# Naturally occurring uranium is composed mostly of 238 U and 235 U, with relative abundances of 99.28% and 0.72%, respectively. The half-life for 238 U is 4.5 × 10 9 years, and the half-life for 235 U is 7.1 × 10 8 years. Assuming that the earth was formed 4.5 billion years ago, calculate the relative abundances of the 238 U and 235 U isotopes when the earth was formed.

### Chemistry: An Atoms First Approach

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

Chapter
Section

### Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243
Chapter 18, Problem 79CP
Textbook Problem
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## Naturally occurring uranium is composed mostly of 238U and 235U, with relative abundances of 99.28% and 0.72%, respectively. The half-life for 238U is 4.5 × 109 years, and the half-life for 235U is 7.1 × 108 years. Assuming that the earth was formed 4.5 billion years ago, calculate the relative abundances of the 238U and 235U isotopes when the earth was formed.

Interpretation Introduction

Interpretation: Half life of 238U and 235U and their relative abundances is given. Their relative abundances when earth was formed 4.5 billion years ago are to be calculated.

Concept introduction: A process through which an unstable nuclide looses its energy due to excess of protons or neutrons is known as radioactive decay. Decay constant is the quantity that expresses the rate of decrease of number of atoms of a radioactive element per second. Half life of radioactive sample is defined as the time required for the number of nuclides to reach half of the original value.

To determine: The relative abundances of 238U and 235U when earth was formed.

### Explanation of Solution

Explanation

The decay constant is calculated by the formula given below.

λ=0.693t1/2

Where

• t1/2 is the half life of nuclide.
• λ is decay constant.

The value of t1/2 for 238U is 4.5×109years .

Substitute the value of half life in the above formula.

λ=0.6934.5×109years-1

The decay constant is 0.6934.5×109years-1 .

The decay constant is calculated by the formula given below.

λ=0.693t1/2

Where

• t1/2 is the half life of nuclide.
• λ is decay constant.

The value of t1/2 for 235U is 7.1×108years .

Substitute the value of half life in the above formula.

λ=0.6937.1×108years-1 .

The decay constant is 0.6937.1×108years-1 .

The relative abundance of 238U is calculated by the formula,

ln(nn0)=λt

Where

• n0 is the amount of 238U when earth was formed.
• n is the amount of 238U after 4.5×109 years.
• t is the decay time.

The value of t is 4.5×109 years.

Substitute the values of t and decay constant in the above equation.

ln(nn0)=λtln(nn0)=0.693×4.5×1094.5×109ln(nn0)=0.693(nn0)=0.50

The value of (nn0) for 238U is 0.50 .

The relative abundance of 235U is calculated by the formula,

ln(nn0)=λt

Where

• n0 is the amount of 235U when earth was formed.
• n is the amount of 235U after 4.5×109 years.
• t is the decay time.

The value of t is 7.1×108 years.

Substitute the values of t and decay constant in the above equation.

ln(nn0)=λtln(nn0)=0.693×4.5×1097.1×108(nn0)=e4.39(nn0)=0

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