Question

The half-life of a radioactive substance, as compared to its mean life, is
A) 30 %
B) 60 %
C) 70 %
D) 100 %

Answer 1

Hint
As here, we need to use the formula for the relation between half-life and mean life which can be derived by using the relation between decay constant and half-life and relation between decay constant and mean life. The required formula will be $halflife = {\log _e}2 \times mean life$.

Complete step by step solution
As we know that rate of decay is $\dfrac{{dN}}{{dt}} = - \lambda N$
Where, N is the original number of atoms and λ is decay constant.
The negative sign indicates that the decay in atoms with time.
Now we also know that, half-life is ${t_{1/2}} = \dfrac{{\ln 2}}{\lambda } = \dfrac{{0.693}}{\lambda }$ …………………..(1)
and relation between mean life and decay constant is $T = \dfrac{1}{\lambda }$………………..(2)
Substitute the value of λ from equation (2) in the equation (1), we get
$\Rightarrow {t_{1/2}} = 0.693 \times T$
$\Rightarrow \dfrac{{{t_{1/2}}}}{T} = 0.693 \times 100 \approx 70\%$.
Thus, the half-life of the radioactive substance as compared to mean life is 70%.
Hence, option (C) is correct.

Note
To solve such a problem one must be thorough with this concept of chemical kinetics and should either know the required formula or how to derive them.
It should also be noticed that the half-life has a probabilistic nature and denotes the time in which on average about half of entities decays. Suppose if we had only one atom, then it is not like after one half life, one half of the atom will decay, so we can say that half-life just describes the decay of distinct entities.