Question

A radioactive substance has a half-life of 60 minutes. After 3 hours, the fraction of atom that have decayed would be
A) $12.5 \%$
B) $87.5 \%$
C) $8.5 \%$
D) $25.1 \%$

Answer 1

Hint: First, we need to find concentration available after time t. Then we need to compare it with the initial value for finding the decayed value.

Formula used:
Here we use the formula, $N_{(t)}=N_{0}\left(\dfrac{1}{2}\right)^{t / T_{1 / 2}}$.
$N_{(t)}$ represents the amount of radioactive substance present after time $t$.
$N_{0}$ represents the initial concentration of the radioactive substance available before starting the reaction.
$T_{1 / 2}$ is the half-life of a radioactive substance.
Finding the fraction of atoms that have decayed the equation is, $\left[\left(N_{0}-N_{(t)}\right) / N_{0}\right] \times 100$. By this equation we get the percentage of atoms decayed.


Complete answer:
Half-life of a radioactive element is the time required by a radioactive element to disintegrate into half its value. Usually, it is represented by $T_{1 / 2}$. Here it is given that the half-life of the element is 60 minutes. By using this we can get a relation between $N_{(t)}$ and $N_{0}$.
That is, $N_{(t)}=N_{0}\left(\dfrac{1}{2}\right)^{t / T_{1 / 2}}$.
$N_{(t)}=N_{0}\left(\dfrac{1}{2}\right)^{t / 60}$
Now we need to find out the amount of atoms left after 3 hours. Here ' $t$ ' is in hour we need to change it to minutes.
We know that 1 hour $=60$ minutes, then 3 hour $=180$ minutes.
So, equation (1) becomes, $N_{(t)}=N_{0}\left(\dfrac{1}{2}\right)^{180 / 60}$
$N_{(t)}=N_{0}\left(\dfrac{1}{2}\right)^{3}$
$N_{(t)}=N_{0} / 8$
Here we get $N_{(t)}$ value after 3 hours. Next step is to find out the fraction of atoms that decayed after 3 hours. For finding this we can use the relation, $\left[\left(N_{0}-N_{(t)}\right) / N_{0}\right] \times 100$
We can put the value of $N_{(t)}$ from equation (2) to equation (a).
Hence, we get $\left[\left(N_{0}-N_{0} / 8\right) / N_{0}\right] \times 100$
$\left[7 N_{0} / 8 N_{0}\right] \times 100$
After 3 hours, the fraction of atom that have decayed would be $=(7 / 8) \times 100$
$(7 / 8) \times 100=0.875 \times 100$
$87.5 \%$ of atoms would decay.

Thus, option (B) is correct.


Note: There may be a little confusion in selecting the equation of half-life because there are more than 2 equations. Finding a fraction of an atom decayed is also a little bit confusing.