Question

1g of a radioactive substance disintegrates at the rate of $3.7 \times {10^{10}}$ disintegration per second. The atomic mass of the substance is 226. Calculate its mean life
(A) $1.2 \times {10^5}s$
(B) $1.39 \times {10^{11}}s$
(C) $2.1 \times {10^5}s$
(D) $7.194 \times {10^{10}}s$

Answer 1

Hint: In order to solve this problem first calculate the number of nuclei
i.e., $N = moles \times {N_A}$
Where
${N_A} = $ Avogadro number
$ = 6.023 \times {10^{23}}$ per mole
After then by putting the value of activity in mean life formula we get desire solution i.e.,
$T = \dfrac{N}{A}$
Where
T $ = $ Mean life
A $ = $ Activity
N $ = $ Number of nuclei

Complete step by step answer:
We know that activity of any substance is the disintegrates rate of substance which is given as
$A = 3.7 \times {10^{10}}$ disintegration per second …..(1)
Let the number of nuclei is N.
So,
N $ = $ Number of moles $ \times {N_A}$
Where
${N_A} = 6.02 \times {10^{23}}$ per mole
Moles $ = $ 1 gram $/$ 226 gram per mole
Moles $ = 0.00442$
So, $N = 0.00442 \times 6.02 \times {10^{23}}$
$N = 0.0266 \times {10^{23}}$ …..(2)
The mean life of radioactive substance is given by following expression
$T = \dfrac{N}{A}$
From equation 1 and 2, putting the values of A and N we get
$T = \dfrac{{0.0266 \times {{10}^{23}}}}{{3.7 \times {{10}^{10}}}}$
$\implies T = 0.007189 \times {10^{23}} \times {10^{ - 10}}$
$\implies T = 0.00719 \times {10^{13}}\sec $
$\therefore T = 7.19 \times {10^{10}}\sec $
Hence, the mean life of substance is $7.19 \times {10^{10}}s$

So, the correct answer is “Option SD”.

Note:
In many problems of radioactivity, they may ask about half life and mean life i.e.,
Half life – Half life measures the time, the radioactive substance takes for a given amount of the substance to become reduced by half as a consequence of decay.
Mean life – The mean life of a particular species of unstable nucleus is always $1.443$ times longer than its half life.