A certain radioactive isotope `._(Z)^(A)X(t_(1//2)=100` days decays to `._(Z-2)^(A-8)Y` . If 1 mole of `._(Z)^(A)X` is kept in sealed container , how much He gas will accumulate at STP in 200 days?

Options

A. 11.2 litres
B. 33.6 litres
C. 22.4 litres
D. 44.8 litres

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Decay Process
The radioactive isotope \( _{Z}^{A}X \) decays to \( _{Z-2}^{A-8}Y \). This indicates that during the decay, 2 helium nuclei (alpha particles) and some beta particles are emitted. Each decay of \( X \) results in the formation of 2 helium atoms.

### Step 2: Determine the Half-Life
The half-life (\( t_{1/2} \)) of the isotope \( X \) is given as 100 days. We need to find out how many half-lives fit into 200 days.

\[ \text{Number of half-lives} = \frac{200 \text{ days}}{100 \text{ days}} = 2 \]

### Step 3: Apply the Decay Formula
Using the decay formula, we can express the relationship between the initial amount \( A_0 \) and the amount remaining \( A_t \) after \( n \) half-lives:

\[ \frac{A_0}{A_t} = 2^n \]

Here, \( A_0 = 1 \) mole (initial amount of \( X \)) and \( n = 2 \) (number of half-lives).

\[ \frac{1}{A_t} = 2^2 = 4 \]

This implies:

\[ A_t = \frac{1}{4} \text{ moles} \]

### Step 4: Calculate the Amount of \( X \) Remaining
Now, we can find the amount of \( X \) that has decayed:

\[ \text{Amount of } X \text{ decayed} = A_0 - A_t = 1 - \frac{1}{4} = \frac{3}{4} \text{ moles} \]

### Step 5: Calculate the Amount of Helium Produced
Since each mole of \( X \) that decays produces 2 moles of helium, the total amount of helium produced from the decay of \( \frac{3}{4} \) moles of \( X \) is:

\[ \text{Moles of He} = 2 \times \frac{3}{4} = \frac{3}{2} \text{ moles} \]

### Step 6: Convert Moles of Helium to Volume at STP
At standard temperature and pressure (STP), 1 mole of any ideal gas occupies 22.4 liters. Therefore, the volume of helium gas produced is:

\[ \text{Volume of He} = \frac{3}{2} \text{ moles} \times 22.4 \text{ L/mole} = 33.6 \text{ L} \]

### Final Answer
The amount of helium gas that will accumulate at STP in 200 days is **33.6 liters**.

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