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Chapter 14: Problem 16

One of the following statements is true and the other is false regarding the first-order reaction $2 \overrightarrow{\mathrm{A}} \longrightarrow \mathrm{B}+\mathrm{C} .$ Identify the true statement and the false one, and explain your reasoning. (a) A graph of [A] versus time is a straight line. (b) The rate of the reaction is one-half the rate of disappearance of A.

Short Answer

Expert verified
Statement (a) is false: A graph of [A] vs. time for a first-order reaction is an exponential decay curve, not a straight line. Statement (b) is true: The rate of the reaction is one-half the rate of disappearance of A, due to the stoichiometry of the reaction.

Step by step solution

01

Analyzing Statement (a)

A first-order reaction is characterized by a constant half-life, and its rate depends only on the concentration of one reactant, in this case A. The equation for a first-order reaction is \([A] = [A]_0 \cdot \exp^{-kt}\), where \([A]\) is the concentration of A at time \(t\), \([A]_0\) is the initial concentration of A, \(k\) is the rate constant, and \(t\) is time. If you graph \([A]\) vs. \(t\), you would get an exponential decay curve, not a straight line. So, statement (a) is false.
02

Analyzing Statement (b)

In the given reaction \(2A \rightarrow B + C\), the rate of disappearance of A is \(−\frac{1}{2} \cdot \frac{d[A]}{dt}\) while the rate of the reaction is normally defined as \(+\frac{1}{d} \cdot \frac{d[B]}{dt} = +\frac{1}{d} \cdot \frac{d[C]}{dt}\). Due to the stoichiometry of the equation (for every 2 particles of A that react, 1 particle of B and 1 particle of C is produced), the rate of the reaction is indeed half the rate of disappearance of A. So, statement (b) is true.

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Most popular questions from this chapter

If even a tiny spark is introduced into a mixture of \(\mathrm{H}_{2}(\mathrm{g})\) and \(\mathrm{O}_{2}(\mathrm{g}),\) a highly exothermic explosive reaction occurs. Without the spark, the mixture remains unreacted indefinitely. (a) Explain this difference in behavior. (b) Why is the nature of the reaction independent of the size of the spark?

We have used the terms order of a reaction and molecularity of an elementary process (that is, unimolecular, bimolecular). What is the relationship, if any, between these two terms?

The following statements about catalysis are not stated as carefully as they might be. What slight modifications would you make in them? (a) A catalyst is a substance that speeds up a chemical reaction but does not take part in the reaction. (b) The function of a catalyst is to lower the activation energy for a chemical reaction.

For the reaction \(\mathrm{A}+2 \mathrm{B} \longrightarrow \mathrm{C}+\mathrm{D},\) the rate law is rate of reaction \(=k[\mathrm{A}][\mathrm{B}]\) (a) Show that the following mechanism is consistent with the stoichiometry of the overall reaction and with the rate law. $$\begin{array}{l} \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{I} \quad(\text { slow }) \\ \mathrm{I}+\mathrm{B} \longrightarrow \mathrm{C}+\mathrm{D} \quad(\text { fast }) \end{array}$$ (b) Show that the following mechanism is consistent with the stoichiometry of the overall reaction, but not with the rate law. $$\begin{array}{c} 2 \mathrm{B} \stackrel{k_{1}}{\mathrm{k}_{1}} \mathrm{B}_{2} \text { (fast) } \\\ \mathrm{A}+\mathrm{B}_{2} \stackrel{k_{2}}{\longrightarrow} \mathrm{C}+\mathrm{D} \text { (slow) } \end{array}$$

For the disproportionation of \(p\)-toluenesulfinic acid, $$3 \mathrm{ArSO}_{2} \mathrm{H} \longrightarrow \mathrm{ArSO}_{2} \mathrm{SAr}+\mathrm{ArSO}_{3} \mathrm{H}+\mathrm{H}_{2} \mathrm{O}$$ (where \(\mathrm{Ar}=p-\mathrm{CH}_{3} \mathrm{C}_{6} \mathrm{H}_{4}-\) ), the following data were obtained: \(t=0 \min ,[\mathrm{ArSO}_{2} \mathrm{H}]=0.100 \mathrm{M} ; 15 \mathrm{min}, 0.0863 \mathrm{M} ; 30 \mathrm{min}, 0.0752 \mathrm{M} ; 45 \mathrm{min}, 0.0640 \mathrm{M} ; 60 \mathrm{min}, 0.0568 \mathrm{M} ; 120 \mathrm{min}, 0.0387 \mathrm{M} ; 180 \mathrm{min}, 0.0297 \mathrm{M}; 300 \mathrm{min}, 0.0196 \mathrm{M}.\) (a) Show that this reaction is second order. (b) What is the value of the rate constant, \(k ?\) (c) At what time would \(\left[\mathrm{ArSO}_{2} \mathrm{H}\right]=0.0500 \mathrm{M} ?\) (d) At what time would \(\left(\mathrm{ArSO}_{2} \mathrm{H}\right)=0.0250 \mathrm{M} ?\) (e) At what time would \(\left[\mathrm{ArSO}_{2} \mathrm{H}\right]=0.0350 \mathrm{M} ?\)
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