Question

First order gas phase reaction \(A \to B + C\). \(p_i = \text{initial pressure of gas A}\), \(P_t = \text{total pressure of the reaction mixture at time t}\). Expression of rate constant (k) is

• A. \(\frac{1}{t} \ln \frac{p_i}{2p_i - P_t}\)
• B. \(\frac{1}{t} \ln \frac{2p_i}{p_i - P_t}\)
• C. \(\frac{1}{t} \ln \frac{p_i}{3p_i - 2P_t}\)
• D. \(\frac{1}{t} \ln \frac{3p_i}{4p_i - P_t}\)

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the rate constant expression for a first-order gas-phase decomposition reaction in terms of initial and total pressures.
Step 2: Key Formula or Approach:
Rate constant for first order: \(k = \frac{1}{t} \ln \frac{p_{initial}}{p_{final}}\).
Step 3: Detailed Explanation: Consider the reaction: \(A(g) \to B(g) + C(g)\)
At \(t = 0\): \(p_i, 0, 0\)
At \(t = t\): \(p_i - x, x, x\)
Total pressure at time \(t\), \(P_t\):
\[ P_t = (p_i - x) + x + x = p_i + x \]
From this, the decrease in pressure \(x\) is:
\[ x = P_t - p_i \]
The partial pressure of A at time \(t\), \(p_A\):
\[ p_A = p_i - x \]
\[ p_A = p_i - (P_t - p_i) = 2p_i - P_t \]
Now substitute the initial and final partial pressures of A into the first-order rate equation:
\[ k = \frac{1}{t} \ln \frac{p_i}{p_A} \]
\[ k = \frac{1}{t} \ln \frac{p_i}{2p_i - P_t} \]
Step 4: Final Answer:
The correct expression is \(k = \frac{1}{t} \ln \frac{p_i}{2p_i - P_t}\).