# Nuclear Binding Energy Calculator - Binding Energy per Nucleon Curve, Semi-Empirical Mass Formula, Q-Value & E=mc²

> Binding energy per nucleon from AME2020 masses, mass defect and SEMF terms for any nuclide, and Q-values for fusion, fission and decay vs TNT and coal.

- URL: https://qilingtools.com/en/math/nuclear
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

This tool carries AME2020 measured atomic masses, to better than a micro-u, for 98 representative nuclides, and has four parts. The binding-energy curve plots measured B/A as dots over the semi-empirical (liquid-drop) mass formula and marks the true peak at ⁶²Ni (8.7946 MeV) as well as the often-quoted ⁵⁶Fe (8.7904 MeV), showing why fusing light nuclei and splitting heavy ones both release energy. The nuclide calculator takes an element and mass number and returns the mass defect, total binding energy and B/A, with a bar chart of the volume, surface, Coulomb, asymmetry and pairing terms; nuclides not in the table are estimated from the formula. The Q-value calculator has 20 presets (D–T and D–D fusion, the proton–proton chain, ²³⁵U fission, ²³⁸U alpha decay and more) or lets you build your own reaction, checks nucleon number and charge, and compares the energy per kilogram with TNT, coal and gasoline. An E = mc² converter rounds it off.

## How to use

1. Pick a panel: binding-energy curve, nuclide calculator, reaction Q-value or E = mc² converter
2. Hover or click a nuclide on the curve, enter an element and mass number, or choose a reaction preset and add or remove particles
3. Read the binding energy, SEMF terms, Q-value, energy per kilogram and how it compares with chemical fuels, together with the formulas used

## Features

- AME2020 atomic masses for 98 nuclides, better than micro-u precision
- Binding energy per nucleon curve: measured points plus the SEMF line, hover to inspect, click to calculate
- Mass defect, total binding energy and B/A, with the five SEMF terms compared against experiment
- Most stable Z for a given mass number (the beta-stability line)
- 20 preset reactions plus custom ones, with electron and positron masses handled from charge conservation
- Energy per kg, TNT equivalent, a log-scale comparison with coal and gasoline, and a multi-unit E = mc² converter

## FAQ

### Is iron-56 the nucleus with the highest binding energy per nucleon?

No. Per nucleon the highest is nickel-62 (about 8.7946 MeV), followed by iron-58 (8.7923 MeV) and iron-56 (8.7904 MeV). Iron-56 is often called the most stable because it has the lowest mass per nucleon (protons are lighter than neutrons and iron-56 has relatively fewer neutrons) and because it is the end point of stellar nucleosynthesis, made from nickel-56 decay, so it is far more abundant. The curve peaks in this region, so fusing lighter nuclei and splitting heavier ones both release energy.

### How do you calculate mass defect and binding energy?

Mass defect Δ = Z·m(¹H) + N·mₙ − M(atom); using the ¹H atomic mass makes the electron masses cancel. Binding energy B = Δc², with 1 u = 931.494 MeV. For ⁴He: 2 × 1.007825 + 2 × 1.008665 − 4.002603 = 0.030377 u, so B = 28.30 MeV and B/A = 7.07 MeV.

### How much energy do D–T fusion and uranium-235 fission release?

²H + ³H → ⁴He + n has Q = 17.59 MeV, about 14.1 MeV of it carried by the neutron; per kilogram of reactants that is about 3.4 × 10¹⁴ J/kg, more than seven million times gasoline. ²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n has Q ≈ 173 MeV, far more per event, but only about 7.1 × 10¹³ J/kg, roughly a fifth of D–T by mass. Counting the later decays of the fragments, a reactor fission releases about 200 MeV in total.

### What do the terms of the semi-empirical mass formula mean?

It treats the nucleus as a charged liquid drop. The volume term aᵥA is the binding of each nucleon to its neighbours; the surface term −aₛA²ᐟ³ removes the binding missing at the surface; the Coulomb term −a꜀Z(Z−1)/A¹ᐟ³ is proton–proton repulsion, increasingly important in heavy nuclei; the asymmetry term −aₐ(A−2Z)²/A is the Pauli-principle cost of unequal neutron and proton numbers; and the pairing term ±aₚ/√A makes even–even nuclei more bound. It is usually within 1% for medium and heavy nuclei but misses near magic numbers and for light nuclei.

### Why does a β⁺ decay Q-value subtract two electron masses?

Tabulated masses are atomic masses, including the electrons. In β⁻ decay the daughter atom has one more electron, exactly supplied by the emitted electron, so Q = [M(parent) − M(daughter)]c². In β⁺ decay the daughter atom has one electron fewer and a positron is also emitted, so Q = [M(parent) − M(daughter) − 2mₑ]c², an extra 1.022 MeV. That is also why p + p → d + e⁺ + ν releases only 0.420 MeV. The tool applies this correction automatically from charge conservation.

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