The initial mass of a radioactive element is `40g`. How many grams of it would be left after 24 years if its half`-` life period I s of 8 years ?

Options

A. 2
B. 5
C. 10
D. 20

Explanation

To solve the problem of how much of a radioactive element remains after a certain period, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial mass of the radioactive element (m₀) = 40 grams
- Half-life period (t₁/₂) = 8 years
- Total time elapsed (t) = 24 years

2. **Calculate the Number of Half-Lives (n):**
- The number of half-lives (n) can be calculated using the formula:
\[ n = \frac{t}{t_{1/2}} \]
- Substituting the values:
\[ n = \frac{24 \text{ years}}{8 \text{ years}} = 3 \]

3. **Use the Formula for Remaining Mass:**
- The remaining mass of the radioactive element after n half-lives can be calculated using the formula:
\[ m = \frac{m_0}{2^n} \]
- Where:
- m = remaining mass
- m₀ = initial mass
- n = number of half-lives
- Substituting the values:
\[ m = \frac{40 \text{ grams}}{2^3} \]

4. **Calculate \(2^3\):**
- Calculate \(2^3\):
\[ 2^3 = 8 \]

5. **Substitute and Calculate Remaining Mass:**
- Now substitute back into the formula:
\[ m = \frac{40 \text{ grams}}{8} = 5 \text{ grams} \]

### Final Answer:
After 24 years, 5 grams of the radioactive element would be left.

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