In a radioactive material the activity at time `t_(1)` is `R_(1)` and at a later time `t_(2)`, it is `R_(2)`. If the decay constant of the material is `lambda`, then

Options

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

Answer

To solve the problem, we need to establish the relationship between the activities \( R_1 \) and \( R_2 \) at two different times \( t_1 \) and \( t_2 \) for a radioactive material with a decay constant \( \lambda \).

### Step-by-Step Solution:

1. **Understand the definition of activity**:
The activity \( R \) of a radioactive material is defined as the rate of decay of radioactive nuclei, which can be expressed mathematically as:
\[ R = -\frac{dN}{dt} = \lambda N \]
where \( N \) is the number of undecayed nuclei at time \( t \).

2. **Relate the number of nuclei over time**:
The change in the number of undecayed nuclei can be described by the differential equation:
\[ \frac{dN}{N} = -\lambda dt \]
Integrating both sides gives:
\[ \int \frac{dN}{N} = -\lambda \int dt \]
This results in:
\[ \ln N = -\lambda t + C \]
where \( C \) is the integration constant.

3. **Exponentiate to solve for \( N \)**:
By exponentiating both sides, we obtain:
\[ N = e^{C} e^{-\lambda t} \]
Let \( N_0 = e^{C} \) be the initial number of nuclei at \( t = 0 \):
\[ N = N_0 e^{-\lambda t} \]

4. **Substitute \( N \) into the activity formula**:
Now, substituting \( N \) back into the activity formula, we get:
\[ R = \lambda N_0 e^{-\lambda t} \]

5. **Express activities at times \( t_1 \) and \( t_2 \)**:
At time \( t_1 \):
\[ R_1 = \lambda N_0 e^{-\lambda t_1} \]
At time \( t_2 \):
\[ R_2 = \lambda N_0 e^{-\lambda t_2} \]

6. **Establish the relationship between \( R_1 \) and \( R_2 \)**:
To find the relationship, we can divide the two equations:
\[ \frac{R_1}{R_2} = \frac{\lambda N_0 e^{-\lambda t_1}}{\lambda N_0 e^{-\lambda t_2}} \]
The \( \lambda N_0 \) terms cancel out:
\[ \frac{R_1}{R_2} = \frac{e^{-\lambda t_1}}{e^{-\lambda t_2}} = e^{-\lambda (t_1 - t_2)} \]

7. **Rearranging the equation**:
We can rearrange this to express \( R_1 \) in terms of \( R_2 \):
\[ R_1 = R_2 e^{-\lambda (t_1 - t_2)} \]

### Final Result:
The relationship between the activities \( R_1 \) and \( R_2 \) is given by:
\[ R_1 = R_2 e^{-\lambda (t_1 - t_2)} \]