A) \[4860\,\,yr\]
B) \[3240\,\,yr\]
C) \[2340\,\,yr\]
D) \[1080\,\,yr\]
Correct Answer: D
Solution :
From Rutherford-Soddy law, the number of atoms left after n half-lives is given by \[N={{N}_{0}}{{\left( \frac{1}{2} \right)}^{n}}\] where, \[{{N}_{0}}\] is original number of atoms. The number of half-life\[n=\frac{time\,\,of\,\,decay}{effective\,\,half-life}\] Relation between effective disintegration constant \[(\lambda )\]and half-life \[(T)\] is \[\lambda =\frac{\ln 2}{T}\] \[\therefore \] \[{{\lambda }_{1}}+{{\lambda }_{2}}=\frac{\ln 2}{{{T}_{1}}}+\frac{\ln 2}{{{T}_{2}}}\] Effective half-life \[\frac{1}{T}=\frac{1}{{{T}_{1}}}+\frac{1}{{{T}_{2}}}=\frac{1}{1620}+\frac{1}{810}\] \[\frac{1}{T}=\frac{1+2}{1620}\Rightarrow T=540\,\,yr\]
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