The decay constant of a radioactive isotope is `lambda`. If `A_1` and `A_2` are its activites at time `t_1` and `t_2` respectively, then the number of nuclei which have decayed the time `(t_1-t_2)`

Options

A. `A_1t_1-A_2t_2`
B. `A_1-A_2`
C. `(A_1-A_2)//lambda`
D. `lambda(A_1-A_2)`

Explanation

To solve the problem, we need to find the number of nuclei that have decayed in the time interval \( (t_1 - t_2) \) given the activities \( A_1 \) and \( A_2 \) at times \( t_1 \) and \( t_2 \) respectively, and the decay constant \( \lambda \).

### Step-by-Step Solution:

1. **Understand the relationship between activity and number of nuclei:**
The activity \( A \) of a radioactive substance is given by the formula:
\[ A = \lambda N \]
where \( N \) is the number of undecayed nuclei and \( \lambda \) is the decay constant.

2. **Express activities at times \( t_1 \) and \( t_2 \):**
At time \( t_1 \):
\[ A_1 = \lambda N_1 \]
At time \( t_2 \):
\[ A_2 = \lambda N_2 \]

3. **Relate the number of undecayed nuclei at \( t_1 \) and \( t_2 \):**
The number of undecayed nuclei at \( t_1 \) and \( t_2 \) can be expressed as:
\[ N_1 = N_0 e^{-\lambda t_1} \]
\[ N_2 = N_0 e^{-\lambda t_2} \]
where \( N_0 \) is the initial number of nuclei.

4. **Subtract the two equations:**
From the expressions for activity, we can write:
\[ A_1 - A_2 = \lambda (N_1 - N_2) \]

5. **Rearranging the equation:**
Rearranging gives us:
\[ N_1 - N_2 = \frac{A_1 - A_2}{\lambda} \]

6. **Interpret the result:**
The term \( N_1 - N_2 \) represents the number of nuclei that have decayed in the time interval \( (t_1 - t_2) \). Thus, the number of decayed nuclei is:
\[ N_{\text{decayed}} = N_1 - N_2 = \frac{A_1 - A_2}{\lambda} \]

### Final Answer:
The number of nuclei that have decayed in the time interval \( (t_1 - t_2) \) is:
\[ N_{\text{decayed}} = \frac{A_1 - A_2}{\lambda} \]