Consider the reaction: 2 O₃(g) → 3 O₂( g) The rate law for this reaction is: Rate = k [O₃]² [O₂] Suppose that a 1.0-L reaction vessel initially contains 1.0 mol of O₃ and 1.0 mol of O₂. What fraction of the O₃ will have reacted when the rate falls to one-half of its initial value?

Verified step by step guidance

1. Identify the initial rate of the reaction using the rate law: \( \text{Rate}_{\text{initial}} = k [\text{O}_3]^2 [\text{O}_2] \).
2. Determine the expression for the rate when it falls to half of its initial value: \( \text{Rate}_{\text{half}} = \frac{1}{2} \times \text{Rate}_{\text{initial}} \).
3. Set up the equation for the rate at half its initial value: \( \frac{1}{2} \times k [\text{O}_3]^2 [\text{O}_2] = k [\text{O}_3']^2 [\text{O}_2'] \), where \([\text{O}_3']\) and \([\text{O}_2']\) are the concentrations when the rate is halved.
4. Assume \( x \) moles of \( \text{O}_3 \) have reacted, then \([\text{O}_3'] = 1.0 - x\) and \([\text{O}_2'] = 1.0 + \frac{3}{2}x\) because for every 2 moles of \( \text{O}_3 \) that react, 3 moles of \( \text{O}_2 \) are produced.
5. Substitute \([\text{O}_3']\) and \([\text{O}_2']\) into the rate equation and solve for \( x \) to find the fraction of \( \text{O}_3 \) that has reacted.

Key Concepts

Rate Law

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. It is typically formulated as Rate = k [A]^m [B]^n, where k is the rate constant, and m and n are the orders of the reaction with respect to reactants A and B. Understanding the rate law is crucial for predicting how changes in concentration affect the reaction rate.

Reaction Order

The reaction order is the sum of the exponents in the rate law and indicates how the rate of reaction depends on the concentration of reactants. In the given reaction, the order with respect to O₃ is 2, meaning that the rate is proportional to the square of its concentration. This concept helps in determining how the concentration of reactants influences the speed of the reaction.

Half-life of a Reaction

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For reactions of different orders, the half-life can vary significantly. In this context, understanding how the rate changes as the concentration of O₃ decreases is essential for calculating the fraction that has reacted when the rate falls to half its initial value.