4.031882
u
0.030376
u
0.504406
×10^-28 kg
28.2951
MeV
4.533378
×10^-12 J
4
7.0738
MeV/nucleon
4.031882
u
0.030376
u
0.504406
×10^-28 kg
28.2951
MeV
4.533378
×10^-12 J
4
7.0738
MeV/nucleon
The Mass Defect Calculator determines the difference between the sum of individual nucleon masses and the actual nuclear mass, along with its energy equivalent via Einstein's famous equation $$E = mc^2$$. The mass defect is direct experimental evidence that mass and energy are interconvertible — the "missing" mass has been converted to the binding energy that holds the nucleus together.
Every stable nucleus in the universe weighs less than the sum of its parts. This mass difference, when multiplied by the speed of light squared, equals the binding energy. Understanding mass defect is fundamental to nuclear physics, explaining why nuclear reactions can release millions of times more energy per atom than chemical reactions, which involve only electron rearrangement.
The mass defect is calculated by subtracting the actual nuclear mass from the total mass of individual nucleons:
$$\Delta m = Z \cdot m_p + N \cdot m_n - M_{\text{nucleus}}$$
where $$m_p = 1.00728$$ u and $$m_n = 1.00866$$ u are the proton and neutron masses respectively. The energy equivalent follows from mass-energy equivalence:
$$E = \Delta m \cdot c^2 = \Delta m \times 931.494 \text{ MeV/u}$$
In SI units: $$E = \Delta m \times 1.66054 \times 10^{-27} \text{ kg/u} \times (2.998 \times 10^8)^2 \text{ m}^2/\text{s}^2$$
Note: this calculator uses nuclear mass (without electrons). If using atomic mass instead, electron masses must be included on both sides, which cancel for the mass defect calculation when using $$Z \cdot m_H + N \cdot m_n - M_{\text{atom}}$$ where $$m_H$$ is the hydrogen atom mass.
The Mass Defect in u shows the "missing" mass in atomic mass units. The Mass Defect in kg gives the same quantity in SI units. The Energy Equivalent in MeV is the nuclear binding energy — the energy released when the nucleons assemble into the nucleus (or equivalently, the energy required to break it apart). The Energy in Joules provides the SI equivalent. A positive mass defect indicates a bound state; the larger the mass defect, the more tightly bound the nucleus.
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Helium-4 has a mass defect of 0.0304 u, equivalent to 28.3 MeV. This large binding energy explains why He-4 (alpha particles) are exceptionally stable.
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Deuterium (H-2) has a relatively small mass defect of 2.22 MeV, reflecting the weak binding of just one proton and one neutron. This is the simplest bound nuclear system.
Mass defect ($$\Delta m$$) is the difference between the total mass of individual protons and neutrons and the actual mass of the nucleus they form. The nucleus weighs less than its components because some mass has been converted to binding energy that holds the nucleus together, as described by $$E = mc^2$$.
The "missing" mass is converted to energy — specifically, the binding energy of the nucleus. When nucleons come together to form a nucleus, energy is released (as gamma rays and kinetic energy). This released energy carries away the equivalent mass, leaving the nucleus lighter. To separate the nucleons again, this same energy must be supplied.
More precisely, the system's invariant mass decreases while energy is released. In relativity, mass and energy are different aspects of the same quantity. The bound nucleus plus the emitted radiation have the same total energy as the original free nucleons; the mass simply shifts between rest mass and radiation energy.
A negative mass defect (nucleus heavier than its constituents) would indicate an unbound system that spontaneously disassembles. In practice, all observed nuclei have positive mass defects, though very unstable nuclei have small values. Truly unbound nuclear systems (resonances) have widths rather than discrete mass defects.
Nuclear masses are measured using Penning trap mass spectrometry, which determines mass from the cyclotron frequency of ions in a magnetic field. Time-of-flight techniques and nuclear reaction kinematics also provide mass measurements. Modern precision reaches parts per billion for stable and some radioactive nuclides.
Atomic mass units (u) provide a convenient scale for nuclear physics. 1 u = 1.66054 × 10⁻²⁷ kg is approximately the mass of one nucleon. Using u keeps numbers manageable (nuclear masses are 1-300 u) and the conversion factor 931.494 MeV/u directly connects mass and energy in nuclear physics units.
One atomic mass unit is equivalent to 931.494 MeV of energy. For perspective, the total mass defect of a U-235 fission event is about 0.215 u, releasing ~200 MeV. A hydrogen atom's mass (1.008 u) contains 938.8 MeV of rest energy, vastly more than any chemical bond energy (~1-10 eV).
The mass defect per nucleon ($$\Delta m / A$$) is equivalent to the binding energy per nucleon divided by 931.494 MeV/u. It peaks at about 0.00943 u/nucleon for iron-56, meaning each nucleon in iron has lost about 0.94% of its free mass to nuclear binding energy.
The Q-value of a nuclear reaction equals the mass defect difference between reactants and products: $$Q = (\Delta m_{\text{products}} - \Delta m_{\text{reactants}}) \times 931.494$$ MeV. Positive Q means the products are more tightly bound and energy is released (exothermic). Negative Q means energy must be supplied (endothermic).
The neutron is about 1.293 MeV/c² (0.001389 u) heavier than the proton due to the mass difference of the constituent quarks and their electromagnetic interactions. The down quark (in neutron) is heavier than the up quark (in proton). This mass difference allows free neutrons to beta-decay to protons but not vice versa.
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