CSIR NET cheat sheet

ΔG = ΔH − TΔS; a process at constant T and p is spontaneous only when ΔG < 0, and equilibrium corresponds to ΔG = 0.

First-order kinetics: t½ = 0.693/k, independent of initial concentration; for an n-th order reaction t½ ∝ [A]₀^(1−n).

Beer–Lambert law A = εcl holds only for dilute solutions, monochromatic radiation and a fixed chromophore environment.

Hückel criterion: a planar, cyclic, fully conjugated ring with 4n+2 π electrons is aromatic; 4n π electrons gives antiaromatic character.

Characteristic IR stretches: O–H ≈ 3300 cm⁻¹ (broad), C–H ≈ 2900 cm⁻¹, C≡N ≈ 2250 cm⁻¹, C=O ≈ 1700 cm⁻¹, C=C ≈ 1650 cm⁻¹.

¹H NMR shift ranges (ppm vs TMS): alkyl 0.9–1.5, allylic 1.6–2.2, vinylic 4.5–6.5, aromatic 6.5–8.0, aldehydic 9–10.

Spectrochemical series (increasing Δo): I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < CN⁻ < CO; strong-field ligands give low-spin complexes.

Irving–Williams stability order for divalent first-row ions: Mn²⁺ < Fe²⁺ < Co²⁺ < Ni²⁺ < Cu²⁺ > Zn²⁺.

The number of ¹H NMR signals equals the number of chemically non-equivalent proton sets; peak areas are proportional to proton counts.

Coriolis parameter f = 2Ω sin φ, zero at the equator and ±2Ω at the poles; the Rossby radius scales as √(gH)/f.

Geostrophic balance: fv = (1/ρ)∂p/∂x and fu = −(1/ρ)∂p/∂y, so geostrophic flow runs parallel to isobars with low pressure to the left in the Northern Hemisphere.

Potential temperature θ = T(1000/p)^(R/cp) is conserved in dry adiabatic motion; dθ/dz > 0 indicates a statically stable column.

Standard atmospheric lapse rate 6.5 K km⁻¹; dry adiabatic lapse rate 9.8 K km⁻¹; saturated adiabatic lapse rate ≈ 5–6 K km⁻¹.

Ekman transport is 90° to the right of the surface wind in the Northern Hemisphere and 90° to the left in the Southern Hemisphere, driving coastal upwelling.

S-waves cannot propagate through liquids, which is the primary evidence for a liquid outer core; the Moho marks the crust–mantle velocity jump.

Milankovitch periodicities: eccentricity ~100 kyr, obliquity ~41 kyr, axial precession ~23 kyr.

Clausius–Clapeyron: saturation vapour pressure increases about 7% per kelvin, amplifying atmospheric water-holding capacity with warming.

Seismic velocities: v_p = √((K + 4μ/3)/ρ) and v_s = √(μ/ρ); mean ocean salinity is about 35 g kg⁻¹.

Hardy–Weinberg: p² + 2pq + q² = 1 for genotype frequencies when mating is random and selection, mutation, migration and drift are absent.

Michaelis–Menten: v = Vmax[S]/(Km + [S]); Km equals the substrate concentration at half-maximal velocity and is a reciprocal affinity measure.

PCR thermal profile: denaturation ≈95 °C, primer annealing ≈55–65 °C (≈5 °C below primer Tm), extension ≈72 °C for Taq polymerase.

Aerobic oxidation of one glucose yields ≈30–32 ATP: 2 net from glycolysis, 2 GTP from the TCA cycle, and the rest from oxidative phosphorylation.

P/O ratios: ≈2.5 ATP per NADH and ≈1.5 ATP per FADH₂ entering the electron transport chain.

Blot mapping: Southern = DNA, Northern = RNA, Western = protein; all use gel separation followed by transfer and probe or antibody detection.

Sanger sequencing uses 2′,3′-dideoxynucleotides to terminate chain elongation, producing a ladder resolved by capillary electrophoresis.

Major cell-cycle checkpoints: the G1/S restriction point, the G2/M DNA-damage checkpoint, and the metaphase spindle-assembly checkpoint.

SpCas9 requires a 5′-NGG-3′ PAM immediately 3′ of the protospacer before it can bind and cleave the target DNA.

Cauchy–Riemann equations: u_x = v_y and u_y = −v_x; necessary for differentiability of f = u + iv, and sufficient when the partials are continuous.

Residue theorem: ∮_C f(z) dz = 2πi Σ Res(f, a_k) for poles inside a positively oriented simple closed contour.

Rank–nullity theorem: rank(A) + nullity(A) = n for any m × n matrix A, i.e. dim(im) + dim(ker) equals the domain dimension.

For any square matrix, trace = sum of eigenvalues and determinant = product of eigenvalues (counted with algebraic multiplicity).

Central limit theorem: standardised sums of iid variables with finite variance converge in distribution to N(0,1).

Lagrange's theorem: the order of a subgroup divides |G|, so the order of every element divides |G| in a finite group.

Newton–Raphson iteration x_{n+1} = x_n − f(x_n)/f′(x_n) converges quadratically near a simple root and fails where f′ ≈ 0.

Fermat's little theorem a^p ≡ a (mod p) for prime p; Euler's theorem a^φ(n) ≡ 1 (mod n) when gcd(a, n) = 1.

Bolzano–Weierstrass theorem: every bounded real sequence possesses a convergent subsequence.

Heisenberg relations: ΔxΔp ≥ ħ/2 and ΔEΔt ≥ ħ/2; the latter sets the natural linewidth of short-lived states.

Blackbody laws: Stefan–Boltzmann P = σAT⁴ with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, and Wien λ_max T = 2.898 × 10⁻³ m·K.

Bragg condition 2d sin θ = nλ governs constructive diffraction from lattice planes of spacing d at glancing angle θ.

Poynting vector S = (1/μ₀) E × B; for a plane wave the time-averaged intensity is E₀²/(2μ₀c).

Lorentz kinematics: γ = 1/√(1 − v²/c²), time dilation Δt = γΔt₀, length contraction L = L₀/γ.

Maxwell relations follow from exactness of thermodynamic differentials, e.g. (∂T/∂V)_S = −(∂p/∂S)_V from dU = TdS − p dV.

Fermi–Dirac occupation f(E) = 1/[exp((E − E_F)/kT) + 1]; at T = 0 all states below E_F are filled, and the 3D free-electron density of states goes as E^{1/2}.

Rydberg formula 1/λ = R(1/n₁² − 1/n₂²) with R = 1.097 × 10⁷ m⁻¹ gives hydrogen spectral lines.

Semiconductor band gaps: Ge ≈ 0.67 eV, Si ≈ 1.1 eV, GaAs ≈ 1.4 eV, diamond ≈ 5.5 eV at room temperature.

Heisenberg uncertainty relations ΔxΔp ≥ ħ/2 and ΔEΔt ≥ ħ/2 set fundamental limits on simultaneous sharpness of conjugate observables.

Blackbody radiation: Stefan–Boltzmann P = σAT⁴ (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴) and Wien's law λ_max T = 2.898 × 10⁻³ m·K.

Bragg's law 2d sin θ = nλ gives the angles of constructive X-ray interference from lattice planes with spacing d.

Poynting vector S = (1/μ₀) E × B gives electromagnetic energy flux; time-averaged intensity for a plane wave is E₀²/(2μ₀c).

Special relativity: γ = 1/√(1 − v²/c²); moving clocks run slow by Δt = γΔt₀ and lengths contract as L = L₀/γ.

Maxwell relations arise from exactness of thermodynamic potentials, e.g. (∂T/∂V)_S = −(∂p/∂S)_V from dU = TdS − p dV.

Fermi–Dirac distribution f(E) = 1/(exp[(E − E_F)/k_BT] + 1); the 3D free-electron density of states scales as E^{1/2}.

Rydberg formula 1/λ = R(1/n₁² − 1/n₂²) with R = 1.097 × 10⁷ m⁻¹ predicts hydrogen line positions.

Room-temperature band gaps: Ge ≈ 0.67 eV, Si ≈ 1.12 eV, GaAs ≈ 1.42 eV, diamond ≈ 5.5 eV.