Question

Explain the terms mass defect and binding energy. How are they related ?

Hint:

A higher binding energy per nucleon indicates a more tightly bound and stable nucleus. When calculating in atomic mass units (u), remember the convenient conversion factor: $1 \text{ u} \approx 931.5 \text{ MeV}$.

Answer 1

Concept:
• The nucleus of an atom is composed of protons and neutrons (nucleons).
• Precise measurements show that the actual mass of a stable nucleus is always less than the sum of the individual masses of its constituent nucleons.
• This missing mass is converted into energy that holds the nucleus together.

Step 1: Explain Mass Defect ($\Delta m$)
Mass defect is defined as the difference between the sum of the resting masses of the individual nucleons (protons and neutrons) making up a nucleus and the actual rest mass of that nucleus. Mathematically, for a nucleus with $Z$ protons and $(A-Z)$ neutrons: \[ \Delta m = [Z \cdot m_p + (A - Z) \cdot m_n] - M \]
where $m_p$ is the mass of a proton, $m_n$ is the mass of a neutron, and $M$ is the actual mass of the nucleus.

Step 2: Explain Binding Energy ($E_b$)
Binding energy is the minimum amount of energy required to completely separate a nucleus into its constituent protons and neutrons to infinite distance. Alternatively, it is the energy released when individual nucleons bind together to form a stable nucleus.

Step 3: Relation between Mass Defect and Binding Energy
The mass defect and binding energy are directly related by Albert Einstein's mass-energy equivalence principle. The "missing" mass ($\Delta m$) is converted entirely into the binding energy ($E_b$). The relationship is given by the formula: \[ E_b = \Delta m \cdot c^2 \]
where $c$ is the speed of light in vacuum.

Step 4: Conclusion
Mass defect is the missing mass upon nucleus formation, and binding energy is the energy equivalent of this missing mass, governing the stability of the nucleus.