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Chapter 14: Problem 43

The half-lives of both zero-order and second-order reactions depend on the initial concentration, as well as on the rate constant. In one case, the half- life gets longer as the initial concentration increases, and in the other it gets shorter. Which is which, and why isn't the situation the same for both?

Short Answer

Expert verified
In the case of zero-order reactions, the half-life increases as the initial concentration increases. For second-order reactions, the half-life decreases as the initial concentration increases. This is because the rate of a zero-order reaction is independent of the concentration of reactant, while the rate of a second-order reaction is highly dependent on it.

Step by step solution

01

Understand zero-order reactions

A zero order reaction has the rate law: rate = k[A]^0 = k. Its half-life equation is \(t_{1/2} = \frac{[A]_0}{2k}\), where [A]_0 is the initial concentration, and k is the rate constant. As we can see, as the initial concentration [A]_0 increases, the half-life also increases, meaning the reaction takes longer.
02

Understand second-order reactions

A second order reaction has the rate law: rate = k[A]^2. Its half-life equation is \(t_{1/2} = \frac{1}{2k[A]_0}\). In this case, as the initial concentration [A]_0 increases, the half-life decreases, meaning the reaction takes less time.
03

Explain why they differ

The reason why these two reactions have opposite dependencies on the initial concentration relates to their rate laws. For a zero-order reaction, the rate is independent of the concentration of reactant, while for a second-order reaction, the rate is highly dependent on the concentration of the reactant. Therefore, having a large initial concentration for a second-order reaction means there are more reactant molecules available to react, thus the reaction completes faster.

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Most popular questions from this chapter

The rate of a chemical reaction generally increases rapidly, even for small increases in temperature, because of a rapid increase in (a) collision frequency; (b) fraction of reactant molecules with very high kinetic energies; (c) activation energy; (d) average kinetic energy of the reactant molecules.

In the first-order reaction \(A \longrightarrow\) products, \([\mathrm{A}]=0.816 \mathrm{M}\) initially and \(0.632 \mathrm{M}\) after \(16.0 \mathrm{min}.\) (a) What is the value of the rate constant, \(k ?\) (b) What is the half-life of this reaction? (c) At what time will \([\mathrm{A}]=0.235 \mathrm{M} ?\) (d) What will [A] be after 2.5 h?

The following data were obtained for the dimerization of 1,3 -butadiene, \(2 \mathrm{C}_{4} \mathrm{H}_{6}(\mathrm{g}) \longrightarrow \mathrm{C}_{8} \mathrm{H}_{12}(\mathrm{g}),\) at 600 K: \(t=0 \min ,\left[\mathrm{C}_{4} \mathrm{H}_{6}\right]=0.0169 \mathrm{M} ; 12.18 \mathrm{min}, 0.0144 \mathrm{M} ; 24.55 \mathrm{min}, 0.0124 \mathrm{M} ; 42.50 \mathrm{min}, 0.0103 \mathrm{M}, 68.05 \min , 0.00845 \mathrm{M}.\) (a) What is the order of this reaction? (b) What is the value of the rate constant, \(k ?\) (c) At what time would \(\left[\mathrm{C}_{4} \mathrm{H}_{6}\right]=0.00423 \mathrm{M} ?\) (d) At what time would \(\left[\mathrm{C}_{4} \mathrm{H}_{6}\right]=0.0050 \mathrm{M} ?\)

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. $$\begin{array}{c} 2 \mathrm{B} \stackrel{k_{1}}{\mathrm{k}_{1}} \mathrm{B}_{2} \text { (fast) } \\\ \mathrm{A}+\mathrm{B}_{2} \stackrel{k_{2}}{\longrightarrow} \mathrm{C}+\mathrm{D} \text { (slow) } \end{array}$$

If the plot of the reactant concentration versus time is linear, then the order of the reaction is (a) zero order; (b) first order; (c) second order; (d) third order.
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