Question

The relation between the half-life $t$ of a radioactive sample and its mean life $T$is:
(A) $t = 0.693T$
(B) $T = 0.693t$
(C) $t = T$
(D) $t = 2.718T$

Answer 1

The half-life of the radioactive substance is the one half of the time period required by the substance to decay which is mean life. The mean lifetime is $1.443$ times the half-life of the radioactive substance and the half-life is $0693$ times the mean life.

Useful formula:
(1) The formula of the half-life of the radioactive sample is given by

$t = 0.693T$

Where $t$ is the half-life of the radioactive sample and $T$ is the mean life of the radioactive sample.

Complete step by step solution:
The mean life of the particular substance is defined as the time taken by the radioactive substance to the process of the complete decay. Hence by writing the formula of the half-life,

$t = 0.693T$

The above formula is derived mathematically as follows.
Half-life is obtained by multiplying the term $\ln \left( 2 \right)$ with the mean life.
Half-life = mean life $\times \ln \left( 2 \right)$
Substituting the considered indications of the half-life and the mean life in the above step.

$t = T \times \ln \left( 2 \right)$

The value of the $\ln \left( 2 \right)$ is $0.693$ , substituting this value in the above equation, we get

$t = 0.693T$

Hence the half-life of the radioactive substance is obtained as $t = 0.693T$ .

Thus the option (A) is correct.

Note: Let us consider an example of the radioactive isotope Uranium $- 238$ . It is the important isotope of the uranium ore and its half life time is $4.5$ billion years and the mean life of the uranium is obtained at $6.493$ billion years for the complete decaying of itself.