Two radioactive samples of different elements (half-lives `t_1` and `t_2` respectively) have same number of nuclei at `t=0`. The time after which their activities are same is

Options

A. `(t_(1)t_(2))/(0.693(t_(2)-t_(1)))In(t_(2))/(t_(1))`
B. `(t_(1)t_(2))/(0.693)In(t_(2))/(t_(1))`
C. `(t_(1)t_(2))/(0.693(t_(2)+t_(1)))In(t_(2))/(t_(1))`
D. None of these

Answer

1

Explanation

After time t activity is same
`A_(1)=A_(2)`
`lamda_(1)N_(1)=lamda_(2)N_(2)`
`(0.693)/(t_(1))N_(1)=(0.693)/(t_(1))N_(2)`
`rArr(N_(1))/(t_(1))=(N_(2))/(t_(1))`
`rArr(N_(oe)^(-lamda_(1)t))/(t_(1))=(N_(oe)^(-lamda_(2)t))/(t_(2))`
`rArre^(lamda_(1)-lamda_(2)t)=(t_(2))/(t_(1))`
`(lamda_(1)-lamda_(2))t=In(t_(2))/(t_(1))`
`0.693((1)/(t_(1))-(1)/(t_(2)))t=In((t_(2))/(t_(1)))`
`t=(t_(1)t_(2))/((0.693)(t_(2)-t_(1)))In (t_(2))/(t_(1))`