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

Options

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

Answer

D

Explanation

(d) The `alpha`-particle emitting radioactive gas, `Thoron-220` decay to `Radium-216` and emits and `alpha`-particle. The reaction can be represented by
`._90^(220) Th rarr ._2^4 He + ._66^126 Ra`
By conservation of momentum, we have momentum of `alpha-`particle = momentum of recoiling nucleus `Ra`
`rArr m_a v_a = m_R v_R`
`rArr (v_R)/(v_a) = (m_a)/(m_R) = (4)/(216) = (1)/(54)`
The kinetic energy of `R_a, E_R` is related to the kinetic energy of alpha particle `E_a` by the reaction below
`(E_R)/(E_a)((1)/(2)m_R v_R^2)/((1)/(2) m_a v_a^2) = ((m_R)/(m_a)) ((v_R)/(m_a))^2 = ((m_R)/(m_a))((m_a)/(m_R))^2`
=`(m_a)/(m_R) = (1)/(54)`
`:. E_R = (E_a)/(54)`.