BARC OCES/DGFS cheat sheet

Binding energy per nucleon peaks near Fe-56 (~8.8 MeV); fission of heavy nuclei and fusion of light nuclei both release energy because the products are closer to this maximum.

Radioactive decay law: N = N0·e^(−λt); half-life T½ = ln2/λ ≈ 0.693/λ; mean life τ = 1/λ = T½/ln2.

Activity A = λN; units: 1 Bq = 1 decay/s, 1 Ci = 3.7×10^10 Bq. Specific activity of a pure radionuclide = λ·NA/M (Bq/g).

Four-factor formula for a thermal reactor: k∞ = η·f·p·ε (reproduction factor × thermal utilization × resonance escape probability × fast fission factor); k_eff = k∞·P_NL, critical when k_eff = 1.

Reactor period relation: reactivity ρ = (k_eff − 1)/k_eff; delayed neutrons (β ≈ 0.0065 for U-235) make the reactor controllable only for ρ < β (prompt critical threshold).

Moderator quality is judged by slowing-down power (ξ·Σs) and moderating ratio (ξ·Σs/Σa); heavy water and graphite excel because of low absorption.

Dose quantities: absorbed dose in gray (1 Gy = 1 J/kg), equivalent dose in sievert (H = D·w_R, w_R = 1 for γ/β, 20 for α), effective dose sums tissue-weighted equivalents.

Thermal neutron flux typical of power reactors ~10^13–10^14 n/cm²·s; 1 MW of fission power corresponds to ~3.1×10^16 fissions/s (≈200 MeV per fission).

Carnot efficiency η = 1 − T_C/T_L (absolute temperatures); no heat engine operating between two reservoirs can exceed it.

Beam bending: σ = M·y/I, with I = b·h³/12 for a rectangular section; maximum stress occurs at the outermost fibre (y = h/2).

Euler buckling load P_cr = π²EI/(K·L)²; K = 1 pinned-pinned, 0.5 fixed-fixed, 2 fixed-free, 0.7 fixed-pinned.

Bernoulli equation along a streamline: p/ρg + v²/2g + z = constant for steady, incompressible, inviscid flow; Reynolds number Re = ρvD/μ decides laminar (Re < 2000 in pipes) versus turbulent.

Thevenin/Norton: any linear two-terminal network reduces to V_th in series with R_th, where V_th is the open-circuit voltage and R_th the resistance seen with sources dead (voltage sources shorted, current sources opened).

First law for a control volume at steady state: Q − W = Σṁ_out·h_out − Σṁ_in·h_in (neglecting KE and PE changes).

Mohr's circle: principal stresses σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]; maximum shear τ_max = (σ1−σ2)/2.

For a DC shunt motor, back emf E_b = V − I_a·R_a and torque T ∝ φ·I_a; speed rises as flux falls, which is why field failure causes overspeed.

Ideal gas law PV = nRT with R = 8.314 J/mol·K; at STP (273.15 K, 1 bar) one mole occupies ~22.7 L (22.4 L at 1 atm).

First law of thermodynamics: ΔU = q + w (IUPAC sign convention, work done on the system positive); enthalpy H = U + PV, so ΔH = q_p at constant pressure.

Gibbs free energy ΔG = ΔH − TΔS; a reaction is spontaneous at constant T and P when ΔG < 0, and equilibrium sets ΔG = 0 with ΔG° = −RT·lnK.

Arrhenius equation k = A·e^(−Ea/RT); a plot of ln k versus 1/T has slope −Ea/R.

pH = −log[H⁺]; for a weak acid pH ≈ ½(pKa − log C); buffer pH follows Henderson–Hasselbalch: pH = pKa + log([A⁻]/[HA]).

Quantum numbers: n = 1,2,…; l = 0…n−1; m_l = −l…+l; m_s = ±½; Pauli exclusion forbids identical sets, so each orbital holds two electrons of opposite spin.

Beer–Lambert law A = ε·c·l; absorbance is linear in concentration only for dilute solutions and monochromatic light.

Rate laws are determined experimentally, not from stoichiometry; for a first-order reaction t½ = ln2/k and concentration decays exponentially.

Percentage change: successive changes of a% and b% give a net change of a + b + ab/100 percent (signs matter).

Compound interest A = P(1 + r/n)^(nt); simple interest A = P(1 + rt). For two years, CI − SI = P(r/100)².

Work–rate: if A alone takes x days and B alone y days, together they take xy/(x+y) days; use the reciprocal-rate method for more workers.

Relative speed: same direction = difference of speeds, opposite direction = sum; time to cross a platform of length L by a train of length l is (L + l)/speed.

Permutations nPr = n!/(n−r)!; combinations nCr = n!/[r!(n−r)!]; nCr = nC(n−r).

Number series: check differences, ratios, squares/cubes, and alternating patterns before assuming a single rule; the odd-one-out is the term breaking the established rule.

Syllogism rule: a conclusion follows only if it is true in every possible Venn arrangement of the premises; 'some A are B' does not imply 'some B are not A' unless existence is asserted.

Average speed for equal distances at speeds u and v is the harmonic mean 2uv/(u+v), not the arithmetic mean.