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Chapter 13: Problem 23

Write the equations relating the half-life of a secondorder reaction to the rate constant. How does it differ from the equation for a first-order reaction?

Short Answer

Expert verified
The equation connecting half-life and rate constant for a first-order reaction is \(T_{1/2} = \frac{0.693}{k}\), and for a second order reaction is \(T_{1/2} = \frac{1}{{k[A]_0}}\). The key difference lies in the dependence on initial concentration - in first order reactions, half-life is independent of initial concentration, while in second order reactions, it is directly proportional to it.

Step by step solution

01

Understanding Half-life

Half-life of a reaction is the time required for the concentration of the reactant to reduce to half its initial value.
02

Half-life for First Order Reactions

In case of a first order reaction, half-life (designated as \(T_{1/2}\)) is given by \(T_{1/2} = \frac{0.693}{k}\), where k is the rate constant of the reaction.
03

Half-life for Second Order Reactions

In case of a second order reaction, the half-life depends not only on the rate constant, but also on the initial concentration of the reactants. It is defined by \(T_{1/2} = \frac{1}{{k[A]_0}}\), where \(k\) is the rate constant and \([A]_0\) is the initial concentration of the reactant.
04

Comparing Half-life Equations

A comparison of the two equations shows that the half-life of a first order reaction is independent of the initial concentration of the reactants (as it is not included in the equation), while in the case of a second-order reaction, it is dependent on the initial concentration.

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Most popular questions from this chapter

Define half-life. Write the equation relating the halflife of a first-order reaction to the rate constant.

Determine the molecularity and write the rate law for each of the following elementary steps: (a) \(\mathrm{X} \longrightarrow\) products (b) \(\mathrm{X}+\mathrm{Y} \longrightarrow\) products (c) \(\mathrm{X}+\mathrm{Y}+\mathrm{Z} \longrightarrow\) products (d) \(\mathrm{X}+\mathrm{X} \longrightarrow\) products (e) \(\mathrm{X}+2 \mathrm{Y} \longrightarrow\) products

The thermal decomposition of phosphine \(\left(\mathrm{PH}_{3}\right)\) into phosphorus and molecular hydrogen is a first-order reaction: $$ 4 \mathrm{PH}_{3}(g) \longrightarrow \mathrm{P}_{4}(g)+6 \mathrm{H}_{2}(g) $$ The half-life of the reaction is \(35.0 \mathrm{~s}\) at \(680^{\circ} \mathrm{C}\). Calculate (a) the first-order rate constant for the reaction and (b) the time required for 95 percent of the phosphine to decompose.

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