#### Question

A free neutron beta-decays to a proton with a half-life of 14 minutes. (a) What is the decay constant? (b) Find the energy liberated in the process.

(Use Mass of proton m_{p} = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_{n} = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c^{2},1 u = 931 MeV/c^{2}.)

#### Solution

Given:

Half-life period of free neutron beta-decays to a proton, `T_"1/2"` = 14 minutes

Half-life period , `T_"1/2" = 0.6931/lambda`

Here, `lambda` = Decay constant

`therefore lambda = 0.693/(14 xx 60)`

= `8.25 xx 10^-4 "S"^-1`

If m_{p} is the mass of proton, let m_{n} and m_{e} be the mass of neutron and mass of electron, respectively.

`therefore "Energy liberated" , E = [m_n - (m_p + m_e)] c^2`

= `[1.008665 "u" - (1.007276 + 0.0005486) "u"]c^2`

= `0.0008404 xx 931 "MeV"`

= `782 "keV"`