In a radioactive material the activity at time `t_(1)` is `R_(1)` and at a later time `t_(2)`, it is `R_(2)`. If the decay constant of the material is `lambda`, then

Options

A. `R_(1)=R_(2)e^(-lambda(t_(1)-t_(2)))`
B. `R_(1)=R_(2)e^(lambda(t_(1)-t_(2)))`
C. `R_(1)=R_(2)(t_(2)//t_(1))`
D. `R_(1)=R_(2)`

Answer

A

Explanation

The decay rate R of a radioactive material is the number of decays per second. From radioactive decay law.
`-(dN)/(dt)alpha N or -(dN)/(dt)=lambdaN`
`"Thus", R=-(dN)/(dt)`
`or R=lambdaN or R=lamdaN_(0)E^(-lambdat).....(i)`
Where, `R_(0)=lambdaN_(0)` is the activity of the radioactive material at time t=0.
At time, `t_(1), R_(1)=R_(0)e^(-lambdat_(1))......(ii)`
At time `t_(2)`, `R_(2)=R_(0)e^(-lambdat_(2))......(iii)`
Dividing Eq, (ii) by (iii), we have
`(R_(1))/(R_(2))=(e^(-lambdat_(1)))/(e^(-lambdat_(2)))=e^(-lambda(t_(1)-t_(2)))`
`or R_(1)=R_(2)e^(-lambda(t_(1)-t_(2)))`