   Chapter 19, Problem 3RQ

Chapter
Section
Textbook Problem

# All radioactive decay processes follow first-order kinetics. What does this mean? What happens to the rate of radioactive decay as the number of nuclides is halved? Write the first-order rate law and the integrated first-order rate law. Define the terms in each equation. What is the half-life equation for radioactive decay processes? How does the half-life depend on how many nuclides are present? Are the half-life and rate constant k directly related or inversely related?

Interpretation Introduction

Interpretation: The various questions based upon kinetics of radioactive decay are to be answered.

Concept introduction: Nuclei of radioactive element decompose in various ways. There are two major categories. One involves a change in mass number of the decaying nucleus while others do not. Types of radioactive processes include α particle production, β particle production, γ ray production, electron capture and many others.

To determine: The answers of various questions based on kinetics of radioactive decay.

Explanation

A reaction is first order if the rate depends only upon the concentration of one reactant. In all the radioactive decay process, the decay rate is proportional to the first power of the radioactive atoms present; therefore, radioactive decay process is a first order kinetics.

The rate law for radioactive decay is,

Rate=ΔNΔt

Where,

ΔN is the change in nuclides.

Δt is the change in time.

The rate is directly proportional to the number of nuclides N in a given sample. Therefore, if the number of nuclides is halved, then the rate will decrease.

The integrated rate law for radioactive decay is given below.

The first order rate law for radioactive decay is,

Rate=ΔNΔt

Where,

ΔN is the change in nuclides.

Δt is the change in time.

The rate is directly proportional to number of nuclides N which is shown in the equation given below.

Rate=ΔNΔtαN

Replacing proportionality sign with rate constant K . Thus the equation becomes,

Rate=ΔNΔt=KN=ΔNN=KΔt

To get the integrated rate law, the above equation is integrated

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