A) \[t={{t}_{1}}+{{t}_{2}}\]
B) \[\sqrt{t}=\sqrt{{{t}_{1}}}+\sqrt{{{t}_{2}}}\]
C) \[{{t}^{-1}}=t_{1}^{-1}+t_{2}^{-1}\]
D) None of these
Correct Answer: C
Solution :
Decay constant for first process, \[{{\lambda }_{1}}=\frac{0.693}{{{t}_{1}}}\] and decay constant for second process,\[{{\lambda }_{2}}=\frac{0.693}{{{t}_{2}}}\]the probability of an active nucleus decays by first process in small time is \[{{\lambda }_{1}}\,dt\], similarly for second process it is \[{{\lambda }_{2}}\,dt\]. Total probability it decays by first process or by second process is \[({{\lambda }_{1}}\,dt\,\,+\,\,{{\lambda }_{2}}\,dt)\] If \[\lambda \] is effective decay constant, then\[\lambda \,dt\,\,=\,\,{{\lambda }_{1}}\,dt\,\,+\,\,{{\lambda }_{2}}\,dt\]\[\Rightarrow \,\,\,\,\,\,\,\lambda ={{\lambda }_{1}}+{{\lambda }_{2}}\] \[\frac{0.693}{t}=\frac{0.693}{{{t}_{1}}}+\frac{0.693}{{{t}_{2}}}\,\,\Rightarrow \,\,\,\frac{1}{t}=\frac{1}{{{t}_{1}}}+\frac{1}{{{t}_{2}}}\]\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,{{t}^{-1}}=t_{1}^{-1}+t_{2}^{-1}\]
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