Structure of Atomic Nuclei, Nuclides, and Isotopes

An atomic nucleus is made of protons and neutrons (nucleons), a nuclide is a nucleus with a specific number of protons Z and neutrons N (mass number A = Z + N), and isotopes are nuclides of the same element that have different numbers of neutrons. In brief: carbon-12, carbon-13 and carbon-14 all have 6 protons but 6, 7 and 8 neutrons, and the nucleus of carbon-12 has a radius of about 2.8 fm, which is roughly 20,000 times smaller than the atom.

Nuclear chemistry begins with the structure of the nucleus: a tiny, extremely dense core that holds most of the mass of the atom. The number of protons decides which element it is, while the number of neutrons decides which isotope and whether the nucleus is stable. This article defines the terms, explains the notation, gives a table of important nuclides, calculates the size and density of nuclei and the average atomic mass from isotope data.

What is in the nucleus?

The protons and neutrons are called nucleons. The atomic number Z is the number of protons and defines the element. The neutron number N is the number of neutrons. The mass number A = Z + N is the total number of nucleons, and is a whole number, in contrast to the atomic mass, which is an average mass in unified atomic mass units (1 u = 1.6605 × 10⁻²⁷ kg, one twelfth of the mass of a carbon-12 atom). A neutral atom has Z electrons.

What is the nuclide notation?

A nuclide is written as ^A_ZX, where X is the element symbol, for example ²³⁵₉₂U. The atomic number is often left out because the symbol tells it, giving ²³⁵U, or “uranium-235”. Special names exist for the hydrogen isotopes: protium (¹H), deuterium (²H or D) and tritium (³H or T). Nuclear equations are balanced in both A and Z: the sum of the mass numbers and the sum of the atomic numbers must be the same on both sides.

Important nuclides and isotopes

What are isotopes, isobars and isotones?

• Isotopes: the same Z, different N and A (¹²C and ¹⁴C). They have the same chemical properties, since the electrons are the same, but they differ in mass and in nuclear properties; small differences in reaction rates (isotope effects) appear mainly for hydrogen and deuterium.
• Isobars: the same A, different Z (¹⁴C and ¹⁴N).
• Isotones: the same N, different Z (¹³C and ¹⁴N, both with 7 neutrons).
• Isomers: the same nuclide in different energy states, such as ^99mTc (the “m” means metastable) and ⁹⁹Tc.

About 250 stable nuclides are known for 80 elements, and in addition a few thousand radioactive nuclides. Tin has the most stable isotopes (ten). The elements with only one stable isotope include beryllium, fluorine, sodium, aluminum and phosphorus.

How big and dense is a nucleus?

The radius of a nucleus is approximately r = r₀A^1/3, with r₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm (femtometers). The volume is thus proportional to A, which means that the density is about the same for all nuclei. The nuclear density is ρ = (A × 1.66 × 10⁻²⁷ kg) / ((4/3)π(1.2 × 10⁻¹⁵ m)³A) = 1.66 × 10⁻²⁷ / 7.24 × 10⁻⁴⁵ = 2.3 × 10¹⁷ kg/m³. A thimbleful (1 cm³) of nuclear matter would have a mass of about 2.3 × 10¹¹ kg, 230 million tonnes. Neutron stars have this density.

How is the atomic mass calculated from isotopes?

The atomic mass listed in the periodic table is the average of the isotope masses weighted by their natural abundance: M = Σ (fraction × mass of isotope). Isotope abundances and masses are measured with a mass spectrometer.

Worked examples

Example 1: counting particles

How many protons, neutrons and electrons are in a neutral atom of ⁶⁵Zn? Z = 30 for zinc, so 30 protons and 30 electrons. N = A − Z = 65 − 30 = 35 neutrons.

Example 2: the average atomic mass of chlorine

Chlorine has two stable isotopes: ³⁵Cl (34.969 u, 75.77 %) and ³⁷Cl (36.966 u, 24.23 %). M = 0.7577 × 34.969 + 0.2423 × 36.966 = 26.496 + 8.957 = 35.453 u. The mass spectrum of Cl₂ accordingly has peaks at m/z 70, 72 and 74 in a ratio of about 9 : 6 : 1.

Example 3: the radius of a nucleus

For ¹²C: r = 1.2 × 12^1/3 = 1.2 × 2.289 = 2.75 fm. For ²³⁸U: r = 1.2 × 238^1/3 = 1.2 × 6.197 = 7.44 fm. The radius of a carbon atom is about 70 pm = 70,000 fm, so the nucleus occupies only about (2.75/70,000)³ ≈ 6 × 10⁻¹⁴ of the volume of the atom.

Example 4: writing the notation

A nuclide has 27 protons and 33 neutrons. Z = 27 (cobalt) and A = 60, so it is ⁶⁰₂₇Co, cobalt-60, which is a radioactive source used in medicine.

Example 5: finding the abundance

Boron has two isotopes, ¹⁰B (10.013 u) and ¹¹B (11.009 u), and an average atomic mass of 10.81 u. Let x be the fraction of ¹⁰B: 10.013x + 11.009(1 − x) = 10.81, so 11.009 − 0.996x = 10.81, x = 0.200. The abundance of ¹⁰B is 20.0 % and of ¹¹B 80.0 %.

How was the nucleus discovered?

In 1909, Hans Geiger and Ernest Marsden, working in Ernest Rutherford’s laboratory, found that a small fraction of alpha particles fired at a gold foil were deflected through large angles. Rutherford concluded in 1911 that the positive charge and most of the mass of an atom are concentrated in a tiny nucleus. Frederick Soddy introduced the concept of isotopes in 1913 to explain the many radioactive “elements” that were chemically identical, and J. J. Thomson found isotopes of neon in the same year, using positive-ray analysis. Francis Aston built the mass spectrograph in 1919 and showed that most elements are mixtures of isotopes (Nobel Prize in Chemistry, 1922). James Chadwick discovered the neutron in 1932, and received the Nobel Prize in Physics in 1935.

Which mistakes are common?

• Confusing mass number and atomic mass: A is a whole number, and atomic mass is an average in u.
• Thinking that isotopes have different chemistry: they behave the same, except for small mass effects.
• Counting electrons for ions as for atoms: the number of electrons depends on the charge.
• Forgetting to balance both A and Z: both must be conserved in nuclear equations.
• Calling the atom mostly empty space and then forgetting the electrons’ role: the volume of an atom is set by the electron cloud.