Chapter 25, Problem 78SCQ

### Chemistry & Chemical Reactivity

10th Edition
John C. Kotz + 3 others
ISBN: 9781337399074

Chapter
Section

### Chemistry & Chemical Reactivity

10th Edition
John C. Kotz + 3 others
ISBN: 9781337399074
Textbook Problem

# You might wonder how it is possible to determine the half-life of long-lived radioactive isotopes such as 238U. With a half-life of more than 109 years, the radioactivity of a sample of uranium will not measurably change in your lifetime. In fact, you can calculate the half-life using the mathematics governing first-order reactions.It can be shown that a 1.0-mg sample of 238U decays at the rate of 12 α emissions per second. Set up a mathematical equation for the rate of decay, ΔN/Δt = −kN, where N is the number of nuclei in the 1.0-mg sample and ΔN/Δt is 12 dps. Solve this equation for the rate constant for this process, and then relate the rate constant to the half-life of the reaction. Carry out this calculation, and compare your result with the literature value, 4.5 × 109 years.

Interpretation Introduction

Interpretation: To calculate the rate constant for the radioactive isotope of uranium-238 by using first order reaction and compared with the literature value.

Concept introduction:

The rate of decay can be calculated by the following formula:

ΔNΔt= -kN

Here,

K is the rate constant

N is the number of nuclei in 1.0mg of sample (Avogadro number=6.02×1023).

Explanation

Given information as follows:

Î”N/Î”tÂ =Â 12dps

The number of nuclei is 1.0mg.

The literature value of rate constant is 4.5Ã—109years

The rate of decay can be calculated by the following formula:

Î”N/Î”t=âˆ’kN

Here,

K=Rate constant

N=Number of nuclei in 1.0mg of sample (Avogadro number=6.02Ã—1023).

Nâ€‰=â€‰1.0â€‰mg238000â€‰mg/molÃ—Â 6.02Â Ã—Â 1023â€‰atom/mol=Â 4.201Ã—10-6Ã—6.02Ã—1023=Â 25

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