Question

A radioisotope, tritium ($_1^3H$) has a half-life of 12.3 years. If the initial amount of tritium is 32 mg, how many milligrams of tritium would remain after 49.2 years?
A. 1 mg
B. 2 mg
C. 4 mg
D. 8 mg

Answer 1

The tritium is a radioactive isotope of hydrogen and emits $\beta$ radiations on decay. Decay of radioactive isotope decreases the amount of that isotope present along with reducing the level of radioactivity. The reduction in the radioactivity of the isotope is measured using the concept of half-life. The half-life of an isotope is considered as the time required for decay of half the amount of radioactive isotope.

Formula Used: The formula we will be using for calculation of half-life of tritium will be following:
$N = \dfrac{{{N_0}}}{{{2^n}}}$
Here, N is the amount of tritium remaining after 49.2 years, ${N_0}$ is the initial amount of tritium and n is the number of half-lives of tritium.

Complete step by step answer:
Tritium has a half-life equal to 12.5 years approximately. This denotes that, after 12.5 years only half the amount of radioactive tritium will decay naturally.
We have to calculate the amount of tritium remaining from 32 milligrams after 49.2 years.
Let us first note down the quantities given in the question.
${t_{1/2}} = 12.3{\text{ }}years$
${N_0} = 32{\text{ mg}}$
Here, ${t_{1/2}}$ is the half-life of tritium and ${N_0}$ expresses the initial amount of tritium present.
Let us start with solving this problem in a stepwise manner.
Step 1: We have to find out the amount of tritium remaining after 49.2 years due to radioactive decay.
We know that, initial amount of tritium is 34 mg. But, we have to find out the value of n that is the number of half-lives for tritium.
The number of half-lives determined by 49. 2 years can be determined using following formula,
$T = {t_{1/2}} \times n$…………….. (1)
Here, n is the number of half-lives of tritium.
As given, $T = 49.2{\text{ years}}$ and ${t_{1/2}} = 12.3{\text{ years}}$
The formula (1) on rearrangement will give rise to the formula for calculating the value of n.
On rearrangement, we get,
$n = \dfrac{T}{{{t_{1/2}}}}$…………….. (2)
On substituting the respective values in formula (2), we get,
$n = \dfrac{{49.2}}{{12.3}} = 4$
The number of half-lives for tritium is 4.
Step 2: Now, we have values for both n and ${N_0}$, we can calculate the amount of tritium remaining after 49.2 years.
$N = \dfrac{{{N_0}}}{{{2^n}}}$…………… (3)
Substitute the corresponding values in the formula (3), we get,
$N = \dfrac{{32}}{{{2^4}}}$
$N = \dfrac{{32}}{{16}}$
$\therefore N = 2{\text{ mg}}]$
Therefore, the amount of tritium remaining after 49.2 years is 2 mg.

So, the correct option is B.

Additional information:
Remember that the half-life of a specific radioisotope is constant. The half-life of a radioisotope is independent of the initial amount of that isotope present. The trace amounts of few elements that are present in the human body are radioactive in nature and it decays constantly.

Note: Without using any formula, the amount of tritium remaining after 49.2 years can be calculated.
As we know, the half-life of tritium is 12.3 years that means after 12.3 years the amount of tritium left behind from 34 mg initial amount is 16 mg. After 24.6 years, it will be 8 mg and so on.
It can be written as,
Initial amount of tritium is 34 mg.
After 12.3 years, the amount of tritium left will be 16 mg.
After 24. 6 years, the amount of tritium left will be 8 mg.
After 36.9 years, the amount of tritium left will be 4 mg.
After 49.2 years, the amount of tritium left will be 2 mg.
Therefore, the correct option is option (B).