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Chapter 12: Problem 107

Which of the following statement(s) is(are) true? a. The half-life for a zero-order reaction increases as the reaction proceeds. b. A catalyst does not change the value of \(\Delta \mathrm{E}\) . c. The half-life for a reaction, aA \(\longrightarrow\) products, that is first order in A increases with increasing \([\mathrm{A}]_{0} .\) d. The half-life for a second-order reaction increases as the reaction proceeds.

Short Answer

Expert verified
The true statements are b and d. Statement b is true because a catalyst does not change the overall energy change of the reaction, only the activation energy. Statement d is true because the half-life for a second-order reaction increases as the reaction proceeds, as it is inversely proportional to the initial concentration of A. Statements a and c are false.

Step by step solution

01

Statement a: The half-life for a zero-order reaction increases as the reaction proceeds.

For a zero-order reaction, the half-life is described by the equation: \[t_{1/2} = \frac{[\mathrm{A}]_{0}}{2k}\] Here, \(t_{1/2}\) is the half-life, \([\mathrm{A}]_{0}\) is the initial concentration of A, and k is the rate constant. Since the half-life depends only on the initial concentration of A and the rate constant, it does not change as the reaction proceeds. This statement is false.
02

Statement b: A catalyst does not change the value of \(\Delta \mathrm{E}\).

A catalyst speeds up the rate of the reaction by lowering the activation energy. However, it does not change the overall energy change of the reaction. In other words, the difference between the potential energy of the reactants and the products remains the same. Therefore, this statement is true.
03

Statement c: The half-life for a reaction, aA \(\longrightarrow\) products, that is first order in A increases with increasing \([\mathrm{A}]_{0}\).

For a first-order reaction, the half-life is defined as: \[t_{1/2} = \frac{0.693}{k}\]. Here, \(t_{1/2}\) is the half-life, and k is the rate constant. In this case, the half-life does not depend on the initial concentration of A, so it remains constant even as \([\mathrm{A}]_{0}\) increases. This statement is false.
04

Statement d: The half-life for a second-order reaction increases as the reaction proceeds.

For a second-order reaction, the half-life is given by: \[t_{1/2} = \frac{1}{k[\mathrm{A}]_{0}}\] In this case, the half-life is inversely proportional to the initial concentration of A, so as the reaction proceeds and \([\mathrm{A}]_{0}\) decreases, the half-life increases. Thus, this statement is true. In summary, statements b and d are true, while statements a and c are false.

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Most popular questions from this chapter

In the gas phase, the production of phosgene from chlorine and carbon monoxide is assumed to proceed by the following mechanism: $$ \mathrm{Cl}_{2} \stackrel{k_{1}}{\rightleftharpoons_{k_{1}}} 2 \mathrm{Cl} $$ $$ \mathrm{Cl}+\mathrm{CO} \stackrel{k_{2}}{\leftrightharpoons_{k-2}} \mathrm{COCl} $$ $$ \mathrm{COCl}+\mathrm{Cl}_{2} \stackrel{k_{3}}{\longrightarrow} \mathrm{COCl}_{2}+\mathrm{Cl} $$ $$ 2 \mathrm{Cl} \stackrel{k}{\longrightarrow} \mathrm{Cl}_{2} $$ Overall reaction: $\mathrm{CO}+\mathrm{Cl}_{2} \longrightarrow \mathrm{COCl}_{2}$ a. Write the rate law for this reaction. b. Which species are intermediates?

Upon dissolving \(\operatorname{In} \mathrm{Cl}(s)\) in $\mathrm{HCl}, \operatorname{In}^{+}(a q)$ undergoes a disproportionation reaction according to the following unbalanced equation: $$ \operatorname{In}^{+}(a q) \longrightarrow \operatorname{In}(s)+\operatorname{In}^{3+}(a q) $$ This disproportionation follows first-order kinetics with a half-life of 667 s. What is the concentration of \(\operatorname{In}^{+}(a q)\) after 1.25 \(\mathrm{h}\) if the initial solution of \(\operatorname{In}^{+}(a q)\) was prepared by dis- solving 2.38 \(\mathrm{g} \operatorname{InCl}(s)\) in dilute \(\mathrm{HCl}\) to make \(5.00 \times 10^{2} \mathrm{mL}\) of solution? What mass of In \((s)\) is formed after 1.25 \(\mathrm{h}\) ?

The reaction $$ 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g) $$ exhibits the rate law $$ \text {Rate} =k[\mathrm{NO}]^{2}\left[\mathrm{O}_{2}\right] $$ Which of the following mechanisms is consistent with this rate law? $$ \begin{array}{l}{\text { a. } \mathrm{NO}+\mathrm{O}_{2} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}} \\ {\mathrm{O}+\mathrm{NO} \longrightarrow \mathrm{NO}_{2}} \\ {\text { b. } \mathrm{NO}+\mathrm{O}_{2} \rightleftharpoons \mathrm{NO}_{3}} \\ {\mathrm{NO}_{3}+\mathrm{NO} \longrightarrow 2 \mathrm{NO}_{2}}\end{array} $$ $$ \begin{array}{l}{\text { c. } 2 \mathrm{NO} \longrightarrow \mathrm{N}_{2} \mathrm{O}_{2}} \\ {\mathrm{N}_{2} \mathrm{O}_{2}+\mathrm{O}_{2} \longrightarrow \mathrm{N}_{2} \mathrm{O}_{4}} \\ {\mathrm{N}_{2} \mathrm{O}_{4} \longrightarrow 2 \mathrm{NO}_{2}} \\ {\text { d. } 2 \mathrm{NO} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2}} \\\ {\mathrm{N}_{2} \mathrm{O}_{2} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}} \\\ {\mathrm{O}+\mathrm{NO} \longrightarrow \mathrm{NO}_{2}}\end{array} $$

A certain first-order reaction is 45.0\(\%\) complete in 65 s. What are the values of the rate constant and the half-life for this process?

A popular chemical demonstration is the "magic genie" procedure, in which hydrogen peroxide decomposes to water and oxygen gas with the aid of a catalyst. The activation energy of this (uncatalyzed) reaction is 70.0 \(\mathrm{kJ} / \mathrm{mol}\) . When the catalyst is added, the activation energy (at \(20 .^{\circ} \mathrm{C} )\) is 42.0 \(\mathrm{kJ} / \mathrm{mol}\) . Theoretically, to what temperature ( \((\mathrm{C})\) would one have to heat the hydrogen peroxide solution so that the rate of the uncatalyzed reaction is equal to the rate of the catalyzed reaction at \(20 .^{\circ} \mathrm{C} ?\) Assume the frequency factor \(A\) is constant, and assume the initial concentrations are the same.
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