GATE MT cheat sheet

Ellingham diagrams plot ΔG° for oxide formation versus temperature; a metal can reduce another metal's oxide only if its line lies below at that temperature.

Gibbs phase rule: F = C − P + 2 for variables T, P, composition; at constant total pressure F = C − P + 1, and at constant T and P F = C − P.

Lever rule: fraction of phase α = (C_β − C_0)/(C_β − C_α); fraction of β = (C_0 − C_α)/(C_β − C_α).

Fick's first law: J = −D (∂C/∂x); Fick's second law: ∂C/∂t = D ∂²C/∂x², with error-function solution for a semi-infinite couple.

Arrhenius diffusivity: D = D₀ exp(−Q/RT); Q is activation energy per mole, R = 8.314 J·mol⁻¹·K⁻¹.

Hall–Petch relation: σ_y = σ₀ + k_y d^(−1/2), where d is grain diameter; finer grains raise yield strength.

Critical resolved shear stress: τ_CRSS = σ cos φ cos λ, with φ the angle to the slip-plane normal and λ the angle to the slip direction.

Crystal structures: FCC has 4 atoms per cell and 12 slip systems; BCC has 2 atoms per cell; HCP has 6 atoms per cell and a c/a ideal ratio of 1.633.

Carbon equivalent (IIW): CE = %C + %Mn/6 + (%Cr + %Mo + %V)/5 + (%Ni + %Cu)/15; values above about 0.4 raise weld cold-cracking risk.

Basic oxygen steelmaking removes carbon by blowing O₂; typical endpoint carbon is below 0.1%, with slag basicity CaO/SiO₂ around 3–4.

Blast furnace reactions: C + O₂ → CO₂ (exothermic), CO₂ + C → 2CO (Boudouard), and Fe₂O₃ reduction proceeds stepwise Fe₂O₃ → Fe₃O₄ → FeO → Fe.

Electrolytic refining uses impure anode and pure cathode in an electrolyte; anode mud collects noble impurities such as Au, Ag, and platinum-group metals.

Froth flotation separates sulfide ores using collectors (xanthates) that render particles hydrophobic; gangue stays wetted and sinks.

Roasting is oxidation of sulfide concentrates in air; calcination is thermal decomposition of carbonates or hydroxides without air.

Ellingham-based carbothermic reduction of alumina is impractical because Al₂O₃ is far more stable than CO above roughly 2000 °C; aluminium is won by Hall–Héroult electrolysis in cryolite.

Cyanidation leaches gold as [Au(CN)₂]⁻ in aerated alkaline cyanide solution; zinc dust cementation recovers the gold.

Tempering of martensite proceeds in stages: epsilon-carbide precipitation near 100–200 °C, retained austenite decomposition near 200–300 °C, and cementite coarsening above 400 °C.

The iron–carbon diagram shows eutectoid at 0.76 wt% C and 727 °C, eutectic at 4.3 wt% C and 1147 °C; maximum solubility of C in austenite is 2.14 wt% at 1147 °C.

TTT and CCT diagrams: the nose of the curve sets the minimum cooling rate for full martensite; CCT noses sit at shorter times than TTT noses.

Jominy end-quench test measures hardenability; hardenability increases with austenite grain size and with alloying elements such as Mn, Cr, Mo, and Ni.

Normalizing refines grain size by air cooling from above A₃ or A_cm; annealing softens by slow furnace cooling; quenching forms martensite.

Pearlite spacing decreases with lower transformation temperature, raising strength by impeding dislocation motion at lamellar boundaries.

Maraging steels harden by precipitation of intermetallic compounds (e.g., Ni₃Ti, Ni₃Mo) in a low-carbon Fe–Ni martensite, not by carbon.

Austenite grain growth is controlled by pinning particles such as AlN or Nb(C,N); solution treatment temperature determines whether pinning survives.

Stress intensity factor: K = Y σ √(πa), with Y a geometry factor; fracture occurs when K reaches K_IC.

Paris law for fatigue crack growth: da/dN = C (ΔK)^m, with m typically 2–4 in the mid-regime.

Hall–Petch and Orowan strengthening are additive contributions to yield strength along with solid-solution and precipitate strengthening.

True stress and strain: σ_t = σ_eng(1 + e), ε_t = ln(1 + e); Considère criterion for necking is dσ_t/dε_t = σ_t.

Resolved shear stress and Schmid factor: slip begins when τ = σ cos φ cos λ reaches the critical value; maximum Schmid factor is 0.5.

Creep: steady-state rate ε̇ = A σⁿ exp(−Q/RT); n ≈ 3–5 for dislocation creep, n ≈ 1 for diffusional creep.

Fatigue limit exists in steels (endurance limit near 10⁶–10⁷ cycles) but not in most aluminium alloys.

Griffith criterion: σ_f = √(2Eγ_s/(πa)) for a crack of length a in a brittle material.

Iron–carbon equilibrium: ferrite (BCC) dissolves at most 0.022 wt% C at 727 °C; cementite Fe₃C contains 6.67 wt% C.

Phase diagram construction: liquidus, solidus, and solvus lines separate single-phase from two-phase regions; tie lines give phase compositions at a temperature.

Eutectic, eutectoid, and peritectic reactions: L → α + β, γ → α + β, and L + α → β respectively.

Binary systems with complete solid solubility require similar atomic radii (within about 15%), same crystal structure, and similar electronegativity (Hume-Rothery rules).

Ternary phase diagrams use an equilateral triangle; composition is read along lines parallel to the side opposite each vertex.

Lever rule in two-phase regions gives phase fractions; the inverse lever rule gives the fraction of a specific phase.

Solidification of a eutectic alloy occurs at constant temperature with coupled lamellar or rod growth, giving fine two-phase structures.

Intermetallic compounds appear as vertical lines at fixed stoichiometric compositions on phase diagrams.

Nernst equation: E = E° − (RT/nF) ln Q; at 25 °C the factor RT/F is 0.0257 V, so E = E° − (0.0592/n) log Q.

Faraday's laws: mass deposited m = (M I t)/(n F), with F = 96485 C·mol⁻¹.

Pourbaix diagrams plot potential versus pH, showing immunity, corrosion, and passivation regions for a metal in water.

Galvanic corrosion accelerates when two metals with a large potential difference are coupled in an electrolyte; the more active metal becomes the anode.

Pilling–Bedworth ratio (oxide volume/metal volume) between 1 and 2 suggests protective scaling; values far outside indicate spalling or porosity.

Cathodic protection: sacrificial anode uses a more active metal; impressed current uses an external DC source to polarize the structure cathodically.

Exchange current density and Tafel slopes: η = a + b log i, with b typically 60–120 mV/decade for activation-controlled corrosion.

Pitting potential marks the breakdown of passivity; chlorides lower it and raise susceptibility to localized attack.