A sample of an element is 10.38 g. If half-life of element is 3.8 days, then after 19 days how much quantity of element remains?

Options

A. 0.151g
B. 0.32 g
C. 1.51 g
D. 0.16 g

Explanation

To solve the problem of how much quantity of an element remains after a certain period given its half-life, we can follow these steps:

### Step 1: Identify the initial quantity and half-life
- The initial quantity of the element is given as \( m_0 = 10.38 \, \text{g} \).
- The half-life of the element is \( t_{1/2} = 3.8 \, \text{days} \).

### Step 2: Determine the total time elapsed
- The total time elapsed is given as \( t = 19 \, \text{days} \).

### Step 3: Calculate the number of half-lives
- To find the number of half-lives that have passed in 19 days, we use the formula:
\[ n = \frac{t}{t_{1/2}} = \frac{19 \, \text{days}}{3.8 \, \text{days}} = 5 \]
- This means that 5 half-lives have passed.

### Step 4: Apply the half-life formula
- The remaining quantity of the element after \( n \) half-lives can be calculated using the formula:
\[ m = m_0 \left(\frac{1}{2}\right)^n \]
- Substituting the values we have:
\[ m = 10.38 \, \text{g} \left(\frac{1}{2}\right)^5 \]

### Step 5: Calculate \( \left(\frac{1}{2}\right)^5 \)
- Calculate \( \left(\frac{1}{2}\right)^5 \):
\[ \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]

### Step 6: Calculate the remaining quantity
- Now substitute this back into the equation:
\[ m = 10.38 \, \text{g} \times \frac{1}{32} = \frac{10.38}{32} \approx 0.324 \, \text{g} \]

### Step 7: Conclusion
- After 19 days, approximately \( 0.324 \, \text{g} \) of the element remains.

### Final Answer
- The remaining quantity of the element after 19 days is approximately \( 0.324 \, \text{g} \).

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