To solve the problem, we need to analyze the dissociation of compound A into products B and C through two parallel first-order reactions. Let's break it down step by step:
### Step 1: Understand the Reactions
We have two parallel reactions:
1. \( A(g) \overset{k_1}{\rightarrow} 2B(g) \) with \( k_1 = 6.93 \times 10^{-3} \, \text{min}^{-1} \)
2. \( A(g) \overset{k_2}{\rightarrow} C(g) \) with \( k_2 = 6.93 \times 10^{-3} \, \text{min}^{-1} \)
### Step 2: Calculate Overall Rate Constant
Since the reactions are parallel, the overall rate constant \( k \) can be calculated as:
\[
k = k_1 + k_2 = 6.93 \times 10^{-3} + 6.93 \times 10^{-3} = 2 \times 6.93 \times 10^{-3} = 1.386 \times 10^{-2} \, \text{min}^{-1}
\]
### Step 3: Calculate Half-Life
For a first-order reaction, the half-life \( t_{1/2} \) is given by:
\[
t_{1/2} = \frac{0.693}{k}
\]
Substituting the value of \( k \):
\[
t_{1/2} = \frac{0.693}{1.386 \times 10^{-2}} \approx 50 \, \text{minutes}
\]
### Step 4: Determine Amount of A Remaining After 50 Minutes
Initially, we have 1 mole of A in a 1-liter container, which corresponds to an initial pressure of 2 atm (using the ideal gas law). After 50 minutes (which is equal to one half-life), the amount of A remaining will be half of the initial amount:
\[
\text{Pressure of } A = 2 \, \text{atm} \times \frac{1}{2} = 1 \, \text{atm}
\]
### Step 5: Calculate Amount of B and C Produced
Since A dissociates into 2B and C:
- From the first reaction, the amount of A that dissociates is \( 1 - 0.5 = 0.5 \, \text{atm} \) (since 0.5 atm of A remains).
- The amount of B produced from the first reaction is:
\[
P_B = 2 \times (0.5 \, \text{atm}) = 1 \, \text{atm}
\]
- The amount of C produced from the second reaction is:
\[
P_C = 0.5 \, \text{atm}
\]
### Step 6: Calculate Total Pressure
The total pressure in the container after 50 minutes is the sum of the pressures of A, B, and C:
\[
P_{\text{total}} = P_A + P_B + P_C = 1 \, \text{atm} + 1 \, \text{atm} + 0.5 \, \text{atm} = 2.5 \, \text{atm}
\]
### Final Answer
The total pressure developed in the container after 50 minutes is **2.5 atm**.
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