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Chapter 12: Problem 53

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

Expert verified
To determine the half-life for the decomposition of compound AB, first find the reaction order by creating plots for each Integrated Rate Law (zero-order, first-order, and second-order) and identify which one provides a straight line. Then, calculate the rate constant (k) using the slope of the best-fit line for the identified reaction order. Finally, use the appropriate half-life equation for the reaction order, plug in the obtained rate constant (k) and the initial concentration of 1.00 M (if required), and calculate the half-life.

Step by step solution

01

Find the Order of the Reaction

To determine the reaction order, we will use the Integrated Rate Laws for zero-order, first-order, and second-order reactions. We'll create a table for each and see which one has a straight line when we plot the values: Zero-order: \([\mathrm{AB}] = -kt+[A]_0\) First-order: \(\ln([\mathrm{A}]_0/[\mathrm{AB}]) = kt\) Second-order: \(1/[\mathrm{AB}] - 1/[\mathrm{A}]_0 = kt\) Plotting these values for each Integrated Rate Laws should give a straight line for one of them. Whichever best fits a straight line represents the true order of the reaction.
02

Calculate the Rate Constant

Once the reaction order has been identified by comparing the plots for Integrated Rate Laws, the rate constant can be determined by calculating the slope of the best-fit line. The rate constant (k) can be found directly from the slope of the graph for the corresponding reaction order. For zero-order: Slope = -k For first-order: Slope = k For second-order: Slope = k
03

Determine the Half-Life

After identifying the reaction order and rate constant (k), use the appropriate half-life equation for the given reaction order. The initial concentration of AB is 1.00 M: For zero-order: \(t_{1/2} = \frac{[\mathrm{AB}]_0}{2k}\) For first-order: \(t_{1/2} = \frac{\ln{2}}{k}\) For second-order: \(t_{1/2} = \frac{1}{k[\mathrm{AB}]_0}\) Plug in the obtained reaction rate constant (k) and initial concentration (if required) into the appropriate equation to find the half-life.

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Most popular questions from this chapter

In the Haber process for the production of ammonia, $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g) $$ what is the relationship between the rate of production of ammonia and the rate of consumption of hydrogen?

Provide a conceptual rationale for the differences in the halflives of zero-, first-, and second-order reactions.

For enzyme-catalyzed reactions that follow the mechanism $$ \begin{aligned} \mathrm{E}+\mathrm{S} & \rightleftharpoons \mathrm{E} \cdot \mathrm{S} \\ \mathrm{E} \cdot \mathrm{S} & \rightleftharpoons \mathrm{E}+\mathrm{P} \end{aligned} $$ a graph of the rate as a function of [S], the concentration of the substrate, has the following appearance: Note that at higher substrate concentrations the rate no longer changes with [S]. Suggest a reason for this.

The reaction $$ \mathrm{A} \longrightarrow \mathrm{B}+\mathrm{C} $$ is known to be zero order in \(\mathrm{A}\) and to have a rate constant of \(5.0 \times 10^{-2} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\) at \(25^{\circ} \mathrm{C}\). An experiment was run at \(25^{\circ} \mathrm{C}\) where \([\mathrm{A}]_{0}=1.0 \times 10^{-3} M\) a. Write the integrated rate law for this reaction. b. Calculate the half-life for the reaction. c. Calculate the concentration of \(\mathrm{B}\) after \(5.0 \times 10^{-3} \mathrm{~s}\) has elapsed assuming \([\mathrm{B}]_{0}=0\).

You and a coworker have developed a molecule that has shown potential as cobra antivenin (AV). This antivenin works by binding to the venom (V), thereby rendering it nontoxic. This reaction can be described by the rate law $$ \text { Rate }=k[\mathrm{AV}]^{1}[\mathrm{~V}]^{1} $$ You have been given the following data from your coworker: $$ \begin{aligned} [\mathrm{V}]_{0} &=0.20 \mathrm{M} \\ [\mathrm{AV}]_{0} &=1.0 \times 10^{-4} \mathrm{M} \end{aligned} $$ A plot of \(\ln [\mathrm{AV}]\) versus \(t\) (s) gives a straight line with a slope of \(-0.32 \mathrm{~s}^{-1}\). What is the value of the rate constant \((k)\) for this reaction?
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