A radioactive element has half life of `4.5 xx 10^(9)` years. If `80 g` of this was taken, the time taken for it to decay to `40 g` will be
a. `2.25 xx 10^(9)` years b. `4.50 xx 10^(9)` years
c. `6.75 xx 10^(9)` years d. `8.75 xx 10^(9)` years

Options

A. a. `2.25 xx 10^(9)` years
B. b. `4.50 xx 10^(9)` years
C. c. `6.75 xx 10^(9)` years
D. d. `8.75 xx 10^(9)` years

Explanation

To solve the problem, we need to determine how long it takes for a radioactive element with a given half-life to decay from 80 grams to 40 grams.

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life:**
The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as \(4.5 \times 10^9\) years.

2. **Initial and Final Amounts:**
We start with an initial amount of \(80 \, \text{g}\) of the radioactive element. We want to find out how long it takes for this amount to decay to \(40 \, \text{g}\).

3. **Determine the Decay:**
The decay from \(80 \, \text{g}\) to \(40 \, \text{g}\) represents a reduction by half. This means that the amount of substance has decreased by 50%.

4. **Relate Decay to Half-Life:**
Since the amount has decreased by half, it will take one half-life for this to occur. Therefore, the time taken to go from \(80 \, \text{g}\) to \(40 \, \text{g}\) is equal to one half-life.

5. **Calculate the Time Taken:**
Given that the half-life is \(4.5 \times 10^9\) years, the time taken for the decay from \(80 \, \text{g}\) to \(40 \, \text{g}\) is:
\[ \text{Time} = 4.5 \times 10^9 \, \text{years} \]

6. **Select the Correct Option:**
From the options provided:
a. \(2.25 \times 10^9\) years
b. \(4.50 \times 10^9\) years
c. \(6.75 \times 10^9\) years
d. \(8.75 \times 10^9\) years

The correct answer is option **b**: \(4.50 \times 10^9\) years.

### Final Answer:
The time taken for the radioactive element to decay from \(80 \, \text{g}\) to \(40 \, \text{g}\) is **\(4.50 \times 10^9\) years**.

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