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)

$R_{1}=N_{1} \lambda$ 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, $\left(N_{1}-N_{2}\right) \propto\left(R_{1}-R_{2}\right) T$