21.1 Nuclear Structure and Stability

Atomic nuclei are made up of protons and neutrons, held together by the nuclear force. These particles define what element you're dealing with and how stable that atom is. Understanding nuclear composition is the foundation for everything else in nuclear chemistry.

Nuclear stability depends on the balance between protons and neutrons. The binding energy and neutron-to-proton ratio are the two big factors that determine whether a nucleus holds together or falls apart through radioactive decay. This topic sets up your understanding of why certain isotopes are radioactive and why nuclear reactions release so much energy.

Atomic Nuclei and Nuclear Stability

Composition of atomic nuclei

The nucleus contains two types of subatomic particles, collectively called nucleons:

• Protons carry a positive charge and define the element. The number of protons is the atomic number ($Z$).
• Neutrons are electrically neutral and add mass to the nucleus without changing the element's identity.

The mass number ($A$) is the total count of protons and neutrons:

$A = Z + N$

where $Z$ is the number of protons and $N$ is the number of neutrons.

Isotopes are atoms of the same element with different numbers of neutrons. They share the same $Z$ but have different $A$ values. You write them using isotope notation: $^A_Z\text{X}$, where X is the element symbol. For example, carbon-12 is written $^{12}_6\text{C}$ (6 protons, 6 neutrons), while carbon-14 is $^{14}_6\text{C}$ (6 protons, 8 neutrons).

The nuclear force (also called the strong nuclear force) is what holds all these protons and neutrons together. This is worth pausing on: protons are all positively charged, so they repel each other electrically. The nuclear force is strong enough to overcome that repulsion, but only at very short distances. That's why nuclei are incredibly tiny compared to the atom as a whole.

Nuclear binding energy and mass defect

If you could pull a nucleus apart into individual protons and neutrons, you'd need to put in a specific amount of energy. That energy is the nuclear binding energy, and it's a direct measure of how stable a nucleus is. Higher binding energy means a more tightly held nucleus.

Here's where it gets interesting: the mass of an intact nucleus is always less than the sum of its individual protons and neutrons. That "missing" mass is called the mass defect.

To calculate it:

$\text{Mass defect} = (\text{sum of individual nucleon masses}) - (\text{actual nuclear mass})$

Mass defect is typically measured in atomic mass units (amu), where 1 amu = $1.66 \times 10^{-27}$ kg.

That missing mass didn't vanish. It was converted into the binding energy that holds the nucleus together, according to Einstein's equation:

$E = mc^2$

where $c$ is the speed of light ($3.00 \times 10^8$ m/s). Even a tiny amount of mass converts to an enormous amount of energy because $c^2$ is such a huge number.

Binding energy per nucleon is the total binding energy divided by the number of nucleons ($A$). This lets you compare the stability of different nuclei on a fair basis. Iron-56 ($^{56}\text{Fe}$) has the highest binding energy per nucleon of any element, making it the most stable nucleus. This fact is central to understanding both fission and fusion.

Patterns of nuclear stability

The neutron-to-proton ratio ($N/Z$) is the main predictor of whether a nucleus is stable or radioactive.

• For light elements (low $Z$), stable nuclei have $N/Z \approx 1$. Helium-4 ($^4_2\text{He}$) has 2 protons and 2 neutrons, a perfect 1:1 ratio.
• As $Z$ increases, stable nuclei need proportionally more neutrons. The extra neutrons help dilute the growing electrical repulsion between protons. Uranium-238 ($^{238}_{92}\text{U}$), for example, has 146 neutrons but only 92 protons, giving $N/Z \approx 1.59$.

The band of stability is the zone on a plot of $N$ vs. $Z$ where stable nuclei are found. Nuclei that fall outside this band are radioactive and will decay to move toward it. Carbon-14 ($^{14}_6\text{C}$), for instance, has too many neutrons for its number of protons, placing it above the band.

Magic numbers (2, 8, 20, 28, 50, 82, 126) represent especially stable arrangements of protons or neutrons, similar to how filled electron shells make atoms chemically stable. Calcium-40 ($^{40}_{20}\text{Ca}$) is "doubly magic" with 20 protons and 20 neutrons, both magic numbers.

Unstable nuclei undergo radioactive decay to reach a more stable configuration. The three main types:

• Alpha decay: the nucleus emits a helium-4 particle ($^4_2\text{He}$), losing 2 protons and 2 neutrons
• Beta decay: a neutron converts to a proton (emitting an electron) or a proton converts to a neutron (emitting a positron), adjusting the $N/Z$ ratio
• Gamma emission: the nucleus releases high-energy photons to shed excess energy, without changing its composition

Stability trends across the periodic table:

1. Light elements (low $Z$) have stable isotopes near $N/Z \approx 1$ (carbon-12, $^{12}_6\text{C}$)
2. Heavy elements (high $Z$) require more neutrons, pushing $N/Z$ well above 1 (uranium-238, $^{238}_{92}\text{U}$)
3. No element beyond lead ($Z = 82$) has any stable isotopes. Elements like polonium and astatine are all radioactive.

Nuclear Processes and Applications

• Radioactivity is the spontaneous emission of particles or energy from unstable nuclei as they decay toward stability.
• Half-life is the time it takes for half of a radioactive sample to decay. Each isotope has its own characteristic half-life, ranging from fractions of a second to billions of years.
• Nuclear fission is the splitting of a heavy nucleus (like uranium-235) into lighter nuclei, releasing large amounts of energy. This is the process behind nuclear power plants.
• Nuclear fusion is the combining of light nuclei (like hydrogen isotopes) to form heavier ones, also releasing energy. Fusion powers the sun and other stars.

Both fission and fusion release energy because the products have higher binding energy per nucleon than the starting materials. Fission works with nuclei heavier than iron-56, and fusion works with nuclei lighter than iron-56. That's why iron sits at the peak of the binding energy curve and is the dividing line between the two processes.