For a radioactive material, its activity `A` and rate of change of its activity of `R` are defined as `A=(-dN)/(dt)` and `R=(-dA)/(dt)`, where `N(t)` is the number of nuclei at time `t`. Two radioactive source `P` (mean life `tau`) and `Q` (mean life `2 tau`) have the same activity at `t=0`. Their rates of activities at `t=2 tau` are `R_(p)` and `R_(Q)`, respectively. If `(R_(P))/(R_(Q))=(n)/(e )`, then the value of `n` is:

Answer

2

Explanation

`N=N_(0)e^(-lambdat)`
`A=lambdaN_(0)e^(-lambdat)`
`R=lambda^(2)N_(0)e^(-lambdat)`
`A_(P)=A_(Q)` at `t=0`
`lambda_(P)N_(P)e^(-lambda_(P)t)=lambda_(Q)N_(Q)e^(-lambda_(Q)t)` at `t=0`
`lambda_(P)N_(P)=lambda_(Q)N_(Q)` …(i)
`R_(P)/R_(Q)=(lambda_(P)/lambda_(Q))^(2) (N_(P)/N_(Q)) e^(-lambda_(P)2tau)/e^(-lambda_(Q)2tau)`
`R_(P)/R_(Q)=(lambda_(P)/lambda_(Q))^(2)(N_(P)/N_(Q))1/e` ...(ii)
`[lambda_(P)=1/("mean life")]`
from equation (i) and equation (ii)
`R_(P)/R_(Q)=2/e`
Ans. 2