The half life period of a radioactive element is 4 s . What fraction of 100 g sample of this radioactive material will remain after 16 s ?

Options

A. `1//64`
B. `1//4`
C. `1//16`
D. `1//32`

Answer

To solve the problem of how much of a 100 g sample of a radioactive element remains after 16 seconds, given that its half-life is 4 seconds, we can follow these steps:

### Step 1: Understand the Concept of Half-Life
The half-life of a radioactive substance is the time required for half of the substance to decay. In this case, the half-life is 4 seconds.

### Step 2: Determine the Number of Half-Lives
To find out how many half-lives fit into the total time of 16 seconds, we can use the following formula:

\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life period}} = \frac{16 \text{ s}}{4 \text{ s}} = 4 \]

### Step 3: Use the Half-Life Formula
The remaining quantity of the radioactive material after a certain number of half-lives can be calculated using the formula:

\[ N = N_0 \left(\frac{1}{2}\right)^n \]

Where:
- \(N\) is the remaining quantity,
- \(N_0\) is the initial quantity (100 g in this case),
- \(n\) is the number of half-lives (which we calculated as 4).

### Step 4: Calculate the Remaining Quantity
Substituting the values into the formula:

\[ N = 100 \text{ g} \left(\frac{1}{2}\right)^4 \]

Calculating \(\left(\frac{1}{2}\right)^4\):

\[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

Now substituting back into the equation:

\[ N = 100 \text{ g} \times \frac{1}{16} = \frac{100}{16} \text{ g} = 6.25 \text{ g} \]

### Step 5: Determine the Fraction Remaining
To find the fraction of the original sample that remains, we can express this as:

\[ \text{Fraction remaining} = \frac{N}{N_0} = \frac{6.25 \text{ g}}{100 \text{ g}} = \frac{1}{16} \]

### Final Answer
Thus, the fraction of the 100 g sample of the radioactive material that will remain after 16 seconds is:

\[ \frac{1}{16} \]