A) 1080 yr
B) 2340 yr
C) 4860 yr
D) 3240 yr
Correct Answer: A
Solution :
Since, from Rutherford-Soddy law, the number of atoms left after half lives is given by \[N={{N}_{0}}{{\left( \frac{1}{2} \right)}^{n}}\] where, \[{{N}_{0}}\] is the original number of atoms. The number of half lives, \[n=\frac{time\text{ }of\text{ }decay}{effective\text{ }half-life}\] Relation between effective disintegration constant \[(\lambda )\] and half life (T) \[\lambda =\frac{\ln \,\,2}{T}\] \[\therefore \] \[{{\lambda }_{1}}+{{\lambda }_{2}}=\frac{\ln \,2}{{{T}_{1}}}+\frac{\ln \,2}{{{T}_{2}}}\] 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 \[\therefore \] \[n=\frac{T}{540}\] \[\therefore \] \[N={{N}_{0}}{{\left( \frac{1}{2} \right)}^{t/540}}\Rightarrow \frac{N}{{{N}_{0}}}={{\left( \frac{1}{2} \right)}^{2}}={{\left( \frac{1}{2} \right)}^{t/540}}\] \[\Rightarrow \] \[\frac{t}{540}=2\Rightarrow t=2\times 540=1080\,\,yr\]
You need to login to perform this action.
You will be redirected in
3 sec
