Half life period of a radio active element A is 10 hours. In certain time 2g of A has become 0.25g. In the same time 4g of B reduced to 0.5g. Half life period of B is

Options

A. 10 hrs
B. 5hrs
C. 2hrs
D. 6 hrs

Answer

To solve the problem, we need to determine the half-life period of radioactive element B based on the decay of element A and the information given. Here’s a step-by-step solution:

### Step 1: Understand the decay of element A
We know that the half-life period of element A is 10 hours. Initially, we have 2 grams of A, which decays to 0.25 grams.

### Step 2: Calculate the number of half-lives for element A
- In the first half-life (10 hours), 2g of A will reduce to:
\[ \text{Remaining A} = \frac{2g}{2} = 1g \]

- In the second half-life (20 hours), 1g of A will reduce to:
\[ \text{Remaining A} = \frac{1g}{2} = 0.5g \]

- In the third half-life (30 hours), 0.5g of A will reduce to:
\[ \text{Remaining A} = \frac{0.5g}{2} = 0.25g \]

Thus, it takes 3 half-lives (30 hours) for element A to decay from 2 grams to 0.25 grams.

### Step 3: Determine the decay of element B
In the same time (30 hours), we know that 4 grams of B has reduced to 0.5 grams.

### Step 4: Calculate the number of half-lives for element B
- In the first half-life, 4g of B will reduce to:
\[ \text{Remaining B} = \frac{4g}{2} = 2g \]

- In the second half-life, 2g of B will reduce to:
\[ \text{Remaining B} = \frac{2g}{2} = 1g \]

- In the third half-life, 1g of B will reduce to:
\[ \text{Remaining B} = \frac{1g}{2} = 0.5g \]

Thus, it also takes 3 half-lives for element B to decay from 4 grams to 0.5 grams.

### Step 5: Relate the time taken to the half-life of B
Since it takes 30 hours for B to decay through 3 half-lives, we can find the half-life of B (\( t_B \)):
\[ 3 \times t_B = 30 \text{ hours} \]
\[ t_B = \frac{30 \text{ hours}}{3} = 10 \text{ hours} \]

### Conclusion
The half-life period of element B is 10 hours.