Question

If the half-life of a radioactive sample is 10 hours, its mean life is?
A. $14.4$ hours
B. $7.2$ hours
C. 20 hours
D. $\quad 6.93$ hours

Answer 1

Hint: Mean life is the expected life time for a radioactive material. Half-life is the time required to reduce the whole concentration to half its value. Half-life is equal to $0.693$ times mean life. In order to solve this problem we have to use the relation between half life and mean life of a radioactive sample.

Formula used:
The relation between half-life and mean life is given by,
$t_{1 / 2}=0.693 \times \tau$
Where $t_{1 / 2}$ is the half-life of the radioactive material and $\boldsymbol{\tau}$ represents the mean life of the radioactive substance.
The exponential decay of a radioactive substance is given by,
$N_{(t)}=N_{0} e^{-\lambda t}$
Where, $N_{(t)}$ represents the concentration of the substance after $t$ time.
$N_{0}$ is the initial concentration. $\lambda$ is the decay constant.
$\mathrm{T}$ is the time.

Complete step by step solution:
Half-life is time required by a radioactive nucleus to disintegrate into half its original value. The exponential decay of a radioactive substance is given by,
$N_{(t)}=N_{0} e^{-\lambda t}$
For calculating half-life, the initial condition is that we need to calculate time required for reducing the concentration to half its value. So, the quantity after $t_{1 / 2}$ will be $\dfrac{N_{0}}{2}$. Therefore, the above equation becomes,
$\dfrac{N_{0}}{2}=N_{0} e^{-\lambda t_{1 / 2}}$
$\Rightarrow \dfrac{1}{2}=e^{-\lambda t_{1 / 2}}$

Taking log on both side we get
$\ln \dfrac{1}{2}=\ln \left(e^{-\lambda t_{1 / 2}}\right) \\
\Rightarrow -0.693=-\lambda t_{1 / 2}$
That is
$t_{1 / 2}=\dfrac{0.693}{\lambda}$............(1)

Mean-life is the average lifetime of a radioactive nucleus. Mean-life time of a radioactive nucleus is given by
$\tau=\dfrac{\text { Total life time of nucleus }}{\text { Total number of nucleus }}$..........(2)
Also, mean life is the reciprocal of decay constant.
$\tau=1 / \lambda$.............(3)

Comparing equation (1) and (3) we get,
$t_{1 / 2}=0.693 \tau$
Here in the question, it is given that half-life is 10 hours.
Then mean-life $\tau=t_{1 / 2} / 0.693$
$\tau=\dfrac{10}{0.693}$
Hence, we get mean-life $\tau=14.4$ hours

Therefore, the correct answer is option A.

Notes:There may be a little confusion about the relation between mean-life and half-life of a radioactive material. Always remember that half-life is $0.693$ times the mean-life.