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.

Answer 1

Given:
Half-life period of free neutron beta-decays to a proton, $T_{1/2}$ = 14 minutes

Half-life period, T_1/2 = $\frac{0.6931}{\lambda }$

Here, $\lambda$
= Decay constant

$$\begin{gathered} \therefore \lambda =\frac{0.693}{14\times 60} \\ =8.25\times {10}^{-4}{\mathrm{s}}^{-1} \end{gathered}$$

If m_p is the mass of proton, let m_n and m_e be the mass of neutron and mass of electron, respectively.

$$\begin{gathered} \therefore \; Energ\mathrm{y}\mathrm{liberated},E=[m_n-\left(m_p+m_e\right)]c^2 \\ =[1.008665\mathrm{u}-\left(1.007276+0.0005486\right)\mathrm{u}]c^2 \\ =0.0008404\times 931\mathrm{MeV} \\ =782\mathrm{keV} \end{gathered}$$