A stationery thorium nucleus `(A=200 , Z=90)` emits an alpha particle with kinetic energy `E_(alpha)`. What is the kinetic energy of the recoiling nucleus

Options

A. `(E_(alpha))/(108)`
B. `(E_(alpha))/(110)`
C. `(E_(alpha))/(55)`
D. `(E_(alpha))/(54)`

Answer

D

Explanation

The `alpha` -particle emitting radioactive gas, thoron `-220`, decays to radium `-216` an emits an `alpha`-particles. The reaction can be represented by `._(90)^(220)Th rarr ._2^4He +._(66)^(216)Ra`
By conservation of momentum of `alpha`-particles = momentum of recoilling nucleus Ra
`rArr m_(alpha) v_(alpha) =M_(R) v_(R)`
`rArr (v_(R))/(v_(alpha)) =(m_(alpha))/( m_(R)) =(4)/(216) =(1)/(54)`
The kinetic energy of Ra, `E_(R)`, is realted to the kinetic energy of alpha particle `E_(alpha)` by
`(E_(R))/(E_(alpha))=((1)/(2)m_(R)v_(R)^(2))/((1)/(2)m_(alpha)v_(alpha)^(2))=((m_(R))/(m_(alpha)))((v_(R))/(m_(alpha)))^(2)=((m_(R))/(m_(alpha)))((m_(alpha))/(m_(R)))^(2)`
`=(m_(alpha))/(m_(R))=(1)/(54)`
`:. E_(R)=(E_(alpha))/(54)`.