question_answer

The radioactivity of the sample is \[{{R}_{1}}\] at time \[{{t}_{1}}\] and \[{{R}_{2}}\] at time \[{{t}_{2}}\]. The mean life of the sample is \[\tau \]. What is the number of nuclei that have disintegrated in the time interval \[({{t}_{1}}-{{t}_{2}})\]?

Answer:

                Let \[{{N}_{1}}\] and \[{{N}_{2}}\] be the number of undecayed nuclei present at times \[{{t}_{1}}\] and \[{{t}_{2}}\] respectively. Then \[{{R}_{1}}=\left| \frac{d{{N}_{1}}}{dt} \right|=\lambda {{N}_{1}}\] and \[{{R}_{2}}=\left| \frac{d{{N}_{2}}}{dt} \right|=\lambda {{N}_{2}}\] \[\therefore \]  \[{{R}_{1}}-{{R}_{2}}=\lambda ({{N}_{1}}-{{N}_{2}})\] or            \[{{N}_{1}}-{{N}_{2}}=\frac{{{R}_{1}}-{{R}_{2}}}{\lambda }\] Clearly, \[({{N}_{1}}-{{N}_{2}})\]is the number of nuclei that have disintegrated in time interval \[({{t}_{1}}-{{t}_{2}})\].