Americium is used in smoke detectors and in medicine for the treatment of certain malignancies. One isotope of americium, 241 Am, has a rate constant, k , for radioactive decay of 0.0016 y ‒1 . In contrast, radioactive iodine-125, which is used for studies of thyroid functioning, has a rate constant for decay of 0.011 d ‒1 . (a) What are the half-lives of these isotopes? (b) Which isotope decays faster? (c) If you are given a dose of iodine-125 containing 1.6 × 10 15 atoms, how many atoms remain after 2.0 days?

Chemistry & Chemical Reactivity

9th Edition
John C. Kotz + 3 others
Publisher: Cengage Learning
ISBN: 9781133949640

Chapter
Section

Chemistry & Chemical Reactivity

9th Edition
John C. Kotz + 3 others
Publisher: Cengage Learning
ISBN: 9781133949640
Chapter 14.4, Problem 5CYU
Textbook Problem
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Americium is used in smoke detectors and in medicine for the treatment of certain malignancies. One isotope of americium, 241Am, has a rate constant, k, for radioactive decay of 0.0016 y‒1. In contrast, radioactive iodine-125, which is used for studies of thyroid functioning, has a rate constant for decay of 0.011 d‒1. (a) What are the half-lives of these isotopes? (b) Which isotope decays faster? (c) If you are given a dose of iodine-125 containing 1.6 × 1015 atoms, how many atoms remain after 2.0 days?

(a)

Interpretation Introduction

Interpretation:

The half-lives of 241Am and 125I isotopes has to be calculated.

Concept Introduction:

The rate of reaction is the quantity of formation of product or the quantity of reactant used per unit time.  The rate of reaction doesn’t depend on the sum of amount of reaction mixture used.

The raise in molar concentration of product of a reaction per unit time or decrease in molarity of reactant per unit time is called rate of reaction and is expressed in units of mol/(L.s).

Integrated rate law for first order reaction:

Consider A as substance, that gives the product based on the equation,

aAproducts

Where a= stoichiometric co-efficient of reactant A.

Consider the reaction has first-order rate law,

Rate=-Δ[A]Δt=k[A]

The integrated rate law equation can be given as,

ln[A]t[A]o=-kt

The above expression is called integrated rate law for first order reaction.

Half-life for first order reactions:

The half-life for the first order reaction is constant and it is independent of the reactant concentration.

Half-life period of first order reaction can be calculated using the equation,

t1/2=0.693k

Explanation of Solution

The half-lives of these reactions are calculated as,

Calculatehalf-lives:_For241Am,k =  0.0016 y-1t1/2 = 0.693k  = 0.6930.0016 y-1 = 433 year.For125I,k =  0

(b)

Interpretation Introduction

Interpretation:

The isotope that decays faster has to be given.

Concept Introduction:

Half-life for first order reactions:

The half-life for the first order reaction is constant and it is independent of the reactant concentration.

Half-life period of first order reaction can be calculated using the equation,

t1/2=0.693k

(c)

Interpretation Introduction

Interpretation:

The number of atoms that remains after two days has to be calculated.

Concept Introduction:

Integrated rate law for first order reaction:

Consider A as substance, that gives the product based on the equation,

aAproducts

Where a= stoichiometric co-efficient of reactant A.

Consider the reaction has first-order rate law,

Rate=-Δ[A]Δt=k[A]

The integrated rate law equation can be given as,

ln[A]t[A]o=-kt

The above expression is called integrated rate law for first order reaction.

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