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Chapter 12: Problem 99

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

Short Answer

Expert verified
First, find the order of the reaction (n) using the provided initial rate data: \(n = \frac{\log{(\frac{Rate_1}{Rate_2})}}{\log{(\frac{\mathrm{AB_1}}{\mathrm{AB_2}})}\) Next, calculate the rate constant (k) using the rate equation: \(k = \frac{Rate_1}{[\mathrm{AB_1}]^n}\) Finally, determine the half-life (\(t_{1/2}\)) using the appropriate formula for the reaction order: For a first-order reaction: \(t_{1/2} = \frac{0.693}{k}\) For a second-order reaction: \(t_{1/2} = \frac{1}{k[\mathrm{AB}]_0}\) Plug in the values of k and \([\mathrm{AB}]_0\) to find the half-life (\(t_{1/2}\)).

Step by step solution

01

Identify the order of the reaction

To determine the order of the reaction, analyze the initial rate data. Suppose the order of the reaction is n. Then, the rate equation will be: \(Rate = k [\mathrm{AB}]^n \) We need to find the value of n.
02

Create a table of initial rate data

Write down the initial concentration of AB and the corresponding initial rate in the form of a table: | Trial | Initial Concentration | Initial Rate | |-------|------------------------|--------------| | 1 | [\(\mathrm{AB}\)_1] | \(Rate_1\) | | 2 | [\(\mathrm{AB}\)_2] | \(Rate_2\) |
03

Determine the order of the reaction

Using the table created in step 2, analyze the change in initial rate corresponding to the change in the initial concentration. Divide the rate equation for the first trial by the rate equation for the second trial: \(\frac{Rate_1}{Rate_2} = \frac{k [\mathrm{AB}_1]^n}{k [\mathrm{AB}_2]^n}\) To determine the order of the reaction (n), you can rearrange the equation and solve for n: \(n = \frac{\log{(\frac{Rate_1}{Rate_2})}}{\log{(\frac{\mathrm{AB_1}}{\mathrm{AB_2}})}\) Plug in the values from the initial rate data table and solve for n.
04

Calculate the rate constant, k

Now that we know the order of the reaction, we can calculate the rate constant k. Using the rate equation and substituting the order found in step 3: \(Rate_1 = k [\mathrm{AB_1}]^n\) Solve for k: \(k = \frac{Rate_1}{[\mathrm{AB_1}]^n}\) Plug in the values from the initial rate data table and solve for k.
05

Determine the half-life of the reaction

Now, we have the rate constant k and the order of the reaction n. We can determine the half-life of the reaction using the following relationship between the half-life and rate constant: For a first-order reaction: \(t_{1/2} = \frac{0.693}{k}\) For a second-order reaction: \(t_{1/2} = \frac{1}{k[\mathrm{AB}]_0}\) Plug in the values of k and \([\mathrm{AB}]_0\) and solve for the half-life, \(t_{1/2}\).

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Most popular questions from this chapter

What are the units for each of the following if the concentrations are expressed in moles per liter and the time in seconds? a. rate of a chemical reaction b. rate constant for a zero-order rate law c. rate constant for a first-order rate law d. rate constant for a second-order rate law e. rate constant for a third-order rate law

Consider the reaction $$ 3 \mathrm{A}+\mathrm{B}+\mathrm{C} \longrightarrow \mathrm{D}+\mathrm{E} $$ where the rate law is defined as $$ -\frac{\Delta[\mathrm{A}]}{\Delta t}=k[\mathrm{A}]^{2}[\mathrm{B}][\mathrm{C}] $$ An experiment is carried out where $[\mathrm{B}]_{0}=[\mathrm{C}]_{0}=1.00 \mathrm{M}$ and \([\mathrm{A}]_{0}=1.00 \times 10^{-4} \mathrm{M}\) a. If after \(3.00 \min ,[\mathrm{A}]=3.26 \times 10^{-5} M,\) calculate the value of \(k .\) b. Calculate the half-life for this experiment. c. Calculate the concentration of \(B\) and the concentration of A after 10.0 min.

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