Question

A certain radioactive isotope \( ^A_Z X \) (half-life = 10 days) decays to give \( ^{A-4}_{Z-2} X \). If 1.0 g of \( X \) is kept in a sealed vessel, find the volume of He accumulated at STP in 20 days. (Molar mass of \( ^A_Z X = 253 \))

• A. 22.40 mL
• B. 33.40 mL
• C. 55.40 mL
• D. 66.40 mL

Hint:

Each alpha decay produces one helium atom, and always use \( 22.4 \, L \) per mole at STP.

Answer 1

The Correct Option is D

Concept: Radioactive decay with half-life: \[ N = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} \] Each decay produces one \( \alpha \)-particle (He nucleus).

Step 1: Find fraction decayed in 20 days. Half-life = 10 days $\Rightarrow$ 2 half-lives \[ \text{Remaining} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \] \[ \text{Decayed fraction} = 1 - \frac{1}{4} = \frac{3}{4} \]

Step 2: Initial moles: \[ \frac{1}{253} \text{ mol} \] Moles decayed: \[ \frac{3}{4} \times \frac{1}{253} = \frac{3}{1012} \]

Step 3: Moles of He formed: \[ = \frac{3}{1012} \]

Step 4: Volume at STP: \[ V = \frac{3}{1012} \times 22400 \] \[ = 66.40 \text{ mL} \] Final Answer: \[ 66.40 \text{ mL} \]