GATE ES cheat sheet

BOD exertion follows first-order kinetics: L_t = L_0(1 - e^(-kt)); BOD_5 at 20 °C uses k = 0.23 /day for domestic sewage (base e).

DO sag curve (Streeter-Phelps): D_t = (k1 L0/(k2 - k1))(e^(-k1 t) - e^(-k2 t)) + D0 e^(-k2 t); critical time t_c = (1/(k2 - k1)) ln[(k2/k1)(1 - D0(k2 - k1)/(k1 L0))].

Henry's law: p = H·x or C_water = K_H·p_gas; low K_H means high volatility and stripping tendency.

Alkalinity is titrated to pH 4.5 (methyl orange) and pH 8.3 (phenolphthalein); P = M means only OH⁻, P = M/2 means only CO3²⁻, P > M means OH⁻ + CO3²⁻.

Hardness (as CaCO3) = 2.497·[Ca²⁺ mg/L] + 4.118·[Mg²⁺ mg/L]; total hardness = carbonate + non-carbonate hardness.

Gibbs free energy: ΔG° = -RT ln K = -nFE°; a reaction is spontaneous when ΔG < 0. Nernst: E = E° - (0.059/n) log Q at 25 °C.

Photolysis, hydrolysis, and sorption onto organic carbon (K_oc) govern fate of organics; log K_ow > 3 indicates bioaccumulation potential.

Langelier Saturation Index = pH - pH_s; positive LSI scales CaCO3, negative LSI is corrosive.

Bacterial growth phases: lag, log (exponential), stationary, death; in log phase dX/dt = μX, doubling time t_d = ln2/μ = 0.693/μ.

Monod kinetics: μ = μ_max S/(K_s + S); combined with yield Y = -dX/dS gives substrate utilization rate.

Nitrification is two-step: NH4+ → NO2- by Nitrosomonas, NO2- → NO3- by Nitrobacter; requires ~4.57 mg O2 per mg N oxidized.

Denitrification converts NO3- to N2 under anoxic conditions by facultative heterotrophs; needs a carbon source (typically 2.86 g COD per g NO3-N reduced).

Anaerobic digestion stages: hydrolysis → acidogenesis → acetogenesis → methanogenesis; optimum mesophilic ~35 °C, pH 6.8-7.4.

Indicator organism is E. coli (thermotolerant coliform); MPN or membrane filtration quantifies coliforms per 100 mL.

Redox potential and electron acceptors sequence: O2 > NO3- > Mn/Fe oxides > SO4²- > CO2 (methanogenesis).

In situ bioremediation relies on biostimulation (nutrients, O2) or bioaugmentation (added cultures); phytoremediation uses plants for metal/organic uptake.

Sedimentation: Stokes law v_s = g(ρ_p - ρ_w)d²/(18μ); ideal removal fraction = v_s/Q/A (overflow rate).

Coagulation destabilizes charge (sweep floc with alum: Al2(SO4)3·18H2O + 3Ca(HCO3)2 → 2Al(OH)3 + 3CaSO4 + 6CO2).

Rapid mix G ~ 300-1000 s⁻¹ for 30-60 s; flocculation G ~ 10-75 s⁻¹ for 15-30 min; G·t (Camp number) is the design criterion.

Filtration head loss (Rose equation or Carmen-Kozeny); backwash when head loss or turbidity limit is reached.

Chlorine disinfection: CT concept (concentration × contact time); Chick-Watson: N/N0 = e^(-kCt).

Activated sludge: F/M = Q·S0/(V·X), SRT = V·X/(Q_w·X_r + Q_e·X_e), sludge volume index SVI = (SV30·1000)/MLSS.

Trickling filter and RBC are attached-growth; MBBR and SBR are modern variants; UASB suits high-strength wastewater.

Sludge thickening (gravity, DAF), dewatering (centrifuge, belt press), and digestion reduce volume before disposal.

Air Quality Index (India CPCB) has six categories: Good (0-50), Satisfactory (51-100), Moderate (101-200), Poor (201-300), Very Poor (301-400), Severe (401-500).

Gaussian plume model for point source: C(x,y,z) = (Q/(2π u σ_y σ_z)) exp(-y²/(2σ_y²)) [exp(-(z-H)²/(2σ_z²)) + exp(-(z+H)²/(2σ_z²))].

Effective stack height H = h + Δh; Briggs buoyancy flux F = g·V_s·d²·(T_s - T_a)/(4·T_s).

Pasquill-Gifford stability classes A (very unstable) to F (stable); σ_y, σ_z grow fastest in class A.

Particulate control: gravity settler, cyclone (cut diameter d50), ESP (Deutsch equation η = 1 - e^(-wA/Q)), baghouse, wet scrubber.

Gaseous control: adsorption (carbon), absorption (scrubber), condensation, incineration; SCR uses NH3 over V2O5/TiO2 for NOx.

Noise: L_p = 20 log(p/p0), p0 = 20 μPa; addition of incoherent sources L_total = 10 log(Σ 10^(Li/10)).

Noise reduction by distance for point source: ΔL = 20 log(r2/r1); for line source: 10 log(r2/r1). OSHA 8-h limit 90 dBA; doubling of sources adds ~3 dB.

Waste hierarchy: prevention > reuse > recycling > recovery (energy) > disposal (landfill).

Moisture content (wet basis) M = (W_w/W_total)·100; dry basis M_d = W_w/W_dry; convert: M_d = M/(100 - M).

Calorific value (dry) can be estimated by Dulong: HCV = 33,800 C + 144,200(H - O/8) + 9,400 S kJ/kg (mass fractions).

Landfill leachate generation: HELP model or water balance P = ET + R + ΔS; clay liner k ≤ 1×10⁻⁹ m/s.

Landfill gas ~50-60% CH4, 40-50% CO2; methane generation follows first-order decay: L0·e^(-kt).

Hazardous waste characteristics (RCRA): ignitability, corrosivity, reactivity, toxicity (TCLP).

Incineration 3T: temperature (>850 °C for MSW, >1100 °C for hazardous), time (>2 s), turbulence.

Sanitary landfill siting: distance from water bodies, groundwater table, and habitation; composite liner (HDPE + clay).

Rational method peak flow Q = C·i·A/3.6 (Q m³/s, i mm/h, A km²); time of concentration via Kirpich: t_c = 0.0195 L^0.77 S^(-0.385) minutes.

Manning equation: Q = (1/n) A R^(2/3) S^(1/2); R = A/P; typical n: concrete 0.013, earth 0.025.

Darcy's law: Q = K·i·A; transmissivity T = K·b; well yield (confined): Q = 2πT(h2 - h1)/ln(r2/r1).

Unit hydrograph: direct runoff hydrograph for 1 cm of rainfall over a basin; S-curve converts duration; SCS curve number Q = (P - 0.2S)²/(P + 0.8S), S = (25400/CN) - 254 mm.

Infiltration: Horton f = f_c + (f_0 - f_c)e^(-kt); Φ-index = (total rainfall - runoff)/duration.

Hydraulic jump: sequent depth ratio y2/y1 = 0.5(√(1 + 8F1²) - 1); energy loss ΔE = (y2 - y1)³/(4 y1 y2).

Weir flow: Q = (2/3) C_d √(2g) L H^(3/2); broad-crested weir coefficient ~1.7 (SI).

Flow measurement in open channel: Parshall flume, V-notch weir Q = (8/15) C_d √(2g) tan(θ/2) H^(5/2).

EIA process in India (EIA Notification 2006, amended): screening, scoping (ToR), baseline, impact prediction, EMP, public hearing, appraisal by EAC/SEAC, EC.

Categories A and B: Category A appraised by EAC (MoEFCC), Category B by SEAC; B1 requires EIA, B2 does not.

Ecological pyramids: numbers, biomass, energy; energy pyramid is always upright (10% law).

Primary productivity measured by light-dark bottle (O2) or 14C method; gross PP - respiration = net PP.

Biodiversity indices: Shannon H' = -Σ p_i ln p_i; Simpson D = Σ p_i²; higher H' means more diversity.

Carrying capacity and logistic growth: dN/dt = rN(1 - N/K); exponential dN/dt = rN.

Rapid and comprehensive EIA tools: matrices (Leopold), checklists, overlays, networks; Leopold matrix uses magnitude and importance 1-10.

Environmental management: ISO 14001 EMS, environmental audit, life cycle assessment (cradle-to-grave).

Standard deviation s = √[Σ(x - x̄)²/(n - 1)]; coefficient of variation CV = s/x̄ × 100%.

Confidence interval of mean: x̄ ± t·s/√n; t from Student's t table with n - 1 df.

Detection limit LOD = 3s (or 3.3s/slope of calibration); LOQ = 10s.

Linear regression y = a + bx; r² measures goodness of fit; r² = 1 is perfect fit.

Hypothesis testing: t-test for means, F-test for variances, chi-square for goodness of fit; p < 0.05 rejects null.

Air sampling: high-volume sampler for SPM/PM10, PM2.5 impactor, gas absorption in impingers (SO2, NOx).

Water quality instruments: pH meter, conductivity meter, spectrophotometer (Beer-Lambert A = εbc), DO meter (Winkler as reference), turbidity nephelometer (NTU).

Chromatography (GC for VOCs, HPLC for non-volatile) and AAS/ICP for metals; mass spectrometry for trace identification.

Solar constant ~1361 W/m²; average insolation at Earth's surface ~200 W/m²; solar cell efficiency η = P_out/(G·A).

Wind power P = 0.5 ρ A v³; Betz limit 16/27 ≈ 59.3% maximum theoretical efficiency.

Hydropower P = ρ g Q H η; typical η 0.85-0.90.

Biomass and biogas: 1 kg of volatile solids yields ~0.5-0.7 m³ biogas (55-70% CH4); calorific value of CH4 ~35.8 MJ/m³.

Greenhouse gases ranked by GWP (100-yr, AR6): CO2 = 1, CH4 ≈ 27-30, N2O ≈ 273, SF6 ≈ 25,200.

Carbon footprint = activity × emission factor; 1 kWh Indian grid electricity ≈ 0.7-0.8 kg CO2.

Energy units: 1 kWh = 3.6 MJ; 1 tonne oil equivalent (toe) ≈ 41.87 GJ; 1 quad = 1.055 EJ.

Sustainable development goals (17 SDGs); triple bottom line: planet, people, profit; circular economy design.

Percentage change: new = old × (1 ± r); successive changes multiply: (1 + a)(1 + b) - 1.

Average speed for equal distances = 2v1v2/(v1 + v2); for equal times = (v1 + v2)/2.

Compound interest A = P(1 + r/n)^(nt); simple interest A = P(1 + rt).

Ratio and proportion: if a:b = c:d then ad = bc; partnership profit shares ∝ capital × time.

Permutation vs combination: nPr = n!/(n - r)!; nCr = n!/(r!(n - r)!).

Probability: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); independent events P(A ∩ B) = P(A)P(B).

Data interpretation: mean, median, mode; pie chart angle = (value/total) × 360°.

Logical reasoning: syllogisms, seating arrangement, blood relations, series completion; verbal ability tests synonyms, antonyms, and grammar.

Percentage change: new = old × (1 ± r); successive changes multiply: (1 + a)(1 + b) - 1.

Average speed for equal distances = 2v1v2/(v1 + v2); for equal times = (v1 + v2)/2.

Compound interest A = P(1 + r/n)^(nt); simple interest A = P(1 + rt).

Ratio and proportion: if a:b = c:d then ad = bc; partnership profit shares ∝ capital × time.

Permutation vs combination: nPr = n!/(n - r)!; nCr = n!/(r!(n - r)!).

Probability: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); independent events P(A ∩ B) = P(A)P(B).

Data interpretation: mean, median, mode; pie chart angle = (value/total) × 360°.

Logical reasoning: syllogisms, seating arrangement, blood relations, series completion; verbal ability tests synonyms, antonyms, and grammar.