A) 3860 year
B) 4240 year
C) 2360 year
D) 1080 year
Correct Answer: D
Solution :
The relation between decay constant \[(\lambda )\] and half-life (T) is \[\lambda =\frac{0.6931}{T}\] Also effective disintegration constant is \[\lambda ={{\lambda }_{1}}+{{\lambda }_{2}}\] \[\therefore \] \[\frac{1}{T}=\frac{1}{{{T}_{1}}}+\frac{1}{{{T}_{2}}}\] Putting the numerical values from the question, we get \[\frac{1}{T}=\frac{1}{1620}+\frac{1}{810}=\frac{3}{1620}\] \[\Rightarrow \] \[T=\frac{1620}{3}=540\,years\] Also the number of atoms left after n half-lives is given by \[N={{N}_{0}}{{\left( \frac{1}{2} \right)}^{n}}\] Given, \[\frac{N}{{{N}_{0}}}=\frac{1}{4}\] Hence, \[t=2\times 540=1080\,years\]
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