Question

Disintegration constant of a radioactive material is λ, then which of following is possible:

A. Its half life is equal to $\dfrac{{{\log }_{e}}}{\lambda }$ .
B. It means life equals $\dfrac{1}{\lambda }$ .
C. At the time equal to mean life,63% of the initial radioactive material is left undecided.
D. After 3-half lives,$\dfrac{1}{3}rd$ of the initial radioactive material is left un-decayed.

Answer 1

Hint: We know that the disintegration constant is the inverse of mean life, now when we equate the formula for half-life we get the value of the whole mean lifetime. The value of the whole mean lifetime is the value of the radioactive compound/element that is still intact.

Complete step by step answer:
Taking option B into consideration, we all know that mean life is equal to $\dfrac{1}{\lambda }$, So option B is correct.
We know that mean life is the inverse of disintegration or decay constant,
That means, mean life =1/ λ.
Now we know that the formula for the half-life is,
${{T}_{1/2}}=\dfrac{\ln 2}{\lambda }$, where ‘ λ’ is the radioactive decay constant.
${{T}_{1/2}}=\dfrac{0.693}{\lambda }$, (ln 2 = 0.693)
${{T}_{1/2}}=0.693\tau $, where $\tau $ is the mean lifetime.
That means at the mean lifetime only 69.3% of the whole nucleus remains undecided.
Therefore option C and option B is the correct option.

Additional Information:
In radioactivity, life is the average lifetime. This time taken can be written as the sum of the lifetimes of all the individual radioactive unstable nuclei samples, that is divided by the total number of unstable nuclei present.
Radioactive decay is the process by which an unstable atomic nucleus radiates all its energy. Any material that contains an unstable nucleus is a radioactive element. The element does not lose its energy all at once it takes an ample amount of time to do so.

Note:
Half-life is nothing but the time required for the nucleus of a radioactive element to get decayed by half of its proportion. is the mean life that is actually the average lifetime of a radioactive unstable substance and not its total lifetime.