The half life period of a radioactive substance is 140 days. After how much time, 15 g will decay from a 16 g sample of the substance?

Options

A. (a)140 days
B. (b)560 days
C. (c)420 days
D. (d)280 days

Answer

To solve the problem, we need to determine how long it will take for 15 grams to decay from a 16-gram sample of a radioactive substance with a half-life of 140 days.

### Step-by-Step Solution:

1. **Understand the Initial and Final Mass**:
- Initial mass (N0) = 16 g
- Final mass (N) = 16 g - 15 g = 1 g

2. **Determine the Mass Decayed**:
- Mass decayed = Initial mass - Final mass = 16 g - 1 g = 15 g

3. **Use the Half-Life Formula**:
The formula for radioactive decay can be expressed as:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
where:
- \( N \) is the remaining quantity of the substance,
- \( N_0 \) is the initial quantity,
- \( n \) is the number of half-lives that have passed.

4. **Set Up the Equation**:
Plugging in the values we have:
\[ 1 = 16 \left( \frac{1}{2} \right)^n \]

5. **Rearranging the Equation**:
Divide both sides by 16:
\[ \frac{1}{16} = \left( \frac{1}{2} \right)^n \]

6. **Expressing 1/16 as a Power of 1/2**:
We know that:
\[ \frac{1}{16} = \left( \frac{1}{2} \right)^4 \]
Therefore, we can equate the exponents:
\[ n = 4 \]

7. **Calculate the Total Time**:
Since each half-life is 140 days, the total time (T) for 4 half-lives is:
\[ T = n \times \text{half-life} = 4 \times 140 \text{ days} = 560 \text{ days} \]

### Final Answer:
The time required for 15 g to decay from a 16 g sample is **560 days**.