Binding Energy Calculator
Compute nuclear binding energy and binding energy per nucleon from mass defect or atomic mass. Supports MeV and joules, with preset examples for common nuclei.
Nuclear Binding Energy Calculator
Presets fill in Z, A and atomic mass. You can still edit any value.
If you already know the mass defect, enter it here.
What is nuclear binding energy?
Nuclear binding energy is the energy required to completely separate a nucleus into its individual protons and neutrons (nucleons), or equivalently, the energy released when those nucleons come together to form the nucleus.
Because of mass–energy equivalence \(E = mc^2\), the mass of a bound nucleus is less than the sum of the masses of its free nucleons. This missing mass is called the mass defect, and it corresponds to the binding energy.
Key formulas used in this calculator
1. Mass number
\[ A = Z + N \]
where \(Z\) is the number of protons and \(N\) is the number of neutrons.
2. Mass defect from nucleon masses
If you provide the atomic mass of the nuclide \(m_{\text{atom}}\) (in atomic mass units u), the calculator computes the mass defect as:
\[ \Delta m = Z m_p + N m_n - m_{\text{nucleus}} \]
For atomic masses (which include electrons), we use:
\[ m_{\text{nucleus}} \approx m_{\text{atom}} - Z m_e \]
where \(m_p\) is the proton mass, \(m_n\) the neutron mass, and \(m_e\) the electron mass, all in u.
3. Binding energy from mass defect
In energy units:
\[ B = \Delta m \, c^2 \]
In practice, we use the conversion
\[ 1 \text{ u} = 931.494 \text{ MeV}/c^2 \]
so if \(\Delta m\) is in atomic mass units:
\[ B(\text{MeV}) = \Delta m(\text{u}) \times 931.494 \]
If \(\Delta m\) is in kilograms:
\[ B(\text{J}) = \Delta m(\text{kg}) \, c^2 \]
4. Binding energy per nucleon
\[ \frac{B}{A} = \frac{\text{binding energy}}{\text{mass number}} \]
This quantity is often used to compare the relative stability of different nuclei. Nuclei around iron-56 have the highest binding energy per nucleon.
Physical constants used
- Speed of light: \(c = 2.99792458 \times 10^8 \,\text{m/s}\)
- 1 atomic mass unit: \(1\,\text{u} = 1.66053906660 \times 10^{-27}\,\text{kg}\)
- Proton mass: \(m_p = 1.007276466621\,\text{u}\)
- Neutron mass: \(m_n = 1.00866491595\,\text{u}\)
- Electron mass: \(m_e = 0.000548579909065\,\text{u}\)
- Energy conversion: \(1\,\text{u}c^2 = 931.494\,\text{MeV}\)
Worked example: Helium-4 binding energy
Consider the alpha particle, helium-4 (\(^4\text{He}\)):
- \(Z = 2\) protons
- \(N = 2\) neutrons
- Atomic mass \(m_{\text{atom}} = 4.002602\,\text{u}\)
-
Compute mass of separated nucleons
\[ m_{\text{nucleons}} = Z m_p + N m_n = 2(1.007276466621) + 2(1.00866491595) \approx 4.0318828\,\text{u} \] -
Approximate nuclear mass (subtract
electrons):
\[ m_{\text{nucleus}} \approx m_{\text{atom}} - Z m_e = 4.002602 - 2(0.00054858) \approx 4.001505\,\text{u} \] -
Mass defect
\[ \Delta m = m_{\text{nucleons}} - m_{\text{nucleus}} \approx 4.0318828 - 4.001505 \approx 0.030378\,\text{u} \] -
Binding energy
\[ B = \Delta m \times 931.494 \approx 0.030378 \times 931.494 \approx 28.3\,\text{MeV} \] -
Binding energy per nucleon
\[ \frac{B}{A} = \frac{28.3}{4} \approx 7.1\,\text{MeV/nucleon} \]
If you select the Helium-4 preset in the calculator, you should obtain values very close to these.
Interpreting binding energy and nuclear stability
A higher binding energy per nucleon generally means a more stable nucleus. The curve of binding energy per nucleon versus mass number peaks around iron and nickel:
- Light nuclei (like hydrogen, helium) have lower \(B/A\).
- Medium-mass nuclei (around iron-56) have the highest \(B/A\) and are very stable.
- Very heavy nuclei (like uranium) have lower \(B/A\) again, which is why they can release energy via fission.
Why fusion and fission release energy
- Fusion: Light nuclei combine to form a heavier nucleus with higher binding energy per nucleon. The increase in total binding energy is released, mostly as kinetic energy of reaction products.
- Fission: A heavy nucleus splits into two medium-mass nuclei that are closer to the peak of the binding energy curve. Again, the increase in total binding energy is released.
FAQ
Is binding energy positive or negative?
By convention, binding energy is reported as a positive number: the amount of energy you must supply to break the nucleus apart. In potential energy terms, the bound state has lower (more negative) energy than the separated nucleons.
Why do we subtract electron mass when using atomic masses?
Tabulated atomic masses include the mass of the electrons. Binding energy is defined for the nucleus only, so we subtract \(Z\) electron masses to approximate the nuclear mass before computing the mass defect.
How accurate is this calculator?
For most stable nuclides, using standard atomic masses and the constants above gives binding energies that agree with nuclear data tables to within a small fraction of a percent. For precision nuclear physics work, always compare against evaluated nuclear data libraries.