VKC Notes · Nuclear chemistry · Geo-Scientist P-I

16.1 Nuclear Chemistry: Decay, Binding Energy and the GM Counter

★★★ 2021–2025: 23 parts, 200 marks, all in Geo-Scientist Paper I, which has set a full question on this unit every year. Numericals carry most of the marks.

Syllabus line. Geo-Scientist Paper I: “Radioactive decay: general characteristics, decay kinetics, parent-daughter decay growth relationships, determination of half-lives. Nuclear stability. Decay theories. Unit of radioactivity. Preparation of artificial radionuclides by bombardment, radiochemical separation techniques. Experimental techniques in the assay of radioisotopes, Geiger-Muller counters. Solid state detectors.” Not in the Civil Services or IFoS syllabus.

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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes

16.1.1 Nuclear equations: balance A and Z ★★★ every year

A nuclide is written : A = mass number (protons + neutrons), Z = atomic number. In every nuclear equation the sum of A and the sum of Z are each the same on both sides; the missing particle is whatever closes both sums.SN

Particle	Symbol	ΔA	ΔZ of nucleus
α particle		−4	−2
β⁻ (electron)		0	+1
β⁺ (positron) or electron capture		0	−1
neutron, proton, deuteron	, ,	as written
γ photon	γ	0	0

Solved PYQs Geo 2025 Q1(i) 5m Geo 2024 Q9(b) 5m Geo 2021 Q1(i) 5m

: 13 = 12 + 1, 6 = 6 + 0 (Chadwick's neutron, 1932).
: Rutherford's first transmutation (1919).
: first artificial radionuclide (Joliot-Curie, 1934).
; .

Also asked: Geo 2024 Q5(b) 10m Al-27 captures , , , 2α, · Geo 2023 Q11(c) 5m · Geo 2022 Q10(b) 5m decay products of U-238, Pa-234, C-14, Na-22 (β⁺), Po-210 (α).

16.1.2 Decay modes, displacement law and the four series ★★★

Group displacement law (Soddy, Fajans, Russell, 1913): α emission: element moves two places left (ΔZ = −2, ΔA = −4); β⁻ emission: one place right (ΔZ = +1, A unchanged); β⁺ or electron capture: one place left Geo 2021 Q10(b) 15m.SN
Which mode? Below the belt of stability (too many neutrons): β⁻. Above it (too many protons): β⁺ or electron capture. Z > 83: α. γ follows when the daughter is left in an excited state.SN
α decay: . The α particle, already formed inside, tunnels through the Coulomb barrier (Gamow, 1928). Geiger–Nuttall rule: log λ rises linearly with log R (range), so higher α energy means much shorter half-life Geo 2021 Q10(c) 10m.SN

Series	Starts at	Ends at (stable)	α, β steps
4n (thorium)			6 α, 4 β
4n + 1 (neptunium)			7 α, 4 β (artificial; extinct in nature)
4n + 2 (uranium)			8 α, 6 β
4n + 3 (actinium)			7 α, 4 β

Count the steps: number of α = ΔA/4; then number of β⁻ = 2 × (α) − ΔZ. For U-238 → Pb-206: α = 32/4 = 8, β = 16 − 10 = 6.SN

16.1.3 Stability: binding energy, magic numbers ★★★

(16.1.1)

Steep rise for A < 20 (surface nucleons have fewer neighbours); ⁴He, ¹²C, ¹⁶O sit above the trend. Nearly flat at about 8 MeV for A ≈ 30–200: the nuclear force is short-range and saturates. Slow fall beyond A ≈ 60: proton–proton repulsion grows as Z² Geo 2025 Q9(a) 15m.SN
Moving towards the maximum releases energy: light nuclei by fusion, heavy nuclei by fission. Fission of ²³⁵U: (8.5 − 7.6) MeV × 236 ≈ 200 MeV per nucleus.SN
Magic numbers 2, 8, 20, 28, 50, 82, 126: closed shells in the nuclear shell model (with spin–orbit coupling). has Z = 82 and N = 126: doubly magic, the most stable heavy nucleus and the end of the thorium series. Even–even nuclei are most numerous (about 160 stable), odd–odd fewest (only 4: ²H, ⁶Li, ¹⁰B, ¹⁴N) Geo 2025 Q4(c) 5m.SN

Fig. 16.1.1 Binding energy per nucleon against mass number (measured values).

Draw it in 60 s: axes A (0–240) and B/A (0–9 MeV); a sharp rise to He-4 (7.1), a broad top at Fe-56 (8.8), a gentle fall to U-238 (7.6); arrows: fusion up from the left, fission up from the right.

Solved PYQ Geo 2023 Q11(a) 15m

Packing fraction, mass defect and binding energy of (55.9375 u); m(¹H) = 1.007825 u, m_n = 1.008665 u.

Mass of free particles: 26 × 1.007825 + 30 × 1.008665 = 26.20345 + 30.25995 = 56.46340 u.
Mass defect Δm = 56.46340 − 55.9375 = 0.52590 u.
Binding energy B = 0.52590 × 931.5 = 489.9 MeV; per nucleon 489.9/56 = 8.75 MeV (near the top of Fig. 16.1.1).
Packing fraction = (55.9375 − 56)/56 × 10⁴ = −11.2; negative: a stable nucleus.SN

16.1.4 Decay kinetics, units, parent and daughter ★★★ numericals

(16.1.2)

Activity A = λN obeys the same law. Units: 1 Bq = 1 disintegration s⁻¹; 1 Ci = 3.7 × 10¹⁰ Bq (activity of 1 g ²²⁶Ra). The decay curve is the falling exponential; a straight line appears only on a ln A plot Geo 2024 Q1(i) 5m.SN
Quick form: A/A₀ = (½)^t/t½. 20 mCi → 1.25 mCi is a factor 16 = 2⁴: 4 × 1600 y = 6400 y (to exactly 1 mCi: 6915 y) Geo 2024 Q9(d) 5m. 74 % left with t½ = 250 y: t = (250/0.693) ln(1/0.74) = 108.6 y, less than one half-life as it must be Geo 2023 Q1(h) 5m.SN
Parent → daughter: dN₂/dt = λ₁N₁ − λ₂N₂. If the parent lives much longer (λ₁ ≪ λ₂), the daughter grows in until secular equilibrium: λ₁N₁ = λ₂N₂ (²²⁶Ra/²²²Rn); if only somewhat longer, transient equilibrium (activities in a fixed ratio). Long half-lives are found from the specific activity, short ones from the slope of ln A against t.

Fig. 16.1.2 First-order decay: equal fractions lost in equal times.

Draw it in 60 s: a falling exponential crossing 1/2, 1/4, 1/8 at 1, 2, 3 half-lives; beside it a small box with a straight line, ln A against t, slope −λ.

Solved PYQs Geo 2022 Q10(a) 15m Geo 2025 Q5(c) 10m

1.0 mol (6.022 × 10²³ atoms). t½ = 12 000 y: λ = 0.693/(12 000 × 3.156 × 10⁷ s) = 1.83 × 10⁻¹² s⁻¹, so 1.10 × 10¹² s⁻¹. t½ = 12 h: λ = 1.60 × 10⁻⁵ s⁻¹, 9.66 × 10¹⁸ in the first second. t½ = 12 s: λN₀ would give 3.48 × 10²², but the count falls noticeably within the second, so use N₀(1 − 2^−1/12) = 3.38 × 10²².
Isotope dilution (²⁴Na, t½ = 15 h): injected 2 × 10³ counts s⁻¹ = 1.2 × 10⁵ counts min⁻¹; after 5 h it has fallen by 2^−5/15 = 0.794 to 9.52 × 10⁴ counts min⁻¹, now spread through the blood. V = 9.52 × 10⁴/16 = 5.95 × 10³ cm³ ≈ 6.0 L.SN

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