Class 12 > Unit # 27: Nuclear Physics > Mass Defect and Binding Energy

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Mass Defect, Binding Energy & Binding Energy Per Nucleon

Binding Energy

Definition: The energy that is necessary to split the nucleus of an atom into its components—namely, neutrons and protons—collectively known as nucleons.

The binding energy of nuclei is a positive value because every nucleus needs net energy supplied to it in order to isolate its constituent neutrons and protons.

Mass Defect

The nuclear binding energy accounts for the significant difference between a nucleus's actual mass and the expected mass calculated from the sum of the masses of its isolated components.

Since energy and mass are related based on Einstein's equation:

$E = m c^2$

Where $c$ is the speed of light. In nuclei, the binding energy is so high that it corresponds to a considerable amount of mass.

The actual mass is less than the sum of the individual masses of the constituent neutrons and protons because energy is ejected when the nucleus is created. This energy equates to the mass ejected from the total mass of the original components, which is called the mass defect ($\Delta m$). This missing mass in the final nucleus describes the energy liberated when the nucleus is formed.

Mass defect is determined as the difference between the expected combined mass of its protons ($m_p = 1.00728 \text{ AMU}$ per proton) and neutrons ($m_n = 1.00867 \text{ AMU}$ per neutron) and the observed atomic mass ($M_0$):

$\Delta m = [Z m_p + (A - Z) m_n] - M_0$

Nuclear Binding Energy Per Nucleon (Packing Fraction)

Definition: It is the ratio of the total binding energy of a nucleus to the total number of its nucleons ($A$).

$E_{bn} = \frac{\text{Total Binding Energy}}{\text{Number of Nucleons (A)}}$

Main Features of the Binding Energy Curve

Figure: Comparison between the combined mass of isolated nucleons and the actual mass of the nucleus.

1. Constant Middle Range ($30 < A < 170$): The binding energy per nucleon ($E_{bn}$) is practically constant and independent of the atomic number for nuclei of middle mass numbers. The curve reaches a maximum of about $8.75 \text{ MeV}$ for $A = 56$ (Iron) and has a value of about $7.6 \text{ MeV}$ for $A = 238$ (Uranium).
2. Lower Values for Extremes: Binding energy per nucleon is lower for both light nuclei ($A < 30$) and heavy nuclei ($A > 170$).

Conclusions Drawn from Observations

• (a) Strong Attractive Force: The nuclear force is attractive and sufficiently strong to produce a binding energy of a few MeV per nucleon.
• (b) Short-Range Nature: The constancy of the binding energy in the range $30 < A < 170$ is a consequence of the fact that the nuclear force is short-ranged (each nucleon interacts only with its immediate neighbors).
• (c) Nuclear Fission: A very heavy nucleus, say $A = 240$, has lower binding energy per nucleon compared to that of a middle nucleus with $A = 120$. Thus, if a nucleus with $A = 240$ breaks into two nuclei with $A = 120$, the nucleons get more tightly bound, implying that energy would be released in the process (Nuclear Fission).
• (d) Nuclear Fusion: Consider two very light nuclei ($A \le 10$) joining to form a heavier nucleus. The binding energy per nucleon of the fused heavier nucleus is more than the binding energy per nucleon of the lighter nuclei. This means the final system is more tightly bound than the initial system, releasing energy (Nuclear Fusion)—which serves as the energy source of the Sun.

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