Question

The half life of polonium is $140$ days. After how many days $16g$ polonium will be $1g$.
A. $700$ days
B. $280$ days
C. $560$ days
D. $420$ days

Answer 1

Half life is the time that is required for one half atoms of a given radioactive substance to disintegrate. As we know the formula of radioactive decay which depends on the initial and final quantities of substance and time, by substituting the values of all the given values, we can easily determine the value of t.

Formula Used:
$N = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{T}}}$
Here N is the quantity of the substance remaining, ${N_0}$ is the initial amount of substance. T is half life and t is the time.

Complete step by step answer:
Half life of polonium is given = $140$ days
We know,
$N = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{T}}}$
Here T is the half life and that is given. We need to calculate the value of t.
Value of N $= 1g$(Quantity of substance remaining)
Value of ${N_0} = 16$ grams (Initial amount of substance)
Now substitute all the given values in the equation,
We get-
$\dfrac{1}{{16}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{140}}}}$
$\Rightarrow{\left( {\dfrac{1}{2}} \right)^4} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{140}}}}$
Now, base at left hand side and right hand side is equal, so-
$4 = \dfrac{t}{{140}}$
$\Rightarrow t = 4\times 140$
$\therefore t=560\,days$

So, option C is correct.

Note: Half-life is also known as biological half life. It is directly computed by the decay constant. By calculating the value of half- lives we can determine whether a sample is safe to handle or not and it also tells us how quickly atoms can undergo or stable atoms able to survive or not the radioactive decay. That’s why it is important to calculate the value of half-lives.