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^(-lamda(t_(1)-t_(2)))`
B. `R_(1)=R_(2)e^(lamda(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)propN or -(dN)/(dt)=lamdaN`
thus, `R=-(dN)/(dt) or R=lamdaN`
or `R=lamdaN_(0)e^(-lamdat)` . . . (i)
Where `R_(0)=lamdaN_(0)` is the activity of the radioactive material at time t=0.
At time `t_(1)," "R_(1)=R_(0)e^(-lamdat_(1))` . .. (ii)
At time `t_(2)" "R_(2)=R_(0)e^(-lamdat_(2))` . . . (iii)
Dividing Eq. (ii) by (iii), we have
`(R_(1))/(R_(2))=(e^(-lamdat_(1)))/(e^(-lamdat_(2)))=e^(-lamda(t_(1)-t_(2))) or R_(1)=R_(2)e^(-lamda(t_(1)-t_(2)))`