Question

Half-life of a radioactive substance is T. The time taken for all the nuclei to disintegrate will be
A. 2T
B. T2
C. 4T
D. Uncertain

Answer 1

Hint:Use the half-life of radioactive decay to calculate the decay constant. The half-life of radioactive decay is the amount of time needed for one half of the atomic nuclei in a radioactive sample to decay. In this case we are asked to find the time taken for all the nuclei to disintegrate. For that, we have to equate the values in the decay law to determine the time for disintegration.

Formula used:
1. \[N = {N_o}{e^{ - \lambda t}}\]
Here, initial concentration \[ = {N_o}\]
Concentration after time \[t\] \[ = N\]
2. The formula for the number of radioactive nuclei remaining after a given amount of time is:
\[N(t) = {N_{0}}\: e^{-λt}\]
Where:
$N(t)$ is the number of radioactive nuclei remaining after time t
$N_{0}$ is the initial number of radioactive nuclei
3. \[\lambda\] is the decay constant, which is related to the half-life by the relationship:
$T_{1/2} = \dfrac {\ln 2}{\lambda}$ t is the elapsed time

Complete answer:
It would be beneficial for us to have knowledge about radioactive disintegration (radioactive decay) and half-life before beginning the mathematical computations.

The spontaneous breakdown of an atomic nucleus, also known as radioactive decay, causes the release of matter and energy from the nucleus.
We are given the data in the question that,
Half-life of a radioactive substance is T
And we have to now determine the time taken for all the nuclei to disintegrate.

The time it takes for the atoms of a given quantity of a radioactive substance to disintegrate is known as the half-life. The biological half-life is another name for half-life.

Therefore, the time taken for all the nuclei to disintegrate will be uncertain, because it is infinite. And also we know that no radioactive element will be disintegrated fully.

Hence, the option D is correct

Note: The word "half-life" was used in this inquiry. We should comprehend this phrase for a better comprehension. The time it takes for the atoms of a given quantity of a radioactive substance to disintegrate is known as the half-life. The biological half-life is another name for half-life.