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Physics Question on Nuclei

The radioactivity of a sample is x at time t1t_1 and is y at time t2t_2. If the mean life of the specimen is τ\tau, the number of atoms that have disintegrated in the time interval (t2−t1)(t_2 - t_1) is

A

x - y

B

(x−y)/τ(x - y) / \tau

C

(x−y)τ(x - y) \tau

D

xt1−yt2x t_1 - y t_2

Answer

(x−y)τ(x - y) \tau

Explanation

Solution

Activity dNdt=−λN \frac{dN}{dt} = - \lambda N
∴x=−λNt1,y=−λNt2\therefore x = - \lambda N_{t_1}, y = - \lambda N_{t_2}
⇒Nt1=−xλ,Nt2=−yλ\Rightarrow N_{t_1} = - \frac{x}{\lambda} , N_{t_2} = - \frac{y}{\lambda}
The number of atoms disintegrated in time interval
⇒Nt2−Nt1=−yλ−(−xλ)\Rightarrow N_{t_2} - N_{t_1} = - \frac{y}{\lambda} - \big(-\frac{x}{\lambda}\big)
=x−yλ=(x−y)τ= \frac{x - y}{\lambda} = (x - y) \tau