What is the difference between nuclear binding energy and ionisation energy?

Nuclear binding energy is the energy to disassemble a nucleus into free protons and neutrons - typically 1–9 MeV per nucleon (millions of electron-volts). Ionisation energy is the energy to remove an electron from an atom - typically 5–25 eV (electron-volts). Nuclear binding energies are about a million times larger than atomic binding energies. This energy difference explains why nuclear reactions (fission, fusion, radioactive decay) release so much more energy than chemical reactions.

What is the semi-empirical mass formula (Bethe-Weizsäcker formula)?

The semi-empirical mass formula estimates nuclear binding energy: Eb = aV·A − aS·A^(2/3) − aC·Z(Z−1)/A^(1/3) − aA·(A−2Z)²/A ± aP·A^(−3/4). The five terms are: volume (proportional to A), surface (negative, proportional to A^(2/3)), Coulomb repulsion (Z²/A^(1/3)), asymmetry (prefers N=Z), and pairing (positive for even-even nuclei, negative for odd-odd). Constants: aV ≈ 15.8, aS ≈ 18.3, aC ≈ 0.714, aA ≈ 23.2, aP ≈ 12 (all in MeV).

How does binding energy explain the energy released in hydrogen fusion?

Deuterium (²H, Eb/A = 1.11 MeV) and tritium (³H, Eb/A = 2.83 MeV) fuse to form helium-4 (Eb/A = 7.07 MeV) plus a neutron. The increase in binding energy per nucleon corresponds to ~17.6 MeV per fusion event. Per unit mass, fusion releases about 4× more energy than U-235 fission - which is why fusion is the ultimate energy source in stars and why controlled fusion power is so actively pursued.

What is the atomic mass unit (u) and how is it defined?

The atomic mass unit (u, also written as Da for dalton) is defined as exactly 1/12 of the mass of a carbon-12 atom in its ground state: 1 u = 1.66053906660 × 10⁻²⁷ kg. The energy equivalent is 1 u = 931.49410242 MeV/c² (per CODATA 2018). This unit is chosen so that atomic and molecular masses are close to integer values, making mass tables convenient. Free proton: 1.007276 u; free neutron: 1.008665 u; electron: 0.000549 u.

Nuclear Binding Energy Calculator

Q: How do you calculate the mass defect of a nucleus?

Mass defect Δm = (Z × mp + N × mn) − M_atom, where Z is the proton count, N = A − Z is the neutron count, mp = 1.007276 u is the free proton mass, mn = 1.008665 u is the free neutron mass, and M_atom is the measured atomic mass in atomic mass units (u). Note: when using atomic masses (which include electron masses), the electron masses cancel in the formula, so no electron mass correction is needed for most purposes.

Q: How is mass defect converted to binding energy in MeV?

Using the conversion factor: 1 atomic mass unit (u) = 931.494 MeV/c². Therefore, Eb = Δm × 931.494 MeV, where Δm is in atomic mass units. For example, if Δm = 0.030 u, then Eb = 0.030 × 931.494 = 27.9 MeV. Alternatively, using SI units: Eb = Δm (kg) × c² = Δm (kg) × (3 × 10⁸)² joules, then convert: 1 MeV = 1.602 × 10⁻¹³ J.

Q: What is binding energy per nucleon and why is it the key stability indicator?

Binding energy per nucleon is Eb/A, where A is the mass number (total nucleon count). It represents the average binding energy of each nucleon in the nucleus. Iron-56 has the highest Eb/A ≈ 8.79 MeV/nucleon and is the most stable nucleus. The curve of Eb/A vs A explains both fusion and fission: light nuclei (H, He) have low Eb/A and gain stability (release energy) by fusing toward Fe; heavy nuclei (U, Pu) also have lower Eb/A than Fe and release energy by splitting.

Q: Why does iron-56 have the highest binding energy per nucleon?

Iron-56 sits at the peak of the binding energy per nucleon curve (~8.79 MeV/nucleon) due to the balance between the attractive strong nuclear force (which increases Eb/A for light nuclei as more nucleons join) and the repulsive Coulomb force between protons (which decreases Eb/A for heavy nuclei as more protons are added). At A ≈ 56, these two competing effects produce the maximum binding energy per nucleon - the most stable nuclear configuration.

Q: How does binding energy explain nuclear fission energy release?

When U-235 fissions into two medium-mass fragments (e.g., Ba-141 and Kr-92), the fragments have higher binding energy per nucleon (~8.5 MeV) than U-235 (~7.6 MeV). The difference ≈ 0.9 MeV/nucleon × 235 nucleons ≈ ~200 MeV is released as kinetic energy of the fragments, neutrons, and gamma radiation. This is the Q-value of the fission reaction - directly calculable from atomic masses via the mass defect approach.

Find mass defect and nuclear binding energy in MeV for any nuclide. Compare stability via binding energy per nucleon.

Quick Select Nuclideoptional - fills Z, A, mass below

Atomic Number Z (protons)

Mass Number A (protons + neutrons)

Atomic Mass Matomic mass units (u)

⚛️ What is Nuclear Binding Energy?

Nuclear binding energy is the energy required to completely separate a nucleus into its constituent protons and neutrons. By Einstein's mass-energy equivalence (E = mc²), this energy corresponds exactly to the mass defect - the difference between the mass of the free constituent nucleons and the actual mass of the assembled nucleus. The nucleus weighs less than the sum of its parts because some mass has been converted to the binding energy that holds the nucleus together.

The concept was established in the early twentieth century through the work of Aston (mass spectrometry), Chadwick (neutron discovery, 1932), and Bethe & Weizsäcker (semi-empirical binding energy formula, 1935-1936). The binding energy per nucleon (Eb/A) is the single most important indicator of nuclear stability. It rises rapidly from hydrogen (0 MeV, unbound proton) through helium-4 (7.07 MeV/nucleon) to a broad peak around iron-56 (8.79 MeV/nucleon), then gradually declines for heavier nuclei like uranium-238 (7.57 MeV/nucleon).

This curve has profound implications. Light nuclei below iron gain stability by fusion - combining two light nuclei releases energy because the product has higher Eb/A. Heavy nuclei above iron gain stability by fission - splitting a heavy nucleus releases energy because the fragments have higher Eb/A than the parent. Both processes release energy because the products are closer to iron on the binding energy curve. The Sun fuses hydrogen to helium, releasing ~26.7 MeV per reaction. A uranium-235 fission releases ~200 MeV.

This calculator computes the mass defect from the experimentally measured atomic mass (from the Atomic Mass Evaluation, AME2020) and converts it to binding energy in MeV using the exact conversion: 1 u = 931.494 MeV/c². It is used in nuclear physics coursework, JEE/NEET modern physics, nuclear engineering, and astrophysics calculations involving nucleosynthesis.

📐 Formula

Step 1 - Mass Defect:

Δm = Z × m_p + N × m_n − M_atom

Z = number of protons (atomic number)

N = number of neutrons = A − Z

m_p = proton mass = 1.007276 u (free proton)

m_n = neutron mass = 1.008665 u

M_atom = measured atomic mass (u) - from AME2020 or NUBASE

Δm = mass defect in atomic mass units (u)

Step 2 - Binding Energy:

E_b = Δm × 931.494 MeV/u

E_b/A = E_b / A (MeV per nucleon)

931.494 MeV/u = energy equivalent of 1 atomic mass unit (exact: 931.49410242 MeV, CODATA 2018)

A = mass number = Z + N (total nucleons)

In joules: E_b (J) = Δm (kg) × c², where c = 2.998 × 10⁸ m/s and 1 u = 1.66054 × 10⁻²⁷ kg

📖 How to Use This Calculator

Optionally, select a nuclide from the Quick Select dropdown to auto-fill Z, A, and atomic mass.

Enter the atomic number Z (protons), mass number A (protons + neutrons), and atomic mass M in u.

Click Calculate - results show Δm, total Eb (MeV and J), and Eb per nucleon.

Compare Eb/A across nuclides - values near 8.79 MeV/nucleon (Fe-56) indicate maximum stability.

💡 Example Calculations

Example 1 - Iron-56 (most stable nucleus)

Calculate the binding energy of ⁵⁶Fe: Z = 26, A = 56, M = 55.934942 u

N = 56 − 26 = 30 neutrons

Δm = 26 × 1.007276 + 30 × 1.008665 − 55.934942 = 26.18917 + 30.25995 − 55.934942 = 0.52818 u

E_b = 0.52818 × 931.494 = 492.26 MeV

E_b/A = 492.26 / 56 = 8.790 MeV/nucleon - the global maximum

Binding energy per nucleon: 8.790 MeV/nucleon - highest of any nuclide

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Example 2 - Helium-4 (exceptionally stable light nucleus)

Calculate the binding energy of ⁴He (alpha particle): Z = 2, A = 4, M = 4.002602 u

N = 4 − 2 = 2 neutrons

Δm = 2 × 1.007276 + 2 × 1.008665 − 4.002602 = 2.014552 + 2.017330 − 4.002602 = 0.029280 u

E_b = 0.029280 × 931.494 = 27.28 MeV

E_b/A = 27.28 / 4 = 6.820 MeV/nucleon

Total binding energy: 28.30 MeV | Per nucleon: 7.074 MeV/nucleon

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Example 3 - Uranium-235 (nuclear fuel)

Calculate binding energy of ²³⁵U: Z = 92, A = 235, M = 235.043930 u

N = 235 − 92 = 143 neutrons

Δm = 92 × 1.007276 + 143 × 1.008665 − 235.043930 = 92.669392 + 144.239095 − 235.043930 = 1.864557 u

E_b = 1.864557 × 931.494 = 1,736.9 MeV

E_b/A = 1,736.9 / 235 = 7.591 MeV/nucleon - lower than Fe-56, enabling fission energy release

Total binding energy: 1,736.9 MeV | Per nucleon: 7.591 MeV/nucleon

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Example 4 - Carbon-12 (mass standard)

Calculate binding energy of ¹²C: Z = 6, A = 12, M = 12.000000 u (exact, by definition)

N = 12 − 6 = 6 neutrons

Δm = 6 × 1.007276 + 6 × 1.008665 − 12.000000 = 6.043656 + 6.051990 − 12.000000 = 0.095646 u

E_b = 0.095646 × 931.494 = 89.09 MeV

E_b/A = 89.09 / 12 = 7.424 MeV/nucleon

Total binding energy: 92.16 MeV | Per nucleon: 7.680 MeV/nucleon

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Frequently Asked Questions

What is nuclear binding energy and what does it represent physically?+

Nuclear binding energy is the energy required to completely disassemble a nucleus into its constituent free protons and neutrons. By Einstein's E = mc², this corresponds to the mass defect: the nucleus weighs less than the sum of its free nucleons. The 'missing' mass has been converted to binding energy. Higher binding energy means more tightly bound, more stable nucleus.

How do you calculate the mass defect of a nucleus?+

Mass defect Δm = (Z × 1.007276 + N × 1.008665) − M_atom, where Z is the proton count, N = A − Z is the neutron count, and M_atom is the measured atomic mass in u. The proton and neutron masses used are the free (unbound) values. The positive result confirms the nucleus is lighter than its free constituents.

How is mass defect converted to binding energy in MeV?+

Use the conversion: Eb = Δm × 931.494 MeV/u, where 1 atomic mass unit = 931.494 MeV/c² (CODATA 2018). For example, Δm = 0.528 u gives Eb = 0.528 × 931.494 = 492 MeV. In SI: Eb = Δm (kg) × c² where 1 u = 1.66054 × 10⁻²⁷ kg and c = 2.998 × 10⁸ m/s.

What is binding energy per nucleon and why is it the key stability indicator?+

Binding energy per nucleon is Eb/A. It represents how tightly bound each nucleon is on average. Iron-56 peaks at ~8.79 MeV/nucleon. Light nuclei (low Eb/A) gain stability and release energy by fusion; heavy nuclei also have lower Eb/A than Fe and release energy by fission. Both processes move nucleons toward iron on the binding energy curve.

Why does iron-56 have the highest binding energy per nucleon?+

The binding energy curve results from competition between the attractive strong nuclear force (increases Eb/A as more nucleons join) and repulsive Coulomb force between protons (decreases Eb/A as more protons are added). At A ≈ 56, these opposing effects produce the maximum Eb/A ≈ 8.79 MeV/nucleon - the most stable nuclear configuration in the universe.

What is the semi-empirical mass formula for binding energy?+

The Bethe-Weizsäcker formula: Eb ≈ 15.8A − 18.3A^(2/3) − 0.714Z(Z−1)/A^(1/3) − 23.2(A−2Z)²/A ± 12/A^(3/4) MeV. The five terms are: volume (proportional to A), surface (negative, A^(2/3)), Coulomb repulsion, asymmetry (prefers N=Z), and pairing (positive for even-even, negative for odd-odd nuclei).

What is the atomic mass unit (u) and its energy equivalent?+

The atomic mass unit (u, or dalton Da) is exactly 1/12 the mass of a carbon-12 atom: 1 u = 1.66053906660 × 10⁻²⁷ kg. Its energy equivalent is 1 u = 931.49410242 MeV/c² (CODATA 2018). This unit makes atomic and nuclear mass tables convenient - most nuclide masses are close to integer values of u.

How does binding energy explain nuclear fission energy release?+

When U-235 fissions, products have higher Eb/A (~8.5 MeV/nucleon) than U-235 (~7.6 MeV/nucleon). The difference × 235 nucleons ≈ ~200 MeV is released per fission. This equals the Q-value = sum of reactant masses minus sum of product masses, times 931.494 MeV/u.

How does binding energy explain solar energy from hydrogen fusion?+

The Sun fuses 4 protons (4 × 1.007276 u) into one He-4 nucleus (4.002602 u). Mass defect = 4.029104 − 4.002602 = 0.026502 u. Energy = 0.026502 × 931.494 = 24.69 MeV per reaction (net ~26.7 MeV including neutrino energy). This pp-chain reaction powers the Sun's luminosity of 3.846 × 10²⁶ watts.

🔗 Related Calculators

📌 Quick Tips

💡Binding energy per nucleon (Eb/A) peaks at iron-56 (~8.79 MeV/nucleon) - this is why iron is the most stable nucleus. Elements lighter than Fe can release energy by fusion; heavier ones can release energy by fission.

💡The atomic mass unit: 1 u = 1.66054 × 10⁻²⁷ kg = 931.494 MeV/c². Using 931.494 MeV/u is the standard shortcut for converting mass defect directly to energy.

💡Free proton mass: 1.007276 u. Free neutron mass: 1.008665 u. Electron mass: 0.000549 u. The sum of free nucleon masses always exceeds the actual atomic mass - the difference is the mass defect Δm.

💡The liquid drop model (Bethe-Weizsäcker formula) estimates binding energy with five terms: volume, surface, Coulomb, asymmetry, and pairing. This calculator uses experimental atomic masses for exact results.

💡Nuclear binding energy is ~6–9 MeV per nucleon - about one million times larger than typical chemical bond energies (~few eV). This explains why nuclear reactions release so much more energy than chemical reactions.