Questions: The breakdown of a certain pollutant X in sunlight is known to follow first-order kinetics. An atmospheric scientist studying the process fills a 35.0 L reaction flask with a sample of urban air and finds that the partial pressure of X in the flask decreases from 0.994 atm to 0.385 atm over 60 minutes. Calculate the initial rate of decomposition of X, that is, the rate at which X was disappearing at the start of the experiment. Round your answer to 2 significant digits. atm/s

Solution Steps

The problem involves the decomposition of a pollutant \( X \) following first-order kinetics. We are given:

• Initial partial pressure of \( X \): \( P_0 = 0.994 \, \text{atm} \)
• Final partial pressure of \( X \): \( P_t = 0.385 \, \text{atm} \)
• Time interval: \( t = 60 \, \text{minutes} = 3600 \, \text{seconds} \)

We need to calculate the initial rate of decomposition of \( X \).

For first-order reactions, the rate law is given by: \[ \ln \left( \frac{P_0}{P_t} \right) = kt \] where \( k \) is the rate constant.

Rearrange the formula to solve for \( k \): \[ k = \frac{\ln \left( \frac{P_0}{P_t} \right)}{t} \]

Substitute the given values: \[ k = \frac{\ln \left( \frac{0.994}{0.385} \right)}{3600} \]

Calculate: \[ k = \frac{\ln(2.582)}{3600} \approx \frac{0.9486}{3600} \approx 0.0002635 \, \text{s}^{-1} \]

The initial rate of decomposition for a first-order reaction is given by: \[ \text{Rate} = k \cdot P_0 \]

Substitute the values: \[ \text{Rate} = 0.0002635 \, \text{s}^{-1} \times 0.994 \, \text{atm} \approx 0.000262 \, \text{atm/s} \]

Final Answer