Question

The radioactivity of a sample is ${{R}_{1}}$ at a time ${{T}_{1}}$ and ${{R}_{2}}$ at times ${{T}_{2}}$ . the half-life of the specimen is $T$, the number of atoms that have disintegrated at the time $({{T}_{2}}-{{T}_{1}})$ is proportional to

• A. ${{R}_{1}}{{T}_{1}}-{{R}_{2}}{{T}_{2}}$
• B. ${{R}_{1}}-{{R}_{2}}$
• C. $\frac{{{R}_{1}}-{{R}_{2}}}{T}$
• D. $({{R}_{1}}-{{R}_{2}})T$

Answer 1

Answer: (D)

$({{R}_{1}}-{{R}_{2}})T$

Answer 2

$R_{1}=N_{1} \lambda$ and $R_{2}=N_{2} \lambda$ Also $T=\frac{\log _{e} 2}{\lambda}$ Or $\lambda=\frac{\log _{e} 2}{T}$ $\therefore R_{1}-R_{2}=\left(N_{1}-N_{2}\right) \lambda$ $=\left(N_{1}-N_{2}\right) \frac{\log _{e} 2}{T}$ $\therefore\left(N_{1}-N_{2}\right)=\frac{\left(R_{1}-R_{2}\right) T}{\log _{e} 2}$ ie, $\left(N_{1}-N_{2}\right) \propto\left(R_{1}-R_{2}\right) T$