A) \[{{A}_{1}}{{t}_{1}}-{{A}_{2}}{{t}_{2}}\]
B) \[\frac{\left( {{A}_{2}}-{{A}_{1}} \right)}{2}\tau \]
C) \[\left( {{A}_{1}}-{{A}_{2}} \right)\left( {{t}_{2}}-{{t}_{1}} \right)\]
D) \[\left( {{A}_{1}}-{{A}_{2}} \right)\tau .\]
Correct Answer: D
Solution :
[d] Let \[{{N}_{0}}\] be the initial number of nuclei, then \[{{N}_{1}}={{N}_{0}}{{e}^{-\lambda {{t}_{1}}}}\] and \[{{N}_{2}}={{N}_{0}}{{e}^{-\lambda {{t}_{2}}}}\] \[\therefore \] Number of nuclei decayed \[={{N}_{1}}-{{N}_{2}}\] \[={{N}_{0}}({{e}^{-\lambda {{t}_{1}}}}-{{e}^{-\lambda {{t}_{2}}}})=\frac{{{A}_{0}}}{\lambda }({{e}^{-\lambda {{t}_{1}}}}-{{e}^{-\lambda {{t}_{2}}}})\] \[=\frac{{{A}_{1}}-{{A}_{2}}}{\lambda }=(.{{A}_{1}}-{{A}_{2}})\tau .\]
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