
\[ N_1 - N_2 \propto (R_1 - R_2) \cdot T \]

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

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