Problem 26 - Chemistry and Chemical Reactivity

Question

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \longrightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})$$ is first order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), and the reaction has a half-life of 245 min at \(600 \mathrm{K}\). If you begin with \(3.6 \times 10^{-3} \mathrm{mol}\) of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in a \(1.0-\mathrm{L}\) flask, how long will it take for the amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to decrease to \(2.00 \times 10^{-4}\) mol?

Step-by-Step Solution

Answer

It takes approximately 1021 minutes for the reaction to reach the target amount.

Step 1: Determine the Initial and Final Concentrations

The initial concentration of \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \) is calculated based on the initial amount of substance and the volume of the flask. This is given by \( \text{Concentration} = \frac{3.6 \times 10^{-3} \text{ mol}}{1.0 \text{ L}} = 3.6 \times 10^{-3} \text{ M} \). The final concentration when \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \) decreases to \( 2.00 \times 10^{-4} \text{ mol} \) is \( \frac{2.00 \times 10^{-4} \text{ mol}}{1.0 \text{ L}} = 2.00 \times 10^{-4} \text{ M} \).

Step 2: Use the First Order Kinetics Equation

The kinetics of the reaction follows a first order reaction, which is represented by the equation \( \ln \left(\frac{[A]_0}{[A]}\right) = kt \), where \( [A]_0 \) is the initial concentration, \([A]\) is the final concentration, \(k\) is the rate constant, and \(t\) is the time needed. We will also use the half-life equation for first-order reactions to find \( k \).

Step 3: Calculate the Rate Constant (k)

The rate constant \( k \) for a first-order reaction is given by the half-life equation \( t_{1/2} = \frac{0.693}{k} \). Given the half-life as 245 min, we can solve for \( k \):\[k = \frac{0.693}{245 \text{ min}} \approx 0.00283 \text{ min}^{-1}.\]

Step 4: Solve for the Time (t) Using the Kinetics Equation

Substitute the values into the first-order kinetics equation:\[\ln \left(\frac{3.6 \times 10^{-3}}{2.00 \times 10^{-4}}\right) = 0.00283t.\]Calculate the left-hand side:\[\ln (18) \approx 2.8904.\]Set the equation as:\[2.8904 = 0.00283t.\]Solve for \( t \):\[t \approx \frac{2.8904}{0.00283} \approx 1021.37 \text{ min}.\]

Step 5: Final Answer

The time it takes for the amount of \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \) to decrease to \( 2.00 \times 10^{-4} \text{ mol} \) is approximately 1021 minutes.

Key Concepts

First-Order Reactions

First-order reactions are a crucial concept in chemical kinetics. In these reactions, the rate is directly proportional to the concentration of one reactant.
This means that if the concentration of the reactant doubles, the rate of the reaction also doubles. The integrated rate law for a first-order reaction is given by the expression:

• \[ ext{ln}([A]_0/[A]) = kt\]

where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), \(k\) is the rate constant, and \(t\) is the elapsed time.
This equation is often used to determine how the concentration of a reactant changes over time, helping us understand how fast a reaction will proceed.

Reaction Rate Calculation

Calculating the rate of a reaction is essential for understanding how quickly a reaction occurs.
For first-order reactions, the rate can be measured by looking at the change in concentration of the reactant over time using the rate constant \(k\).
This is expressed as the differential rate equation:

• \[ ext{Rate} = k[A]\]

Here, \(k\) is a constant characteristic for the reaction at a given temperature, and \([A]\) is the concentration of the reactant.
By knowing \(k\), you can predict how the concentration of a substance decreases over time, helping in reaction time estimation and process control.

Half-Life of Reactions

The half-life of a reaction refers to the time needed for half of the reactant to be consumed. For first-order reactions, the half-life \(t_{1/2}\) is a constant and is independent of the initial concentration.
The formula for the half-life is:

• \[t_{1/2} = \frac{0.693}{k}\]

where \(k\) is the rate constant.
This equation can be used to quickly find \(k\) when the half-life is known, and it simplifies understanding how long it will take for a significant portion of the reactant to react in these processes.
In the context of practical applications, such as calculating the time needed for a reactant concentration to drop to a certain level, knowing the half-life can be very useful.