Half life of a radioactive sample is 4 days. After 16 days how much quantity of matter remain undecayed :

Options

A. `1/4`
B. `1/8`
C. `1/16`
D. `1/32`

Answer

To solve the problem of how much quantity of a radioactive sample remains undecayed after 16 days given that its half-life is 4 days, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Half-Life and Total Time**:
- The half-life (t_half) of the radioactive sample is given as 4 days.
- The total time (t) after which we want to find the remaining quantity is 16 days.

2. **Calculate the Number of Half-Lives (n)**:
- We can calculate the number of half-lives that have passed using the formula:
\[ n = \frac{t}{t_{half}} = \frac{16 \text{ days}}{4 \text{ days}} = 4 \]
- So, 4 half-lives have passed.

3. **Use the Decay Formula**:
- The amount of radioactive material remaining after n half-lives can be calculated using the formula:
\[ N_t = N_0 \left(\frac{1}{2}\right)^n \]
- Here, \(N_t\) is the remaining quantity, \(N_0\) is the initial quantity, and \(n\) is the number of half-lives.
- Substituting the values, we get:
\[ N_t = N_0 \left(\frac{1}{2}\right)^4 \]

4. **Calculate the Remaining Quantity**:
- We know that \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\).
- Therefore, we can express the remaining quantity as:
\[ N_t = N_0 \cdot \frac{1}{16} \]

5. **Conclusion**:
- After 16 days, the quantity of the radioactive sample that remains undecayed is \(\frac{1}{16} N_0\), where \(N_0\) is the initial quantity of the sample.

### Final Answer:
The quantity of matter that remains undecayed after 16 days is \(\frac{1}{16} N_0\).

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