The mass of one atom is tiny – a carbon-12 atom weighs 1.993 × 10⁻²⁶ kg – so chemists measure it in unified atomic mass units (u, also called daltons, Da). One u is defined as one twelfth of the mass of a carbon-12 atom, 1.66054 × 10⁻²⁷ kg. On this scale protons and neutrons each weigh just over 1 u and an electron only 0.00055 u, which is why the mass number A = Z + N is already a good first guess for the atomic mass.
The guess is not exact because a nucleus weighs less than its separate protons and neutrons. When nucleons bind together, energy is released, and by E = mc² that energy is missing from the mass. This mass defect is about 0.8–0.9% of the total for most nuclei – roughly 0.5 u for iron-56.
This calculator gives all three numbers: the mass number, the sum of the free hydrogen atoms and neutrons, and a corrected atomic mass that subtracts the binding energy estimated with the semi-empirical (Weizsäcker) mass formula. For medium and heavy nuclei the estimate is usually within about 0.01 u of the measured value.
How to use the Atomic Mass Calculator
- Number of protons (Z) — The atomic number – 26 for iron.
- Number of neutrons (N) — Mass number minus atomic number.
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the number of protons Z and neutrons N. The calculator adds Z hydrogen-atom masses (which include the electrons) and N neutron masses, estimates the binding energy B from the volume, surface, Coulomb, asymmetry and pairing terms of the liquid-drop model, and subtracts B/931.494 MeV per u. Use the unit menu to show the result in kilograms, grams or MeV/c².
Worked example
Iron-56 has Z = 26 and N = 30, so A = 56 and the quick estimate is 56 u. The separate particles weigh 26 × 1.007825 + 30 × 1.008665 = 56.4634 u. The binding-energy terms (MeV) are: volume 15.8 × 56 = 884.80, surface −18.3 × 56^(2/3) = −267.87, Coulomb −0.714 × 26 × 25 / 56^(1/3) = −121.31, asymmetry −23.2 × (56 − 52)² / 56 = −6.63 and pairing (even–even) +12/√56 = +1.60, so B ≈ 490.6 MeV (measured: 492.3 MeV) or 8.76 MeV per nucleon. Dividing by 931.494 MeV/u gives a mass defect of 0.5267 u, and the atomic mass is 56.4634 − 0.5267 = 55.9367 u – only 0.0018 u above the measured 55.9349 u. In kilograms that is 9.2885 × 10⁻²⁶ kg.
Constants used
| Quantity | Value |
|---|---|
| Mass of a hydrogen-1 atom | 1.00782503223 u |
| Neutron mass | 1.00866491595 u |
| 1 u in kg | 1.66053906660 × 10⁻²⁷ kg |
| 1 u in energy | 931.49410242 MeV/c² |
| Liquid-drop coefficients (MeV) | aᵥ 15.8 · aₛ 18.3 · a_c 0.714 · a_a 23.2 · a_p 12.0 |
Tips
- Type formulas with normal capitals and numbers, e.g. Ca(OH)2 or CuSO4·5H2O – brackets and hydrates work.
- Atomic masses are IUPAC standard atomic weights; radioactive elements use the mass number of their longest-lived isotope.
- Electronegativity values are on the Pauling scale.
- Everything is calculated in your browser.
Common mistakes to avoid
- Writing CO (carbon monoxide) when you mean Co (cobalt) – capitals matter.
- Forgetting to multiply the bracket contents by the subscript after the bracket.
- Mixing up atomic number (protons) and mass number (protons + neutrons).
Inputs explained
| Field | What to enter |
|---|---|
| Number of protons (Z) | A number. The atomic number – 26 for iron. |
| Number of neutrons (N) | A number. Mass number minus atomic number. |
What each result means
- Mass number (A = Z + N) = [Number of protons (Z)] + [Number of neutrons (N)]
- Quick estimate (A × 1 u) = [Number of protons (Z)] + [Number of neutrons (N)]
- Sum of free hydrogen atoms and neutrons = [Number of protons (Z)]×1.00782503223 + [Number of neutrons (N)]×1.00866491595
- Binding energy per nucleon = [Nuclear binding energy (estimate)]÷([Number of protons (Z)] + [Number of neutrons (N)])
- Mass defect = [Nuclear binding energy (estimate)]÷931.49410242
- Mass defect as a share of the free mass = [Nuclear binding energy (estimate)]÷931.49410242÷[Sum of free hydrogen atoms and neutrons]×100
Formula
A = Z + N · m ≈ A u (quick estimate)m = Z·m(¹H) + N·mₙ − B/c² · B = aᵥA − aₛA^(2/3) − a_c·Z(Z−1)/A^(1/3) − a_a(A − 2Z)²/A ± δWorked example
Inputs — Number of protons (Z): 26; Number of neutrons (N): 30.
Results — Estimated atomic mass (with binding energy) = 55.936709 u (Da); Element = Iron (Fe); Mass number (A = Z + N) = 56; Quick estimate (A × 1 u) = 56 u (Da); Sum of free hydrogen atoms and neutrons = 56.463398 u (Da); Nuclear binding energy (estimate) = 490.61 MeV; Binding energy per nucleon = 8.761 MeV; Mass defect = 0.52669 u; Mass defect as a share of the free mass = 0.933 %.
Key terms
- Unified atomic mass unit (u, Da)
- One twelfth of the mass of a carbon-12 atom, 1.66054 × 10⁻²⁷ kg.
- Mass defect
- The difference between the mass of the separate nucleons and the mass of the nucleus, Δm = B/c².
- Binding energy
- The energy needed to pull a nucleus apart into free protons and neutrons.
- Semi-empirical mass formula
- The liquid-drop model of the nucleus (Weizsäcker, 1935) that estimates binding energy from five terms.
Frequently asked questions
What is the atomic mass of carbon-12?
Exactly 12 u by definition – the dalton is one twelfth of a carbon-12 atom. Its 6 protons, 6 neutrons and 6 electrons weigh 12.0989 u when separate; the 0.0989 u difference is the binding energy (92.2 MeV).
How do I calculate atomic mass from protons and neutrons?
A quick answer is the mass number in daltons: Z + N. Iron with 26 protons and 30 neutrons is about 56 u. The measured value is 55.9349 u; this calculator's binding-energy estimate gives 55.9367 u.
Why is atomic mass not a whole number?
Protons and neutrons each weigh slightly more than 1 u, electrons add a little, and the binding energy removes some mass. Only carbon-12 is exactly whole by definition.
What is the difference between atomic mass and atomic weight?
Atomic mass belongs to one isotope (chlorine-35: 34.969 u). The standard atomic weight on the periodic table averages all natural isotopes by abundance (chlorine: 35.45).
How much does one atomic mass unit weigh?
1 u = 1.66053906660 × 10⁻²⁷ kg, which is also 931.494 MeV/c² of energy.
How accurate is the semi-empirical estimate?
For nuclei heavier than about oxygen it is usually within 0.01 u (a few MeV of binding energy). For very light nuclei such as ²H, ³He or ⁴He it is poor, because the liquid-drop model assumes a large nucleus.
Which nucleus has the most binding energy per nucleon?
Nickel-62, with about 8.79 MeV per nucleon, closely followed by iron-58 and iron-56. That peak is why fusion releases energy below iron and fission above it.
What formula does the Atomic Mass Calculator use?
A = Z + N · m ≈ A u (quick estimate).