GTLE Subject Specialism Chemistry cheat sheet

Quantum numbers: n = 1,2,3…; l = 0…n−1 (s,p,d,f); m_l = −l…+l; m_s = ±½. Pauli exclusion forbids identical sets of all four.

Orbital capacity = 2(2l+1); subshell order by (n+l) rule, with lower n filling first when (n+l) ties (4s before 3d).

Effective nuclear charge Z_eff = Z − S (Slater); across a period Z_eff rises, so atomic radius falls and ionisation energy generally rises.

Exceptions to filling: Cr = [Ar]3d⁵4s¹ and Cu = [Ar]3d¹⁰4s¹; cations lose ns electrons before (n−1)d electrons (Fe²⁺ = [Ar]3d⁶).

First ionisation energy drops at group 13→14 (np electron shielded by ns²) and at group 15→16 (half-filled np³ → paired np⁴).

Electron affinity is most exothermic for Group 17; noble gases and alkaline earths have near-zero or endothermic values.

Periodic trends: radius increases down a group and decreases across; electronegativity (Pauling) increases up and across, F = 3.98 highest.

Paramagnetism requires unpaired electrons; count them from the orbital filling diagram, not from total electron number.

VSEPR electron-domain count (bonding + lone pairs) sets geometry: 2 linear, 3 trigonal planar, 4 tetrahedral, 5 trigonal bipyramidal, 6 octahedral; lone pairs compress angles (CH₄ 109.5° → NH₃ 107° → H₂O 104.5°).

Formal charge = valence e⁻ − nonbonding e⁻ − ½(bonding e⁻); the dominant resonance form minimises formal charge and places negative charge on the more electronegative atom.

Bond order from MO theory = (bonding − antibonding)/2; O₂ has bond order 2 and two unpaired π* electrons (paramagnetic), N₂ has bond order 3.

Hybridisation: steric number 2 → sp, 3 → sp², 4 → sp³, 5 → sp³d, 6 → sp³d²; π bonds form from unhybridised p orbitals.

Polarity requires polar bonds plus an asymmetric arrangement; CO₂ (linear) is nonpolar, H₂O (bent) is polar despite similar bond dipoles.

Hydrogen bonding needs H bonded to N, O or F; it explains water's high boiling point, high surface tension and ice's lower density.

Dispersion (London) forces scale with polarisability and surface area, so boiling points rise with molar mass and with less branching.

Lattice energy (Born–Landé, ∝ q₁q₂/r) rises with charge and falls with ionic radius; Fajans' rules: more covalent character with high cation charge, small cation, large anion.

Mole relations: n = m/M, n = N/N_A (N_A = 6.022×10²³ mol⁻¹), n = V/22.7 dm³ at STP (273 K, 100 kPa) or 22.4 dm³ at 273 K, 1 atm.

Ideal gas law PV = nRT with R = 8.314 J mol⁻¹ K⁻¹; for gas mixtures, partial pressure = mole fraction × total pressure.

Limiting reagent: divide moles of each reactant by its stoichiometric coefficient; the smallest quotient limits and sets theoretical yield.

Percentage yield = actual/theoretical × 100; percentage atom economy = (M of desired product / ΣM of reactants) × 100.

Concentration: c = n/V (mol dm⁻³); dilution obeys c₁V₁ = c₂V₂; mass concentration (g dm⁻³) = c × M.

Empirical formula from percentage composition: divide each % by atomic mass, then divide by the smallest ratio; molecular formula = empirical × (M_actual/M_empirical).

Titration relation at equivalence: c_aV_a/n_a = c_bV_b/n_b, where n_a and n_b are the stoichiometric coefficients from the balanced equation.

Combustion/elemental analysis: mass of CO₂ × 12/44 gives carbon mass, mass of H₂O × 2/18 gives hydrogen mass; oxygen is found by difference.

Kinetic theory assumptions: negligible molecular volume, no intermolecular forces except elastic collisions, random motion; average kinetic energy ∝ T (Kelvin).

Root-mean-square speed u_rms = √(3RT/M); at equal T, lighter gases move faster (Graham's law: rate₁/rate₂ = √(M₂/M₁)).

Maxwell–Boltzmann distribution: raising T shifts the peak to higher speed, lowers its height and broadens the curve while total area stays 1.

Real gases deviate most from ideality at high pressure and low temperature; van der Waals equation: (P + an²/V²)(V − nb) = nRT.

Critical temperature is the highest temperature at which a gas can be liquefied by pressure alone; above it no liquid phase exists.

Phase diagrams: the triple point is where all three phases coexist; the solid–liquid line usually slopes positively, but water's slopes negatively.

Heating curves show flat plateaus at melting and boiling points where added energy changes potential, not kinetic, energy.

Diffusion and effusion rates follow Graham's law; mean free path shortens as pressure rises because molecular collisions become more frequent.

First law: ΔU = q + w; for work against constant external pressure w = −PΔV, so at constant pressure q_p = ΔH = ΔU + PΔV.

Enthalpy of formation ΔH_f° is defined for one mole of compound from elements in their standard states; ΔH_f° of an element in its standard state is zero.

Hess's law: ΔH_rxn° = ΣΔH_f°(products) − ΣΔH_f°(reactants), or from bond enthalpies Σ(bonds broken) − Σ(bonds formed).

Calorimetry: q = mcΔT for solutions (c ≈ 4.18 J g⁻¹ K⁻¹ for water); q = CΔT for a bomb calorimeter, giving ΔU at constant volume.

Entropy: ΔS > 0 for gas formation, increased moles of gas, and increased temperature; ΔS°_rxn = ΣS°(products) − ΣS°(reactants).

Gibbs energy: ΔG° = ΔH° − TΔS°; spontaneous when ΔG < 0, and ΔG° = −RT ln K links thermodynamics to equilibrium.

ΔG° = −nFE°_cell and ΔG = ΔG° + RT ln Q; at equilibrium ΔG = 0 and Q = K.

Second law: ΔS_universe = ΔS_system + ΔS_surroundings > 0 for a spontaneous process; ΔS_surroundings = −ΔH/T.

K_p = K_c(RT)^Δn, where Δn = moles gaseous products − moles gaseous reactants; K is temperature-dependent only.

Reaction quotient Q: if Q < K the reaction proceeds forward, Q > K reverse, Q = K at equilibrium.

Le Chatelier: raising pressure favours the side with fewer gas moles; adding an inert gas at constant volume changes nothing.

Catalysts lower E_a by providing an alternative pathway and speed both directions equally; they do not change K or ΔH.

Arrhenius: k = Ae^(−E_a/RT); ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂); a plot of ln k vs 1/T has slope −E_a/R.

Rate laws are determined experimentally: rate = k[A]^m[B]^n; orders are not read from stoichiometric coefficients.

Integrated forms: zero order [A] = [A]₀ − kt; first order ln[A] = ln[A]₀ − kt (t½ = 0.693/k); second order 1/[A] = 1/[A]₀ + kt.

Rate-determining step is the slowest elementary step; the overall rate law must match the mechanism with intermediates eliminated.

Brønsted–Lowry: acids donate protons, bases accept; conjugate pairs differ by one H⁺. Lewis acids accept electron pairs (e.g. BF₃, AlCl₃).

K_w = [H⁺][OH⁻] = 1.0×10⁻¹⁴ at 25 °C; pH + pOH = 14; pH = −log[H⁺] for dilute aqueous solutions.

Strong acid/base: [H⁺] or [OH⁻] taken directly from concentration; weak acid uses K_a with x² ≈ K_a·c when dissociation is under ~5%.

Henderson–Hasselbalch: pH = pK_a + log([A⁻]/[HA]); a buffer works best when pH ≈ pK_a, effective over roughly pK_a ± 1.

Buffer capacity depends on absolute concentrations of the conjugate pair, not only their ratio; dilution lowers capacity but barely changes pH.

Salt hydrolysis: strong acid + strong base → neutral; weak acid + strong base → basic; strong acid + weak base → acidic; weak + weak depends on relative K_a and K_b.

Titration curves: equivalence pH equals 7 only for strong–strong; the half-equivalence point of a weak acid gives pH = pK_a.

Polyprotic acids: K_a1 ≫ K_a2, so the first dissociation dominates; amphiprotic species use pH ≈ ½(pK_a1 + pK_a2).

Oxidation is loss of electrons (anode); reduction is gain (cathode); in a galvanic cell oxidation occurs at the anode and electrons flow anode → cathode.

Cell potential E°_cell = E°_cathode − E°_anode using standard reduction potentials; a positive E°_cell means spontaneous as written.

Nernst equation: E = E° − (0.0592/n) log Q at 25 °C; E° = (0.0592/n) log K at equilibrium.

ΔG° = −nFE° with F = 96,485 C mol⁻¹; a spontaneous cell has E° > 0 and ΔG° < 0.

Balancing in acid: balance atoms, add H₂O for O, H⁺ for H, then electrons; in base add OH⁻ to neutralise H⁺ and combine to water.

Disproportionation is simultaneous oxidation and reduction of the same element, e.g. Cl₂ in cold dilute NaOH giving Cl⁻ and ClO⁻.

Electrolysis: moles of substance = It/(nF); the quantity of charge Q = It in coulombs, with n electrons per species.

Standard hydrogen electrode (H⁺/H₂, 1 M, 1 bar, 298 K) is assigned E° = 0.00 V; more positive E° means stronger oxidising agent.

Homologous series differ by CH₂; functional group determines reactivity while chain length mainly affects physical properties such as boiling point.

Alcohols (R–OH) hydrogen-bond and boil higher than isomeric ethers; primary, secondary and tertiary alcohols oxidise to aldehyde, ketone, or resist oxidation respectively.

Aldehydes (RCHO) reduce Tollens' and Fehling's reagents; ketones (R₂C=O) do not, which distinguishes them.

Carboxylic acids (RCOOH) are stronger acids than alcohols and phenols because the carboxylate anion is resonance-stabilised.

Amines are basic (lone pair on N); base strength in water follows R₂NH > RNH₂ > NH₃ > ArNH₂, with aromatic amines weakened by ring delocalisation.

Isomerism: structural (chain, position, functional) versus stereoisomerism (geometric cis/trans and optical, requiring a chiral centre with four different groups).

Aromaticity requires a planar cyclic conjugated system with 4n + 2 π electrons (Hückel); benzene has 6 π electrons and delocalised stability.

Nomenclature priority: carboxylic acid > ester > amide > nitrile > aldehyde > ketone > alcohol > amine > ether > alkene > alkyne > alkane > halide.

Nucleophilic substitution: S_N2 is one step, second order, backside attack giving inversion; S_N1 is two steps via carbocation, first order, giving racemisation.

S_N1/S_N2 competition: tertiary and benzylic/allylic halides favour S_N1; primary and methyl favour S_N2; strong bulky base or heat favours E2.

Carbocation stability: 3° > 2° > 1° > methyl; allylic and benzylic are stabilised by resonance; rearrangements (hydride/methyl shifts) can occur.

Electrophilic aromatic substitution: substituents that are ortho/para directing and activating have lone pairs on the atom attached to the ring (−OH, −NH₂, −OR); meta directors are deactivating (−NO₂, −COOH, −SO₃H).

Markovnikov addition of HX to alkenes puts H on the carbon with more hydrogens via the more stable carbocation; peroxide conditions give anti-Markovnikov.

Aldehyde/ketone nucleophilic addition: CN⁻ gives cyanohydrin, Grignard gives alcohol, NaBH₄/LiAlH₄ gives alcohol; the mechanism is nucleophilic attack on the carbonyl carbon.

Esterification is acid-catalysed Fischer esterification, reversible; hydrolysis under base (saponification) is irreversible because the carboxylate forms.

Oxidation/reduction toolkit: KMnO₄ or K₂Cr₂O₇/H⁺ oxidises primary alcohols to acids and aldehydes to acids; LiAlH₄ reduces acids and esters to alcohols.

Coordination number and geometry: 2 linear, 4 tetrahedral or square planar, 6 octahedral; square planar is common for d⁸ ions such as Pt(II) and Ni(II) with strong-field ligands.

Ligand field splitting: strong-field ligands (CN⁻, CO, NO₂⁻) give low-spin and larger Δ_o; weak-field (I⁻, Br⁻, Cl⁻, F⁻) give high-spin in the spectrochemical series.

Crystal field stabilisation and colour: d–d transitions give colour; d⁰ and d¹⁰ ions such as Zn²⁺ and Sc³⁺ are colourless.

Magnetic moment μ ≈ √(n(n+2)) Bohr magnetons for n unpaired d electrons; high-spin Fe³⁺ (d⁵) gives μ ≈ 5.9 BM.

Chelate effect: multidentate ligands such as EDTA increase stability and entropy, favouring complex formation.

Group trends: thermal stability of carbonates and nitrates increases down Group 2; hydroxide solubility increases down Group 2 but decreases down Group 17.

Transition metals show variable oxidation states, catalytic activity (Fe in Haber, V₂O₅ in contact process, MnO₂ in peroxide decomposition) and complex formation.

Qualitative tests: flame colours (Li crimson, Na yellow, K lilac, Ca brick-red, Cu blue-green); NH₃ solution dissolves Cu²⁺ giving deep blue [Cu(NH₃)₄]²⁺ and AgCl dissolves in excess ammonia.