If `N_(t) = N_(o)e^(lambdat)` then number of disintegrated atoms between `t_(1)` to `t_(2) (t_(2) gt t_(1))` will ve :-

Options

A. `N_(o)[e^(lambdat_(2))-e^(lambdat_(3))]`
B. `N_(o)[-e^(lambdat_(2)) - e^(-lambdat_(1))]`
C. `N_(o)[e^(-lambdat_(1)) - e^(-lambdat_(2))]`
D. None of these

Explanation

To find the number of disintegrated atoms between time \( t_1 \) and \( t_2 \), we start with the given equation for the number of active radioactive atoms at time \( t \):

\[ N(t) = N_0 e^{-\lambda t} \]

where:
- \( N(t) \) is the number of active atoms at time \( t \),
- \( N_0 \) is the initial number of atoms,
- \( \lambda \) is the decay constant.

### Step 1: Calculate the number of active atoms at \( t_2 \)

Using the formula, the number of active atoms at time \( t_2 \) is:

\[ N(t_2) = N_0 e^{-\lambda t_2} \]

### Step 2: Calculate the number of active atoms at \( t_1 \)

Similarly, the number of active atoms at time \( t_1 \) is:

\[ N(t_1) = N_0 e^{-\lambda t_1} \]

### Step 3: Calculate the number of disintegrated atoms at \( t_2 \)

The number of disintegrated atoms at time \( t_2 \) is the initial number of atoms minus the number of active atoms at \( t_2 \):

\[ \text{Disintegrated atoms at } t_2 = N_0 - N(t_2) = N_0 - N_0 e^{-\lambda t_2} \]

This simplifies to:

\[ \text{Disintegrated atoms at } t_2 = N_0 (1 - e^{-\lambda t_2}) \]

### Step 4: Calculate the number of disintegrated atoms at \( t_1 \)

Similarly, the number of disintegrated atoms at time \( t_1 \) is:

\[ \text{Disintegrated atoms at } t_1 = N_0 - N(t_1) = N_0 - N_0 e^{-\lambda t_1} \]

This simplifies to:

\[ \text{Disintegrated atoms at } t_1 = N_0 (1 - e^{-\lambda t_1}) \]

### Step 5: Calculate the number of disintegrated atoms between \( t_1 \) and \( t_2 \)

To find the number of disintegrated atoms between \( t_1 \) and \( t_2 \), we subtract the number of disintegrated atoms at \( t_1 \) from the number at \( t_2 \):

\[ \text{Disintegrated atoms between } t_1 \text{ and } t_2 = \text{Disintegrated atoms at } t_2 - \text{Disintegrated atoms at } t_1 \]

Substituting the expressions we found:

\[ = N_0 (1 - e^{-\lambda t_2}) - N_0 (1 - e^{-\lambda t_1}) \]

This simplifies to:

\[ = N_0 e^{-\lambda t_1} - N_0 e^{-\lambda t_2} \]

Factoring out \( N_0 \):

\[ = N_0 (e^{-\lambda t_1} - e^{-\lambda t_2}) \]

### Final Answer

Thus, the number of disintegrated atoms between \( t_1 \) and \( t_2 \) is:

\[ N_0 (e^{-\lambda t_1} - e^{-\lambda t_2}) \]