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.

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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.

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.