Question

Chlorine oxide \((\mathrm{ClO}),\) which plays an important role in the depletion of ozone, decays rapidly at room temperature according to the equation: $$ 2 \mathrm{ClO}(g) \longrightarrow \mathrm{Cl}_{2}(g)+\mathrm{O}_{2}(g) $$ From the following data, determine the reaction order and calculate the rate constant of the reaction. $$ \begin{array}{ll} \text { Time (s) } & {[\mathrm{ClO}](M)} \\ \hline 0.12 \times 10^{-3} & 8.49 \times 10^{-6} \\ 0.96 \times 10^{-3} & 7.10 \times 10^{-6} \\ 2.24 \times 10^{-3} & 5.79 \times 10^{-6} \\ 3.20 \times 10^{-3} & 5.20 \times 10^{-6} \\ 4.00 \times 10^{-3} & 4.77 \times 10^{-6} \end{array} $$

Short answer

The reaction is second-order with a rate constant of \(2.37 \times 10^7 \text{ M}^{-1} \text{s}^{-1}\).

Step-by-step solution

Determine the Reaction Order

To determine the reaction order, we need to analyze how the concentration of \( \text{ClO} \) changes with time. A general method is to plot various graphs and analyze their linearity: 1. **Zero Order:** Plot \([\text{ClO}] \) vs. time. 2. **First Order:** Plot \(\ln [\text{ClO}] \) vs. time. 3. **Second Order:** Plot \(1/[\text{ClO}] \) vs. time. After plotting these, we find that the plot of \(1/[\text{ClO}] \) vs. time is a straight line, indicating the reaction is second-order.

Write the Rate Equation

For a second-order reaction, the rate law can be written as: \[ \text{Rate} = k [\text{ClO}]^2 \]where \( k \) is the rate constant.

Use Linear Plot to Find Rate Constant

For a second-order reaction, we have the formula: \[ \frac{1}{[\text{ClO}]} = kt + \frac{1}{[\text{ClO}]_0} \]Here, the slope of the line from the plot of \(1/[\text{ClO}] \) vs. time gives us the rate constant \( k \). Calculate the slope using the data points from the plot:- Initial concentration \( \frac{1}{8.49 \times 10^{-6}} = 1.18 \times 10^{5} \)- Final concentration \( \frac{1}{4.77 \times 10^{-6}} = 2.10 \times 10^{5} \)- Change in time: \( 4.00 \times 10^{-3} - 0.12 \times 10^{-3} = 3.88 \times 10^{-3} \)sThe slope \( k = \frac{2.10 \times 10^{5} - 1.18 \times 10^{5}}{3.88 \times 10^{-3}} \approx 2.37 \times 10^{7} \text{ M}^{-1} \text{s}^{-1} \).

Key concepts

Reaction Order

Understanding reaction order is crucial to determine how the concentration of a reactant affects the rate of a chemical reaction. It's a number that tells us to what power the concentration of a reactant is raised in the rate equation.
For instance, if the reaction order with respect to chlorine oxide (ClO) is two, the rate of reaction depends on the square of the concentration of ClO.

• To identify the reaction order, different plots are analyzed: zero-order plots \([\text{ClO}] \ vs. \ \text{time}\), first-order plots \(\ln [\text{ClO}] \ vs. \ \text{time}\), and second-order plots \(\frac{1}{[\text{ClO}]} \ vs. \ \text{time}\).
• The linearity of the plot determines the reaction order. For ClO, the second-order plot was linear, confirming the reaction is second-order.

Rate Constant

The rate constant \( k \) gives insight into the speed of a chemical reaction. It's a proportionality factor in the rate equation, which describes how the rate depends on reactant concentration.
The value of \( k \) can vary with temperature and other conditions. In our chlorine oxide reaction, the rate constant was calculated from the slope of the linear plot.

• For second-order reactions, the plot of \( \frac{1}{[\text{ClO}]} \) vs. time has a slope equal to the rate constant \( k \).
• Finding the rate constant involves calculating the change in \( \frac{1}{[\text{ClO}]} \) over time, providing insights into how rapidly the reaction proceeds.
• A higher \( k \) value implies a faster reaction.

Second-Order Reaction

A second-order reaction means that the rate is proportional to the square of the reactant concentration.
In the reaction involving ClO, it's described by the rate law: \( \text{Rate} = k [\text{ClO}]^2 \).

• Second-order reactions often involve the interaction between two molecules or the doubling effect of a single reactant.
• These reactions show a distinctive quadratic relationship in their kinetics, as seen in the straight-line plot of \( \frac{1}{[\text{ClO}]} \) vs. time.
• Understanding this helps predict how changes in concentration affect reaction rates over time.

Chlorine Oxide

Chlorine oxide (ClO) is a reactive gas that plays a significant role in atmospheric chemistry, particularly in ozone layer depletion.
In this exercise, we observe its decomposition into chlorine gas \( (\text{Cl}_2) \) and oxygen gas \( (\text{O}_2) \).

• ClO is part of a series of reactions that lead to the breakdown of ozone molecules, a process vital for understanding environmental impacts.
• Since it decays rapidly at room temperature, studying ClO helps understand both natural and man-made impacts on our atmosphere.
• This knowledge is key to devising strategies to protect the ozone layer.

Ozone Depletion

Ozone depletion refers to the thinning of the ozone layer in Earth's stratosphere, caused by chemical reactions.
ClO, as observed in our reaction, contributes significantly to this process.

• Depletion results from complex interactions involving chlorine compounds, such as ClO, in the atmosphere.
• ClO reacts with ozone (\( \text{O}_3 \)) leading to the release of oxygen molecules and further contributing to ozone layer thinning.
• Understanding the kinetics of reactions involving ClO is essential in assessing environmental policies and the health of our atmosphere.