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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# A certain radionuclide is being manufactured in a cyclotron at a constant rate R. It is also decaying with disintegration constant ${\mathbit{\lambda }}$. 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${\mathbit{N}}{\mathbf{=}}\frac{\mathbf{R}}{\mathbf{\lambda }}$. (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 $N=\frac{R}{\lambda }$.

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

See the step by step solution

## Step 1: The given data

The time taken by the production process $t>{T}_{\frac{1}{2}}\left(\mathrm{halflife}\mathrm{of}\mathrm{the}\mathrm{radionuclide}\right)$,

## 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:

$\frac{dN}{dt}=R-\lambda N$ …… (i)

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

R is the rate of production by the cyclotron,

$\lambda$ 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:

${\int }_{{\mathrm{N}}_{0}}^{\mathrm{N}}\frac{dN}{R-\lambda N}={\int }_{0}^{t}dt\phantom{\rule{0ex}{0ex}}-\frac{1}{\lambda }In\frac{R-\lambda N}{R-\lambda {N}_{0}}=t\phantom{\rule{0ex}{0ex}}N=\frac{R}{\lambda }+\left({N}_{0}-\frac{R}{\lambda }\right){e}^{-\lambda t}$

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

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

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

The result $N=\frac{R}{\lambda }$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.