The half life of a radioactive element is 30 min. One sixteenth of the original quantity of element will be left after
a. 1 hr b. 16 hr c. 4 hr d. 2 hr

Options

A. a. 1 hr
B. b. 16 hr
C. c. 4 hr
D. d. 2 hr

Explanation

To solve the problem, we need to determine how long it takes for a radioactive element with a half-life of 30 minutes to decay to one-sixteenth of its original quantity.

### Step-by-Step Solution:

1. **Understanding Half-Life**:
The half-life of a radioactive element is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life (t_half) is given as 30 minutes.

2. **Determine the Decay Factor**:
We need to find out how many half-lives it takes for the element to decay to one-sixteenth of its original quantity.
- One-sixteenth can be expressed as \( \frac{1}{16} \).
- Since \( \frac{1}{2} \) is the amount left after one half-life, we can express the decay as follows:
- After 1 half-life: \( \frac{1}{2} \)
- After 2 half-lives: \( \frac{1}{4} \)
- After 3 half-lives: \( \frac{1}{8} \)
- After 4 half-lives: \( \frac{1}{16} \)

Thus, it takes 4 half-lives to reach one-sixteenth of the original quantity.

3. **Calculate Total Time**:
Since each half-life is 30 minutes, we can calculate the total time for 4 half-lives:
\[ \text{Total Time} = 4 \times \text{Half-life} = 4 \times 30 \text{ minutes} = 120 \text{ minutes} \]

4. **Convert Minutes to Hours**:
To convert the total time from minutes to hours:
\[ 120 \text{ minutes} = \frac{120}{60} \text{ hours} = 2 \text{ hours} \]

5. **Final Answer**:
Therefore, the time taken for one-sixteenth of the original quantity of the radioactive element to remain is **2 hours**.

### Answer:
The correct option is **d. 2 hr**.

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