   Chapter 25, Problem 78SCQ

Chapter
Section
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:

ΔN/Δt = 12dps

The number of nuclei is 1.0mg.

The literature value of rate constant is 4.5×109years

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

ΔN/Δt=kN

Here,

K=Rate constant

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

N=1.0mg238000mg/mol× 6.02 × 1023atom/mol= 4.201×10-6×6.02×1023= 25

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