Electrolysis

Two definitions coexist, and knowing both prevents confusion. Arrhenius: acids increase [H⁺] in water, bases increase [OH⁻]. Brønsted–Lowry: an acid *donates* a proton, a base *accepts* one — which makes water amphiprotic (it can do either) and explains why ammonia, with no OH of its own, is basic in water.

Water self-ionises: Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ at 25 °C. That constant ties the two concentrations together at all times, and it is what the pH scale reports:

pH = −log₁₀[H⁺], and likewise pOH = −log₁₀[OH⁻], with pH + pOH = 14.00 at 25 °C.

Because it is a logarithm, the scale is factors of ten: pH 3 has one hundred times more H⁺ than pH 5; vinegar near pH 3 versus coffee near pH 5 differs by ~100×, while a pH change of 0.3 corresponds to a factor of two (log₁₀ 2 ≈ 0.301). Laboratory pH meters routinely read to ±0.01 — but the precision of the *reading* does not improve the precision of the underlying chemistry.

Strong acids (HCl, HNO₃, HClO₄, H₂SO₄ first proton) dissociate fully; weak acids dissociate by an equilibrium described by Ka, and pKa = −log Ka measures strength conveniently: acetic acid's pKa of 4.76 means at pH = pKa the acid and conjugate base are exactly 50/50. That equality is the backbone of the Henderson–Hasselbalch equation:

pH = pKa + log ([A⁻]/[HA])

It designs buffers — mix a weak acid with its conjugate base near the pKa, and the solution resists pH change because added H⁺ or OH⁻ is consumed by the partner species. Buffer capacity peaks at pH ≈ pKa and fades within ±1 pH unit, which is why real buffers are chosen with pKa values close to their working pH.

Titration curves follow: before the equivalence point the buffer region dominates, at the equivalence point moles of acid equal moles of base (for a weak acid–strong base, slightly above pH 7 because the conjugate base hydrolyses), and beyond it excess titrant controls the pH. Indicators change colour within ±1 pH unit of that jump.

Not everything dissolves, and the leftovers are governed by an equilibrium constant too. A sparingly soluble salt sets Ksp = [A⁺]ᵇ[B⁻]ᶜ from the solid's dissolution equation; comparing the ion product Q to Ksp predicts what happens: Q > Ksp precipitates, Q < Ksp stays clear, Q = Ksp is exactly saturated. The common-ion effect (adding a second source of the same ion) shifts the dissolution equilibrium left, lowering solubility — the reason CaSO₄ is less soluble in sulphate-rich water, and the tactic used to precipitate one ion while leaving another in solution. Temperature matters: most solids become more soluble as temperature rises, but gases do the opposite.

Electrochemistry converts the same bookkeeping of electrons into voltage. Split any redox reaction into two half-reactions and the electron transfer becomes measurable:

• Oxidation numbers track electron ownership: they assign formal ownership to atoms so that change in the number *is* electron flow. Sum of oxidation numbers equals the compound's charge.
• A galvanic cell converts spontaneous electron transfer (ΔG < 0) into electrical work, with E°cell = E°cathode − E°anode and ΔG° = −n F E°cell (n = electrons transferred, F = 96 485 C/mol).
• Standard potentials assume 1 M concentrations, 1 bar, 25 °C — real cells do not. Nernst's equation corrects: E = E° − (RT/nF) ln Q, often written E = E° − (0.0592/n) log Q at 25 °C. Each tenfold concentration change shifts the potential by 0.0592/n volts — small, measurable, and the basis of pH electrodes and sensors.

An electrolytic cell runs the other way: an applied voltage drives a non-spontaneous reaction, and the charge passed relates to material moved through Faraday's laws — mass = (current × time × molar mass) / (n F), the calculation behind electroplating and electrolysis throughput estimates (this is a quantitative relationship; actual cells need overpotential, cell voltage and safety engineering).

Conductivity follows from ion mobility: more ions or higher charge means higher conductivity — which is how water purity and salinity are monitored industrially.

Why does sodium give away an electron while chlorine grabs one? The answer sits in electron configurations, built from four quantum numbers: principal (n, energy shell), angular momentum (ℓ = 0…n−1, subshell shape), magnetic (mℓ, orbital orientation) and spin (ms = ±½). Subshell capacities follow 2(2ℓ + 1): s holds 2, p holds 6, d holds 10, f holds 14.

Filling rules: Aufbau order (1s 2s 2p 3s 3p 4s 3d …), Hund's rule — electrons occupy degenerate orbitals singly with parallel spins before pairing, because pairing costs repulsion energy — and Pauli exclusion. Cr and Cu break the naive pattern (3d⁵4s¹ and 3d¹⁰4s¹) because half-filled and filled subshells gain exchange-energy stability; remembering the exceptions saves whole families of mistakes.

From configuration flow the rest:

• Formal charge = valence electrons − (lone-pair electrons + ½ bonding electrons) — a Lewis-structure bookkeeping tool, not a real charge.
• Lewis structures place electrons to satisfy the octet rule (expanding it for period-3+ elements), revealing connectivity and lone pairs.
• Hybridisation (sp, sp², sp³) describes the orbital mixing that matches observed molecular geometry — count electron domains first, then assign hybridisation; geometry does not follow from hybridisation alone.
• Bond order = ½(bonding − antibonding) electrons — predicting bond strength and length (O₂'s bond order 2, O₂⁺ 2.5, O₂⁻ 1.5), and N₂'s triple bond explaining its inertness.
• Coordination compounds add a second counting game: ligands donate electron pairs to a metal centre, and crystal-field splitting explains why some complexes are coloured and why d⁵ high-spin vs low-spin cases differ. Splitting energy Δ relates to absorbed wavelength by Δ = hc/λ, linking colour directly to numbers.

Atomic spectra tie it together: electron transitions emit or absorb photons with ΔE = h f = hc/λ, producing discrete spectral lines; the Rydberg equation reproduces hydrogen's series, and molar absorptivity in Beer–Lambert work (A = ε b c) quantifies how strongly a species absorbs at a chosen wavelength.

Common errors: stopping at the noble-gas core without checking 4s/3d order; treating formal charge as real charge; confusing hybridisation and geometry; and using log and ln interchangeably in pH or Nernst work — pH uses log₁₀, thermodynamics often uses ln, and mixing them shifts results systematically.

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Key references for this formula

• NIST Chemistry WebBook — reference data for acid–base and thermochemical quantities of chemical species. ↗
• IUPAC Gold Book — official compendium of terminology used in chemistry, including acid–base, redox and equilibrium definitions. ↗
• OpenStax, Chemistry 2e — free peer-reviewed textbook covering acid–base equilibrium, electrochemistry and atomic structure. ↗