Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Fundamentals Of Physics
Found in: Page 1304

Answers without the blur.

Just sign up for free and you're in.


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 byN=Rλ. (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=Rλ.

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

See the step by step solution

Step by Step Solution

Step 1: The given data

The time taken by the production process t>T12halflife 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.


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

dNdt=R-λN …… (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:

N0NdNR-λN=0tdt-1λIn R-λNR-λN0=tN=Rλ+N0-Rλe-λt

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

Thus, the above equation becomes, N=Rλ

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

The result N=Rλholds regardless of the initial value N0, because the dependence on N0shows up only in the second term, which is exponentially suppressed at large t.

Recommended explanations on Physics Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.