Question: 30: Given below are two statements :
Statement I : The law of radioactive decay states that the number of nuclei undergoing the decay per unit time is inversely proportional to the total number of nuclei in the sample.
Statement II : The half life of a radionuclide is the sum of the life time of all nuclei, divided by the initial concentration of the nuclei at time \mathrm{t}=0.
In the light of the above statements, choose the most appropriate answer from the options given below :
(1) Both Statement I and Statement II are correct
(2) Both Statement I and Statement II are incorrect
(3) Statement I is correct but Statement II is incorrect
(4) Statement I is incorrect but Statement II is correct
Answer: Option (2)
Explanation:
Statement I is incorrect. According to the law of radioactive decay, the rate of decay (number of nuclei decaying per unit time) is directly proportional to the number of undecayed nuclei present at that time, not inversely proportional.
Mathematically,
\frac{dN}{dt} = -\lambda Nwhere N is the number of undecayed nuclei and \lambda is the decay constant.
Statement II is also incorrect. The quantity defined as the sum of the lifetimes of all nuclei divided by the initial number of nuclei corresponds to the mean life, not the half life.
Mean life is given by
\tau = \frac{1}{\lambda}Half life is related to mean life as
T_{1/2} = \frac{\ln 2}{\lambda}Therefore, both Statement I and Statement II are incorrect.