   # Consider a reaction of the type aA ⟶ products, in which the rate law is found to he rate = k [A] 3 (termolecular reactions are improbable but possible). If the first half-tife of the reaction is found to he 40. s, what is the time for the second half-life? Hint: Using your calculus knowledge, derive the integrated rate law from the differential rate law for a tennolecular reaction: Rate = − d [ A ] d t = k [ A ] 3 ### Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243

#### Solutions

Chapter
Section ### Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243
Chapter 11, Problem 102CP
Textbook Problem
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## Consider a reaction of the type aA ⟶ products, in which the rate law is found to he rate = k[A]3 (termolecular reactions are improbable but possible). If the first half-tife of the reaction is found to he 40. s, what is the time for the second half-life? Hint: Using your calculus knowledge, derive the integrated rate law from the differential rate law for a tennolecular reaction: Rate = − d [ A ] d t = k [ A ] 3

Interpretation Introduction

Interpretation: The rate law for a reaction and first half-life is given. By using these values, second half-life is to be determined for the reaction.

Concept introduction: Integrated rate law is an expression that combines the rate constant, time and change in concentrations in the form of a mathematical equation. The half-life of the third order reaction can be calculated by using the formula,

t1/2=32k[A]02

To determine: The time for the second half-life of the given reaction.

### Explanation of Solution

Given

First half-life of the reaction is 40s.

Rate law for third order reaction is,

Rate=k[A]3 (1)

Differential rate law for third order reaction is,

Rate=d[A]dt (2)

From equation (1) and equation (2),

d[A]dt=k[A]3d[A][A]3=kdt

Integrate the above equation,

k0tdt=[A]0[A]d[A][A]3k(t)0t=(12[A]2)[A]0[A]kt=(12[A]2+12[A]02)kt=(12[A]212[A]02) (3)

Where,

• [A]0 is the initial concentration of reactant A

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