The half-life of radioactive sodium is 15.0 hours. How many hours would it taken for 64 gms of sodium to decay one-eighth of its original value

Options

A. 3
B. 15
C. 30
D. 45

Explanation

To solve the problem of how long it will take for 64 grams of radioactive sodium to decay to one-eighth of its original value, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Amount and Final Amount**:
- The initial amount of sodium is 64 grams.
- We need to find out how long it takes for this amount to decay to one-eighth of its original value.
- One-eighth of 64 grams is calculated as:
\[ \text{Final Amount} = \frac{64}{8} = 8 \text{ grams} \]

2. **Determine the Decay Factor**:
- The decay from 64 grams to 8 grams represents a decay factor of:
\[ \text{Decay Factor} = \frac{\text{Initial Amount}}{\text{Final Amount}} = \frac{64}{8} = 8 \]

3. **Use the Half-Life Formula**:
- The half-life (t½) of sodium is given as 15.0 hours.
- The relationship between the number of half-lives (n) and the decay factor can be expressed as:
\[ \text{Decay Factor} = 2^n \]
- To find n for a decay factor of 8:
\[ 8 = 2^n \implies n = 3 \]
- This means it takes 3 half-lives to decay from 64 grams to 8 grams.

4. **Calculate the Total Time**:
- Since each half-life is 15.0 hours, the total time (T) taken for 3 half-lives is:
\[ T = n \times t_{1/2} = 3 \times 15.0 \text{ hours} = 45.0 \text{ hours} \]

### Final Answer:
The time taken for 64 grams of sodium to decay to one-eighth of its original value is **45.0 hours**.

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