1. PUNEQE

Binding Energy and Mass Defect

This material is referenced from (Lamarsh, et. al. 2001).

Einstein postulated that the rest mass energy of an atom was defined by

\[E_{rest}=m_{o}c^{2}\]

where \(c\) is the speed of light, and \(m_{o}\) is the atomic mass of the atom. This establishes the equivalence of mass and energy.

Consequently, the kinetic energy (KE) is the difference between total and rest mass energy

\[E=mc^{2}-m_{o}c^{2}=m_{o}c^{2}\left[ \frac{1}{\sqrt{1-\beta^{2}}} - 1 \right]\]

and this, when non-relativistics (\(v\ll c\)), leads to

\[E=\frac{1}{2}mv^{2}\]

Binding energy

We can remember that the mass defect is defined by

\[Q=\left[\left(M_{a}+M_{b}\right) - \left(M_{c}+M_{d}\right)\right]\underbrace{\left(931.5\unit{\nicefrac{MeV}{amu}}\right)}_{c^{2}}\]

so the mass defect is the difference between the constituents and the actual atom

\[ \begin{align}\begin{aligned}\Delta=ZM_{p}+NM_{n}-M_{A}\\which leads to\end{aligned}\end{align} \]
\[BE\left( a \right) = Z_{a}M\left(\ce{^{1}H}\right) - N_{a}\left(M_{n}\right) - M_{a}\]

To determine the binding energy of the last neutron, we determine the least bound neutron - i.e. the difference in binding energy between the isotope of interest and the isotope of interest with one less neutron plus the rest mass of an unbound neutron:

\[E_{s}=\left[M_{n}+M\left(\ce{^{A-1}Z}\right) - M\left(\ce{^{A}Z}\right)\right]931.5\unit{\nicefrac{MeV}{amu}}\]

Bibliography