Question

If the half-life of a substance is \[5\] yrs, then the total amount of substance left after $15$ yrs, when the initial amount is $64$ grams is-
A.$16{\text{ grams}}$
B.$2{\text{ grams}}$
C.${\text{32 grams}}$
D.$8{\text{ grams}}$

Answer 1

Hint: Half-life of a substance is the time at which the substance’s weight or moles or concentration is reduced to one-half of their initial values. So reduced the given amount of substance to half of its initial value to get the amount of substance left after $5$ yrs. Then again reduce the obtained value to its half to get the amount of substance left after $10$ yrs. Then to calculate the total amount of substance left after $15$ yrs, further reduce the obtained value to its half and you’ll get the answer.

Complete step by step answer:
Given, the half-life of a substance =\[5\] yrs
The initial amount of the substance is =$64$ grams
We have to find the amount of substance left after $15$ yrs
We know that the half-life of a substance is the time in which the substance is reduced to one half of its initial value.
So here the amount of substance will be reduced by one-half after a half-life period.
Then after \[5\] yrs, the amount of substance will be reduced to half of the $64$ grams
$ \Rightarrow $ The amount of substance left=$\dfrac{{64}}{2} = 32$
So after \[5\] yrs, the amount of substance left is $32{\text{ grams}}$.
Now after $10{\text{ }}$yrs, the $32{\text{ grams}}$will be reduced to half and we get,
$ \Rightarrow $ The amount of substance left=$\dfrac{{32}}{2} = 16$
So after $10{\text{ }}$yrs, the amount of substance left is $16{\text{ grams}}$.
Now after $15$yrs, the amount of substance will be further reduced to half of $16{\text{ grams}}$ and we will get-
$ \Rightarrow $ The amount of substance left= $\dfrac{{16}}{2} = 8$
Hence after $15$yrs, the amount of substance left is $8{\text{ grams}}$.

Hence the correct answer is D.

Note:
The concept of half-life is generally associated with radioactive decay. The time one-half of the atoms of radioactive material take to disintegrate is the half-life. It is important for the following purposes-
1.It can be used to determine the approximate age of organic objects, by determining how much $C - 14$ has transformed.
2.It is also used to calculate how much time the radioactive waste should be stored until it is safe for disposal.
3.The radioactive isotopes are used as medical tracers by doctors to treat the condition of patients. For this, the isotopes used must have a short half-life so it can be used to treat the condition and not to injure the healthy cells.