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Chapter 13: Problem 83

The integrated rate law for the zero-order reaction \(\mathrm{A} \longrightarrow \mathrm{B}\) is \([\mathrm{A}]_{t}=[\mathrm{A}]_{0}-k t .\) (a) Sketch the follow- ing plots: (i) rate versus \([\mathrm{A}]_{t}\) and (ii) \([\mathrm{A}]_{t}\) versus \(t\). (b) Derive an expression for the half-life of the reaction. (c) Calculate the time in half-lives when the integrated rate law is no longer valid, that is, when \([\mathrm{A}]_{t}=0\)

Short Answer

Expert verified
For a zero-order reaction, (a) the plot of rate versus [A]ₜ is a straight line while the plot of [A]ₜ versus t is a straight line with a negative slope. (b) The half-life of the reaction can be expressed as t₁/₂=[A]₀/2k. (c) When [A]ₜ=0, the time in terms of half-lives is 2.

Step by step solution

01

Understanding the plots

For a zero-order reaction, the rate of reaction is independent of the concentration of the reactant, i.e., it remains constant. So, when we plot rate against [A]ₜ, we get a horizontal straight line. Conversely, when [A]ₜ is plotted against time (t), the graph will be a straight line with a negative slope since the concentration of A decreases with time.
02

Deriving the half-life expression

Half-life (t₁/₂) of a reaction is the time during which the concentration of the reactant gets reduced to half its original concentration. For a zero-order reaction, we can derive it using the equation [A]ₜ=[A]₀-kt. Let the concentration at t=t₁/₂ be [A]₀/2. Thus, we have [A]₀/2=[A]₀-kt₁/₂. Simplifying this gives us t₁/₂=[A]₀/2k.
03

Time at which rate becomes invalid

The integrated rate law of a zero-order reaction becomes invalid when [A]ₜ = 0. Plugging these into the integrated rate law gives us 0=[A]₀-kt. Upon rearrangement, t=[A]₀/k. To express this in half-lives, divide this by the half-life, t₁/₂=[A]₀/2k, to get t/t₁/₂= 2.

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Most popular questions from this chapter

The bromination of acetone is acid-catalyzed: \(\mathrm{CH}_{3} \mathrm{COCH}_{3}+\mathrm{Br}_{2} \frac{\mathrm{H}^{+}}{\text {cually }} \mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{Br}+\mathrm{H}^{+}+\mathrm{Br}^{-}\) The rate of disappearance of bromine was measured for several different concentrations of acetone, bromine, and \(\mathrm{H}^{+}\) ions at a certain temperature: $$ \begin{array}{lclll} \hline & &{\text { Rate of }} \\ & & & & \text { Disappearance } \\ & {\left[\mathrm{CH}_{3} \mathrm{COCH}_{3}\right]} & {\left[\mathrm{Br}_{2}\right]} & {\left[\mathrm{H}^{+}\right]} & \text {of } \mathrm{Br}_{2}(M / \mathrm{s}) \\ \hline(1) & 0.30 & 0.050 & 0.050 & 5.7 \times 10^{-5} \\ (2) & 0.30 & 0.10 & 0.050 & 5.7 \times 10^{-5} \\ (3) & 0.30 & 0.050 & 0.20 & 1.2 \times 10^{-4} \\ (4) & 0.40 & 0.050 & 0.20 & 3.1 \times 10^{-4} \\ (5) & 0.40 & 0.050 & 0.050 & 7.6 \times 10^{-5} \\ \hline \end{array} $$ (a) What is the rate law for the reaction? (b) Determine the rate constant. (c) The following mechanism has been proposed for the reaction: Show that the rate law deduced from the mechanism is consistent with that shown in (a).

The reaction \(\mathrm{H}+\mathrm{H}_{2} \longrightarrow \mathrm{H}_{2}+\mathrm{H}\) has been studied for many years. Sketch a potential energy versus reaction progress diagram for this reaction.

Is the rate constant \((k)\) of a reaction more sensitive to changes in temperature if \(E_{\mathrm{a}}\) is small or large?

Distinguish between homogeneous catalysis and heterogeneous catalysis. Describe three important industrial processes that utilize heterogeneous catalysis.

What are the units of the rate constant for a thirdorder reaction?
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