If half-life of a substance is `3.8` days and its quantity is `10.38 gm`. Then substance quantity remaining left after `19` days will be

Options

A. `2.151 g`
B. `0.31` g
C. `1.51` g
D. `0.16` g

Explanation

To solve the problem, we need to determine the quantity of a substance remaining after a certain period, given its half-life and initial quantity. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Half-life (\( t_{1/2} \)) = 3.8 days
- Initial quantity (\( N_0 \)) = 10.38 grams
- Time elapsed (\( t \)) = 19 days

2. **Calculate the Number of Half-Lives:**
- To find the number of half-lives (\( n \)), we use the formula:
\[ n = \frac{t}{t_{1/2}} \]
- Substituting the values:
\[ n = \frac{19 \text{ days}}{3.8 \text{ days}} = 5 \]

3. **Use the Decay Formula:**
- The quantity remaining after \( n \) half-lives is given by:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
- Substituting the known values:
\[ N = 10.38 \text{ grams} \left( \frac{1}{2} \right)^5 \]

4. **Calculate \( \left( \frac{1}{2} \right)^5 \):**
- \( \left( \frac{1}{2} \right)^5 = \frac{1}{32} \)

5. **Calculate the Remaining Quantity:**
- Now substituting back into the equation:
\[ N = 10.38 \text{ grams} \times \frac{1}{32} \]
- Performing the multiplication:
\[ N = \frac{10.38}{32} \approx 0.324375 \text{ grams} \]

6. **Round the Result:**
- Rounding to two decimal places, we get:
\[ N \approx 0.32 \text{ grams} \]

### Final Answer:
The quantity of the substance remaining after 19 days is approximately **0.32 grams**.