Question

The isotope Francium $ 224 $ has a half life of $ 20 $ minutes. A sample of the isotope has an initial activity of $ 9 $ disintegrations per second. The approximate activity of the sample an hour later will be what? Multiple choice question. Please explain why.

Answer 1

Hint: Some unstable nucleus undergoes decomposition to form a stable nucleus. This process is known as radioactive decay or radioactive decomposition. The given half life is $ 20 $ minutes. Given time is one hour, by substituting this time in the below formula gives the approximate activity of the sample an hour later.
 $ N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{1/2}}}}}} $
 $ N\left( t \right) $ is the quantity of the substance remaining
 $ {N_0} $ is the initial amount
 $ t $ is the time given
 $ {t_{1/2}} $ is half -life.

Complete Step By Step Answer:
Francium is a radioactive element that can undergo radioactive decay or decomposition to form a stable nucleus and has a half-life of $ 20 $ minutes.
Given that the isotope Francium $ 224 $ has a half-life of $ 20 $ minutes.
The given time is an hour which means one hour means $ 60 $ minutes. Which means the sample is passed through three half-lives.
As the initial quantity is same, substitute the half lime in $ {t_{1/2}} $ and $ 60 $ minutes in $ t $
 $ N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{{60}}{{20}}}} $
By simplifying the above formula,
 $ N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^3} $
Thus, the approximate activity of the sample an hour later which is known as disintegration is $ \dfrac{{{N_0}}}{8} $ .

Note:
The initial we have taken is the same, but the half-life time and given time is $ 60 $ minutes, from these both the number of half-lives that sample passes can be known. From the half-lives passed the amount of activity of the sample can be determined. For three one half-life the value is $ \dfrac{{{A_0}}}{2} $ , and for the two half-lives the value is $ \dfrac{{{A_0}}}{4} $ . These constant values are given the activity of the sample.