Question

One loss mechanism for ozone in the atmosphere is the reaction with the \(\mathrm{HO}_{2} \cdot\) radical: Using the following information, determine the rate law expression for this reaction: $$\begin{array}{ccc}\text { Rate }\left(\mathrm{cm}^{-3} \mathrm{s}^{-1}\right) & {\left[\mathrm{HO}_{2} \cdot\right]\left(\mathrm{cm}^{-3}\right)} & {\left[\mathrm{O}_{3}\right]\left(\mathrm{cm}^{-3}\right)} \\ \hline 1.9 \times 10^{8} & 1.0 \times 10^{11} & 1.0 \times 10^{12} \\\9.5 \times 10^{8} & 1.0 \times 10^{11} & 5.0 \times 10^{12} \\\5.7 \times 10^{8} & 3.0 \times 10^{11} & 1.0 \times 10^{12}\end{array}$$

Short answer

The rate law expression for the reaction between ozone and the HO₂· radical in the atmosphere is: Rate = k [HO₂·][O₃]

Step-by-step solution

Analyzing the concentration effect on the rate for HO₂· radical

We will first inspect the effect of the concentration of HO₂· on the reaction rate. To do this, we will compare experiments 1 and 3, in which we have the same [O₃] but different [HO₂·] concentrations. From the table, we have: For experiment 1: \(\) Rate = 1.9 × 10⁸ cm³/s, [HO₂·] = 1.0 × 10¹¹ cm⁻³ For experiment 3: \(\) Rate = 5.7 × 10⁸ cm³/s, [HO₂·] = 3.0 × 10¹¹ cm⁻³ Now we will determine the relationship between the reaction rate and the concentration of HO₂·.

Calculating the order with respect to HO₂· radical

To find the order of the reaction with respect to the HO₂· radical, we will use the following relationship: \( \frac{\text{Rate}_3}{\text{Rate}_1} = \left(\frac{[\text{HO}_{2}\text{·}]_3}{[\text{HO}_{2}\text{·}]_1}\right)^{m} \) Plugging in the given values: \( \frac{5.7 \times 10^{8}}{1.9 \times 10^{8}} = \left(\frac{3.0 \times 10^{11}}{1.0 \times 10^{11}}\right)^{m} \) \( \Rightarrow 3 = 3^m \) Thus, m = 1, which implies the reaction is first order with respect to the concentration of the HO₂· radical.

Analyzing the concentration effect on the rate for O₃

Next, we will investigate the effect of the concentration of O₃ on the reaction rate. To do this, we will compare experiments 1 and 2, in which we have the same [HO₂·] but different [O₃] concentrations. From the table, we have: For experiment 1: \(\) Rate = 1.9 × 10⁸ cm³/s, [O₃] = 1.0 × 10¹² cm⁻³ For experiment 2: \(\) Rate = 9.5 × 10⁸ cm³/s, [O₃] = 5.0 × 10¹² cm⁻³ Now we will determine the relationship between the reaction rate and the concentration of O₃.

Calculating the order with respect to O₃

To find the order of the reaction with respect to O₃, we will use the same relationship we used to derive the reaction order for the HO₂· radical: \( \frac{\text{Rate}_2}{\text{Rate}_1} = \left(\frac{[\text{O}_{3}]_2}{[\text{O}_{3}]_1}\right)^{n} \) Plugging in the given values: \( \frac{9.5 \times 10^{8}}{1.9 \times 10^{8}} = \left(\frac{5.0 \times 10^{12}}{1.0 \times 10^{12}}\right)^{n} \) \( \Rightarrow 5 = 5^n \) Thus, n = 1, which implies the reaction is first order with respect to the concentration of O₃.

Writing the rate law expression

Now that we know the reaction is first order with respect to both the HO₂· radical and O₃, we can write the rate law expression as follows: Rate = k [HO₂·]^1 [O₃]^1 Or simply: Rate = k [HO₂·][O₃] This is the rate law expression for the given reaction between ozone and the HO₂· radical in the atmosphere.

Key concepts

Chemical Kinetics and Reaction Rate Law

Chemical kinetics is the study of the rates at which chemical reactions proceed and the factors that influence these rates. The reaction rate law, also known as the rate equation, describes this behavior quantitatively, indicating how the reaction rate depends on the concentration of the reactants. By measuring how different concentrations affect the reaction speed, chemists can deduce the rate law for a specific reaction.

For instance, consider a reaction where the rate changes as the concentration of a reactant is altered. Suppose the rate doubles when the concentration of the reactant doubles; this suggests that the rate is directly proportional to the concentration of that reactant. This proportionality can be converted into a mathematical equation, which is the rate law. It captures the essence of kinetics by allowing predictions of how fast a reaction will occur under different conditions.

Determining Reaction Order

The reaction order is another key concept in chemical kinetics. It specifies the power to which the concentration of each reactant is raised in the rate law equation. For each reactant, the reaction order tells us how the rate of reaction will respond to a change in that reactant's concentration.

To determine the reaction order, scientists conduct a series of experiments altering reactant concentrations while observing the resulting change in reaction rates. Calculating the reaction order requires a mathematical analysis where experimental data is used to find a relationship between concentration changes and rate changes, as shown in the solution provided for the reaction between ozone and the HO₂· radical. This is a fundamental step in developing a deep understanding of a reaction's mechanics.

The Phenomenon of Ozone Depletion

Understanding chemical kinetics is not just essential for laboratory reactions but also for environmental phenomena, such as ozone depletion. Ozone depletion refers to the thinning of the Earth's ozone layer, which is largely attributed to chemicals like chlorofluorocarbons (CFCs) and other ozone-depleting substances. The kinetics of ozone-depleting reactions involve various radicals, including the HO₂· radical as illustrated in the exercise.

Scientists use the kinetics of these reactions to understand the rates at which ozone molecules are destroyed in the stratosphere. This understanding is crucial for developing strategies to manage and prevent further depletion of the ozone layer, which serves as a protective shield against the sun's harmful ultraviolet radiation.

Concentration Effect on Reaction Rates

The concentration effect is a fundamental principle illustrating how variations in the concentration of reactants influence the rate of a chemical reaction. When reactant concentrations are high, there are more particles available to collide and react, leading to a higher reaction rate. Conversely, a reduction in concentration typically results in a slower reaction.

The rate law for a reaction includes terms that mathematically describe the concentration effect. By analyzing how altering concentrations affects the reaction rate, we can deduce these terms, which are critical in predicting how a reaction will proceed under various conditions. This concept is pivotal in processes ranging from industrial synthesis to environmental chemistry.

Writing the Rate Law Expression

The rate law expression is a mathematical representation that relates the reaction rate to the concentration of each reactant, each raised to a power corresponding to their reaction order. For a given reaction, the rate law can only be determined experimentally and cannot be predicted solely by the reaction's stoichiometry.

The rate constant (k) in the expression is specific to the reaction and can also depend on temperature and the presence of a catalyst. Determining the rate law expression is a crucial step toward controlling a chemical process, whether for industrial application or understanding natural phenomena. The exercise showcased how to deduce the rate law for the reaction of ozone with the HO₂· radical by analyzing experimental data, leading to the expression Rate = k [HO₂·][O₃].