Mass Defect and Nuclear Binding Energy

The mass defect is the disparity between the aggregate mass of each protons and neutrons in an atomic nucleus when evaluated separately and the actual mass of the nucleus.

Experimental observations contradict the logical assumption that a nucleus’s mass would be equal to the sum of the masses of its constituent nucleons. The mass defect, denoted as Δm, is given by:

 Δm = Z_mp + N_mn− m_nucleus (Equation 1)

Here:

• Z is the number of protons,
• N is the number of neutrons,
• m_p and m_n denote the masses of a proton and a neutron, accordingly.
• m_nucleus​ is the actual measured mass of the nucleus.

The mass defect occurs when some of the nucleons’ mass is transformed into energy during the nucleus’ formation, which is in accordance with Einstein’s mass-energy relation (E=mc²).

Nuclear binding energy sustains a nucleus’s everlasting vitality. It stands for the energy needed to break down a nucleus into its constituent protons and neutrons, or, conversely, the energy released during the formation of a nucleus from free nucleons.

It can be obtained using Einstein’s equation:

Eb = (Δm)c² ( Equation 2)

In equation (2), Δm represents the mass defect, and c denotes the speed of light (3×10⁸ m/s). 

The link here illustrates how the nucleus’s “missing” mass is converted into binding energy rather than being lost.

The most fascinating and significant occurrences in the physical universe can be found in the core of an atom known as the nucleus, which is encircled by an electron cloud. Protons, which are positively charged particles, and neutrons, which are neutral particles, make up the atom’s nucleus. Electrons belong to negatively charged entities that travel in different energy levels surrounding the nucleus. The nucleus is glued together by a powerful nuclear pull that wins over the electrostatic repulsion among protons. The number of neutrons contributes to nuclear stability.

Nuclear binding energy and mass defect are two examples of such phenomena that are crucial to grasping the stability of matter and the mechanisms underlying the universe. These ideas explain how enormous amounts of energy are produced in nuclear reactions in addition to exposing the complexities of atomic structures.

Understanding the Relationship Between Mass and Energy

A tiny portion of the nucleus’ mass is transformed into energy during nuclear fission or fusion. The enormous energy produced during nuclear processes originates from this mass differing, also known as the mass defect. Einstein’s well-known formula mentioned in equation (2) captures the connection between mass and energy. Because the speed of light squared is so high, the c² factor allows a tiny amount of mass to produce an immense amount of energy. As consequence, energy may manifest as matter, and mass can be converted into energy.

The eternal strength of a nucleus is preserved by the nuclear binding energy. While lower values imply less stability and a higher probability of nuclear reactions or radioactive decay, larger binding energies signify more stable nuclei. An object’s energy greatly increases at high speeds i.e. around the speed of light. The direct correlation between energy and mass can be seen by the fact that this increase in energy also raises the object’s relativistic mass. In accelerators such as the Large Hadron Collider, collisions of high-energy particles exhibit mass-energy equivalence because collision energy can produce new particles (mass).

Representing Nuclear Reactions

The major point in understanding the energy shifts in nuclear reactions is the mass defect (Δm). It shows the difference between the total mass of the participating reactants (the nuclei prior to the reaction) and the entire mass of the reaction’s products, or nuclei.

According to Einstein’s mass-energy connection, the “lost” mass is transformed into energy rather than being annihilated. Whether energy is released or absorbed during a nuclear reaction depends on the difference in binding energy between reactants and products. When a neutron strikes uranium-235, it breaks down, generating energy that is used in nuclear reactors and weapons by breaking into smaller nuclei.

⁹²₂₃₅​U + ⁰₁​n → ³⁶₉₂​Kr + ⁵⁶₁₄₁​Ba + 3⁰₁​n + Energy

As part of the Carbon-Nitrogen-Oxygen cycle (CNO cycle), nitrogen can fuse with helium nuclei (alpha particles). Energy production in stars larger than the Sun is dominated by this cycle.For instance,

¹⁴N + ⁴He → ¹⁸F + γ

Followed by:

¹⁸F → ¹⁸O + e⁺ + ν_e

Here, a helium nucleus and nitrogen-14 combine to create the unstable fluorine-18, which then beta decays to oxygen-18.

Balanced nuclear equations are useful for researching nuclear reactions because they guarantee the conservation of basic quantities and offer crucial details about the processes involved. Fundamental conservation laws control nuclear reactions.Whether prior to or after the reaction, the mass number is constant. The proper distribution of energy and momentum between reactants and products is ensured when the equation is balanced.

Variation of Binding Energy per Nucleon

Calculating the binding energy per nucleon, or E_b/A, where A=Z+N is the mass number, is a helpful method for comparing stability between nuclei. Nuclei with larger binding energies are more stable, whereas those with smaller values are less stable and more likely to undergo nuclear processes or radioactive decay.

Certain trends of binding energy are mentioned below:

• Light Nuclei

Because of increased attractive nuclear attractions, the binding energy per nucleon in light nuclei rises with mass when more nucleons are added. Nevertheless, this pattern does not last forever. 

• Intermediate Nuclei

The binding energy per nucleon peaks for intermediate nuclei. This stability peak can be seen in nuclei such as ⁵⁶Fe and ⁶²Ni. 

• Heavy Nuclei

The binding energy per nucleon diminishes as the repellent effect among protons in dense nuclei increases. As a result, heavier nuclei such as uranium are prone to fission.

Nuclear Fission and Nuclear Fusion

Nuclear Fission

A heavy nucleus splitting into smaller nuclei and releasing a large quantity of energy is known as nuclear fission. When a neutron is absorbed by the nuclei of heavy isotopes like uranium-235, nuclear fission generally takes place. A passive (thermal) neutron is absorbed by a massive nucleus, resulting in an extremely volatile intermediate state. The unbalanced nuclear and electrostatic forces cause the energized nucleus to break down. The repellent electrostatic interaction between protons overcomes the enormous nuclear force that holds the nucleus intact. Free neutrons are released when the nucleus divides into two smaller nuclei, known as fission fragments. There is a notable energy release that coincides with the split. A great deal of the energy generated during fission takes the form of kinetic energy of the released neutrons, gamma radiation, and fission particles.

For example: Uranium-236 (²³⁶U) is produced when uranium-235 captures a neutron.

²³⁵U + 1n → ²³⁶U∗

Uranium-236 now undergoes a breakdown involving the following reaction.

²³⁶U → ⁹²Kr + ¹⁴¹Ba + 3¹n + Energy

The resultant particles’ overall mass is lower than the initial mass. Einstein’s equation (2) is used to turn this mass difference (termed as mass defect) into energy.

Nuclear Fusion

Nuclear fusion involves the combination of light nuclei, such as hydrogen isotopes, to form a heavier nucleus, releasing energy. The process of nuclear fusion releases enormous quantities of energy as two light atomic nuclei unite to form a more powerful nucleus. This phenomena has tremendous potential as a neat, practically unlimited source of energy on Earth and powers stars, notably the Sun.

The Coulomb barrier, which is the electrical repulsion between positively charged nuclei, must be overcome for fusion processes to occur. Extreme circumstances like high temperature and high pressure are necessary for nuclei to smash with enough energy for fusion to take place. The mass variation between the reactants and the product is the source of the energy generated during fusion. The previous Einstein equation (2) explains this mass defect that gets converted into energy.

For example we have Deuterium-Deuterium (D-D) fusion,

2H + 2H → 3He + ¹n + Energy

Relevance of Binding Energy in Nuclear Reactions

The binding energy per nucleon of heavy nuclei, such uranium-235 and plutonium-239, is smaller than that of their fission fragments. The difference is released as energy when these nuclei split because the ensuing fragments have a larger total binding energy. Nuclear reactors and weapons are powered by the energy emitted per fission process, which is around 200 MeV.

Similar to this, light nuclei—like the hydrogen isotopes deuterium and tritium—have a lower binding energy per nucleon in fusion reactions than their fusion products. The increase in the binding energy per nucleon is released as energy when they join to form a heavier nucleus, such as helium. While unstable nuclei are more prone to decay or take part in reactions, stable nuclei have a high binding energy per nucleon and are less likely to decay.

Binding energy plays a pivotal role in astrophysical processes. The energy that affords stars their radiance and heat is released when light components like hydrogen are fused into helium. Binding energy differences between parent and daughter nuclei also govern radioactive decay. Nuclear reactors generate electricity from the energy released by binding energy differences in fission reactions. The conception behind atomic and hydrogen bombs is the quick release of binding energy in uncontrolled fission or fusion events. The creation of isotopes for cancer treatment and medical imaging is facilitated by binding energy differences.

Calculating Energy Released in Nuclear Reactions

The energy analogous of the mass defect is computed as follows:

E=Δmc²,

Where, c² = 9.00 × 1016 × (m²/s²). Atomic mass units (amu) are frequently used to express nuclear masses for simplicity, and the energy released can be computed as:

1 amu = 931.5 MeV/c²

Thus,

E (MeV) = Δm(amu) × 931.5 MeV.

The energy potential of nuclear fission and fusion is illustrated by these sample calculations below.

When exposed to a neutron, uranium-235 initiates fission in the manner shown:

²³⁵U + ¹n → ⁹²Kr + ¹⁴¹Ba + 3¹n + Energy

Step 1: Masses of Reactants and Products

• Mass of ²³⁵U: 235.0439 amu,
• Mass of neutron (¹n): 1.0087 amu,
• Mass of ⁹²Kr: 91.9262 amu,
• Mass of ¹⁴¹Ba: 140.9144 amu,
• Mass of 3 neutrons: 3×1.0087=3.0261 amu

Step 2: Calculate Total Mass of Reactants

Mass of Reactants=235.0439+1.0087=236.0526 amu.

Step 3: Calculate Total Mass of Products

Mass of Products=91.9262+140.9144+3.0261=235.8667 amu.

Step 4: Calculate Mass Defect

Δm=Mass of Reactants−Mass of Products

Δm=236.0526−235.8667=0.1859 amu

Step 5: Convert Mass Defect to Energy

E=Δm⋅c².

Using 1 amu=931.5 MeV/c²:

E=0.1859×931.5 MeV.

E=173.1 MeV

Result:

The energy released per fission reaction is approximately 173 MeV.:

The merging of deuterium (2H) and tritium (3H) into helium and a neutron

2H + 3H → 4He + 1n + Energy

Step 1: Masses of Reactants and Products

• Mass of 2H: 2.0141 amu,
• Mass of 3H: 3.0160 amu,
• Mass of 4He: 4.0026 amu,
• Mass of neutron (¹n): 1.0087 amu.

Step 2: Calculate Total Mass of Reactants

Mass of Reactants=2.0141+3.0160=5.0301 amu

Step 3: Calculate Total Mass of Products

Mass of Products=4.0026+1.0087=5.0113 amu

Step 4: Calculate Mass Defect

Δm=Mass of Reactants−Mass of Products

Δm=5.0301−5.0113=0.0188 amu

Step 5: Convert Mass Defect to Energy

E=Δm⋅c²

Using 1 amu=931.5 MeV/c²:

E=0.0188×931.5 MeV.

E=17.5 MeV

Result:

The energy emitted during a fusion process is around 17.5 MeV.

Nuclear reactors now use fission, whereas fusion, despite its experimental promise, still needs to be developed further before it can be used as a practical energy source. Both procedures demonstrate the tremendous power of nuclear collisions and are based on the mass defect and binding energy theories.

Video on Mass Defect and Binding Energy

Conclusions

The stability and energy properties of atomic nuclei are explained by the fundamental ideas of mass defect and binding energy in nuclear physics. The conversion of mass into energy during nuclear creation is the cause of the mass defect, which is the discrepancy between the total mass of nucleons and the actual mass of the nucleus. Despite the repulsive forces between protons, this energy—also referred to as the binding energy—keeps the nucleus together.

These ideas are important in ways that go beyond theoretical physics. They are essential for comprehending the energy dynamics of nuclear fusion and fission, which form the foundation for astrophysical processes such as star nucleosynthesis as well as energy generation. While nuclei with lower binding energies are more likely to disintegrate or go through energy-releasing processes, those with higher binding energies per nucleon are more stable.

Binding energy has practical uses in understanding the workings of nuclear reactors, the possibility of fusion as a clean energy source, and the natural processes that drive stars. These ideas also highlight how crucial mass-energy equivalency is in describing how slight changes in mass result in significant energy releases. Knowing mass defect and binding energy helps us better understand nuclear forces and emphasize how important they are to developing technologies and solving the world’s energy problems.

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