Mass–energy equivalence

Mass and energy are equivalent: when the energy of a system changes, its mass changes too. A system that gains energy gains mass; a system that loses energy loses mass. The relationship between the changes is:

Here, $$\Delta E$$ is the change in energy, $$\Delta m$$ is the corresponding change in mass, and $$c$$ is the speed of light in a vacuum, approximately $$3.0 \times 10^8\ \text{m/s}$$. If energy is measured in joules and mass in kilograms, the equation gives the corresponding change in those units.

Because $$c^2$$ is very large, a small change in mass corresponds to a substantial change in energy. Conversely, the mass change associated with an everyday energy transfer is usually too small to notice.

The relationship applies to all energy changes, not only those in nuclear reactions. For example, when a sealed object is heated and gains energy, its mass increases slightly. If it later loses energy as it cools, its mass decreases.

Nuclear binding energy

A nucleus contains protons and neutrons, collectively called nucleons. Energy must be supplied to separate a nucleus completely into its individual, free nucleons. This required energy is the nuclear binding energy.

The reverse process releases the same amount of energy: when free nucleons form a nucleus, the bound nucleus has less energy than the separated nucleons. This energy is released to the surroundings.

Binding energy and mass defect

The mass of a bound nucleus is less than the sum of the masses of its separate protons and neutrons. The difference is called the mass defect. It corresponds to the energy released when the nucleus forms, through mass–energy equivalence.

For a nucleus containing $$Z$$ protons and $$N$$ neutrons, the binding energy is:

Here $$m_{\mathrm{p}}$$ and $$m_{\mathrm{n}}$$ are the masses of a free proton and neutron, $$m_{\mathrm{nucleus}}$$ is the mass of the nucleus, and $$c$$ is the speed of light in a vacuum. The expression in brackets is the mass defect.

Binding energy in MeV

The atomic mass unit, written $$\mathrm{u}$$, is one twelfth of the mass of a carbon-12 atom. A mass of $$1,\mathrm{u}$$ has an energy equivalent of approximately $$931.5,\mathrm{MeV}$$. Therefore, if the mass defect is calculated in $$\mathrm{u}$$:

Atomic and nuclear masses

A nuclear mass is the mass of the nucleus alone; an atomic mass also includes the atom’s electrons. The mass defect must compare masses that account consistently for those electrons.

If nuclear masses are used, subtract the nuclear mass from the sum of the free proton and neutron masses. If neutral atomic masses are used instead, the mass of the electrons must also be included on the separated-particle side. One way is to use $$Z$$ neutral hydrogen-atom masses in place of $$Z$$ proton masses: the electron masses then approximately cancel. This neglects the small difference between the electronic binding energies of the atoms.

Nuclear fission

A heavy atomic nucleus can split into smaller nuclei. This process is called nuclear fission. It can begin when a suitable heavy nucleus absorbs a neutron and becomes unstable. The unstable nucleus then splits, typically releasing energy and additional neutrons.

Fission chain reactions

The neutrons released by one fission can be absorbed by other suitable nuclei, causing them to split and release further neutrons. This sequence is a chain reaction. It can continue if enough of the released neutrons cause further fissions; otherwise, it dies out.

Energy released in fission

The total rest mass of the original nucleus and the absorbed neutron is greater than the total rest mass of the fission products, including the released neutrons. The difference in mass corresponds to the energy released:

Here, $$E$$ is the energy released, $$\Delta m$$ is the decrease in total rest mass, and $$c$$ is the speed of light in a vacuum. Much of the released energy appears as kinetic energy of the fission products.

Nuclear fusion

Nuclear fusion occurs when light atomic nuclei join to form a heavier nucleus. Fusion of suitable light nuclei releases energy.

Why fusion requires high temperatures

Nuclei are positively charged, so they repel one another. To fuse, they must approach closely enough for the attractive nuclear force to overcome this repulsion and bind them together.

At very high temperatures, particles have high kinetic energies. This helps nuclei approach one another closely enough for fusion to occur.

Energy released by fusion

In fusion of suitable light nuclei, the total mass of the products is less than the total mass of the reacting nuclei. The mass difference corresponds to the energy released, according to $$E = mc^2$$. Here, $$m$$ is the mass difference and $$c$$ is the speed of light.

Energy release from nuclear reaction masses

A nuclear reaction can produce particles whose total rest mass is less than the total rest mass of the particles that reacted. The mass difference corresponds to energy released, carried by the reaction products and any emitted radiation.

For fission or fusion, the energy released is calculated from:

Here, $$Q$$ is the energy released, $$m_{\mathrm{initial}}$$ is the total rest mass of the reactants, $$m_{\mathrm{final}}$$ is the total rest mass of the products, and $$c$$ is the speed of light. If the masses are in kilograms, the equation gives $$Q$$ in joules.

A positive mass difference gives a positive $$Q$$: energy is released. If the products have a greater total rest mass than the reactants, $$Q$$ is negative and energy must be supplied.

Calculating energy in MeV

If the mass difference $$\Delta m$$ is in atomic mass units, denoted by $$\mathrm{u}$$, the energy can be found directly in megaelectronvolts, or $$\mathrm{MeV}$$:

For example, suppose the total reactant mass is $$2.0000,\mathrm{u}$$ and the total product mass is $$1.9980,\mathrm{u}$$. The mass difference is $$0.0020,\mathrm{u}$$, so:

Choosing and adding reaction masses

Include every reactant and product when finding the two total masses, including any emitted particles. Use either atomic masses or nuclear masses consistently rather than mixing them.

Atomic masses include electrons, whereas nuclear masses do not. When using atomic masses, check that the electrons are accounted for consistently on both sides of the reaction.

Binding energy per nucleon graph

A nucleus is held together by forces between its nucleons: its protons and neutrons. Its binding energy is the energy needed to separate the nucleus completely into individual nucleons.

The average binding energy per nucleon is the total binding energy divided by the nucleon number:

Shape of the binding energy per nucleon graph

The graph plots average binding energy per nucleon on the vertical axis against nucleon number on the horizontal axis. It rises steeply for light nuclei, reaches a broad maximum around iron, then falls gradually for heavier nuclei.

Nuclei near the maximum have a greater average binding energy per nucleon than nuclei at either extreme. More energy per nucleon is therefore needed, on average, to separate them into individual nucleons.

Energy release and the graph

In fusion, light nuclei join to form a heavier nucleus. For suitable light nuclei, the product has a greater average binding energy per nucleon than the starting nuclei. The increase in total binding energy corresponds to energy released. This is why the steeply rising, left-hand region of the graph indicates where fusion can release energy.

In fission, a heavy nucleus splits into smaller nuclei. For suitable heavy nuclei, the products have a greater average binding energy per nucleon than the original nucleus. Again, the increase in total binding energy corresponds to energy released. The heavy-nucleus region of the graph therefore indicates where fission can release energy.

The graph shows why these processes can release energy; it does not mean that every fusion or fission reaction releases energy. Knowledge of nuclear processes and their energy release provides scientific evidence that can inform societal decisions about nuclear energy.