Two substances A and B are present such that `[A_(0)]=4[B_(0]` and half-life of A is 5 minutes and that of B is 15 minutes. If they start decaying at the same time following first order kinetics after how much time the concentration of both of them would be same ?

Options

A. `15 min`
B. `10 min`
C. `5 min`
D. `12 min`

Answer

A

Explanation

Amount of `A` left in `n_(1)` halves`= ((1)/(2))^(n_(1)) [A_(0)]`
Amount of `B` left in `n_(2)` halves `= ((1)/(2))^(n_(2))[B_(0)]`
At the end, according to the question
`([A_(0)])/(2^(n_(1))) ([B_(0)])/(2^(n_(2))) rArr (4)/(2^(n_(1))) = (1)/(2^(n_(2))), ([A_(0)] = 4[B_(0)])`
`(2^(n_(1)))/(2^(n_(2))) = 4 rArr 2^(n_(1)-n_(2)) = (2)^(2) rArr n_(1) - n_(2) = 2`
`:. n_(2) = (n_(1) - 2)` ....(i)
Also `t = n_(1) t_(1//2(A)), t = n_(2)t_(1//2(B))`
(Let concentration of both become equal after time `t`)
`:. (n_(1) xx t_(1//2(A)))/(n_(2) xx t_(1//2(B))) = 1 rArr (n_(1) xx 5)/(n_(2) xx 15) = 1 rArr (n_(1))/(n_(2)) = 3` ...(ii)
form Eqs. (i) and (ii), we get
`n_(1) = 3, n_(2) = 1`
`t = 3 xx 5 = 15 min`