A compound A dissociate by two parallel first order paths at certain temperature
`A(g)overset(k_(1)("min"^(-1))rarr 2B(g)k_(1)=6.93xx10^(-3)"min"^(-1)`
`A(g)overset(k_(2)("min"^(-1))rarrC(g) k_(2)=6.93xx10^(-3)"min"^(-1)`
If reaction started with pure 'A' with 1 mole of A in 1 litre closed container with initial pressure (in atm) developed in container after 50 minutes from start of experiment?

Options

A. 1.25
B. 0.75
C. `1.50`
D. `2.50`

Answer

To solve the problem, we need to calculate the total pressure developed in the container after 50 minutes, given the dissociation of compound A into products B and C through two parallel first-order reactions.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Initial amount of A = 1 mole
- Volume of the container = 1 L
- Initial pressure of A (P_A0) = 2 atm
- Rate constants: k₁ = k₂ = 6.93 × 10^(-3) min^(-1)

2. **Calculate the Overall Rate Constant (K):**
Since the reactions are parallel, the overall rate constant K is the sum of the individual rate constants:
\[ K = k_1 + k_2 = 6.93 \times 10^{-3} + 6.93 \times 10^{-3} = 2 \times 6.93 \times 10^{-3} = 1.386 \times 10^{-2} \text{ min}^{-1} \]

3. **Calculate the Half-Life (t₁/₂):**
The half-life for a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{K} \]
Substituting the value of K:
\[ t_{1/2} = \frac{0.693}{1.386 \times 10^{-2}} \approx 50 \text{ minutes} \]

4. **Determine the Amount of A Remaining After 50 Minutes:**
After one half-life (50 minutes), half of the initial amount of A will remain:
\[ \text{Remaining A} = \frac{1 \text{ mole}}{2} = 0.5 \text{ moles} \]

5. **Calculate the Amount of B and C Produced:**
Since A dissociates into 2B and C:
- Let x be the amount of A that dissociates.
- After 50 minutes, x = 0.5 moles (since 0.5 moles of A remain, 0.5 moles of A have dissociated).
- From the reaction A → 2B, the amount of B produced is 2x = 2 × 0.5 = 1 mole.
- From the reaction A → C, the amount of C produced is x = 0.5 moles.

6. **Calculate the Total Moles in the Container:**
Total moles in the container after 50 minutes:
\[ \text{Total moles} = \text{Remaining A} + \text{B produced} + \text{C produced} = 0.5 + 1 + 0.5 = 2 \text{ moles} \]

7. **Calculate the Total Pressure:**
Using the ideal gas law, the pressure is proportional to the number of moles (at constant temperature and volume):
\[ P_{\text{total}} = \text{Total moles} \times \text{Pressure per mole} \]
Since the initial pressure was 2 atm for 1 mole, the total pressure after 50 minutes is:
\[ P_{\text{total}} = 2 \text{ atm} \]

### Final Answer:
The total pressure developed in the container after 50 minutes is **2.5 atm**.