Problem 60 - Chemistry The Science in Context

Question

Hydroperoxyl Radicals in the Atmosphere During a smog event, trace amounts of many highly reactive substances are present in the atmosphere. One of these is the hydroperoxyl radical, \(\mathrm{HO}_{2},\) which reacts with sulfur trioxide, \(\mathrm{SO}_{3} .\) The rate constant for the reaction $$ 2 \mathrm{HO}_{2}(g)+\mathrm{SO}_{3}(g) \rightarrow \mathrm{H}_{2} \mathrm{SO}_{3}(g)+2 \mathrm{O}_{2}(g) $$ at \(298 \mathrm{K}\) is \(2.6 \times 10^{11} M^{-1} \mathrm{s}^{-1} .\) The initial rate of the reaction doubles when the concentration of \(\mathrm{SO}_{3}\) or \(\mathrm{HO}_{2}\) is doubled. What is the rate law for the reaction?

Step-by-Step Solution

Answer

Based on the given information, the rate law for the reaction between hydroperoxyl radicals (HO₂) and sulfur trioxide (SO₃) is: $$rate = k[\mathrm{HO}_{2}][\mathrm{SO}_{3}]$$

Step 1: Write general rate law

To start, we'll need to write the general rate law for the reaction. $$rate = k[\mathrm{HO}_{2}]^x[\mathrm{SO}_{3}]^y$$ We are given the rate constant, k, and we need to find the values of x and y.

Step 2: Analyze rate change with respect to HO₂ concentration

We know that the initial rate of the reaction doubles when the concentration of HO₂ is doubled. Let's see how that relates to the rate law: $$2 \times rate = k[\mathrm{2 \times HO}_{2}]^x[\mathrm{SO}_{3}]^y$$ We can simplify this to: $$2 = [\mathrm{2 \times HO}_{2}]^x$$ Since the rate doubles, this implies that x=1.

Step 3: Analyze rate change with respect to SO₃ concentration

Now let's analyze how the reaction rate changes with respect to the concentration of SO₃: $$2 \times rate = k[\mathrm{HO}_{2}]^x[\mathrm{2 \times SO}_{3}]^y$$ Again, we are given that the rate doubles, so we can simplify this to: $$2 = [\mathrm{2 \times SO}_{3}]^y$$ This implies that y=1 as well since the rate doubles.

Step 4: Write the final rate law

Now that we've determined the values of x and y, we can write the final rate law for the reaction: $$rate = k[\mathrm{HO}_{2}]^1[\mathrm{SO}_{3}]^1$$ Or, simplifying further: $$rate = k[\mathrm{HO}_{2}][\mathrm{SO}_{3}]$$ The rate law for this reaction is: $$rate = k[\mathrm{HO}_{2}][\mathrm{SO}_{3}]$$

Key Concepts

Hydroperoxyl Radical

The hydroperoxyl radical, denoted as \( \mathrm{HO}_2 \), is a key component in atmospheric chemistry. It plays a significant role during smog events when many reactive substances are present in the atmosphere.
This radical is a part of the hydrogen-oxygen reaction network and can influence the formation and depletion of ozone. Its reactive nature makes it essential for various atmospheric reactions.

• Characteristics: Highly reactive, often short-lived due to fast reactions with other chemicals.
• Roles: Involves in chain reactions that affect atmospheric gases such as ozone.
• Implication in Pollution: These radicals can lead to the creation of other pollutants under certain conditions.

Understanding hydroperoxyl radicals is crucial for studying environmental chemistry, especially in urban areas with high pollution levels.

Sulfur Trioxide

Sulfur trioxide, \( \mathrm{SO}_3 \), is a significant compound in various chemical reactions, particularly those involving air pollutants. It reacts readily with water to form sulfuric acid, contributing to acid rain.
In this atmospheric scenario, \( \mathrm{SO}_3 \) interacts with hydroperoxyl radicals to produce new chemical species.

• Properties: A potent oxidizing agent that can react with a variety of chemicals.
• Environmental Impact: Plays a critical role in atmospheric reactions, especially in pollution events.
• Industrial Relevance: Used in manufacturing processes, making it prevalent in industrial areas.

Recognizing the behavior of sulfur trioxide in reactions, such as with \( \mathrm{HO}_2 \) radicals, is essential for understanding industrial impacts on the environment.

Reaction Rate

The reaction rate is a fascinating concept that explains how fast a chemical reaction occurs. It depends on various factors, including the concentration of reactants and the temperature.
For the reaction between hydroperoxyl radicals and sulfur trioxide, the rate doubles when the concentration of either reactant is increased. This information helps derive the rate law.

• Mathematics of Reaction Rate: Given by the equation \( rate = k[\mathrm{HO}_2][\mathrm{SO}_3] \), where \( k \) is the rate constant.
• Influence on Reaction: Affected by concentration changes, which can be observed experimentally.
• Kinetic Studies: Understanding these rates helps in predicting how long a reaction will take under different conditions.

This knowledge is particularly useful for chemists working with atmospheric reactions and pollution control.

Chemistry Education

In the realm of chemistry education, understanding the fundamentals of reaction kinetics and rate laws is essential. These topics provide the foundation for how chemical reactions are perceived and analyzed.
Students learn to write rate laws, determine reaction orders, and understand the implications of changes in concentration and temperature.

• Significance: Fundamental for advanced studies in chemistry, informing both theoretical and practical skills.
• Approach: Courses often emphasize hands-on experiments to underscore theoretical concepts.
• Applications: Applied in fields like environmental science, engineering, and pharmacology.

Through studying chemistry, students garner insights into the processes that govern chemical changes, equipping them with analytical and problem-solving skills.