Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

A certain radionuclide is being manufactured in a cyclotron at a constant rate R. It is also decaying with disintegration constant $λ$ . Assume that the production process has been going on for a time that is much longer than the half-life of the radionuclide. (a) Show that the numbers of radioactive nuclei present after such time remains constant and is given by $N = \frac{R}{λ}$ . (b) Now show that this result holds no matter how many radioactive nuclei were present initially. The nuclide is said to be in secular equilibrium with its source; in this state its decay rate is just equal to its production rate.

a) The number of radioactive nuclei present after such time remains constant and is thus given by $N = \frac{R}{λ}$ .

b) The results does not change for any amount of radioactive nuclei present initially.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

The time taken by the production process $t > T_{\frac{1}{2}} (halflife of the radionuclide)$ ,

Step 2: Determine the concept of decay and production rate

The radioactive decay is due to the loss of the elementary particles from an unstable nucleus to convert them into a more stable one. The radioactive decay constant or the disintegration constant represents the fraction of radioactive atoms that disintegrates in a unit of time. The production rate of the atoms of a given isotope about the disintegration constant and time will give us the required relation. At times that are long compared to the half-life, the rate of production equals the rate of decay, and N is a constant. The nuclide is in secular equilibrium with its source.

Formula:

The rate of undecayed nuclei for the given time is as follows:

$\frac{d N}{d t} = R - λ N$ …… (i)

Here,N is the number of undecayed nuclei present at time .

R is the rate of production by the cyclotron,

$λ$ is the disintegration constant.

The second term gives the rate of decay t.

Step 3: a) Calculate the number of the remaining nuclei

Rearranging equation (i) and integrating it as per the problem, we can get the equation to number of nuclei as follows:

$\int_{N_{0}}^{N} \frac{d N}{R - λ N} = \int_{0}^{t} d t - \frac{1}{λ} I n \frac{R - λ N}{R - λ N_{0}} = t N = \frac{R}{λ} + (N_{0} - \frac{R}{λ}) e^{- λ t}$

After many half-lives, the exponential is small and the second term can be neglected.

Thus, the above equation becomes, $N = \frac{R}{λ}$

Step 4: b) Determine the known behavior of the result for all radioactive nuclei

The result $N = \frac{R}{λ}$ holds regardless of the initial value $N_{0}$ , because the dependence on $N_{0}$ shows up only in the second term, which is exponentially suppressed at large t.

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Determine the concept of decay and production rate

Step 3: a) Calculate the number of the remaining nuclei

Step 4: b) Determine the known behavior of the result for all radioactive nuclei

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Fundamentals Of Physics

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Step by Step Solution

Step 1: The given data

Step 2: Determine the concept of decay and production rate

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Step 4: b) Determine the known behavior of the result for all radioactive nuclei

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