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Walk-through

How to Use This Calculator

3 steps

1

Enter protons, neutrons, and atomic mass

Fill in the number of protons (Z) and neutrons (N) in the nucleus, then the isotope's measured atomic mass (M) in unified atomic mass units (u). Nuclide data tables (NIST, NNDC) list atomic mass to 6+ decimal places — use as much precision as you have, since the binding energy comes from a tiny difference between two large numbers.

2

Read the binding energy, mass defect, or per-nucleon value

The Binding Energy tab shows the total energy (in MeV) released if the nucleus were assembled from separate protons and neutrons. Switch to Mass Defect to see the underlying mass difference in atomic mass units, or Per Nucleon to see binding energy divided across all nucleons — the number nuclear physicists use to compare stability between isotopes of different sizes.

3

Compare isotopes

Try a light nucleus (He-4) against a mid-mass one (Fe-56) and a heavy one (U-238). Per-nucleon binding energy rises from light elements, peaks around iron/nickel, then declines for the heaviest elements — that curve is why fusion releases energy for light elements and fission releases energy for heavy ones.

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Reference

Formula & Methodology

3 formulas

Mass defect

Δm = Z·m(¹H) + N·mₙ − M

Z is the proton count, N the neutron count, m(¹H) = 1.007825032 u is the atomic mass of hydrogen-1 (used instead of the bare proton mass so the Z electrons implicit in the measured atomic mass M cancel out), mₙ = 1.008664916 u is the neutron mass, and M is the isotope's measured atomic mass in u.

Binding energy

E = Δm × 931.494 MeV/u

Mass defect converts to energy via E = Δm·c². Expressed with mass in unified atomic mass units, 1 u·c² = 931.494 MeV, so binding energy in MeV is the mass defect times that constant.

Binding energy per nucleon

E/A = E ÷ (Z + N)

Total binding energy divided by the mass number A = Z + N (total nucleon count). This is the standard measure for comparing nuclear stability across isotopes of different sizes — it peaks near iron-56 and nickel-62.

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Glossary

Key Terms Explained

7 terms

Binding energy The energy that would be released if a nucleus were assembled from its separate, unbound protons and neutrons — equivalently, the energy required to break the nucleus apart into individual nucleons. Measured in MeV (mega-electron-volts).

Mass defect The difference between the summed mass of a nucleus's separate protons and neutrons and the nucleus's actual measured mass. The nucleus is always lighter than its parts because some mass converted to binding energy when it formed.

Nucleon A proton or neutron — the particles that make up an atomic nucleus. The total nucleon count is the mass number, A = Z + N.

Atomic mass unit (u) A unit of mass defined as 1/12 the mass of a carbon-12 atom, approximately 1.6605 × 10⁻²⁷ kg. Also called the dalton (Da). Atomic masses of isotopes are conventionally reported in u.

MeV Mega-electron-volt, a unit of energy equal to 1 million electron-volts (1.602 × 10⁻¹³ joules). The standard unit for nuclear binding energies, which typically run from a few MeV to about 1,800 MeV for the heaviest stable nuclei.

E = mc² Einstein's mass-energy equivalence: mass and energy are interchangeable, related by the speed of light squared. In nuclear physics this is usually applied per unit atomic mass: 1 u of mass corresponds to 931.494 MeV of energy.

Nuclear stability A nucleus's resistance to spontaneous decay. Binding energy per nucleon is the standard proxy for stability — it rises through light elements, peaks around iron-56 (about 8.79 MeV/nucleon), and gradually declines for heavier elements, which is why both fusing light nuclei and splitting heavy nuclei release energy.

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Scenarios

Real-World Examples

3 worked examples

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Helium-4 nucleus

Binding energy of an alpha particle

Protons (Z) 2 Neutrons (N) 2 Atomic mass (M) 4.002602 u

Mass defect Δm = 2(1.007825032) + 2(1.008664916) − 4.002602 ≈ 0.030378 u. Multiplying by 931.494 MeV/u gives a binding energy of about 28.3 MeV — an unusually high value for such a light nucleus, which is why the helium-4 nucleus (an alpha particle) is exceptionally stable and shows up as a building block in heavier-element fusion chains.

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Helium-4 nucleus

Per-nucleon binding energy

Protons (Z) 2 Neutrons (N) 2 Atomic mass (M) 4.002602 u

With A = 4 nucleons and total binding energy ≈ 28.3 MeV, the per-nucleon value is 28.3 ÷ 4 ≈ 7.07 MeV/nucleon. That is well above the roughly 7 MeV/nucleon average for light nuclei, confirming He-4's outsized stability relative to its neighbors on the nuclide chart.

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Iron-56 nucleus

Mass defect at the peak of the stability curve

Protons (Z) 26 Neutrons (N) 30 Atomic mass (M) 55.934936 u

Mass defect Δm ≈ 0.528462 u, giving a total binding energy of about 492.3 MeV and roughly 8.79 MeV/nucleon — the highest per-nucleon value of any common isotope. This is why iron-56 sits at the peak of the nuclear binding energy curve: fusing lighter elements toward iron releases energy, and splitting heavier elements toward iron also releases energy.

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Deep Dive

Understanding the Binding Energy Calculator

3 chapters · 4 min read

Every atomic nucleus weighs slightly less than the sum of its individual protons and neutrons — that missing mass converted to energy when the nucleus formed, and it's what holds the nucleus together against the electrostatic repulsion between its protons. This calculator turns proton count, neutron count, and measured atomic mass into that mass defect, the total binding energy in MeV, and the per-nucleon value physicists use to compare nuclear stability.

How the Binding Energy Calculator works

The calculator applies Δm = Z·m(¹H) + N·mₙ − M, then converts the mass defect to energy via E = Δm × 931.494 MeV/u (the mass-energy equivalence 1 u·c² = 931.494 MeV). A subtle but important detail: the isotope's measured atomic mass M already includes the mass of its Z orbital electrons, so the calculation uses the atomic mass of hydrogen-1 (1.007825032 u, which itself includes 1 electron) in place of the bare proton mass — that way the electron masses cancel correctly on both sides of the equation, rather than being double-counted or dropped. The neutron mass used is 1.008664916 u. Dividing total binding energy by the nucleon count A = Z + N gives binding energy per nucleon, plotted on the classic nuclear-stability curve that peaks near iron and nickel.

Inputs and what they mean

Protons (Z) is the atomic number — it identifies the element and determines the isotope's chemistry. Neutrons (N) varies between isotopes of the same element and mainly affects nuclear stability. Atomic mass (M) is the measured mass of the whole neutral atom in unified atomic mass units (u), typically sourced from a nuclide table like the NIST Atomic Weights and Isotopic Compositions database or the NNDC's Nuclear Wallet Cards — small errors in the atomic mass input have an outsized effect on the result, since binding energy comes from a tiny difference between two much larger numbers.

Limits and edge cases

This calculator computes the classical liquid-drop-style mass-defect result from measured atomic mass — it does not model nuclear shell structure, pairing effects, or predict masses for isotopes you haven't measured yet; for that, semi-empirical mass formulas (like the Weizsäcker formula) or full nuclear data evaluations are the right tool. Because binding energy is a small difference of two large numbers, results are only as precise as the atomic mass you enter — use at least 6 significant figures from an authoritative nuclide table for research-grade accuracy. The calculator also assumes Z and N are non-negative and the atomic mass is positive; it will not validate that a given (Z, N) combination corresponds to a real, physically observed isotope.

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Questions

Frequently Asked Questions

6 questions

What is the formula for binding energy?+

Binding energy E equals the mass defect Δm times 931.494 MeV per atomic mass unit: E = Δm × 931.494 MeV/u, where Δm = Z·m(¹H) + N·mₙ − M is the difference between the summed mass of the nucleus's separate protons and neutrons and its actual measured atomic mass.

What is mass defect?+

Mass defect is the sum of the masses of a nucleus's individual protons and neutrons minus the nucleus's actual measured mass. The nucleus is always lighter than the sum of its parts — the missing mass was converted to the binding energy that holds the nucleus together, per Einstein's E = mc².

What does binding energy per nucleon mean?+

It's the total binding energy divided by the number of nucleons (protons plus neutrons), giving an average energy-per-particle figure that's comparable across isotopes of different sizes. It is the standard measure of nuclear stability: higher per-nucleon binding energy means a more tightly bound, more stable nucleus.

What units does this calculator use?+

Protons and neutrons are plain counts (no units). Atomic mass is entered in unified atomic mass units (u), also called daltons. Binding energy and binding energy per nucleon are both output in MeV (mega-electron-volts), the standard energy unit in nuclear physics.

What is 1 atomic mass unit in MeV?+

1 u of mass is equivalent to 931.494 MeV of energy, via Einstein's mass-energy equivalence E = mc². This calculator uses that conversion factor to turn the mass defect (in u) into binding energy (in MeV).

Which isotope has the highest binding energy per nucleon?+

Nickel-62 has the single highest binding energy per nucleon at about 8.7945 MeV/nucleon, with iron-56 extremely close behind at about 8.7903 MeV/nucleon — iron-56 is more commonly cited as the 'peak' because it's far more abundant, produced in large quantities by stellar nucleosynthesis. This peak is why nuclear fusion releases energy for elements lighter than iron, while fission releases energy for elements heavier than iron.

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Binding Energy Calculator

Inputs

Binding Energy

Inputs

Mass Defect

Inputs

Binding Energy Per Nucleon

How to Use This Calculator

Enter protons, neutrons, and atomic mass

Read the binding energy, mass defect, or per-nucleon value

Compare isotopes

Formula & Methodology

Key Terms Explained

Real-World Examples

Helium-4 nucleus

Helium-4 nucleus

Iron-56 nucleus

Understanding the Binding Energy Calculator

How the Binding Energy Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Binding Energy Calculator

Inputs

Binding Energy

Inputs

Mass Defect

Inputs

Binding Energy Per Nucleon

How to Use This Calculator

Enter protons, neutrons, and atomic mass

Read the binding energy, mass defect, or per-nucleon value

Compare isotopes

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Binding Energy Calculator

How the Binding Energy Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

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