Chemistry — Complete Formula Sheet

$z$: Number of particles (atoms or ions) per unit cell · $N_c$: Particles at the corners · $N_f$: Particles at face centres · $N_e$: Particles at edge centres · $N_b$: Particles at the body centre

A corner particle is shared by 8 unit cells, a face-centre particle by 2 and an edge-centre particle by 4; a body-centre particle belongs wholly to one cell. z is a pure number.

These three numbers start every density and packing numerical. Coordination numbers: simple cubic 6, bcc 8, fcc (ccp) and hcp 12.

$a$: Edge length of the unit cell (pm) · $r$: Atomic radius (pm)

Neighbouring atoms touch along the cube edge. Nearest-neighbour distance = a.

Atoms touch along the body diagonal, not along the edge. Nearest-neighbour distance = 2r = (√3/2)a.

Atoms touch along the face diagonal. Nearest-neighbour distance = 2r = a/√2.

$z$: Number of atoms per unit cell · $r$: Atomic radius · $a^3$: Volume of the cubic unit cell

Percentage of the unit-cell volume occupied by the spheres. Substitute the r–a relation of the lattice so that r cancels. Empty space = 100% − packing efficiency.

The least efficient cubic packing: 47.64% of the space is empty.

32% of the space in a bcc crystal is empty.

The most efficient packing of identical spheres; hcp has the same 74%. Only 26% of the space is empty.

$\rho$: Density (g cm⁻³) · $z$: Atoms per unit cell (1, 2 or 4) · $M$: Molar mass (g mol⁻¹) · $a$: Edge length (cm); 1 pm = 10⁻¹⁰ cm · $N_A$: Avogadro number = 6.022 × 10²³ mol⁻¹

z is the number of atoms per unit cell (the Maharashtra textbook writes n). With M in g mol⁻¹, a in cm and N_A = 6.022 × 10²³ mol⁻¹, ρ is in g cm⁻³. Rearrange the same equation to find z, a or M.

$x$: Mass of the metal sample (g) · $\rho a^3$: Mass of one unit cell (g)

ρa³ is the mass of one unit cell. The number of unit cells in a volume V of crystal is V/a³.

For N spheres in ccp or hcp. A tetrahedral void is surrounded by 4 spheres, an octahedral void by 6. To find a formula, count ions in the lattice and in the fraction of voids occupied, then take the simplest whole-number ratio.

Schottky: equal numbers of cations and anions missing (NaCl, KCl) — density decreases. Frenkel: the smaller ion, usually the cation, moves to an interstitial site (ZnS, AgCl) — density unchanged. Both keep the crystal electrically neutral. AgBr shows both.

Moles of solute per litre of solution. Temperature-dependent because volume changes with T.

Moles of solute per kilogram of solvent. Independent of temperature.

Dimensionless. Used in Raoult's law.

Partial vapour pressure = pure vapour pressure × mole fraction. Total pressure p = p_A + p_B.

Relative lowering of vapour pressure = mole fraction of solute. Directly gives molar mass.

K_b = molal elevation constant of solvent (K·kg/mol). Colligative.

K_f = molal depression constant of solvent (K·kg/mol). Colligative.

Van't Hoff equation for solutions (analogue of PV = nRT for gases). Useful for macromolecules.

i > 1 for dissociation (NaCl → 2); i < 1 for association (dimer → ½).

Multiply each colligative expression by van't Hoff factor to include electrolyte effect.

n = number of ions per formula unit of salt. e.g. NaCl → n = 2.

Partial pressure of a gas above liquid = K_H × its mole fraction in liquid.

α ≈ 1 for strong electrolytes and α ≪ 1 for weak electrolytes. Percent dissociation = 100α. No unit.

$K_a$: Acid dissociation (ionisation) constant · $\alpha$: Degree of dissociation · $c$: Initial molar concentration of the acid (mol L⁻¹)

For a weak monobasic acid HA. The approximation 1 − α ≈ 1 is valid only when α is small. α increases as the solution is diluted (c decreases). Not applicable to strong electrolytes.

For a weak base BOH such as NH₄OH: BOH ⇌ B⁺ + OH⁻. The algebra is the same as for a weak acid.

Substitute α = √(Ka/c) into [H₃O⁺] = αc. Then pH = −log₁₀[H₃O⁺].

Find pOH from [OH⁻] first, then pH = 14 − pOH at 298 K.

Holds for pure water and every aqueous solution. In pure water at 298 K, [H₃O⁺] = [OH⁻] = 1.0 × 10⁻⁷ M. Kw increases with temperature.

For a strong acid, [H₃O⁺] = c × number of H⁺ released (0.01 M HCl gives pH 2). For a strong base, [OH⁻] = c × number of OH⁻ released (Ba(OH)₂ gives 2c).

Obtained by taking −log₁₀ of both sides of Kw. At 298 K: pH < 7 acidic, pH = 7 neutral, pH > 7 basic.

The smaller the pKa, the stronger the acid; the smaller the pKb, the stronger the base.

Strong acid + strong base salt (NaCl): no hydrolysis, neutral. Weak acid + strong base salt (CH₃COONa): the anion hydrolyses, CH₃COO⁻ + H₂O ⇌ CH₃COOH + OH⁻, basic. Strong acid + weak base salt (NH₄Cl): the cation hydrolyses, NH₄⁺ + H₂O ⇌ NH₃ + H₃O⁺, acidic. pH values are for 298 K.

Both ions hydrolyse, so the relative strengths of the parent acid and base decide. CH₃COONH₄ is nearly neutral because Ka of CH₃COOH and Kb of NH₄OH are both about 1.8 × 10⁻⁵.

For a weak acid with its salt of a strong base (e.g. CH₃COOH + CH₃COONa). When [salt] = [acid], pH = pKa.

For a weak base with its salt of a strong acid (e.g. NH₄OH + NH₄Cl). Convert to pH = 14 − pOH at 298 K.

Product of the molar concentrations of the ions in a saturated solution, each raised to its coefficient. The solid does not appear. Ksp of a salt changes only with temperature.

$K_{sp}$: Solubility product · $S$: Molar solubility (mol L⁻¹) · $x,y$: Numbers of cations and anions in the formula AxBy

S in mol L⁻¹ (divide g L⁻¹ by molar mass). AB (AgCl, BaSO₄): Ksp = S². AB₂ or A₂B (CaF₂, PbI₂, Ag₂CrO₄): Ksp = 4S³. AB₃ (Fe(OH)₃): Ksp = 27S⁴. A₃B₂ (Ca₃(PO₄)₂): Ksp = 108S⁵.

Ionic product (IP) has the same form as Ksp but uses the actual concentrations after mixing; mixing equal volumes halves each concentration. Adding a common ion raises IP, so solubility decreases (common ion effect).

$W$: Work (J) · $P_{ext}$: Constant external pressure (Pa or bar) · $\Delta V$: Change in volume V₂ − V₁ (m³ or L)

Work against a constant external pressure. Expansion (ΔV > 0) gives W < 0; compression gives W > 0. Free expansion into vacuum (P_ext = 0) gives W = 0. 1 L bar = 100 J.

$n$: Amount of gas (mol) · $R$: Gas constant = 8.314 J K⁻¹ mol⁻¹ · $T$: Constant temperature (K) · $V_1,V_2$: Initial and final volumes

For n mol of an ideal gas at constant T on a reversible path. The work done by the gas has its largest possible magnitude. Use R = 8.314 J K⁻¹ mol⁻¹ to get W in joules.

At constant temperature Boyle's law gives V₂/V₁ = P₁/P₂. Note that the pressure ratio is initial over final.

$\Delta U$: Change in internal energy of the system (J) · $Q$: Heat exchanged; + when absorbed by the system (J) · $W$: Work; + when done on the system (J)

Sign convention used in the Maharashtra textbook (IUPAC): Q is positive when heat is absorbed by the system and negative when heat is released; W is positive when work is done ON the system and negative when work is done BY the system. Do not mix this with the older form ΔU = Q − W, in which W meant work done by the system.

U of an ideal gas depends only on T, so in isothermal expansion the heat absorbed equals the work done by the gas. In an adiabatic process work done on the system raises U (temperature rises); adiabatic expansion lowers U (temperature falls).

At constant volume no pressure–volume work is done, so the heat absorbed equals ΔU. At constant pressure part of the heat absorbed is used for expansion work.

At constant pressure the heat absorbed equals the enthalpy change. ΔH < 0 for exothermic and ΔH > 0 for endothermic processes.

$\Delta n_g$: Change in moles of gas (products − reactants) · $R$: 8.314 J K⁻¹ mol⁻¹ = 8.314 × 10⁻³ kJ K⁻¹ mol⁻¹

Δn_g = moles of gaseous products − moles of gaseous reactants; ignore solids and liquids. If Δn_g = 0, ΔH = ΔU. Equivalent form: Q_P = Q_V + Δn_g RT.

At constant T and P. If Δn_g > 0 the system expands and W is negative (work done by the system). If Δn_g < 0, W is positive (work done on the system).

Multiply each ΔfH° by its stoichiometric coefficient. ΔfH° of an element in its standard state (H₂(g), O₂(g), C(graphite)) is zero.

For reactions in the gas phase. The order is reactants − products, the opposite of the formation formula, because breaking bonds absorbs energy and forming bonds releases it.

Enthalpy is a state function, so ΔH is the same whether the change takes place in one step or several. Thermochemical equations can be added, subtracted and multiplied like algebraic equations. Example: ΔsubH = ΔfusH + ΔvapH.

Heat exchanged reversibly divided by the absolute temperature. Unit J K⁻¹ (J K⁻¹ mol⁻¹ for molar values). For a phase change at its transition temperature, ΔS = ΔH/T, e.g. ΔvapS = ΔvapH/T_b.

A process is spontaneous when the total entropy of system and surroundings increases; at equilibrium ΔS_total = 0. At constant T and P, ΔS_surr = −ΔH_sys/T.

At constant T and P. Use T in kelvin and express ΔS in kJ K⁻¹ when ΔH is in kJ. Standard form: ΔG° = ΔH° − TΔS°.

ΔH < 0, ΔS > 0: spontaneous at all T. ΔH > 0, ΔS < 0: non-spontaneous at all T. ΔH < 0, ΔS < 0: spontaneous below T = ΔH/ΔS. ΔH > 0, ΔS > 0: spontaneous above T = ΔH/ΔS. Assumes ΔH and ΔS do not change with T.

ΔG° < 0 means K > 1 (products favoured); ΔG° > 0 means K < 1. With R = 8.314 J K⁻¹ mol⁻¹, ΔG° must be in J mol⁻¹.

Use standard reduction potentials for both. Cell is spontaneous if E°_cell > 0.

Links thermodynamics to electrochemistry. n = number of moles of electrons, F = 96500 C/mol.

Applies to a single half-cell. In Balbharati form base 10 log with prefactor 0.0591/n V.

For general cell reaction. At equilibrium E_cell = 0 and Q = K.

Larger E° ⇒ larger K ⇒ more spontaneous reaction.

Mass deposited proportional to charge passed. Z = electrochemical equivalent (kg/C).

For same charge, masses deposited proportional to equivalent weights.

$\kappa$: Specific conductivity (S/m) · ${\Lambda }_m$: Molar conductivity (S·cm²·mol⁻¹) · $c$: Concentration (mol/L)

κ in S·cm⁻¹ (or S·m⁻¹). Λ_m in S·cm²·mol⁻¹ if c in mol/L, κ in S/cm.

At infinite dilution, the limiting molar conductivity is the sum of ionic contributions.

Fraction of molecules dissociated; ratio of measured to infinite-dilution molar conductivity.

For a weak monobasic acid HA. When α ≪ 1, K_a ≈ cα² ⇒ α = √(K_a/c).

Only geometric; unit cm⁻¹ or m⁻¹.

For aA → bB. Sign is chosen so that rate is positive.

m, n are determined experimentally, NOT from stoichiometry. m + n = order.

Concentration decreases linearly with time. k has units mol·L⁻¹·s⁻¹.

Log form. Slope of log[A] vs t gives −k/2.303.

Independent of initial concentration. Signature of first-order kinetics.

For a single reactant with rate = k[A]². Plot 1/[A] vs t gives straight line.

Depends on initial concentration — distinguishes it from first-order.

A = frequency factor, E_a = activation energy. Fundamental temperature dependence of rate.

Straight line: log k vs 1/T with slope −E_a/(2.303R).

Two-point form of Arrhenius equation. Useful when calculating E_a.

Rate roughly doubles for every 10 K rise near room temperature.

Primary valency ⇌ oxidation state; secondary valency ⇌ coordination number.

Common CNs: 2 (linear), 4 (tetrahedral/square planar), 6 (octahedral).

Neutral ligands don't add charge; anionic ligands add their negative charge.

Neutral: aqua, ammine, carbonyl, nitrosyl. Cationic: −ium suffix.

d-orbitals split: t2g (lower, 3 orbitals) and eg (higher, 2 orbitals). Δ_o depends on ligand strength.

For same metal and ligand. Tetrahedral splitting is smaller and pairing rarely occurs.

Sum electron contributions in Δ_o units. Ignore pairing energy in this simple form.

n = number of unpaired electrons. BM = Bohr magneton.

Sidgwick's rule. Stable complexes often have EAN = noble-gas configuration.

Strong field ligands (CN⁻, CO) → large Δ_o → low-spin. Weak field → small Δ_o → high-spin.

P = pairing energy. Compare Δ_o with P to predict spin state.

Second-order overall (bimolecular). Rate depends on both substrate and nucleophile concentrations.

First-order in substrate only. Rate-determining step is carbocation formation.

Reverse of SN2. Tertiary carbocations most stable.

Steric hindrance blocks back-side attack for bulky substrates.

Weaker C–X bond ⇒ better leaving group. Iodide leaves easiest.

E2 gives Zaitsev alkene (more substituted). E1 also usually Zaitsev but may rearrange.

Two alkyl halides coupled by sodium. Used to make symmetric alkanes.

Alkyl chloride/bromide → alkyl iodide via SN2 with NaI in acetone.

Alkyl fluoride prep from alkyl chloride using AgF, Hg₂F₂, or SbF₃.

Alkyl magnesium halide; C in R–MgX has partial negative charge; strong nucleophile.

Aryl halides are ~10⁵ times less reactive under normal SN conditions.

Amide bond formed by condensation between −COOH and −NH₂ of two amino acids.

At isoelectric pH, amino acid exists as dipolar zwitterion. Explains high melting point and water solubility.

pH at which amino acid carries no net charge. Precipitates most easily here.

Formed between anomeric −OH of one sugar and −OH of another. Basis of polysaccharides.

Glucose forms pyranose (6-membered), fructose forms furanose (5-membered). Anomeric C carries new −OH.

Chargaff's rule: %A = %T and %G = %C in any DNA.

DNA lacks 2′-OH; RNA has 2′-OH. Base uracil replaces thymine in RNA.

Enzyme reaction rate. K_m = substrate concentration at half V_max. Not always in Maharashtra syllabus but appears in HOTS.

Mutarotation: α-glucose (+112°) and β-glucose (+18.7°) equilibrate to give +52.7°.

Amylose forms complex with iodine giving characteristic blue colour. Cellulose does not.