Cyclopropane (C3H6) reacts to form propene (C3H6) in the gas phase. The reaction is first order in cyclopropane and has a rate constant of 5.87 * 10^-4 s^-1 at 485 °C. If a 2.5-L reaction vessel initially contains 722 torr of cyclopropane at 485 °C, how long will it take for the partial pressure of cyclopropane to drop to below 1.00 * 10^2 torr?

Verified step by step guidance

1. Identify the reaction order and relevant equation: The reaction is first order, so we use the first-order rate equation: \( \ln \left( \frac{[A]_0}{[A]} \right) = kt \), where \([A]_0\) is the initial concentration (or pressure), \([A]\) is the concentration (or pressure) at time \(t\), \(k\) is the rate constant, and \(t\) is the time.
2. Convert pressures to the same units: The initial pressure \([A]_0\) is 722 torr, and the final pressure \([A]\) is 100 torr. Ensure both are in the same units for calculation.
3. Substitute the known values into the first-order rate equation: \( \ln \left( \frac{722}{100} \right) = (5.87 \times 10^{-4} \text{ s}^{-1}) \times t \).
4. Solve for \(t\): Rearrange the equation to solve for \(t\): \( t = \frac{\ln \left( \frac{722}{100} \right)}{5.87 \times 10^{-4}} \).
5. Calculate \(t\): Perform the calculation to find the time \(t\) it takes for the pressure to drop to below 100 torr.

Key Concepts

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. In this case, the rate of cyclopropane's reaction is dependent solely on its concentration. The integrated rate law for a first-order reaction can be expressed as ln([A]0/[A]) = kt, where [A]0 is the initial concentration, [A] is the concentration at time t, k is the rate constant, and t is time.

Rate Constant

The rate constant (k) is a proportionality factor in the rate law that provides insight into the speed of a reaction. It varies with temperature and is specific to each reaction. In this scenario, the rate constant is given as 5.87 * 10^-4 s^-1 at 485 °C, indicating how quickly cyclopropane reacts under these conditions. A higher rate constant signifies a faster reaction.

Partial Pressure

Partial pressure is the pressure exerted by a single component of a gas mixture. It is crucial in understanding gas reactions, as the total pressure is the sum of the partial pressures of all gases present. In this question, the initial partial pressure of cyclopropane is 722 torr, and the goal is to determine the time required for it to decrease to below 100 torr, which involves applying the first-order kinetics to the change in pressure over time.