Question

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

Hint:

Remember: Separated Nucleons weigh MORE than the joined Nucleus! \[ \text{Mass Defect } (\Delta m) = \text{Mass}_{\text{parts}} - \text{Mass}_{\text{whole}} \] Multiply \(\Delta m\) in \(\text{u}\) by \(931.5 \text{ MeV}\) to instantly find the Binding Energy.

Answer 1

Concept: The nucleus of an atom consists of bound states of protons and neutrons, collectively referred to as nucleons. Precision measurements reveal that the rest mass of any stable, bound nucleus is invariably strictly less than the total combined rest masses of its individual separated constituent protons and neutrons. This missing mass is known as the mass defect. According to Einstein's mass-energy equivalence principle, this lost mass is converted into binding energy during the formation of the nucleus.

Step 1: Detailed definition and formula for Mass Defect (\(\Delta m\))
Let us consider a nucleus represented by the standard notation \({}_{Z}^{A}\text{X}\), where:
• \(Z\) represents the atomic number, which is the total number of protons inside the nucleus.
• \(A\) represents the mass number, which is the total number of nucleons (protons + neutrons).
• \((A - Z)\) represents the total number of neutrons inside the nucleus.
• \(m_p\) represents the rest mass of a single isolated proton.
• \(m_n\) represents the rest mass of a single isolated neutron.
• \(M_{\text{nucleus}}\) represents the actual experimentally measured rest mass of the bound nucleus. The theoretical mass of the individual unbonded constituent parts is calculated by: \[ M_{\text{constituents}} = Z \cdot m_p + (A - Z) \cdot m_n \] The mass defect (\(\Delta m\)) is defined as the mathematical difference between the total mass of individual nucleons and the actual mass of the formed stable nucleus: \[ \Delta m = \left[ Z \cdot m_p + (A - Z) \cdot m_n \right] - M_{\text{nucleus}} \] Because the bound system is stable and in a lower energy state, \(\Delta m > 0\) always holds true for stable bound nuclei.

Step 2: Detailed definition of Binding Energy (\(E_b\))
Nuclear Binding Energy is defined as the total net energy released when its constituent individual nucleons are brought together from infinity to form the bound nucleus. Alternatively, it can be defined as the minimum external energy that must be supplied to a completely stable nucleus in order to break it apart completely into its individual constituent nucleons and separate them to infinity so they no longer exert nuclear forces on one another.

Step 3: Relationship between Mass Defect and Binding Energy
The connection between mass defect and binding energy is directly governed by Albert Einstein’s special theory of relativity, which establishes the fundamental equivalence of mass and energy via the equation: \[ E = m \cdot c^2 \] Where \(c\) represents the constant speed of light in a vacuum (\(c \approx 3 \times 10^8 \text{ m/s}\)). When individual nucleons fuse to form a stable nucleus, the mass defect \(\Delta m\) disappears as mechanical mass and transforms completely into the equivalent amount of nuclear binding energy (\(E_b\)). Therefore, the relationship is expressed explicitly as: \[ E_b = \Delta m \cdot c^2 \] Substituting our expanded formula for the mass defect from Step 1 into this relation provides: \[ E_b = \left( \left[ Z \cdot m_p + (A - Z) \cdot m_n \right] - M_{\text{nucleus}} \right) \cdot c^2 \] In atomic and nuclear physics, mass is often expressed in atomic mass units (\(\text{amu}\) or \(\text{u}\)). One atomic mass unit corresponds to an energy equivalent of approximately \(931.5 \text{ MeV}\). Thus, if the mass defect \(\Delta m\) is given directly in units of \(\text{u}\), the binding energy can be computed simply as: \[ E_b = \Delta m \times 931.5 \text{ MeV} \]