The nucleus `._(92)U^(238)` is unstable angainst alpha decay with half-life of `4.5 xx 10^(9)` years. Estimate the kinetic energy of the emitted `alpha`-particle.
Mass of the `._(92)U^(238)` atom `= 238.05081 u`
Mass of the `._(2)He^(4)` atom `= 4.00260 u`
Mass of the `._(90)Th^(234)` atom `= 234.04363 u`

Explanation

`._(92)U^(238)rarr._(90)Th^(234)+._(2)He^(4)`
Mass defect: `Deltam = [(238.05081)-(4.00260+234.04363)]`
`= 4.58 xx 10^(-3)u`
Equivalent energy `= 4.58 xx 10^(-3) xx 931.5 = 4.27MeV = K`
`K_(alpha) = (p^(2))/(2m_(alpha)),K_(Th) = (p^(2))/(2m_(Th))`
Momentum of `Th` and `alpha` particle is equal and opposite, due to momentum conservation.
`K_(alpha)+K_(Th) = K`
`(p^(2))/(2)((1)/(m_(alpha))+(1)/(m_(Th)))=K`
`(p^(2))/(2m_(alpha)) (1+(m_(alpha))/(m_(Th))) = K`
`K_(alpha) = (K)/(1+(m_(alpha))/(m_(Th)))=(4.27)/(1+(4.00260)/(234.04363))=4.198MeV`