A) \[\lambda \]
B) \[\frac{1}{2}\lambda \]
C) \[\frac{1}{4\lambda }\]
D) \[\frac{e}{\lambda }\]
Correct Answer: C
Solution :
| If N is the number of radioactive nuclei present at some instant, then \[N={{N}_{0}}{{e}^{-\lambda t}}\] |
| The constant \[{{N}_{0}}\] represents the number of radioactive nuclei at \[t=0\] |
| Now, \[\frac{{{N}_{1}}}{{{N}_{2}}}=\frac{{{e}^{-{{\lambda }_{1}}t}}}{{{e}^{-{{\lambda }_{2}}t}}}\] |
| or \[\frac{{{N}_{1}}}{{{N}_{2}}}=\frac{{{e}^{-5\lambda t}}}{{{e}^{-\lambda t}}}={{e}^{-4\lambda t}}\] |
| but \[\frac{{{N}_{1}}}{{{N}_{2}}}=\frac{1}{e}\] (as provided) |
| Therefore, \[\frac{1}{e}=\frac{1}{{{e}^{4\lambda t}}}\] |
| or \[4\lambda \,t=1\] |
| or \[t=\frac{1}{4\lambda }\] |
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