Problem 3 - Chemistry and Chemical Reactivity

Question

In the reaction \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g}),\) the rate of for- mation of \(\mathrm{O}_{2}\) is \(1.5 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}\). What is the rate of decomposition of \(\mathrm{O}_{3} ?\)

Step-by-Step Solution

Answer

The rate of decomposition of \(\mathrm{O}_{3}\) is \(1.0 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}\).

Step 1: Identify the Reaction Stoichiometry

The balanced chemical equation is \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})\). This tells us that for every 2 moles of \(\mathrm{O}_{3}\) that decompose, 3 moles of \(\mathrm{O}_{2}\) are formed.

Step 2: Relate the Rates of Reaction

The rate of decomposition of \(\mathrm{O}_{3}\) is related to the rate of formation of \(\mathrm{O}_{2}\) by the stoichiometric coefficients. According to the equation, the rate of formation of \(\mathrm{O}_{2}\) is \(\frac{3}{2}\) times the rate of decomposition of \(\mathrm{O}_{3}\). This can also be written as \(\text{Rate of } \mathrm{O}_{3} \text{ decomposition} = \frac{2}{3} \times \text{Rate of } \mathrm{O}_{2} \text{ formation}\).

Step 3: Calculate the Rate of \(\mathrm{O}_{3}\) Decomposition

The given rate of formation of \(\mathrm{O}_{2}\) is \(1.5 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}\). Use the relation from the stoichiometry to find the rate of decomposition of \(\mathrm{O}_{3}\). Therefore, the rate of \(\mathrm{O}_{3}\) decomposition is \(\frac{2}{3} \times 1.5 \times 10^{-3} \).

Step 4: Perform the Calculation

Calculate \(\frac{2}{3} \times 1.5 \times 10^{-3} \):\[\text{Rate of } \mathrm{O}_{3} \text{ decomposition} = \frac{2}{3} \times 1.5 \times 10^{-3} = 1.0 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}\]

Key Concepts

Reaction Stoichiometry

Understanding reaction stoichiometry is crucial in chemical kinetics. It involves the quantitative relationship between reactants and products in a chemical reaction. In our example with \(2 \mathrm{O}_{3}(\mathrm{g}) \rightarrow 3 \mathrm{O}_{2}(\mathrm{g})\), the stoichiometry tells us that every 2 moles of ozone (\(O_3\)) decompose to produce 3 moles of oxygen gas (\(O_2\)). This relationship is expressed with the coefficients 2 and 3 in the balanced equation.

To use stoichiometry in problem-solving, refer to these coefficients to understand how changing one component in a reaction affects the others. By knowing how much of a substance reacts or forms, one can predict consumption or production levels. This is why stoichiometry is foundational to determining the rate of reactions.

Rate of Reaction

The rate of a chemical reaction is an indicator of how fast the reactants are transformed into products. It is typically expressed in terms of concentration changes over time, such as moles per liter per second (mol/L·s). In our example, the rate of formation of \(O_2\) is provided as \(1.5 \times 10^{-3} \space \text{mol/L·s}\).

Understanding the rate involves the stoichiometric coefficients from the balanced chemical equation. Since 3 moles of \(O_2\) form from 2 moles of \(O_3\), the rate of decomposition of \(O_3\) is related to the formation rate of \(O_2\) by the factor of \( \frac{2}{3} \). Hence, finding the rate of \(O_3\) decomposition becomes crucial to predicting how quickly the reaction progresses.

Decomposition Reaction

A decomposition reaction involves the breakdown of a single compound into two or more products. In our case, ozone \(O_3\) is decomposing into oxygen gas \(O_2\). Such reactions are often driven by heat, light, or chemical changes and are fundamental in processes ranging from atmospheric chemistry to industrial applications.

Key characteristics of decomposition reactions include:

• They frequently involve complex substances breaking down into simpler substances.
• They may be endothermic, requiring energy to proceed.
• Analyzing the decomposition rate provides insights into the stability and half-life of compounds under specific conditions.

Understanding the decomposition rate in this context helps chemists predict reaction behaviors, which is important for both academic studies and practical chemical processes.