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) \[{{R}_{1}}={{R}_{2}}\]7
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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