Nuclei: NEET notes

1. Atomic mass unit and isotopes

NCERT §13.1, §13.2

• A nucleus is about 10⁴ times smaller in radius than its atom, so its volume is about 10⁻¹² of the atom's. Blow an atom up to the size of a classroom and the nucleus would be a pinhead, yet it holds more than 99.9% of the atom's mass.
• A ¹²C atom has a mass of 1.992647 × 10⁻²⁶ kg, too tiny for the kilogram to be handy. Atomic masses are instead given in the atomic mass unit u, one-twelfth of the mass of a ¹²C atom: 1 u = 1.660539 × 10⁻²⁷ kg (Eq. 13.1).
• Atomic masses in u come out close to whole-number multiples of the hydrogen atom's mass, but with notable exceptions: chlorine's atomic mass is 35.46 u.
• A mass spectrometer measures atomic masses precisely. It revealed isotopes: atoms of one element with the same chemistry but different masses. The Greek word means 'same place', since isotopes sit in one place in the periodic table. Nearly every element is a mixture of isotopes.
• Chlorine is a mix of two isotopes: 34.98 u, making up 75.4%, and 36.98 u, making up 24.6%. Weighting each mass by its share, 0.754 × 34.98 + 0.246 × 36.98 ≈ 35.47 u, which is chlorine's atomic mass.
• Hydrogen has three isotopes. The lightest, of mass 1.0078 u, makes up 99.985% of hydrogen; the others weigh 2.0141 u and 3.0160 u. Removing one electron (0.00055 u) from a light hydrogen atom (1.00783 u) leaves its nucleus, the proton, of mass 1.00727 u, or 1.67262 × 10⁻²⁷ kg (Eq. 13.2).
• The heavier two are deuterium and tritium. Tritium nuclei are unstable, so tritium is not found in nature and is made artificially.

2. Protons, neutrons and nuclides

NCERT §13.2

• The proton carries one unit of positive charge and is stable. Quantum arguments rule out electrons inside the nucleus; all Z electrons of a neutral atom are outside it. So the nucleus has charge +Ze and contains exactly Z protons.
• Deuterium and tritium each have one proton, yet the nuclear masses of hydrogen, deuterium and tritium are in the ratio 1 : 2 : 3. The extra mass must be neutral matter, about one and two proton masses respectively.
• In 1932 Chadwick bombarded beryllium with alpha-particles and found a neutral radiation that could knock protons out of helium, carbon and nitrogen. If it were photons, energy and momentum conservation would demand far more energy than the reaction could supply.
• Chadwick resolved this by proposing a new neutral particle, the neutron, whose mass he found to be very nearly the proton's. Today mn = 1.00866 u = 1.6749 × 10⁻²⁷ kg (Eq. 13.3). He received the Nobel Prize in Physics in 1935.
• Left on its own, a neutron is unstable and turns into a proton, an electron and an antineutrino; its mean life is about 1000 s. Bound in a nucleus, it does not decay.
• Z (atomic number) is the number of protons, N (neutron number) the number of neutrons, and A = Z + N (mass number) the total number of nucleons, a nucleon being a proton or a neutron.
• A nuclide is written ᴬ_ZX, with X the chemical symbol. Gold, ¹⁹⁷₇₉Au, has 197 nucleons: 79 protons and 118 neutrons.
• Isotopes have equal Z but different N: deuterium ²₁H has one proton and one neutron, tritium ³₁H one proton and two neutrons. Gold has 32 isotopes, from A = 173 to A = 204. Their electron structure is the same, so their chemistry is too.
• Isobars share the mass number A, for example ³₁H and ³₂He. Isotones share the neutron number N but differ in Z, for example ¹⁹⁸₈₀Hg and ¹⁹⁷₇₉Au.

3. Size of the nucleus

NCERT §13.3

• Geiger and Marsden's alpha-particles of 5.5 MeV came no closer than about 4.0 × 10⁻¹⁴ m to a gold nucleus. Pure Coulomb repulsion explained the scattering, so the nucleus must be smaller than that.
• Faster alpha-particles come closer. At some energy the short-range nuclear force starts to act and the scattering departs from Rutherford's pure-Coulomb prediction; where that happens gives the nuclear size.
• Nuclear sizes of many elements have been measured accurately by scattering fast electrons, instead of alpha-particles, off targets.
• Radii from electron scattering differ slightly from those from alpha-particle scattering: electrons probe how the nuclear charge is spread, while alpha-particles and similar projectiles probe the nuclear matter itself.
• A nucleus of mass number A has radius R = R₀A^(1/3) (Eq. 13.5), with R₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm (1 fm = 10⁻¹⁵ m).
• Volume goes as R³, so it is proportional to A: the density of nuclear matter is the same for all nuclei, like drops of one liquid. It is about 2.3 × 10¹⁷ kg m⁻³, against 10³ kg m⁻³ for water, because ordinary matter is mostly empty space.
• Example 13.1: an iron nucleus of mass 55.85 u (9.27 × 10⁻²⁶ kg) and A = 56 has density 9.27 × 10⁻²⁶/[56 × (4π/3)(1.2 × 10⁻¹⁵)³] = 2.29 × 10¹⁷ kg m⁻³.
• Matter in neutron stars has a density comparable to this: it is squeezed so hard that the star resembles one huge nucleus.

4. Mass–energy equivalence

NCERT §13.4.1

• Before special relativity, mass and energy were each assumed to be conserved on their own in a reaction. Einstein showed that mass is itself a form of energy, and can be converted into other forms, such as kinetic energy, and back.
• The mass–energy relation is E = mc² (Eq. 13.6), with c ≈ 3 × 10⁸ m s⁻¹ the speed of light in vacuum.
• Example 13.2: the energy equivalent of 1 g is 10⁻³ × (3 × 10⁸)² = 9 × 10¹³ J, an enormous amount for one gram.
• The relation has been verified in reactions between nucleons, nuclei, electrons and other particles. Energy is conserved in a reaction only when the energy tied up in mass is counted too, so there is one combined law of conservation of mass and energy.

5. Mass defect and binding energy

NCERT §13.4.2

• A nucleus always weighs less than the sum of its separate protons and neutrons. Take ¹⁶₈O, with 8 protons and 8 neutrons: 8 × (1.00727 + 1.00866) u = 8 × 2.01593 u = 16.12744 u.
• Mass spectroscopy gives the ¹⁶O atom 15.99493 u. Taking off 8 electrons (8 × 0.00055 u) leaves the nucleus at 15.99053 u, which is 0.13691 u less than its parts.
• This shortfall is the mass defect: ΔM = [Zmp + (A − Z)mn] − M (Eq. 13.7).
• Because mass is energy, the bound nucleus has less energy than its free nucleons. To pull the nucleus apart into free protons and neutrons, the energy ΔMc² must be supplied: this is the binding energy, Eb = ΔMc² (Eq. 13.8). The same energy is released when the nucleons come together.
• Example 13.3: 1 u = 1.6605 × 10⁻²⁷ kg, and multiplying by c² = (2.9979 × 10⁸)² gives 1.4924 × 10⁻¹⁰ J = 931.5 MeV. So 1 u = 931.5 MeV/c².
• For ¹⁶O, ΔM = 0.13691 u = 0.13691 × 931.5 MeV/c² = 127.5 MeV/c², so 127.5 MeV is needed to separate it into its nucleons. (NCERT writes this energy with the unit MeV/c²; as an energy it is 127.5 MeV.)
• A nucleus cannot actually be torn apart like this, but Eb is still a handy measure of how firmly it holds together. A better measure is the binding energy per nucleon, Ebn = Eb/A (Eq. 13.9): the average energy per nucleon needed to take the nucleus apart. For ¹⁶O it is 127.5/16 ≈ 8.0 MeV.

6. Binding energy curve

NCERT §13.4.2

• Fig. 13.1 plots the binding energy per nucleon Ebn against the mass number A for many nuclei.
• For middle-mass nuclei, 30 < A < 170, Ebn hardly changes with A. The curve peaks at about 8.75 MeV for A = 56 and falls to 7.6 MeV at A = 238.
• Ebn is lower for light nuclei (A < 30) and for heavy nuclei (A > 170).
• Conclusion (i): the force binding nucleons is attractive and strong enough to give a few MeV per nucleon.
• Conclusion (ii): the flat middle means the force is short-ranged. A nucleon deep inside a big nucleus feels only the neighbours within range. If it can have at most p of them, its binding is about pk, k being a constant energy. Adding more nucleons far away does not change it, and most nucleons of a large nucleus are inside, not on the surface.
• This is the saturation property of the nuclear force: each nucleon affects only nucleons close to it.
• Conclusion (iii): a nucleus of A = 240 has lower Ebn than one of A = 120. If it splits into two A = 120 nuclei, the nucleons end up more tightly bound and energy is released: fission.
• Conclusion (iv): two very light nuclei (A ≤ 10) joining into a heavier one also end up with higher Ebn, again releasing energy: fusion, the energy source of the sun.
• The curve is smooth, but it shows peaks at nuclides such as ⁴He and ¹⁶O, which is taken as evidence of an atom-like shell structure in nuclei.

7. Nuclear force

NCERT §13.5

• Electrons in atoms move under the Coulomb force. Nuclei of average mass have roughly 8 MeV of binding for every nucleon, much more than any atomic binding energy, so something other than electric attraction must hold a nucleus together: a strong attraction of an entirely new kind.
• It has to beat the repulsion between the protons and hold protons and neutrons inside the tiny nuclear volume. Its properties were worked out from many experiments between 1930 and 1950.
• (i) It is far stronger than the electric repulsion among the protons, which it has to overpower to keep the nucleus together. Gravity is weaker again than the electric force.
• (ii) Between two nucleons the force drops rapidly to zero once they are more than a few femtometres apart. This short range gives saturation in medium and large nuclei, and so the near-constant binding energy per nucleon.
• Fig. 13.2 roughly plots the potential energy of two nucleons against separation. It is lowest at r₀ ≈ 0.8 fm: beyond 0.8 fm the force is attractive, and closer than 0.8 fm it is strongly repulsive.
• (iii) The force between neutron and neutron, proton and neutron, and proton and proton is roughly the same: the nuclear force does not depend on electric charge.
• Unlike Coulomb's law or Newton's law of gravitation, the nuclear force has no simple mathematical formula.

8. Radioactivity

NCERT §13.6

• Becquerel discovered radioactivity by accident in 1896. He lit pieces of uranium-potassium sulphate with visible light, wrapped them in black paper and put a piece of silver between the package and a photographic plate.
• After several hours the developed plate was blackened: the compound had given off something that passed through the black paper and the silver.
• Later experiments showed radioactivity to be a nuclear process: an unstable nucleus decays. This is radioactive decay.
• Three kinds occur in nature: α-decay, in which a helium nucleus ⁴₂He is emitted; β-decay, in which electrons or positrons are emitted; and γ-decay, in which high-energy photons (hundreds of keV or more) are emitted.
• A positron has the same mass as an electron and an exactly opposite charge; the two are a particle–antiparticle pair. When an electron and a positron meet they annihilate, giving gamma-ray photons.
• Radioactivity signals an unstable nucleus. Stable light nuclei have neutrons and protons roughly 1 : 1; for heavy nuclei the ratio rises to about 3 : 2, as extra neutrons offset the repulsion between protons. Nuclei far from these ratios are unstable.
• Only about 10% of known isotopes are stable. The rest have been made in laboratories, by bombarding stable nuclei with α, p, d, n or other particles, or identified in astronomical observations.

9. Nuclear fission

NCERT §13.7, §13.7.1

• The Ebn curve is flat at about 8.0 MeV for 30 < A < 170 and lower on either side. The more tightly bound a system, the less its total mass, so turning less tightly bound nuclei into more tightly bound ones releases energy: in fission of a heavy nucleus or fusion of light ones.
• Coal and petroleum rely on chemical reactions involving energies of electron volts; nuclear reactions involve MeV. So, for the same amount of matter, nuclear sources give about a million times more energy: fission of 1 kg of uranium gives about 10¹⁴ J, burning 1 kg of coal about 10⁷ J.
• Bombarding nuclei with particles such as protons, neutrons or alpha-particles opens up reactions beyond natural decay. The most important neutron-induced one is fission.
• A neutron striking ²³⁵₉₂U forms ²³⁶₉₂U, which splits into two intermediate-mass fragments, for example ¹⁴⁴₅₆Ba + ⁸⁹₃₆Kr + 3 neutrons (Eq. 13.10), or ¹³³₅₁Sb + ⁹⁹₄₁Nb + 4 neutrons (Eq. 13.11). Another outcome is ¹⁴⁰₅₄Xe + ⁹⁴₃₈Sr + 2 neutrons (Eq. 13.12).
• The fragments are radioactive and emit β particles one after another until they reach stable end products.
• The energy released (Q value) is about 200 MeV per fission. Estimate: a nucleus of A = 240 (Ebn ≈ 7.6 MeV) splits into two of A = 120 (Ebn ≈ 8.5 MeV), a gain of about 0.9 MeV per nucleon, so 240 × 0.9 = 216 MeV in all.
• This energy appears first as kinetic energy of the fragments and neutrons, and ends up as heat in the surroundings. Nuclear reactors that make electricity run on fission; uncontrolled fission is the source of an atom bomb's energy.
• In a nuclear reaction the numbers of protons and of neutrons are each conserved (Example 13.4). The energy comes from the difference in total binding energy between the two sides, which shows up as a difference in mass.

10. Nuclear fusion in stars

NCERT §13.7.2, §13.7.3

• When two light nuclei fuse into a larger, more tightly bound nucleus, energy is released. Examples: ¹₁H + ¹₁H → ²₁H + e⁺ + ν + 0.42 MeV; ²₁H + ²₁H → ³₂He + n + 3.27 MeV; ²₁H + ²₁H → ³₁H + ¹₁H + 4.03 MeV (Eq. 13.13).
• To fuse, the nuclei must come close enough for the short-range nuclear force to act, but both are positive and repel. They need enough energy to get over this Coulomb barrier, whose height depends on their charges and radii: about 400 keV for two protons, and more for more highly charged nuclei.
• For the average proton in a gas to have that energy, (3/2)kT = 400 keV, which needs T ≈ 3 × 10⁹ K. Fusion driven by high temperature is called thermonuclear fusion, and it powers the interior of stars.
• The sun's core is at 1.5 × 10⁷ K, well below that estimate, so the fusion there involves protons whose energies are far above average.
• The sun burns hydrogen into helium through the proton–proton cycle: ¹H + ¹H → ²H + e⁺ + ν + 0.42 MeV; e⁺ + e⁻ → γ + γ + 1.02 MeV; ²H + ¹H → ³He + γ + 5.49 MeV; ³He + ³He → ⁴He + ¹H + ¹H + 12.86 MeV (Eq. 13.14).
• The first three steps must happen twice for the fourth to occur. The net result is that four hydrogen atoms make one ⁴He atom and release 26.7 MeV (Eq. 13.15).
• When the core's hydrogen has turned to helium, the core cools, the star contracts under gravity and heats up. At about 10⁸ K helium fuses into carbon, and so on to heavier elements, but not beyond those near the peak of the Ebn curve.
• The sun is about 5 × 10⁹ years old and has hydrogen for about another 5 billion years. Then hydrogen burning stops, the core collapses and heats, and the outer envelope swells: the sun becomes a red giant.
• Controlled fusion reactors aim to heat fuel to about 10⁸ K, where it is a plasma of positive ions and electrons. No container can withstand such heat, so the challenge is confining the plasma. Several countries, India among them, are developing ways to do it.
• Chemical reactions also convert mass to energy, since chemical binding also lowers mass, but the mass changes are about a million times smaller than in nuclear reactions (Example 13.4).

Must-know facts

1. 1 u = 1/12 of the mass of a ¹²C atom = 1.660539 × 10⁻²⁷ kg; its energy equivalent is 931.5 MeV.
2. Chlorine: isotopes 34.98 u (75.4%) and 36.98 u (24.6%), average 35.47 u.
3. mp = 1.00727 u = 1.67262 × 10⁻²⁷ kg; mn = 1.00866 u = 1.6749 × 10⁻²⁷ kg; me = 0.00055 u.
4. Chadwick found the neutron in 1932 (Nobel Prize 1935); a free neutron decays with a mean life of about 1000 s.
5. Z protons, N neutrons, A = Z + N nucleons; ¹⁹⁷₇₉Au has 79 protons and 118 neutrons.
6. Isotopes: same Z. Isobars: same A (³₁H, ³₂He). Isotones: same N (¹⁹⁸₈₀Hg, ¹⁹⁷₇₉Au).
7. R = R₀A^(1/3), R₀ = 1.2 fm, so volume ∝ A and nuclear density is the same for all nuclei, about 2.3 × 10¹⁷ kg m⁻³.
8. E = mc²: 1 g of matter is equivalent to 9 × 10¹³ J.
9. Mass defect ΔM = [Zmp + (A − Z)mn] − M; binding energy Eb = ΔMc².
10. ¹⁶O: ΔM = 0.13691 u, Eb = 127.5 MeV.
11. Ebn is nearly constant (about 8 MeV) for 30 < A < 170, peaks at about 8.75 MeV at A = 56 and is 7.6 MeV at A = 238.
12. Nuclear force: attractive, much stronger than the Coulomb force, short-ranged (a few fm), repulsive below about 0.8 fm, charge-independent.
13. Becquerel discovered radioactivity in 1896; the decays are α (⁴₂He), β (electrons or positrons) and γ (photons).
14. Fission of ²³⁵U by a neutron releases about 200 MeV per nucleus; 1 kg of uranium gives about 10¹⁴ J against 10⁷ J from 1 kg of coal.
15. Two protons face a Coulomb barrier of about 400 keV; the sun's core is at 1.5 × 10⁷ K.
16. Proton–proton cycle: four hydrogen atoms make one helium atom and release 26.7 MeV.

Common traps

Formulas

Key terms

Atomic mass unit

One-twelfth of the mass of a carbon-12 atom, 1.660539 × 10⁻²⁷ kg.

Isotopes

Nuclides of one element: same Z, different N and A.

Isobars

Nuclides with the same mass number A.

Isotones

Nuclides with the same neutron number N but different Z.

Nucleon

A proton or a neutron.

Nuclide

A nuclear species, written ᴬ_ZX.

Mass defect

How much less a nucleus weighs than its separate protons and neutrons.

Binding energy

Energy needed to pull a nucleus fully apart into free nucleons, Eb = ΔMc².

Saturation

A nucleon interacts only with its close neighbours, so Ebn stays nearly constant.

Fission

A heavy nucleus splitting into two intermediate-mass fragments.

Fusion

Light nuclei joining into a heavier, more tightly bound nucleus.

Coulomb barrier

The electric repulsion two nuclei must overcome to come within reach of the nuclear force.

Thermonuclear fusion

Fusion driven by raising the temperature so that nuclei move fast enough to overcome the barrier.