Question

The first order reaction, \(\mathrm{SO}_{2} \mathrm{Cl}_{2} \rightarrow \mathrm{SO}_{2}+\mathrm{Cl}_{2}\), has a rate constant equal to \(2.20 \times 10^{-5} \mathrm{~s}^{-1}\) at \(593 \mathrm{~K}\). What percentage of the initial amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) will remain after \(2.00\) hours? a. \(45.8 \%\) b. \(85.4 \%\) c. \(15.4 \%\) d. \(75.6 \%\)

Short answer

After 2 hours, 85.4% of the initial amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) remains.

Step-by-step solution

Convert Time to Seconds

First, we need to convert the time given in hours to seconds because the rate constant is in units of per second. There are 3600 seconds in an hour. Thus, for 2 hours:\[\text{Time in seconds} = 2 \text{ hours} \times 3600 \text{ seconds/hour} = 7200 \text{ seconds} \]

First Order Reaction Formula

For a first order reaction, the amount of reactant remaining at time \( t \) is given by:\[ [A]_t = [A]_0 e^{-kt} \]where \( [A]_t \) is the concentration at time \( t \), \( [A]_0 \) is the initial concentration, \( k \) is the rate constant, and \( t \) is time.

Substitute Values into the Formula

Substitute the given values into the first order reaction formula:\[ [A]_t = [A]_0 e^{-(2.20 \times 10^{-5} \text{ s}^{-1})(7200 \text{ s})} \]

Calculate Exponential Term

Calculate the exponential term:\[ -(2.20 \times 10^{-5})(7200) = -0.1584 \]Thus, the exponential part of the formula becomes:\[ e^{-0.1584} \approx 0.8530 \]

Calculate Concentration Ratio

The concentration ratio \( \frac{[A]_t}{[A]_0} \) is equal to the calculated exponential value:\[ \frac{[A]_t}{[A]_0} = 0.8530 \]

Convert to Percentage

Convert the concentration ratio to a percentage to find out what percentage of the initial reactant remains:\[ 0.8530 \times 100\% = 85.3\% \]

Match Closest Option

Compare the calculated percentage to the provided options.The closest value is \( 85.4\% \), which corresponds to option b.

Key concepts

Rate Constant

The rate constant is a crucial part of understanding chemical reactions, especially first-order reactions. It determines how quickly a reaction proceeds. In this particular case, we are considering the decomposition of sulfuryl chloride, \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \), into sulfur dioxide and chlorine. The rate constant \( k \) is given as \( 2.20 \times 10^{-5} \text{ s}^{-1} \) at a temperature of 593 K. This unit is essential because it indicates that the reaction rate is measured per second. Therefore, to work with this rate constant, we need to ensure that time is also in seconds. This constant is specific to the reaction and temperature, underlining how it is affected by temperature changes and other conditions. Understanding the rate constant allows us to predict the behavior of the reaction over time.

Exponential Decay

Exponential decay is a principle seen clearly in first-order reactions. It describes the decreasing concentration of a reactant over time. The equation employed here is \( [A]_t = [A]_0 e^{-kt} \), where \( [A]_t \) is the concentration at time \( t \), \( [A]_0 \) is the initial concentration, and \( k \) is the rate constant. Over time, the exponential function \( e^{-kt} \) shows that the concentration decreases exponentially, never truly reaching zero but dwindling continuously. This decay is controlled by the rate constant \( k \), which, as discussed, measures how fast the reaction progresses. It's this predictable pattern that lets us determine the remaining percentage of the initial compound after a certain period.

Time Conversion

In calculations involving rate constants, aligning the units of time is vital. Given that our rate constant \( k \) is in terms of seconds, the time for the reaction must also be expressed in seconds. Initially given as 2 hours, we need to convert time for compatibility. Knowing there are 3600 seconds in an hour is helpful here. Therefore, multiplying the number of hours (2) by 3600, we can convert 2 hours into 7200 seconds. Accurately matching these units ensures the calculations with the first-order reaction formula are correct and meaningful.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that investigates the rates of chemical reactions and the factors affecting them. It provides a quantitative understanding of reaction rates based on conditions such as concentration, temperature, and the presence of catalysts. In our specific sulfuryl chloride decomposition reaction, chemical kinetics helps understand how long it takes for the concentration to reduce to a specific level. It involves employing mathematical models, like the first-order reaction formula, to predict behavior over time. By studying these kinetics, we can deduce important information about the mechanism of the reaction and how different conditions might alter its course.