The activity of a radioactive susbtance is `R_(1)` at time `t_(1)` and `R_(2)` at time `t_(2)`(>t1). its decay constant is λ. Then, number of atoms decayed between time interval `t_(1)` and `t_(2)` are

Options

A. `(ln(2))/(lambda)(R_(1)-R_(2))`
B. `R_(1)e^(-lambdat_(1))-R_(2)e^(-lambdat_(2))`
C. `lambda(R_(1)-R_(2))`
D. `((R_(1)-R_(2))/(lambda))`

Answer

To find the number of atoms decayed between time intervals \( t_1 \) and \( t_2 \) for a radioactive substance, we can follow these steps:

### Step-by-Step Solution:

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

2. **Define the activities at times \( t_1 \) and \( t_2 \):**
Let:
- \( R_1 \) be the activity at time \( t_1 \)
- \( R_2 \) be the activity at time \( t_2 \)
- \( N_1 \) be the number of atoms at time \( t_1 \)
- \( N_2 \) be the number of atoms at time \( t_2 \)

From the activity formula, we can express:
\[ R_1 = \lambda N_1 \]
\[ R_2 = \lambda N_2 \]

3. **Relate the number of atoms at \( t_1 \) and \( t_2 \):**
The number of atoms decayed between \( t_1 \) and \( t_2 \) can be expressed as:
\[ \text{Atoms decayed} = N_1 - N_2 \]

4. **Express \( N_1 \) and \( N_2 \) in terms of activity:**
Rearranging the equations for \( N_1 \) and \( N_2 \):
\[ N_1 = \frac{R_1}{\lambda} \]
\[ N_2 = \frac{R_2}{\lambda} \]

5. **Substitute \( N_1 \) and \( N_2 \) into the decay formula:**
Now substituting these into the expression for atoms decayed:
\[ N_1 - N_2 = \frac{R_1}{\lambda} - \frac{R_2}{\lambda} \]
\[ = \frac{R_1 - R_2}{\lambda} \]

6. **Final expression for the number of atoms decayed:**
Thus, the number of atoms that have decayed between the time intervals \( t_1 \) and \( t_2 \) is:
\[ \text{Atoms decayed} = \frac{R_1 - R_2}{\lambda} \]

### Conclusion:
The number of atoms decayed between time intervals \( t_1 \) and \( t_2 \) is given by:
\[ \text{Atoms decayed} = \frac{R_1 - R_2}{\lambda} \]