A) \[{{R}_{1}}={{R}_{2}}\,{{e}^{-\lambda ({{t}_{1}}-{{t}_{2}})}}\]
B) \[{{R}_{1}}={{R}_{2}}\,{{e}^{\lambda ({{t}_{1}}-{{t}_{2}})}}\]
C) \[{{R}_{1}}={{R}_{2}}\,({{t}_{2}}/{{t}_{1}})\]
D) \[{{R}_{1}}={{R}_{2}}\]
Correct Answer: A
Solution :
The decay rate R of a radioactive materials the number of decays per second. From radioactive decay law. \[-\frac{dN}{dt}\,\propto \,\,N\,or\,-\frac{dN}{dt}=\lambda N\] Thus \[R=-\frac{dN}{dt}or\,\,R\,\propto \,\,N\] or \[R=\lambda N\,or\,R=\lambda \,{{N}_{0}}{{e}^{-\lambda t}}\] ...(i) where \[{{R}_{0}}=\lambda {{N}_{0}}\] is the activity of the radioactive material at time t = 0. At time \[{{t}_{1}},{{R}_{1}}={{R}_{0}}\,{{e}^{-\lambda {{t}_{1}}}}\] ....(ii) At time \[t & {{ & }_{2}}\], \[{{R}_{2}}={{R}_{0}}\,{{e}^{-\lambda {{t}_{2}}}}....(iii)\] Dividing Eq. (ii) by (iii), we have \[\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{{{e}^{-\lambda {{t}_{1}}}}}{{{e}^{-\lambda {{t}_{2}}}}}={{e}^{-\lambda ({{t}_{1}}-{{t}_{2}})}}\] \[or{{R}_{1}}={{R}_{2}}\,{{e}^{-\lambda \,({{t}_{1}}-{{t}_{2}}}}\]
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