Master Formula Sheet

# Result Formula
1 Atomic mass unit 1 u = 112m(12C)\frac{1}{12}m(^{12}C) = 1.660539×10−271.660539 \times 10^{-27} kg = 931.5 MeV/c2c^2
2 Nucleon masses mpm_p = 1.00727 u; mnm_n = 1.00866 u; mem_e = 0.00055 u
3 Vocabulary isotopes: same Z; isobars: same A; isotones: same N
4 Radius R=R0A1/3R = R_0A^{1/3}, R0R_0 = 1.2 fm; V ∝ A
5 Density ρ=3m4πR03≈2.3×1017\rho = \dfrac{3m}{4\pi R_0^3} \approx 2.3 \times 10^{17} kg/m³ — same for ALL nuclei
6 Mass defect ΔM=[Zmp+(A−Z)mn]−M\Delta M = [Zm_p + (A-Z)m_n] - M
7 Binding energy Eb=ΔMc2=ΔM(u)×931.5E_b = \Delta Mc^2 = \Delta M(u) \times 931.5 MeV; Ebn=Eb/AE_{bn} = E_b/A
8 Curve landmarks plateau ~8.0 MeV (30 < A < 170); max 8.75 MeV at A = 56; 7.6 MeV at A = 238
9 Q value Q=[Σmi−Σmf]c2Q = [\Sigma m_i - \Sigma m_f]c^2 = final KE - initial KE
10 Decay law N=N0e−λtN = N_0e^{-\lambda t}; after n half-lives: N0/2nN_0/2^n
11 Time constants T1/2=0.693λT_{1/2} = \dfrac{0.693}{\lambda}; τ=1λ\tau = \dfrac{1}{\lambda}; T1/2=0.693τT_{1/2} = 0.693\tau
12 Activity R=λNR = \lambda N; 1 Bq = 1 decay/s; 1 Ci = 3.7×10103.7 \times 10^{10} Bq
13 Fission ~200 MeV per 235^{235}U fission; 101410^{14} J/kg (coal: 10710^7 J/kg)
14 Fusion (p-p) 41H+2e−→4He+2ν+6γ+26.74^1H + 2e^- \to {}^4He + 2\nu + 6\gamma + 26.7 MeV
15 Coulomb barrier ~400 keV (p + p) → T ~ 3×1093 \times 10^9 K; Sun's core 1.5×1071.5 \times 10^7 K

The Decay Table & Curve Anatomy

Alpha / beta / gamma at a glance:

Alpha Beta (β−\beta^- / β+\beta^+) Gamma
Emitted 24^4_2He nucleus electron / positron (+ antineutrino / neutrino) photon (100s of keV+)
Charge +2e -e / +e 0
(A, Z) change (-4, -2) (0, +1) / (0, -1) (0, 0)
Periodic table 2 left 1 right / 1 left unmoved
Penetration least (paper) middle (Al foil) most (lead)

Binding-energy curve — the four features and four conclusions:

  1. Flat ~8.0 MeV plateau (30 < A < 170) → short-range, saturating force.
  2. Peak 8.75 MeV at A = 56 (iron) → most stable matter; spikes at 4^4He, 16^{16}O → shell structure.
  3. Heavy droop (7.6 MeV at 238) → fission pays ~0.9 MeV/nucleon ≈ 200 MeV.
  4. Light droop → fusion pays; the Sun's 26.7 MeV per helium.

Nuclear force checklist: strongest force (≫ Coulomb ≫ gravity); range a few fm; pair-potential minimum at r0r_0 = 0.8 fm (attract outside, hard-core repel inside); charge-independent (n-n ≈ n-p ≈ p-p); no simple formula. Don't confuse r0r_0 = 0.8 fm with R0R_0 = 1.2 fm.

Stability rules: N/Z ≈ 1 (light) → ~3/2 (heavy); neutron-rich → β−\beta^-; proton-rich → β+\beta^+; only ~10% of isotopes stable; free neutron decays (~1000 s mean life), free proton stable.

Fission vs Fusion & Exam Traps

The comparison table:

Fission Fusion
What happens heavy nucleus splits (U-235 + n → Ba + Kr + 3n) light nuclei merge (4p → He-4)
Energy/event ~200 MeV 3.27-26.7 MeV
Energy/nucleon ~0.85 MeV ~6.7 MeV (≈ 8× richer per kg)
Trigger slow neutron (no barrier — neutral!) ~400 keV Coulomb barrier → 10710^7-10910^9 K
Where reactors (controlled), atom bomb (uncontrolled) stars (p-p cycle); controlled fusion in development (plasma confinement; India participating)
Products radioactive fragments (β−\beta^- chains) mostly clean (He)

Trap list (each costs marks yearly):

  • Density questions: answer is ALWAYS 'constant / 1 : 1'.
  • Radius ratios need the cube root; volume ratios don't (V ∝ A).
  • 'Decayed' vs 'remaining' — complementary fractions; read the last word.
  • Mean life vs half-life: τ=1.44T1/2\tau = 1.44T_{1/2}; 1/e ↔ τ, 1/2 ↔ T1/2T_{1/2}.
  • Stability index is EbnE_{bn}, never total EbE_b (uranium's huge total is bait).
  • Two-body decay: light particle takes mheavymtotal\dfrac{m_{heavy}}{m_{total}} of Q.
  • Atomic-mass bookkeeping: use mHm_H and atomic masses together (electrons cancel).
  • Nuclear equations balance protons AND neutrons separately — not elements.

One-Glance Revision Flow

The chapter in eight steps:

  1. Bookkeeping: 1 u (carbon-12 standard) = 931.5 MeV/c2c^2; isotopes/isobars/isotones; Z, N, A; Chadwick's neutron (1932, beryllium + alpha, Nobel 1935).
  2. Size: R=1.2A1/3R = 1.2A^{1/3} fm → V ∝ A → density 2.3×10172.3 \times 10^{17} kg/m³, universal (neutron stars!).
  3. Mass defect: ΔM=[Zmp+(A−Z)mn]−M\Delta M = [Zm_p + (A-Z)m_n] - M; O-16's 0.13691 u ↔ 127.5 MeV.
  4. The curve: plateau 8.0, peak 8.75 at Fe-56, droops both ends → fission AND fusion release energy rolling toward iron.
  5. The force: strongest, few-fm range, saturating, 0.8 fm hard core, charge-blind, formula-less.
  6. Radioactivity: Becquerel 1896; α/β/γ\alpha/\beta/\gamma displacement rules; N=N0e−λtN = N_0e^{-\lambda t}, T1/2=0.693/λT_{1/2} = 0.693/\lambda, R=λNR = \lambda N.
  7. Fission: n + U-235 → ~200 MeV, fragments beta-active, chain via 2-4 neutrons; reactors vs bombs = control.
  8. Fusion: d-d reactions (3.27/4.03 MeV), p-p cycle 26.7 MeV, 400 keV barrier, Sun's 1.5×1071.5 \times 10^7 K core burning tail protons for 5 more billion years; plasma confinement the reactor challenge.

Morning-of-exam checklist: 931.5 converts u ↔ MeV … cube root for radii, never for volumes … density constant, 1 : 1 always … EbnE_{bn} = stability index, peak at A = 56 … binding energy = ΔM×931.5\Delta M \times 931.5 … 200 MeV/fission, 101410^{14} J/kg vs coal's 10710^7 … 26.7 MeV per helium in the Sun … 2n2^n halvings, decayed = 1 - remaining … τ=1.44T1/2\tau = 1.44T_{1/2}; 1/e ↔ mean life … balance protons and neutrons separately … neutron = barrier-free trigger, fusion needs 10710^7-10910^9 K … r0r_0 = 0.8 fm ≠ R0R_0 = 1.2 fm. Go score.