A) 249 years
B) 449 years
C) 133 years
D) 99 years
Correct Answer: B
Solution :
The decay constant is the reciprocal of the mean life\[\tau \]. Thus, \[{{\lambda }_{a}}=\frac{1}{1620}\] per year and \[{{\lambda }_{\beta }}=\frac{1}{405}\] per year \[\therefore \] Total decay constant, \[\lambda ={{\lambda }_{\alpha }}+{{\lambda }_{\beta }}\] or \[\lambda =\frac{1}{1620}+\frac{1}{405}=\frac{1}{324}\] per year We know that \[N={{N}_{0}}\,{{e}^{-\lambda t}}\] When \[\frac{3}{4}\] th part of the sample has disintegrated, \[\therefore \,\,\,\frac{{{N}_{0}}}{4}={{N}_{0}}{{e}^{-\lambda t}}\] or \[{{e}^{\lambda t}}=4\] Taking logarithm of both sides, we get \[\lambda t={{\log }_{e}}\,4\] or \[t=\frac{1}{\lambda }{{\log }_{e}}{{2}^{2}}=\frac{2}{\lambda }\,{{\log }_{e}}2\] \[=2\times 324\times 0.693\] = 449 year
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