Every second, the Sun emits enough energy to power human civilization many times over, and yet its fuel is "just" hydrogen. Where does all that energy really come from? The answer lies deep in the atomic nucleus, where powerful forces and tiny mass differences add up to enormous energy changes.
This lesson explores how nuclear processes—radioactive decay, fission, and fusion—change nuclei and involve the release or absorption of energy. A key theme is that while energy may change dramatically, the total number of protons plus neutrons stays the same in every nuclear reaction.
To understand nuclear processes, we first need to recall the basic structure of matter. An atom has a tiny, dense nucleus at its center, surrounded by electrons. The nucleus contains positively charged protons and neutral neutrons. Together, protons and neutrons are called nucleons.
The atomic number, usually written as a lower left subscript in nuclear notation, is the number of protons, often written as \(Z\). The mass number, an upper left superscript, is the total number of protons plus neutrons, written as \(A\).
For example, a carbon-14 nucleus is written as \(^{14}_{6}\textrm{C}\). This means:
Different versions of the same element with different numbers of neutrons are called isotopes. For example, \(^{12}_{6}\textrm{C}\) and \(^{14}_{6}\textrm{C}\) are both carbon but have different mass numbers.
As shown in [Figure 1], the atomic number and mass number in the nuclear symbol provide a quick way to count protons and neutrons in any nucleus, which is crucial for checking conservation in nuclear equations.
Why do nuclear processes involve so much energy? The key idea is binding energy. Binding energy is the energy that holds the nucleus together, overcoming the repulsion between positively charged protons.
When nucleons come together to form a nucleus, the total mass of the nucleus is actually slightly less than the sum of the separate proton and neutron masses. This "missing" mass is called the mass defect, and it corresponds to the binding energy that has been released when the nucleus formed.
Mass–energy equivalence is expressed by the famous equation \(E = mc^2\), where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light in vacuum.
Because \(c\) is so large (about \(3.0 \times 10^8\,\textrm{m/s}\)), even a tiny mass defect corresponds to a huge amount of energy.
When a nuclear process causes the final products to have a lower total mass than the initial reactants, the difference in mass \(\Delta m\) is released as energy \(E\): \(E = \Delta m c^2\). If the final mass is higher, energy must be absorbed from outside.
For example, suppose a certain nuclear reaction has a mass defect of \(\Delta m = 1.0 \times 10^{-30}\,\textrm{kg}\). The associated energy is:
\[E = \Delta m c^2 = \left(1.0 \times 10^{-30}\right) \left(3.0 \times 10^8\right)^2 = 9.0 \times 10^{-14}\,\textrm{J}\]
This might not seem like much, but that energy comes from an incredibly small mass change. In real reactors or stars, enormous numbers of nuclei are involved, so the total energy becomes huge.
Some nuclei are unstable; they spontaneously change into more stable nuclei by emitting particles or energy. This is called radioactive decay. During decay, the nucleus changes its composition, and energy is released because the products usually have higher binding energy per nucleon.
The main types of radioactive decay are alpha, beta, and gamma decay, as summarized in [Figure 2].
In alpha decay, an unstable heavy nucleus emits an alpha particle, which is the same as a helium-4 nucleus \(^{4}_{2}\textrm{He}\) (2 protons and 2 neutrons).
A general alpha decay looks like:
\[^{A}_{Z}\textrm{X} \rightarrow ^{A-4}_{Z-2}\textrm{Y} + ^{4}_{2}\textrm{He}\]
For example, uranium-238 undergoes alpha decay:
\[^{238}_{92}\textrm{U} \rightarrow ^{234}_{90}\textrm{Th} + ^{4}_{2}\textrm{He}\]
Check conservation of nucleon number: before, the total \(A\) is \(238\). After, \(234 + 4 = 238\). The total number of nucleons is the same, even though one nucleus has become two.
In beta-minus decay, a neutron in the nucleus converts into a proton, releasing an electron (beta particle) and an antineutrino (symbol \(\bar{\nu}_e\)). The mass number \(A\) stays the same, but the atomic number \(Z\) increases by 1.
A general beta-minus decay looks like:
\[^{A}_{Z}\textrm{X} \rightarrow ^{A}_{Z+1}\textrm{Y} + ^{0}_{-1}e + \bar{\nu}_e\]
For example, carbon-14 decays to nitrogen-14:
\[^{14}_{6}\textrm{C} \rightarrow ^{14}_{7}\textrm{N} + ^{0}_{-1}e + \bar{\nu}_e\]
Check nucleon number: left side has \(A = 14\). Right side, nitrogen-14 has \(A = 14\), and the emitted electron and antineutrino have negligible mass number \(0\). So the total number of protons plus neutrons stays \(14\).
In beta-plus decay (positron emission), a proton turns into a neutron, emitting a positron (the antimatter version of the electron, symbol \(^{0}_{+1}e\)) and a neutrino \(\nu_e\). Here, \(A\) is constant, while \(Z\) decreases by 1.
A general beta-plus decay looks like:
\[^{A}_{Z}\textrm{X} \rightarrow ^{A}_{Z-1}\textrm{Y} + ^{0}_{+1}e + \nu_e\]
For example, sodium-22 decays to neon-22:
\[^{22}_{11}\textrm{Na} \rightarrow ^{22}_{10}\textrm{Ne} + ^{0}_{+1}e + \nu_e\]
Again, nucleon number is conserved: \(22\) before and \(22\) after.
After a nucleus undergoes alpha or beta decay, it may be left in an excited energy state. It can drop to a lower energy state by emitting a high-energy photon called a gamma ray \(\gamma\).
In gamma decay, the nucleus does not change its number of protons or neutrons. Only its energy state changes:
\[^{A}_{Z}\textrm{X}^{*} \rightarrow ^{A}_{Z}\textrm{X} + \gamma\]
The asterisk indicates the excited state. Since \(A\) and \(Z\) stay the same, gamma decay is a clear example where nucleon number and charge are obviously conserved, but energy is released as electromagnetic radiation.
Nuclear fission is the splitting of a heavy nucleus into two (occasionally more) lighter nuclei, plus several free neutrons and a large amount of energy. As shown in [Figure 3], a typical fission reaction starts when a heavy nucleus absorbs a slow neutron.
A classic example is the fission of uranium-235:
\[^{235}_{92}\textrm{U} + ^{1}_{0}n \rightarrow ^{141}_{56}\textrm{Ba} + ^{92}_{36}\textrm{Kr} + 3\,^{1}_{0}n + \textrm{energy}\]
Let us check conservation of nucleon number and atomic number.
The total number of nucleons, as well as total charge (atomic number), is conserved. However, the binding energy per nucleon in the products is higher, so the reaction releases energy. This energy appears mostly as kinetic energy of the fission fragments and emitted neutrons, which then transfer their energy as heat in a reactor.
The emitted neutrons can be absorbed by other \(^{235}_{92}\textrm{U}\) nuclei, causing more fission events. This leads to a chain reaction. In a controlled nuclear power plant, control rods and moderators manage how many neutrons go on to cause further fissions, keeping the power output steady. In a nuclear weapon, the chain reaction is uncontrolled and extremely rapid, releasing enormous energy in a fraction of a second.
Nuclear fusion is the process where two light nuclei combine to form a heavier nucleus, releasing energy. This is the process that powers the Sun and other stars. In fusion, the products again have higher binding energy per nucleon, so mass is converted to energy.
As depicted in [Figure 4], one common fusion reaction studied for future power plants combines deuterium and tritium, both isotopes of hydrogen:
\[^{2}_{1}\textrm{H} + ^{3}_{1}\textrm{H} \rightarrow ^{4}_{2}\textrm{He} + ^{1}_{0}n + \textrm{energy}\]
Check conservation of nucleon number:
Total nucleon number is unchanged. Total charge is also conserved: \(Z = 1 + 1 = 2\) before and \(Z = 2 + 0 = 2\) after. Yet the reaction releases a very large amount of energy per reaction compared to chemical reactions.
Fusion requires extremely high temperatures and pressures so that positively charged nuclei can get close enough to overcome their electrostatic repulsion and allow the strong nuclear force to bind them. In stars, gravity provides these conditions. On Earth, experimental reactors like tokamaks and inertial confinement systems attempt to achieve these conditions for sustainable energy generation.
Across alpha decay, beta decay, gamma decay, fission, and fusion, certain quantities are always conserved:
The main learning goal here is to see that even though nuclei are changing and energy may be released or absorbed, the total number of protons plus neutrons remains constant. For example:
Whether a nuclear process releases or absorbs energy depends on the relative binding energies of the reactants and products:
Nuclear processes are not just abstract ideas—they underlie important technologies and natural phenomena that affect everyday life.
Nuclear fission is used in power plants to generate electricity. In a reactor core, fuel rods containing \(^{235}_{92}\textrm{U}\) or \(^{239}_{94}\textrm{Pu}\) undergo controlled fission. The kinetic energy of the fission fragments heats water, producing steam that spins turbines and generators.
Designers must carefully control the chain reaction so that the rate of fission is steady and safe. Control rods made of materials like cadmium or boron absorb excess neutrons. Even though energy output is huge, every fission event still respects nucleon-number conservation.
Radioactive isotopes are used in medicine for both diagnosis and therapy. For example, positron emission tomography (PET) scans use isotopes that undergo beta-plus decay. The emitted positron eventually meets an electron, and they annihilate, producing gamma rays. Detectors form images from these gamma rays, helping doctors see metabolic activity in tissues.
In cancer treatment, beams of gamma rays, X-rays, or charged particles can damage the DNA of tumor cells more than healthy cells, shrinking or destroying tumors. The radiation comes from nuclear processes (such as cobalt-60 decay), where again nucleon number is conserved even as energy is transferred to biological tissue.
The predictable rate of certain nuclear decays provides a "clock" for dating ancient materials. In carbon dating, the ratio of \(^{14}_{6}\textrm{C}\) to \(^{12}_{6}\textrm{C}\) in once-living materials reveals how long it has been since the organism died. Carbon-14 decays via beta-minus decay to \(^{14}_{7}\textrm{N}\). The half-life and decay law are based on large numbers of nuclei undergoing statistically predictable transformations, each one conserving nucleon number.
Similar methods use uranium-lead or potassium-argon decay chains to estimate the age of rocks and Earth's crust.
The light and heat from stars, including our Sun, come from nuclear fusion in their cores. Hydrogen nuclei fuse into helium, releasing enormous energy that travels through space and eventually reaches Earth, driving climate, weather, and life. The long-term evolution of stars (from main sequence to red giants, supernovae, and neutron stars) is governed by which fusion reactions are possible at different temperatures and pressures.
To strengthen the idea of nucleon-number conservation, consider a few mental checks, similar to the visual shown in [Figure 1], where atomic and mass numbers are clearly separated.
Thought check 1: Suppose a nucleus written as \(^{60}_{27}\textrm{Co}\) emits a beta-minus particle. What is the daughter nucleus? Beta-minus decay increases \(Z\) by 1 but leaves \(A\) unchanged. So the daughter is \(^{60}_{28}\textrm{Ni}\). Total nucleons: \(60\) before and \(60\) after.
Thought check 2: A fictional fission reaction is written as:
\[^{240}_{94}\textrm{Pu} + ^{1}_{0}n \rightarrow ^{140}_{54}\textrm{Xe} + ^{99}_{40}\textrm{Zr} + 2\,^{1}_{0}n\]
Check nucleon number: left side \(240 + 1 = 241\). Right side \(140 + 99 + 2\times 1 = 241\). Even though three nuclei appear on the right, the total nucleon number remains the same.
Thought check 3: Imagine a fusion reaction in a star where four protons (hydrogen nuclei) combine through several steps to form one \(^{4}_{2}\textrm{He}\) nucleus and some positrons, neutrinos, and gamma rays. Counting nucleons: four protons mean \(A = 4\) total to start. The final helium nucleus also has \(A = 4\). The neutrinos, positrons, and photons do not contribute to \(A\). So nucleon number is conserved, while the mass of the final products is slightly less than the initial mass, and the difference appears as energy.
Across all these processes, one rule holds: even as energy changes form and nuclear identities transform, the total number of neutrons plus protons remains constant, reflecting deep conservation laws in nature.
