A) \[{{t}_{1/2}}=\frac{0.693}{k}\]
B) \[N={{N}_{0}}{{e}^{-kt}}\]
C) \[\frac{1}{N}-\frac{1}{{{N}_{0}}}=\ln \,k{{t}_{t/2}}\]
D) None of the above
Correct Answer: C
Solution :
Key Idea According to the law of radioactive decay The quantity of a radioactive element which disappears in unit time (rate of disintegration) is directly proportional to the amount present \[\frac{-d{{N}_{t}}}{dt}=\lambda N\] where, \[\lambda =\] radioactive constant \[{{N}_{t}}={{N}_{0}}{{e}^{-\lambda t}}\] \[\ln \,\frac{N}{{{N}_{0}}}=-kt\] \[\ln \,\frac{N}{{{N}_{0}}}=kt\] \[\therefore \] \[\frac{1}{N}-\frac{1}{{{N}_{0}}}=\ln \,\,k\,\,{{t}_{1/2}}\] is not correct. Half-life period is defined as the time required by a given amount of the element to decay to one-half of its initial value. \[{{t}_{1/2}}=\frac{0.693}{\lambda }\]
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