Question
The relationship between disintegration constant $$\left( \lambda \right)$$ and half-life $$\left( T \right)$$ will be
A.
$$\lambda = \frac{{{{\log }_{10}}2}}{T}$$
B.
$$\lambda = \frac{{{{\log }_e}2}}{T}$$
C.
$$\lambda = \frac{T}{{{{\log }_e}2}}$$
D.
$$\lambda = \frac{{{{\log }_2}e}}{T}$$
Answer :
$$\lambda = \frac{{{{\log }_e}2}}{T}$$
Solution :
The time required for the number of parent nuclei to fall to $$50\% $$ is called half-life $$T$$ and may be related to disintegration constant $$\lambda $$ as follows.
Since,
$$0.5\,{N_0} = {N_0}{e^{ - \lambda t}}\,\left[ {_{\lambda = \,{\text{decay constant}}}^{N = {\text{ Final No}}{\text{. of nuclei}}\, = 0.5\,{N_0}}} \right]$$
$${N_0} = $$ Initial No. of nuclei
$$\lambda = $$ decay constant
we have, $$\lambda T = {\log _e}2$$
$$\therefore \lambda = \frac{{{{\log }_e}2}}{T}$$