NCERT Plus Two (Class 12) Physics • Chapter 13

Nuclei

Q: Two nuclei have mass numbers in the ratio 1 : 64. What is the ratio of their (i) nuclear radii and (ii) nuclear densities?

(i) Radius ratio: R₁ / R₂ = (A₁ / A₂) 1/3 = (1 / 64) 1/3 = 1 : 4 . (ii) Density ratio: Nuclear density is independent of mass number A ⇒ 1 : 1 .

Q: Prove that the density of a nucleus is independent of its mass number A.

Nuclear radius R = R₀ A 1/3 . Nuclear volume V = 4 3 πR³ = 4 3 πR₀³ A . Mass of nucleus M ≈ A · m (where m is average nucleon mass). Density ρ = M V = A · m (4/3)πR₀³ A = 3m 4πR₀³ ≈ 2.3 × 10¹⁷ kg/m³ , which contains no A and is hence constant for all nuclei.

Q: Sketch the binding energy per nucleon (E_bn) vs mass number (A) curve. Explain using this curve why energy is released during both nuclear fission and nuclear fusion.

Curve Features: Rises rapidly for light nuclei ( A 230 ) split into medium nuclei with higher BE/A . In both cases, total binding energy increases (total mass decreases), releasing energy Q = Δm c² .

Q: Calculate the binding energy of an alpha particle (⁴₂He) in MeV. Given: m_p = 1.007825 u, m_n = 1.008665 u, M(⁴He) = 4.002603 u, and 1 u = 931.5 MeV.

Δm = [2 m p + 2 m n ] − M(⁴He) = [2(1.007825) + 2(1.008665)] − 4.002603 = 4.032980 − 4.002603 = 0.030377 u . E_b = 0.030377 × 931.5 ≈ 28.3 MeV (or 7.07 MeV/nucleon ).

Q: Write any four characteristic properties of the strong nuclear force.

1. Strongest Force: Much stronger than Coulomb repulsion between protons at nuclear separations. 2. Short Range: Rapidly drops to zero beyond ~2–3 fm . 3. Repulsive Core: Strongly attractive for 0.8 < r < 2.5 fm , yet turns strongly repulsive for r < 0.8 fm . 4. Charge-Independent & Saturating: Acts equally on p–p, n–n, p–n pairs and binds only nearest neighbours.

Q: Define isotopes, isobars, and isotones with one example each.

• Isotopes: Same atomic number Z , different A (e.g. ¹₁H, ²₁H, ³₁H ). • Isobars: Same mass number A , different Z (e.g. ¹⁴₆C, ¹⁴₇N ). • Isotones: Same neutron number N = A − Z (e.g. ¹⁹⁸₈₀Hg, ¹⁹⁷₇₉Au with N = 118 , or ³₁H, ⁴₂He with N = 2 ).

Q: Complete the nuclear decay reaction and identify particles X and Y: ²³⁴₉₀Th → ²³⁴₉₁Pa + X + ν̄ and ²²₁₁Na → ²²₁₀Ne + Y + ν.

• First decay: Z increases 90 → 91 while A = 234 is unchanged ⇒ X = e⁻ (Beta-minus / electron) . • Second decay: Z decreases 11 → 10 while A = 22 is unchanged ⇒ Y = e⁺ (Beta-plus / positron) .

Q: Write a balanced nuclear fission reaction for ²³⁵U with thermal neutrons. State the function of (a) moderator and (b) control rods in a nuclear reactor.

¹₀n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + ~200 MeV . (a) Moderator (heavy water D₂O / graphite): Slows fast fission neutrons down to thermal energies so they efficiently trigger further ²³⁵U fission. (b) Control Rods (Cadmium / Boron): Absorb excess neutrons to maintain multiplication factor k = 1 for steady controlled power.

Atomic mass unit, isotopes/isobars/isotones, nuclear size & constant density, mass defect & binding energy curve, strong nuclear force, radioactivity overview, nuclear fission & solar fusion.

📄 Official NCERT Textbook PDF

Topic 13.2: Atomic Mass Unit (u), Isotopes, Isobars & Isotones

Would a Palakkad spice trader weigh a single cardamom pod on a lorry weighbridge? Never — you need a jeweller's milligram balance! A proton weighs around 10⁻²⁷ kg, so measuring nuclei in kilograms is ridiculous. Physicists built a tiny atomic scale called the atomic mass unit (u) — defined cleanly as one-twelfth the mass of a single Carbon-12 atom!

1 ATOMIC MASS UNIT (u)1 u = m(¹²C)12 = 1.6605 × 10⁻²⁷ kg • Energy equiv: 931.5 MeV

NUCLIDE CODE ᴬ_Z XZ = protons, N = neutrons, Mass number A = Z + N (nucleons)

3 NUCLEAR FAMILIESIsotopes: same Z • Isobars: same A • Isotones: same N = A − Z

🧠 Memory Trick: Check the letter clue! IsoPes → Wait: IsoToPes = same Z (bottom number, same element); IsoBars = same A (top bar); IsoNes = same N (neutrons)!

💡 Simple Kid-Check: Inside every atomic nucleus sit positively charged protons (+e) and uncharged neutrons (discovered by James Chadwick in 1932). Together they are called nucleons — just count protons for Z and add neutrons for A!

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Mass Unit & Nucleon Masses: mₚ = 1.00727 u, mₙ = 1.00866 u (neutron is slightly heavier than proton!), and mₑ = 0.00055 u. Using Einstein's E = mc², 1 u equals 931.5 MeV!
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Isotopes (Same Z, different N): Identical element & chemistry! E.g. Hydrogen family: Protium (¹₁H), Deuterium (²₁H), Tritium (³₁H); Chlorine ³⁵₁₇Cl (75.4%) & ³⁷₁₇Cl (24.6%) give average atomic mass 35.47 u.
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Isobars (Same A) & Isotones (Same N = A − Z):
• Isobars: Different elements sharing same nucleon count A, e.g. ³₁H & ³₂He (A=3), or ¹⁴₆C & ¹⁴₇N (A=14).
• Isotones: Same neutron count N, e.g. ¹⁹⁸₈₀Hg & ¹⁹⁷₇₉Au (both have N = 118) or ³₁H & ⁴₂He (N = 2).

🧒 Super-Easy Example: Classify these pairs: (a) ¹⁴₆C & ¹²₆C, (b) ⁴⁰₁₈Ar & ⁴⁰₂₀Ca, (c) ¹⁴₆C & ¹⁶₈O:
• (a) Same Z = 6 → Isotopes.
• (b) Same A = 40 → Isobars.
• (c) Neutrons: 14 − 6 = 8 and 16 − 8 = 8 → same N = 8 → Isotones!

Watch two classic Kerala HSE traps: (1) Students confuse Isobars and Isotones — always compute N = A − Z quickly before answering! (2) Average atomic mass of an element is a weighted average of its isotopes (e.g. Chlorine 35.5 u because of 35 and 37 in 3 : 1 ratio), NOT a whole number!

Topic 13.3: Nuclear Radius (R = R₀ A^(1/3)) & Constant Nuclear Density

Pack cooked rice tightly into a small Kochi steel cup versus a giant temple-festival cauldron — the big cauldron occupies more space, sure, yet one spoonful scooped from either vessel has the EXACT same density! Incredibly, whether you inspect tiny Helium (A = 4) or giant Uranium (A = 238), nuclear matter packs nucleons to the exact same staggering density!

NUCLEAR RADIUSR = R₀ A^1/3 where R₀ ≈ 1.2 × 10⁻¹⁵ m = 1.2 fm

NUCLEAR VOLUMEV = 43πR³ = 43πR₀³ A → strictly V ∝ A!

CONSTANT DENSITYρ ≈ 2.3 × 10¹⁷ kg/m³ — INDEPENDENT of A! Ratio is ALWAYS 1 : 1!

🧠 Memory Trick: Nuclear radius grows slowly as cube root A^1/3, so V ∝ A cancels M ∝ A → Nuclear density is ALWAYS 1 : 1 for EVERY nucleus!

💡 Simple Kid-Check: Why does R go as A^1/3 instead of A? Think of packing identical marbles into a tight ball without squashing them: 8 marbles only double the radius (8^1/3 = 2), and 27 marbles only triple it (27^1/3 = 3)!

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Radius Scaling: Electron-scattering experiments give R = R₀ A^1/3 with R₀ ≈ 1.2 fm (1 fm = 10⁻¹⁵ m). Comparing two nuclei: R₁R₂ = (A₁A₂)^1/3.
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Proof That Nuclear Density Is Constant:
Mass M ≈ A · m, Volume V = 43πR³ = 43πR₀³ A.
Divide: ρ = MV = A · m(4/3)πR₀³ A = 3m4πR₀³ ≈ 2.3 × 10¹⁷ kg/m³. Notice A vanishes completely!
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Mind-Boggling Fact: Ordinary water is 10³ kg/m³; a single teaspoon of pure nuclear matter (like a neutron star!) weighs billions of tonnes!

🧒 Super-Easy Example: Compare the radii and densities of ²⁷₁₃Al (A₁ = 27) and ¹²⁵₅₂Te (A₂ = 125):
• Radius ratio: R₁ / R₂ = (27 / 125)^1/3 = 3 / 5 (or 3 : 5).
• Density ratio: Since nuclear density is independent of A, ρ₁ / ρ₂ = 1 : 1!

Kerala DHSE's #1 favourite 1-mark trap: "Two nuclei have mass numbers in the ratio 1 : 27 (or 8 : 64). What is the ratio of their nuclear densities?" Students reflexively write 1 : 3 or 1 : 27 — WRONG! The ratio of nuclear densities is ALWAYS 1 : 1! Only the radius ratio takes the cube root (A₁/A₂)^1/3!

Topic 13.4: Mass-Energy Equivalence, Mass Defect & Binding Energy Curve

Put two unattached Lego blocks on an ultra-sensitive scale and read 100.0 g. Snap them tightly together and re-weigh — shockingly, the joined block reads 99.9 g! Einstein's E = mc² solves that magic trick: the missing mass (mass defect Δm) vanished into pure glue energy (Binding Energy) locking the protons and neutrons tightly inside the nucleus!

MASS DEFECT (Δm)Δm = [Z mₚ + (A − Z) mₙ] − M (always > 0 for bound nucleus!)

BINDING ENERGYE_b = Δm c² = Δm × 931.5 MeV • Stability metric: E_bn = E_b / A

BE/A CURVE HIGHLIGHTSPeak at ⁵⁶Fe (~8.75 MeV) • Flat plateau 30 < A < 170 • Fusion (A<20) & Fission (A>170) climb UP!

🧠 Memory Trick: Higher BE/A = Deeper Glue = More Stable! Iron-56 sits on the summit (8.75 MeV) — both light fusion and heavy fission release energy by climbing toward Iron!

💡 Simple Kid-Check: Think of Binding Energy like the cash needed to break a locked piggy bank! A bound nucleus has LESS mass-energy than its separated parts, so you must supply E_b = Δm c² to rip every proton and neutron completely free!

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Mass-Energy & Mass Defect: A bound nucleus weighs LESS than the sum of its free protons and neutrons: Δm = [Z mₚ + (A − Z) mₙ] − M_nuc. Note: when given neutral atomic masses, substitute m_H for mₚ and subtract M_atom so electron masses cancel cleanly!
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Binding Energy per Nucleon (E_bn = E_b / A):
• Flat Plateau (30 < A < 170): Practically constant at ~8.0–8.5 MeV (proves nuclear force saturates!). Peak stability is ⁵⁶Fe (8.75 MeV).
• Light Nuclei (A < 20): Low E_bn (except sharp magic spikes at ⁴He, ¹²C, ¹⁶O). Two light nuclei fuse to gain higher BE/A → energy out!
• Heavy Nuclei (A > 170): Lower E_bn (²³⁸U ≈ 7.6 MeV). Splitting into medium nuclei raises BE/A to ~8.5 MeV → Fission energy out!

🧒 Super-Easy Example: Compute the mass defect and binding energy per nucleon of an alpha particle (⁴₂He), given m_H = 1.007825 u, mₙ = 1.008665 u, and M(⁴He) = 4.002603 u:
• Constituents: 2(1.007825) + 2(1.008665) = 4.032980 u.
• Δm = 4.032980 − 4.002603 = 0.030377 u.
• E_b = 0.030377 × 931.5 ≈ 28.3 MeV → BE/A = 28.3 / 4 = 7.07 MeV/nucleon (super-stable!).

Big exam misconception alert: Which is more stable — ²³⁸U (total BE ≈ 1800 MeV) or ⁵⁶Fe (total BE ≈ 492 MeV)? ⁵⁶Fe is far more stable! Stability depends on Binding Energy PER NUCLEON (BE/A), NOT total binding energy! Also remember: when products have HIGHER BE/A than reactants, final total mass is LOWER and energy Q = Δm c² is RELEASED!

Topic 13.5: Strong Nuclear Force Characteristics

Pack a dozen positive protons together inside a space ten thousand times smaller than an atom — electrostatic repulsion should blow the whole nucleus apart faster than lightning! The only reason you and the Earth exist in one piece is a superhero ultra-short-range glue called the Strong Nuclear Force that overpowers Coulomb repulsion inside 2 fm — yet vanishes completely outside the nucleus!

STRENGTH & RANGEStrongest force in Nature at ~1 fm, yet vanishes beyond 2–3 fm (short-range!).

REPULSIVE COREAttractive for 0.8 < r < 2.5 fm (min V(r) at r₀ ≈ 0.8 fm); STRONGLY REPULSIVE for r < 0.8 fm!

CHARGE-INDEPENDENT & SATURATIONEqual pull for p–p, n–n, p–n • Each nucleon pulls only nearest neighbours (BE/A ≈ const)!

🧠 Memory Trick: Think of Nuclear Force as "Velcro with a Spring Bumper" — pulls neighbours fiercely between 0.8–2 fm, turns repulsive below 0.8 fm, and stops working outside 3 fm!

💡 Simple Kid-Check: Gravity and electricity follow gentle 1/r² long-range rules reaching across galaxies. Nuclear force acts like heavy-duty Velcro: nucleons must practically touch (~1–2 fm) to stick, and pushing them closer than 0.8 fm triggers an iron spring that stops the nucleus from crushing to zero size!

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Potential Energy Curve V(r): At separation r₀ ≈ 0.8 fm, potential energy hits a deep negative trough (~−100 MeV). For r > r₀, slope is positive → attractive force (fading to zero by ~3 fm). For r < 0.8 fm, V(r) shoots steeply positive → strong repulsive force preventing collapse!
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Saturation Property (Why BE/A Is Flat): Because range is only ~2 fm, a nucleon in a large nucleus only feels its immediate shell of neighbours, NOT distant nucleons across the nucleus — keeping BE/A ≈ 8 MeV constant for 30 < A < 170!
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Charge Independence & Neutron Helper: Strong force pulls p–p, n–n, and p–n equally. That is why heavy nuclei (Z > 82) need extra neutrons (N > Z) to add extra nuclear glue without adding long-range proton repulsion!

Two classic board-exam misconceptions: (1) Is nuclear force ALWAYS attractive? NO! Below r ≈ 0.8 fm, it turns strongly repulsive (which is why nuclear density stays constant instead of collapsing!). (2) Nuclear force is non-central and does NOT obey an inverse-square 1/r² law!

Topic 13.6: Radioactivity Overview (Alpha, Beta & Gamma Emissions)

Hold a water balloon stuffed way too full — sooner or later it ejects excess droplets to settle peacefully! Unstable heavy nuclei discovered by Henri Becquerel and Marie Curie behave the exact same way, firing out three invisible nuclear messengers — heavy Alpha packets, nimble Beta electrons/positrons, and ultra-pure Gamma light flashes — until they calm down!

ALPHA (α = ⁴₂He)Z → Z − 2, A → A − 4 • Charge +2e, highest ionisation, stopped by paper!

BETA (β⁻ / β⁺)β⁻ (n → p + e⁻ + ν̄): Z → Z+1, ΔA=0 • β⁺: Z → Z−1, ΔA=0 • Stopped by thin Al!

GAMMA (γ PHOTON)Z & A unchanged! Neutral EM ray, zero field deflection, needs thick Lead!

🧠 Memory Trick: Charge & Mass Rules! Alpha drops (−2, −4) • Beta-minus nudges (+1, 0) • Gamma keeps (0, 0)! And Ionisation (α > β > γ) runs opposite to Penetration (γ > β > α)!

💡 Simple Kid-Check: Every nuclear equation obeys two ironclad counting checks: total bottom numbers (charge Z) on left = total on right, AND total top numbers (nucleons A) on left = total on right! Spontaneous decay also requires Q = (M_parent − M_products)c² > 0.

Ray	Identity	Charge & Mass	`ΔZ`, `ΔA`	Field Bend	Penetration
Alpha (α)	`⁴₂He` nucleus	`+2e`, ~`4 u`	`−2`, `−4`	Small bend (heavy)	Low (sheet of paper)
Beta⁻ (β⁻)	Fast nuclear `e⁻`	`−e`, `0.00055 u`	`+1`, `0`	Sharp opposite bend	Medium (~mm Al foil)
Gamma (γ)	High-energy photon	`0`, rest mass `0`	`0`, `0`	ZERO bend (straight)	Very High (thick Pb)

🧒 Super-Easy Decay Balance: Trace ²³⁸₉₂U emitting one α followed by one β⁻:
• After α (⁴₂He): ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He (90 + 2 = 92; 234 + 4 = 238 ✓).
• Next after β⁻ (e⁻ + ν̄): ²³⁴₉₀Th → ²³⁴₉₁Pa + e⁻ + ν̄ (91 − 1 = 90; 234 + 0 = 234 ✓).

Watch two classic DHSE traps: (1) Where does the electron in β⁻ decay come from? It is created right inside the nucleus when a neutron converts into a proton (n → p + e⁻ + ν̄) — NOT from outer atomic shells! (2) Why is energy spectrum of β particles continuous unlike sharp α lines? Because the decay energy is shared randomly between the electron and the elusive antineutrino (ν̄)!

Topic 13.7.1 – 13.7.2: Nuclear Fission Chain Reaction & Solar Nuclear Fusion

Look at two opposite ways to light up the cosmos! Inside a nuclear power plant, nudge one heavy Uranium-235 giant with a slow neutron and it splits into two mid-sized fragments plus three fresh neutrons — triggering a self-sustaining chain domino effect! Meanwhile inside the Sun's blazing 1.5 × 10⁷ K core, tiny Hydrogen protons slam together so hard they fuse into Helium, bathing Kerala in sunlight!

NUCLEAR FISSION (²³⁵U)¹₀n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + ~200 MeV (~0.85 MeV/nucleon)

REACTOR PARTSModerator (D₂O / graphite) slows neutrons • Control rods (Cd / B) absorb extras to hold k = 1!

SOLAR FUSION (p-p CYCLE)4 ¹₁H → ⁴₂He + 2e⁺ + 2ν + 26.7 MeV (~6.7 MeV/nucleon — much higher per kg!)

🧠 Memory Trick: Moderator = Speed Breaker (slows fast neutrons to thermal speed); Control Rods = Sponge (absorbs excess neutrons to lock multiplication factor k = 1)!

💡 Simple Kid-Check: Why do BOTH splitting a giant nucleus (Fission) AND gluing tiny nuclei (Fusion) give out energy? Look back at the BE/A mountain: Uranium sits on the right slope and Hydrogen sits on the left slope — BOTH move products HIGHER toward Iron-56!

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Controlled Fission Chain Reaction: Each ²³⁵U fission releases ~200 MeV and ~2.5 fast neutrons. Because thermal (slow) neutrons trigger fission best, a Moderator (water, heavy water D₂O, or graphite) slows neutrons down, while Boron/Cadmium Control Rods keep reproduction factor k = 1 (steady power; k > 1 explodes, k < 1 shuts down).
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Thermonuclear Solar Fusion: Protons repel each other by Coulomb barrier! Only at extreme temperatures (~10⁷ K inside stars) do protons move fast enough to fuse via the proton-proton cycle: 4 ¹₁H → ⁴₂He + 2e⁺ + 2ν + 26.7 MeV (in D-T fusion: ²H + ³H → ⁴He + ¹n + 17.6 MeV).

🧒 Super-Easy Comparison: Compare energy per nucleon:
• 1 Fission of ²³⁵U: 200 MeV / 236 ≈ 0.85 MeV/nucleon.
• 1 Solar Fusion of 4 ¹H: 26.7 MeV / 4 ≈ 6.68 MeV/nucleon → Fusion packs nearly 8× MORE energy per gram of fuel and produces clean Helium instead of long-lived radioactive waste!

Two super-common exam traps: (1) Students confuse Moderator with Control Rods — Moderator (D₂O, graphite) only slows down fast neutrons so ²³⁵U captures them easily; Control Rods (Cd, B) actually absorb neutrons! (2) Total energy per reaction is larger in ²³⁵U fission (200 MeV vs 26.7 MeV), YET energy per unit mass (or per nucleon) is MUCH larger in fusion!

Solved Previous Year Questions (Kerala DHSE & NCERT)

Q1 (Kerala HSE March • 1 Marks): Two nuclei have mass numbers in the ratio 1 : 64. What is the ratio of their (i) nuclear radii and (ii) nuclear densities?

(i) Radius ratio: R₁ / R₂ = (A₁ / A₂)^1/3 = (1 / 64)^1/3 = 1 : 4.
(ii) Density ratio: Nuclear density is independent of mass number A ⇒ 1 : 1.

Q2 (Kerala HSE Say • 2 Marks): Prove that the density of a nucleus is independent of its mass number A.

Nuclear radius R = R₀ A^1/3.
Nuclear volume V = 43πR³ = 43πR₀³ A.
Mass of nucleus M ≈ A · m (where m is average nucleon mass).
Density ρ = MV = A · m(4/3)πR₀³ A = 3m4πR₀³ ≈ 2.3 × 10¹⁷ kg/m³, which contains no A and is hence constant for all nuclei.

Q3 (Kerala HSE March • 3 Marks): Sketch the binding energy per nucleon (E_bn) vs mass number (A) curve. Explain using this curve why energy is released during both nuclear fission and nuclear fusion.

Curve Features: Rises rapidly for light nuclei (A < 20, sharp spikes at ⁴He, ¹²C, ¹⁶O), forms a broad plateau (~8.0–8.5 MeV) peaking at ⁵⁶Fe (8.75 MeV), and dips gradually for heavy nuclei (²³⁸U ≈ 7.6 MeV).
• Fusion: Very light nuclei (low BE/A) fuse into heavier nuclei with higher BE/A.
• Fission: Heavy nuclei (A > 230) split into medium nuclei with higher BE/A.
In both cases, total binding energy increases (total mass decreases), releasing energy Q = Δm c².

Q4 (Kerala HSE Model • 2 Marks): Calculate the binding energy of an alpha particle (⁴₂He) in MeV. Given: m_p = 1.007825 u, m_n = 1.008665 u, M(⁴He) = 4.002603 u, and 1 u = 931.5 MeV.

Δm = [2 m_p + 2 m_n] − M(⁴He)
= [2(1.007825) + 2(1.008665)] − 4.002603 = 4.032980 − 4.002603 = 0.030377 u.
E_b = 0.030377 × 931.5 ≈ 28.3 MeV (or 7.07 MeV/nucleon).

Q5 (Kerala HSE March • 2 Marks): Write any four characteristic properties of the strong nuclear force.

1. Strongest Force: Much stronger than Coulomb repulsion between protons at nuclear separations.
2. Short Range: Rapidly drops to zero beyond ~2–3 fm.
3. Repulsive Core: Strongly attractive for 0.8 < r < 2.5 fm, yet turns strongly repulsive for r < 0.8 fm.
4. Charge-Independent & Saturating: Acts equally on p–p, n–n, p–n pairs and binds only nearest neighbours.

Q6 (Kerala HSE Say • 2 Marks): Define isotopes, isobars, and isotones with one example each.

• Isotopes: Same atomic number Z, different A (e.g. ¹₁H, ²₁H, ³₁H).
• Isobars: Same mass number A, different Z (e.g. ¹⁴₆C, ¹⁴₇N).
• Isotones: Same neutron number N = A − Z (e.g. ¹⁹⁸₈₀Hg, ¹⁹⁷₇₉Au with N = 118, or ³₁H, ⁴₂He with N = 2).

Q7 (Kerala HSE March • 2 Marks): Complete the nuclear decay reaction and identify particles X and Y:
²³⁴₉₀Th → ²³⁴₉₁Pa + X + ν̄ and ²²₁₁Na → ²²₁₀Ne + Y + ν.

• First decay: Z increases 90 → 91 while A = 234 is unchanged ⇒ X = e⁻ (Beta-minus / electron).
• Second decay: Z decreases 11 → 10 while A = 22 is unchanged ⇒ Y = e⁺ (Beta-plus / positron).

Q8 (Kerala HSE Model • 3 Marks): Write a balanced nuclear fission reaction for ²³⁵U with thermal neutrons. State the function of (a) moderator and (b) control rods in a nuclear reactor.

¹₀n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + ~200 MeV.
(a) Moderator (heavy water D₂O / graphite): Slows fast fission neutrons down to thermal energies so they efficiently trigger further ²³⁵U fission.
(b) Control Rods (Cadmium / Boron): Absorb excess neutrons to maintain multiplication factor k = 1 for steady controlled power.

Q9 (NCERT / DHSE Numerical • 2 Marks): Write the overall nuclear equation for the proton-proton fusion cycle powering the Sun. Why is fusion energy per unit mass higher than fission?

Overall reaction: 4 ¹₁H → ⁴₂He + 2e⁺ + 2ν + 26.7 MeV.
• Fusion releases 26.7 / 4 ≈ 6.68 MeV/nucleon, whereas ²³⁵U fission releases 200 / 236 ≈ 0.85 MeV/nucleon — nearly 8 times more energy per kg of fuel!