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 radioactive substance is 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 \[{{R}_{0}}\] is \[\lambda {{N}_{0}}\] is the activity of the radioactive substance at time t is 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 Eq. (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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