Calculate binding energy from mass defect and explore nuclear stability
Mass Defect: The difference between the mass of separated nucleons and the mass of the nucleus.
Binding Energy: The energy required to separate all nucleons in a nucleus, calculated using Einstein's equation E = mc².
Formula: BE = Δm × c² = Δm × 931.494 MeV (where Δm is in amu)
Stability: Higher binding energy per nucleon means greater nuclear stability. Iron-56 has the highest BE/nucleon (~8.79 MeV).
Nuclear binding energy is the energy required to disassemble a nucleus into its individual protons and neutrons. It represents the "glue" that holds the nucleus together, overcoming the electrostatic repulsion between positively charged protons.
The binding energy arises from the strong nuclear force, one of the four fundamental forces of nature. This force is much stronger than the electromagnetic force at very short distances (within the nucleus), allowing nucleons to bind together despite proton-proton repulsion.
Key Concept:
The mass of a nucleus is always less than the sum of the masses of its individual nucleons. This "missing mass" is called the mass defect, and it has been converted to binding energy according to Einstein's famous equation: E = mc².
The mass defect (Δm) is the difference between the theoretical mass of separated nucleons and the actual mass of the nucleus:
Δm = (Z × mp + N × mn) - mnucleus
Where:
The binding energy is calculated using Einstein's mass-energy equivalence:
BE = Δm × c²
In nuclear physics units:
BE (MeV) = Δm (amu) × 931.494 MeV/amu
The conversion factor 931.494 MeV/amu comes from c² = (2.998 × 10⁸ m/s)²
Important Note:
The mass defect is always positive because energy is released when nucleons combine to form a nucleus. This released energy corresponds to the binding energy, and the system becomes more stable (lower energy state).
Calculate the binding energy of a helium-4 nucleus (⁴He). The atomic mass of ⁴He is 4.002603 amu.
Step 1: Identify the Components
⁴He has:
Step 2: Calculate Theoretical Mass
Theoretical mass = (Z × mp) + (N × mn)
= (2 × 1.007276 amu) + (2 × 1.008665 amu)
= 2.014552 + 2.017330
= 4.031882 amu
Step 3: Calculate Mass Defect
Δm = Theoretical mass - Actual mass
= 4.031882 amu - 4.002603 amu
= 0.030279 amu
Step 4: Calculate Binding Energy
BE = Δm × 931.494 MeV/amu
= 0.030279 × 931.494
= 28.20 MeV
Step 5: Calculate Binding Energy per Nucleon
BE/nucleon = Total BE / A
= 28.20 MeV / 4
= 7.05 MeV/nucleon
Answer:
The binding energy of ⁴He is 28.20 MeV, or 7.05 MeV per nucleon. This means you would need 28.20 MeV of energy to completely disassemble a helium-4 nucleus into 2 separate protons and 2 separate neutrons.
The binding energy per nucleon (BE/A) is a measure of nuclear stability. The higher the binding energy per nucleon, the more stable the nucleus.
Light Nuclei (A < 20):
BE/A increases rapidly with mass number. Very light nuclei are relatively unstable.
Medium Nuclei (20 < A < 60):
Peak stability occurs around ⁵⁶Fe (Iron-56) with BE/A ≈ 8.79 MeV/nucleon. This is the most stable nucleus in nature.
Heavy Nuclei (A > 60):
BE/A gradually decreases. Heavy nuclei are less stable due to increased electrostatic repulsion between protons.
| Nuclide | Z | A | BE/nucleon (MeV) | Stability |
|---|---|---|---|---|
| ²H (Deuterium) | 1 | 2 | 1.112 | Low |
| ⁴He | 2 | 4 | 7.074 | Moderate |
| ¹²C | 6 | 12 | 7.680 | Moderate |
| ¹⁶O | 8 | 16 | 7.976 | High |
| ⁵⁶Fe (Iron-56) | 26 | 56 | 8.790 | Maximum |
| ²³⁵U | 92 | 235 | 7.591 | Moderate-Low |
| ²³⁸U | 92 | 238 | 7.570 | Moderate-Low |
Why Iron-56 is Most Stable:
⁵⁶Fe has the optimal balance between the strong nuclear force (which binds nucleons) and electrostatic repulsion (which pushes protons apart). Elements lighter than iron can release energy through fusion(combining to form heavier elements), while elements heavier than iron can release energy through fission(splitting into lighter elements). This is why stars fuse elements up to iron, but not beyond.
Combining light nuclei to form heavier nuclei releases energy because the product has higher BE/A.
Example: Deuterium fusion
²H + ²H → ⁴He + energy
Initial BE/A: 1.112 MeV
Final BE/A: 7.074 MeV
Energy released ≈ 24 MeV
This powers the sun and hydrogen bombs.
Splitting heavy nuclei into lighter nuclei releases energy because products have higher BE/A.
Example: Uranium-235 fission
²³⁵U + n → ⁹⁰Sr + ¹⁴³Xe + 3n + energy
Initial BE/A: 7.591 MeV
Final BE/A: ~8.5 MeV (avg)
Energy released ≈ 200 MeV
This powers nuclear reactors and atomic bombs.
Energy Calculation:
For any nuclear reaction, the energy released (Q-value) equals the difference in total binding energies:Q = BEproducts - BEreactantsIf Q > 0, energy is released (exothermic). If Q < 0, energy must be supplied (endothermic).
Nuclear reactors use fission of ²³⁵U or ²³⁹Pu to generate electricity. Understanding binding energy helps optimize fuel efficiency and predict energy output from fission reactions.
Stars generate energy through fusion reactions. Binding energy calculations explain why stars can fuse elements up to iron, but heavier elements require supernova conditions to form.
Radioisotopes for medical imaging and treatment are selected based on their binding energies and decay modes. PET scans use positron emission from unstable nuclei with specific BE characteristics.
Both fission bombs (atomic bombs) and fusion bombs (hydrogen bombs) exploit binding energy differences. The massive energy release comes from conversion of mass defect to energy.
Understanding nuclear stability helps explain radioactive decay. Carbon-14 dating, uranium-lead dating, and other techniques rely on known decay rates of unstable nuclei with lower binding energies.
Projects like ITER aim to harness fusion energy for clean power generation. Binding energy calculations are crucial for determining optimal fuel combinations and predicting energy yields.
Atomic mass includes electrons. For precise calculations, use nuclear mass (atomic mass - electron mass).
Correction: For most problems, atomic mass is sufficient, but be aware of the difference in high-precision calculations.
Students often forget that N = A - Z, not just Z.
Correct: Always calculate N = A - Z before finding theoretical mass
Mass defect should be positive: Δm = (separated mass) - (nucleus mass).
Correct: If you get negative Δm, you've subtracted in the wrong order
The conversion from amu to MeV is 931.494, not 931.5 or other approximations in precise work.
Correct: Use 931.494 MeV/amu for accurate results
mp = 1.007276 amu
mn = 1.008665 amu
me = 0.000549 amu
1 amu = 931.494 MeV/c²
1 amu = 1.66054 × 10⁻²⁷ kg
N = A - Z
Δm = (Zmp + Nmn) - mnucleus
BE = Δm × 931.494 MeV
BE/nucleon = BE / A
Q = BEproducts - BEreactants
• Higher BE/A = more stable
• Maximum stability: ⁵⁶Fe (~8.79 MeV/nucleon)
• Light nuclei: gain stability by fusion
• Heavy nuclei: gain stability by fission
²H: 1.11 MeV
⁴He: 7.07 MeV
¹²C: 7.68 MeV
⁵⁶Fe: 8.79 MeV (max)
²³⁸U: 7.57 MeV