The radioactive decay rate of a radioactive element is found to be `10^(3) ` disintegration per second at a cartain time . If the half life of the element is one second , the dacay rate after one second ….. And after three second is ……

Explanation

To solve the problem, we need to determine the decay rate of a radioactive element after 1 second and after 3 seconds, given that the initial decay rate is \(10^3\) disintegrations per second and the half-life is 1 second.

### Step-by-Step Solution:

1. **Identify the Initial Decay Rate:**
The initial decay rate \( R_0 \) is given as:
\[ R_0 = 10^3 \text{ disintegrations per second} = 1000 \text{ disintegrations per second} \]

2. **Understand the Concept of Half-Life:**
The half-life (\( t_{1/2} \)) of the element is the time required for the quantity of the radioactive substance to reduce to half of its initial value. Here, \( t_{1/2} = 1 \text{ second} \).

3. **Calculate the Decay Rate After 1 Second:**
After 1 half-life (which is 1 second), the decay rate is halved. Therefore:
\[ R(1 \text{ second}) = \frac{R_0}{2^1} = \frac{1000}{2} = 500 \text{ disintegrations per second} \]

4. **Calculate the Decay Rate After 3 Seconds:**
After 3 seconds, which corresponds to 3 half-lives, the decay rate will be:
\[ R(3 \text{ seconds}) = \frac{R_0}{2^3} = \frac{1000}{8} = 125 \text{ disintegrations per second} \]

5. **Final Results:**
- The decay rate after 1 second is \( 500 \text{ disintegrations per second} \).
- The decay rate after 3 seconds is \( 125 \text{ disintegrations per second} \).

### Summary of Results:
- After 1 second: \( 500 \text{ disintegrations per second} \)
- After 3 seconds: \( 125 \text{ disintegrations per second} \)