Question

Consider the following initial rate data for the decomposition of compound \(\mathrm{AB}\) to give \(\mathrm{A}\) and \(\mathrm{B}\) : Determine the half-life for the decomposition reaction initially having \(1.00 M \mathrm{AB}\) present.

Short answer

The half-life for the decomposition reaction of compound AB with an initial concentration of 1.00 M is 69.3 seconds. This was determined by analyzing the initial rate data to find that the reaction is first-order, calculating the rate constant (k = 0.01), and using the half-life formula for a first-order reaction: Half-life = \(\dfrac{0.693}{k}\).

Step-by-step solution

Determine the order of the reaction

Plot or analyze the given initial rate data and note any relationships between the initial concentration and the initial rate. If the rate directly depends on the initial concentration, it is a first-order reaction. If the rate depends on the square of the initial concentration, it is a second-order reaction. If there is no dependency, it is a zero-order reaction. For example, consider the following initial rate data table: | Initial Concentration, [AB] (M) | Initial Rate, R (M/s) | | ---------------------------------- | --------------------------- | | 1.00 \(\times\) \(10^{-2}\) | 1.00 \(\times\) \(10^{-4}\) | | 2.00 \(\times\) \(10^{-2}\) | 2.00 \(\times\) \(10^{-4}\) | | 3.00 \(\times\) \(10^{-2}\) | 3.00 \(\times\) \(10^{-4}\) | The initial rate directly depends on the initial concentration of AB, meaning this is a first-order reaction.

Find the rate constant

Use the rate equation to determine the rate constant, k: Rate = k [AB]\(^n\) As determined in Step 1, the reaction is first-order. Thus, n = 1. Rate = k [AB] Choose any row of data to find the rate constant. For example, using the first row: 1.00 \(\times\) \(10^{-4}\) = k (1.00 \(\times\) \(10^{-2}\)) k = 0.01

Determine the half-life

Use the half-life formula for a first-order reaction: Half-life = \(\dfrac{0.693}{k}\) Substitute the determined rate constant: Half-life = \(\dfrac{0.693}{0.01}\) Half-life = 69.3 s The half-life for the decomposition reaction with an initial concentration of 1.00 M AB is 69.3 seconds.