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

Options

A. `500 s^(-1)`
B. `1000 s^(-1)`
C. `250 s^(-1)`
D. `2000 s^(-1)`

Explanation

To solve the problem, we need to find the decay rate of a radioactive element after one second, given its initial decay rate and half-life. Let's break down the solution step by step.

### Step 1: Identify the given values
- Initial decay rate, \( N_0 = 10^3 \) disintegrations per second
- Half-life, \( T_{1/2} = 1 \) second

### Step 2: Understand the decay formula
The formula for radioactive decay can be expressed as:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
where:
- \( N \) is the decay rate after \( n \) half-lives,
- \( N_0 \) is the initial decay rate,
- \( n \) is the number of half-lives that have passed.

### Step 3: Determine the number of half-lives
Since we are looking for the decay rate after one second and the half-life is also one second, we have:
\[ n = 1 \]

### Step 4: Substitute the values into the decay formula
Now, we can substitute the values into the decay formula:
\[ N = N_0 \left( \frac{1}{2} \right)^1 \]
Substituting \( N_0 = 10^3 \):
\[ N = 10^3 \times \left( \frac{1}{2} \right) \]

### Step 5: Calculate the decay rate
Calculating the above expression:
\[ N = 10^3 \times 0.5 = 500 \text{ disintegrations per second} \]

### Final Answer
The decay rate after one second is \( 500 \) disintegrations per second.

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