Question

Define mass defect and binding energy.

Hint:

Mass defect is the difference between the summed mass of free nucleons and the actual nuclear mass; the binding energy is that mass defect converted to energy via E = (mass defect) c squared.

Answer 1

Step 1: Mass defect.
The actual mass of a nucleus is always found to be less than the total mass of its individual nucleons (protons and neutrons) when they are free. This difference in mass is called the mass defect, denoted \( \Delta m \).
For a nucleus with atomic number \( Z \), mass number \( A \), and nuclear mass \( M \):
\[ \Delta m = \big[\,Z\,m_p + (A-Z)\,m_n\,\big] - M \]
where \( m_p \) is the mass of a proton and \( m_n \) the mass of a neutron.

Step 2: Binding energy.
The binding energy is the energy released when the free nucleons combine to form the nucleus, or equivalently the energy required to break the nucleus completely into its separate nucleons. It arises from the mass defect through Einstein's mass-energy relation:
\[ E_b = \Delta m \cdot c^2 \]

Step 3: Practical unit form.
If \( \Delta m \) is expressed in atomic mass units (u), then
\[ E_b = \Delta m \times 931.5 \ \text{MeV} \]
A larger binding energy means a more stable nucleus.
\[\boxed{\Delta m = [Zm_p+(A-Z)m_n]-M,\quad E_b=\Delta m\,c^2}\]