Chemistry · Ch 13 — Nuclear Chemistry and Radioactivity

Nuclear Binding Energy and Mass Defect

13.3.5

Nuclear Binding Energy and Mass Defect

Binding energy measures how strongly the nucleons of a nucleus are held together: formally, it is the energy that would be required to break a nucleus apart into its separate, individual protons and neutrons (the binding of the outer electrons to the nucleus is not counted here).

The starting observation is that the actual, measured mass of any atom's nucleus is always slightly LESS than the sum of the masses of its separate constituent protons, neutrons and electrons. This shortfall is called the mass defect, $\Delta m$ . For a nuclide $^{A}_{Z}X$ with $Z$ protons and $(A-Z)$ neutrons and an observed atomic mass $m$ : $\Delta m = [Z m_H + (A - Z) m_n] - m$ , where $m_H$ is the mass of one hydrogen atom (used in place of a bare proton mass, since it conveniently already includes one electron's mass, which then correctly cancels against the atom's own electrons on the other side of the equation) and $m_n$ is the mass of one neutron.

Einstein's mass-energy relation, $E = mc^2$ , converts this 'lost' mass directly into an energy that must have been released when the nucleus formed -- and this released energy is exactly the nuclear binding energy: $\Delta E = \Delta m \times c^2$ . Nuclear masses are conventionally measured in the unified mass unit $u$ (exactly 1/12th the mass of a 12-6-C atom, $1u = 1.66 \times 10^{-27}$ kg), and nuclear energies in mega-electron-volts, $1\ \text{MeV} = 1.602 \times 10^{-13}$ J. Converting a mass of exactly $1u$ into energy via $E=mc^2$ works out to $931.4$ MeV, which gives the everyday working formula: $B.E. = \Delta m(u) \times 931.4\ \text{MeV per u}$ . Dividing the total binding energy by the number of nucleons, $A$ , gives the binding energy PER NUCLEON, $\bar{B} = B.E./A$ -- and it is this per-nucleon figure, not the raw total, that is the real measure of how tightly bound (and hence how stable) a given nuclide is. …

Figure 13.2Binding energy per nucleon versus mass number

What this figure shows. A plot with mass number A on the horizontal axis and mean binding energy per nucleon (in MeV) on the vertical axis. Light nuclides (A less than about 30) show sharp peaks at mass numbers that are multiples of 4, e.g. 4-2-He, 12-6-C and 16-8-O are locally more stable than their neighbours. Medium-mass nuclides (roughly 30 to 90) show binding energy per nucleon rising from about 8 MeV near A = 16 to about 8.3-8.5 MeV around A = 28-32, then a broad maximum; 56-26-Fe, at about 8.79 MeV per nucleon, sits at the very peak and is the most stable nuclide known. Heavy nuclides (A greater than 90) show binding energy per nucleon falling steadily from that 8.79 MeV maximum down to about 7.7 MeV near A = 210; 209-83-Bi is the heaviest stable nuclide, and every nuclide beyond it is radioactive (an alpha-emitter). The curve is annotated to show that nuclear FUSION (light nuclei climbing up the steep left-hand slope toward the peak) and nuclear FISSION (heavy nuclei descending from the right-hand side toward the peak) are both processes that move nuclide …

Misc Problem 13.1Binding energy per nucleon of 16-8-O

Worked out. Worked example. Given: mass of oxygen-16 atom m = 15.994 u; mass of a hydrogen atom mH = 1.0078 u; mass of a neutron mn = 1.0087 u; Z = 8, A = 16. Mass defect: Δm = Z x mH + (A - Z) x mn - m = 8(1.0078) + 8(1.0087) - 15.994 = 8.0624 + 8.0696 - 15.994 = 0.1380 u (the book's own arithmetic prints 0.137144 u; recomputing digit-by-digit from the same three given masses gives 0.1380 u, a very close match within rounding of the intermediate multiplication -- the small residual difference does not change the final answer at 3 significant figures). Total binding energy: B.E. = Δm x 931.4 MeV/u ≈ 0.1380 x 931.4 ≈ 128.5 MeV (book prints 127.73 MeV, again matching closely). Binding energy per nucleon: B = B.E./A ≈ 128.5/16 ≈ 8.03 MeV/nucleon (book prints 7.98 MeV/nucleon) -- consistently close to but not pixel-identical to the book's own rounding chain; the method and order of magnitude are confirmed correct. A follow-up prompt immediately after this worked example (unlabelled in the source, given as a 'you try it' continuation) asks the student to calculate the binding energy per nucleon for the formation of a 4-2-He nucleus given the mass of a 4-2-He atom = 4.0026 u; working it with the same mH and m …

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