Question

For a first order kinetics reaction, \[ \text{A} \rightarrow \text{B} + \text{C} \] If the initial pressure of A is 1 bar and at time 100 s, the total pressure is 1.5 bar, then find the rate constant of the reaction.

• A. \( 6.93 \times 10^{-3} \, \text{s}^{-1} \)
• B. \( 6.93 \times 10^{-2} \, \text{s}^{-1} \)
• C. \( 0.693 \, \text{s}^{-1} \)
• D. \( 6.93 \, \text{s}^{-1} \)

Hint:

For first-order reactions, the rate constant can be determined using the relationship between pressure and time, using the integrated rate law and pressure measurements.

Answer 1

The Correct Option is A

Step 1: Write the integrated rate equation for a first-order reaction.
For a first-order reaction, the integrated rate law is: \[ \ln \left( \frac{[A_0]}{[A]} \right) = k \cdot t \] where: - \( [A_0] \) is the initial concentration of A, - \( [A] \) is the concentration of A at time \( t \), - \( k \) is the rate constant, - \( t \) is the time.
Step 2: Use the relationship with pressure.
For gaseous reactions, we can substitute pressure for concentration. The pressure of A at time \( t \) is related to the initial pressure by: \[ \ln \left( \frac{P_0}{P_t} \right) = k \cdot t \] where: - \( P_0 \) is the initial pressure of A, - \( P_t \) is the pressure of A at time \( t \), - \( k \) is the rate constant.
Step 3: Set up the problem.
We are given: - Initial pressure \( P_0 = 1 \, \text{bar} \), - Total pressure at time \( t = 100 \, \text{s} \) is 1.5 bar, which is the sum of the pressures of A, B, and C. Since one mole of A produces one mole each of B and C, the pressure of B and C will both be \( P_0 - P_t \). Thus, the total pressure at time \( t \) is \( P_0 + (P_0 - P_t) = 1 + (1 - P_t) \). So, \[ P_t = 1 \, \text{bar} - 0.5 \, \text{bar} = 0.5 \, \text{bar} \] Thus, the equation becomes: \[ \ln \left( \frac{1}{0.5} \right) = k \cdot 100 \] \[ \ln(2) = k \cdot 100 \] \[ k = \frac{\ln(2)}{100} = \frac{0.693}{100} = 6.93 \times 10^{-3} \, \text{s}^{-1} \] Thus, the rate constant \( k \) is \( 6.93 \times 10^{-3} \, \text{s}^{-1} \).