We have two radioactive nuclei `A` and `B`. Both convert into a stable nucleus `C`. Nucleus `A` converts into `C` after emitting two two `alpha`-particles and three `beta`-particles. Nucleus `B` converts into `C` after emitting one.
`alpha`-particle anf five `beta`-particles. A time `t = 0`, nuclei of `A` are `4N_(0)` and that of `B` are `N_(0)`. Half-life of `A`(into the conversion of `C`) is `1min` and that of `B` is `2min`. Initially number of nuclei of `C` are zero.
At what time rate of disintergrations of `A` and `B` are equal.

Options

A. `4 min`
B. `6 min`
C. `8 min`
D. `2 min`

Answer

B

Explanation

`((dN)/(dt))= - lambda_(A)N_(A)= -((ln 2)/(1))4N_(0)e^(-lambda t)`----(1)
`((dN)/(dt))= -lambda_(B)N_(B)= - ((ln 2)/(2))N_(0)e^(-lambda_(B)t)`----(2)
Given
`((dN)/(dt))_(A) = ((dN)/(dt))_(B)rArr N_(0)e^(-((ln2)/(1)))=N_(0)/(2)e^(-((ln 2)/(2))t)`
`rArr 8=e^((t ln 2)/2)rArrt=6 min`