A) \[\frac{1}{t}=\frac{1}{{{t}_{1}}}-\frac{1}{{{t}_{2}}}\]
B) \[\frac{1}{t}=\frac{1}{{{t}_{1}}\,{{t}_{2}}}\]
C) \[\frac{1}{t}=\frac{1}{\sqrt{{{t}_{1}}\,{{t}_{2}}}}\]
D) \[\frac{1}{t}=\frac{{{t}_{2}}}{{{t}_{1}}}+1\]
E) \[\frac{1}{t}=\frac{1}{{{t}_{1}}}+\frac{1}{{{t}_{2}}}\]
Correct Answer: E
Solution :
For first type of disintegration \[{{\lambda }_{1}}=\frac{{{(\ln )}^{2}}}{{{t}_{1}}}\] while for second type of disintegration, \[{{\lambda }_{2}}=\frac{{{(\ln )}^{2}}}{{{t}_{2}}}\] The probability to active nucleus decay according to first type of disintegration in time interval \[dt\] is \[{{\lambda }_{1}}\,dt\] while the same for second type of disintegration is\[{{\lambda }_{2}}dt\]. Thus, probability to either decay is\[{{\lambda }_{1}}dt+{{\lambda }_{2}}dt\]. If effective decay constant is \[\lambda ,\]then \[\lambda \,dt={{\lambda }_{1}}dt+{{\lambda }_{2}}dt\] \[\Rightarrow \] \[\lambda ={{\lambda }_{1}}+{{\lambda }_{2}}\] \[\Rightarrow \] \[\frac{1}{t}=\frac{1}{{{t}_{1}}}+\frac{1}{{{t}_{2}}}\]
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