13.8 Summary

• An atom has a nucleus. The nucleus is positively charged. The radius of the nucleus is smaller than the radius of an atom by a factor of \(10^4\). More than \(99.9 \%\) mass of the atom is concentrated in the nucleus.
• On the atomic scale, mass is measured in atomic mass units (u). By definition, 1 atomic mass unit ( 1 u ) is \(1 / 12^{\text {th }}\) mass of one atom of \({ }^{12} \mathrm{C}\); \(1\mathrm{~u}=1.660563 \times 10^{-27} \mathrm{~kg}\).
• A nucleus contains a neutral particle called neutron. Its mass is almost the same as that of proton.
• The atomic number \(Z\) is the number of protons in the atomic nucleus of an element. The mass number \(A\) is the total number of protons and neutrons in the atomic nucleus; \(A=Z+N\); Here \(N\) denotes the number of neutrons in the nucleus.
  A nuclear species or a nuclide is represented as \({ }_Z^A \mathrm{X}\), where X is the chemical symbol of the species.
  Nuclides with the same atomic number \(Z\), but different neutron number \(N\) are called isotopes. Nuclides with the same \(A\) are isobars and those with the same \(N\) are isotones.
  Most elements are mixtures of two or more isotopes. The atomic mass of an element is a weighted average of the masses of its isotopes and calculated in accordance to the relative abundances of the isotopes.
• A nucleus can be considered to be spherical in shape and assigned a radius. Electron scattering experiments allow the determination of the nuclear radius; it is found that radii of nuclei fit the formula
  \(
  R=R_0 A^{1 / 3}
  \)
  where \(R_0=\) a constant \(=1.2 \mathrm{fm}\). This implies that the nuclear density is independent of \(A\). It is of the order of \(10^{17} \mathrm{~kg} / \mathrm{m}^3\).
• Neutrons and protons are bound in a nucleus by the short-range strong nuclear force. The nuclear force does not distinguish between neutron and proton.
• The nuclear mass \(M\) is always less than the total mass, \(\Sigma m\), of its constituents. The difference in mass of a nucleus and its constituents is called the mass defect,
  \(
  \Delta M=\left(Z m_p+(A-Z) m_n\right)-M
  \)
  Using Einstein’s mass energy relation, we express this mass difference in terms of energy as
  \(
  \Delta E_b=\Delta M c^2
  \)
  The energy \(\Delta E_b\) represents the binding energy of the nucleus. In the mass number range \(A=30\) to 170 , the binding energy per nucleon is nearly constant, about \(8 \mathrm{MeV} /\) nucleon.
• Energies associated with nuclear processes are about a million times larger than chemical process.
• The \(Q\)-value of a nuclear process is
  \(
  Q=\text { final kinetic energy – initial kinetic energy. }
  \)
  Due to conservation of mass-energy, this is also,
  \(
  Q=(\text { sum of initial masses }- \text { sum of final masses }) c^2
  \)
• Radioactivity is the phenomenon in which nuclei of a given species transform by giving out \(\alpha\) or \(\beta\) or \(\gamma\) rays; \(\alpha\)-rays are helium nuclei; \(\beta\)-rays are electrons. \(\gamma\)-rays are electromagnetic radiation of wavelengths shorter than \(X\)-rays.
• Energy is released when less tightly bound nuclei are transmuted into more tightly bound nuclei. In fission, a heavy nucleus like \({ }_{92}^{235} \mathrm{U}\) breaks into two smaller fragments, e.g., \({ }_{92}^{235} \mathrm{U}+{ }_0^1 \mathrm{n} \rightarrow{ }_{51}^{133} \mathrm{Sb}+{ }_{41}^{99} \mathrm{Nb}+4{ }_0^1 \mathrm{n}\)
• In fusion, lighter nuclei combine to form a larger nucleus. Fusion of hydrogen nuclei into helium nuclei is the source of energy of all stars including our sun.

POINTS TO PONDER

• The density of nuclear matter is independent of the size of the nucleus. The mass density of the atom does not follow this rule.
• The radius of a nucleus determined by electron scattering is found to be slightly different from that determined by alpha-particle scattering. This is because electron scattering senses the charge distribution of the nucleus, whereas alpha and similar particles sense the nuclear matter.
• After Einstein showed the equivalence of mass and energy, \(E=m c^2\), we cannot any longer speak of separate laws of conservation of mass and conservation of energy, but we have to speak of a unified law of conservation of mass and energy. The most convincing evidence that this principle operates in nature comes from nuclear physics. It is central to our understanding of nuclear energy and harnessing it as a source of power. Using the principle, \(Q\) of a nuclear process (decay or reaction) can be expressed also in terms of initial and final masses.
• The nature of the binding energy (per nucleon) curve shows that exothermic nuclear reactions are possible when two light nuclei fuse or when a heavy nucleus undergoes fission into nuclei with intermediate mass.
• For fusion, the light nuclei must have sufficient initial energy to overcome the coulomb potential barrier. That is why fusion requires very high temperatures.
• Although the binding energy (per nucleon) curve is smooth and slowly varying, it shows peaks at nuclides like \({ }^4 \mathrm{He},{ }^{16} \mathrm{O}\), etc. This is considered as evidence of atom-like shell structure in nuclei.
• Electrons and positron are a particle-antiparticle pair. They are identical in mass; their charges are equal in magnitude and opposite. (It is found that when an electron and a positron come together, they annihilate each other giving energy in the form of gamma-ray photons.)
• Radioactivity is an indication of the instability of nuclei. Stability requires the ratio of neutron to proton to be around \(1: 1\) for light nuclei. This ratio increases to about \(3: 2\) for heavy nuclei. (More neutrons are required to overcome the effect of repulsion among the protons.) Nuclei which are away from the stability ratio, i.e., nuclei which have an excess of neutrons or protons are unstable. In fact, only about \(10 \%\) of knon isotopes (of all elements), are stable. Others have been either artificially produced in the laboratory by bombarding \(\alpha, \mathrm{p}, \mathrm{d}, \mathrm{n}\) or other particles on targets of stable nuclear species or identified in astronomical observations of matter in the universe.