A radioactive element reduces to 25% of its initial value in 1000 years. What is half-life of the element ?

Explanation

To find the half-life of a radioactive element that reduces to 25% of its initial value in 1000 years, we can follow these steps:

### Step 1: Understand the given information
We know that the radioactive element reduces to 25% of its initial value (N₀) in 1000 years. This means:
\[ N = \frac{N_0}{4} \]
where \( N \) is the remaining quantity after 1000 years.

### Step 2: Relate the remaining quantity to half-lives
The relationship between the remaining quantity and the number of half-lives can be expressed as:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
where \( n \) is the number of half-lives that have passed.

### Step 3: Set up the equation
Since we know that \( N = \frac{N_0}{4} \), we can set up the equation:
\[ \frac{N_0}{4} = N_0 \left( \frac{1}{2} \right)^n \]

### Step 4: Simplify the equation
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^n \]

### Step 5: Rewrite \( \frac{1}{4} \) as a power of \( \frac{1}{2} \)
We know that:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^2 \]
Thus, we can equate the exponents:
\[ n = 2 \]

### Step 6: Relate the number of half-lives to time
The number of half-lives \( n \) can also be expressed in terms of total time \( T \) and half-life \( t_{1/2} \):
\[ n = \frac{T}{t_{1/2}} \]
Given that \( T = 1000 \) years and \( n = 2 \):
\[ 2 = \frac{1000}{t_{1/2}} \]

### Step 7: Solve for half-life
Rearranging the equation gives:
\[ t_{1/2} = \frac{1000}{2} = 500 \text{ years} \]

### Conclusion
The half-life of the radioactive element is **500 years**.

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