Atomic Masses and Composition of the Nucleus

The Atomic Mass Unit

Atomic masses are far too small for the kilogram to be a comfortable unit. The mass of a carbon-12 atom, for instance, is $1.992647 \times 10^{-26}\ \text{kg}$ — a number that is awkward to carry through calculations. To handle such quantities cleanly, physicists adopt the atomic mass unit, written $\mathrm{u}$, defined as exactly one-twelfth of the mass of a single $^{12}\mathrm{C}$ atom.

With that definition, dividing the carbon-12 mass by twelve gives the value of one unit:

$$1\,\mathrm{u} = \frac{\text{mass of one } ^{12}\mathrm{C}\text{ atom}}{12} = \frac{1.992647 \times 10^{-26}\ \text{kg}}{12} = 1.660539 \times 10^{-27}\ \text{kg}$$

Because mass and energy are equivalent through $E = mc^2$, the atomic mass unit also has an energy equivalent that recurs constantly in the rest of the chapter. Multiplying $1\,\mathrm{u}$ by $c^2$ and converting joules to electron-volts gives the standard result $1\,\mathrm{u} = 931.5\ \text{MeV}/c^2$, which NCERT lists among the data for the chapter's exercises.

Accurate atomic masses are measured with a mass spectrometer, an instrument that sorts ionised atoms by their mass-to-charge ratio. It was precisely these measurements that revealed something the simple "whole-number" picture cannot explain: atoms of one and the same element can differ in mass. The numbers given in $\mathrm{u}$ turn out to be close to integral multiples of the hydrogen-atom mass, but with striking exceptions — the atomic mass of chlorine is $35.46\,\mathrm{u}$, nowhere near a whole number. Resolving that puzzle requires the idea of isotopes, taken up below.

Protons, Neutrons and Nucleons

The nucleus is built from two kinds of particle. The proton carries one unit of positive fundamental charge and is stable; its mass is $m_p = 1.00727\,\mathrm{u} = 1.67262 \times 10^{-27}\ \text{kg}$. This equals the mass of the hydrogen atom ($1.00783\,\mathrm{u}$) minus the mass of one electron ($m_e = 0.00055\,\mathrm{u}$), which is consistent with the lightest hydrogen nucleus being a single proton.

The neutron is electrically neutral, with mass $m_n = 1.00866\,\mathrm{u} = 1.6749 \times 10^{-27}\ \text{kg}$ — very slightly heavier than the proton. It was discovered in 1932 by James Chadwick, who bombarded beryllium with alpha-particles and observed a neutral radiation that could knock protons out of light nuclei. Conservation of energy and momentum showed the radiation could not be photons; instead it consisted of neutral particles whose mass was very nearly that of the proton. Chadwick received the 1935 Nobel Prize in Physics for the discovery.

A proton or a neutron is generically called a nucleon. The total number of nucleons therefore equals the mass number of the atom.

Z, N and A — The Composition Numbers

Three integers fix the composition of any nucleus completely. They are defined as follows:

Since the atom is neutral and its electrons carry a total charge $-Ze$, the nucleus must carry $+Ze$, and the number of protons is therefore exactly $Z$. The mass number $A$, being the nucleon count, is the integer closest to the atomic mass in $\mathrm{u}$. For most nuclei $N > Z$, and the excess $(N - Z)$ grows as $A$ increases — heavier nuclei need proportionally more neutrons to remain bound.

Nuclide Notation

A specific nuclear species — a nuclide — is written by placing the mass number $A$ as a superscript and the atomic number $Z$ as a subscript, both to the left of the chemical symbol $\mathrm{X}$:

$$^{A}_{Z}\mathrm{X}$$

For example, gold is denoted $^{197}_{79}\mathrm{Au}$: it contains 197 nucleons, of which 79 are protons and the remaining $197 - 79 = 118$ are neutrons. The notation is partly redundant, since the symbol $\mathrm{X}$ already fixes $Z$, but writing both numbers makes nucleon bookkeeping in reactions transparent.

Isotopes, Isobars and Isotones

The mass spectrometer revealed atoms of the same element that share chemical properties but differ in mass — these are isotopes (Greek for "same place," since they sit at one location in the periodic table). Isotopes of an element have the same number of protons but differ in the number of neutrons. Deuterium $^{2}_{1}\mathrm{H}$ holds one proton and one neutron; tritium $^{3}_{1}\mathrm{H}$ holds one proton and two neutrons. Because isotopes have identical electronic structure, they are chemically indistinguishable.

Two further families group nuclides by their other shared numbers. Nuclides with the same mass number $A$ but different $Z$ are isobars; nuclides with the same neutron number $N$ but different $Z$ are isotones.

The isotone pair above shares $N$: mercury has $198 - 80 = 118$ neutrons and gold has $197 - 79 = 118$ neutrons. NIOS adds further examples — argon ($A=40$, $Z=18$) is an isobar of calcium ($A=40$, $Z=20$), while sodium $^{23}_{11}\mathrm{Na}$ is an isotone of magnesium $^{24}_{12}\mathrm{Mg}$ (each has 12 neutrons).

Atomic Mass as a Weighted Average

Isotopes explain why tabulated atomic masses are usually not whole numbers. Practically every element is a mixture of isotopes in fixed relative abundances, so the atomic mass quoted in the periodic table is a weighted average of the isotopic masses.

Chlorine is the textbook case. Its two isotopes have masses $34.98\,\mathrm{u}$ and $36.98\,\mathrm{u}$, with relative abundances of $75.4\%$ and $24.6\%$. The weighted average is:

$$\bar{m} = \frac{75.4 \times 34.98 + 24.6 \times 36.98}{100} = 35.47\,\mathrm{u}$$

This agrees with the measured atomic mass of chlorine, $35.46\,\mathrm{u}$ — so the non-integer value is simply the signature of an isotopic mixture, not a contradiction of the whole-number rule. Even hydrogen, the lightest element, has three isotopes of masses $1.0078\,\mathrm{u}$, $2.0141\,\mathrm{u}$ and $3.0160\,\mathrm{u}$, the lightest (the proton) having a relative abundance of $99.985\%$.