Solutions, Phase Equilibria and Electrochemistry

In an ideal solution every component obeys Raoult’s law over the whole composition range: its partial vapour pressure is p_A = x_A p_A, where p_A is the vapour pressure of pure A. This happens when A–A, B–B and A–B interactions are alike (benzene–toluene), and then ΔmixH = 0 and ΔmixV = 0. The total pressure P = x_A p_A* + x_B p_B* is linear in x, and the vapour is richer in the more volatile component: y_A = x_A p_A*/P. Henry’s law describes a dilute solute: p_B = K_H x_B, with K_H not equal to p_B*. In an ideal-dilute solution the solvent obeys Raoult and the solute Henry, and Gibbs–Duhem guarantees that one implies the other.

Non-ideal solutions deviate. Positive deviation (p above Raoult, γ > 1) arises when A–B attractions are weaker than the average of A–A and B–B — ethanol–hexane, acetone–CS₂ — with ΔmixH > 0 and ΔmixV > 0. Negative deviation (p below Raoult, γ < 1) arises when A–B attractions are stronger — acetone–chloroform, which hydrogen-bond to each other, and water–HCl — with ΔmixH < 0. Large deviations produce an extremum in the vapour-pressure curve, which is an azeotrope.

The colligative properties of a dilute solution of a non-volatile solute depend only on the number of solute particles: the relative lowering of vapour pressure Δp/p_A* = x_B; the boiling-point elevation ΔT_b = iK_b m and freezing-point depression ΔT_f = iK_f m, with m the molality and K_b, K_f solvent constants (water: K_f = 1.86 and K_b = 0.512 K kg mol⁻¹); and the osmotic pressure π = icRT (van ’t Hoff). The van ’t Hoff factor i counts particles per formula unit: about 2 for NaCl, 3 for CaCl₂, below 1 for a solute that dimerises (acetic acid in benzene).

Gibbs’ phase rule F = C − P + 2 gives the number of intensive variables that can be changed independently, with C the number of components (independent species after subtracting reactions and fixed stoichiometric constraints) and P the number of phases. For one component, a single phase is bivariant (an area), two phases univariant (a line) and three phases invariant (a triple point). When pressure is fixed, as for most condensed systems, the reduced rule is F′ = C − P + 1.

Along any two-phase line the Clapeyron equation dP/dT = ΔH/(TΔV) holds. For liquid–vapour or solid–vapour equilibrium, neglecting the condensed-phase volume and treating the vapour as ideal, it becomes the Clausius–Clapeyron equation d ln P/dT = ΔH/(RT²), or ln(P₂/P₁) = −(ΔvapH/R)(1/T₂ − 1/T₁). For water’s solid–liquid line ΔV is negative (ice is less dense than water), so the melting line slopes backwards: pressure lowers the melting point.

Liquid–vapour. On a temperature–composition diagram at fixed pressure the liquid (bubble-point) curve lies below the vapour (dew-point) curve, and a tie line joins a liquid to the vapour in equilibrium with it; the lever rule gives the relative amounts of the two phases from the lengths of the tie-line segments. Each vaporisation–condensation step enriches the vapour in the lower-boiling component, and a column with many such theoretical plates performs fractional distillation, delivering the more volatile component at the top.

An azeotrope is a mixture that boils without change of composition, because at the extremum the liquid and vapour compositions coincide, so fractional distillation cannot pass it. Minimum-boiling azeotropes come from positive deviation — ethanol–water at about 95.6% ethanol by mass boils at 78.2 °C, below either component — and distillation gives the azeotrope at the top and the excess component at the bottom. Maximum-boiling azeotropes come from negative deviation — HCl–water at about 20% HCl near 109 °C — and the azeotrope then collects in the still.

Liquid–liquid. Partially miscible liquids form two layers inside a solubility curve. Phenol–water has an upper critical solution temperature near 66 °C, above which they mix in all proportions; triethylamine–water has a lower one near 18.5 °C; nicotine–water shows a closed loop with both. Solid–liquid. When two components are miscible as liquids but not as solids, adding either lowers the other’s freezing point, and the two liquidus curves meet at the eutectic point, the lowest temperature at which liquid exists; there the liquid freezes at constant temperature into an intimate mixture of the two solids, and with pressure fixed F′ = 2 − 3 + 1 = 0. Systems that form a compound with a congruent melting point show two eutectics, one on each side of the compound.

A galvanic cell is written anode | anode solution || cathode solution | cathode, with oxidation on the left. Electrode potentials are reduction potentials measured against the standard hydrogen electrode (Pt | H₂(1 bar) | H⁺(a = 1), E° = 0 by definition), and E°_cell = E°_cathode − E°_anode. A positive E_cell means the cell reaction as written is spontaneous. The link to thermodynamics is ΔᵣG = −nFE, so ΔᵣG° = −nFE° and, since ΔᵣG° = −RT ln K, ln K = nFE°/RT (log K = nE°/0.05916 at 298 K). The temperature coefficient of the emf gives the rest: ΔᵣS = nF(∂E/∂T)_P and ΔᵣH = −nF[E − T(∂E/∂T)_P].

The Nernst equation gives the emf away from standard conditions: E = E° − (RT/nF) ln Q = E° − (0.05916/n) log Q at 298 K, Q written in activities. Applications: cell potentials at any concentration; concentration cells, in which E° = 0 and the emf comes entirely from the activity ratio (M | Mⁿ⁺(a₁) || Mⁿ⁺(a₂) | M gives E = (0.05916/n) log(a₂/a₁)); pH measurement with the glass or hydrogen electrode (E changes 59.2 mV per pH unit); and solubility products and equilibrium constants from E°. Reference electrodes — the saturated calomel electrode (+0.241 V vs SHE) and Ag/AgCl — replace the SHE in practice.

In a potentiometric titration the emf of an indicator electrode is followed against titrant volume and the end point is the steepest point of the curve (the maximum of dE/dV); it needs no coloured indicator and works in turbid or coloured solutions, for acid–base, redox (Fe²⁺ with Ce⁴⁺) and precipitation (Cl⁻ with Ag⁺) reactions alike.

The conductivity κ of a solution (S m⁻¹) is found from its resistance and the cell constant l/A; the molar conductivity Λm = κ/c. For strong electrolytes Λm falls slightly with concentration according to Kohlrausch’s law Λm = Λm° − K√c, so Λm° comes from extrapolating against √c. Kohlrausch’s law of independent migration states that at infinite dilution each ion contributes independently: Λm° = ν₊λ₊° + ν₋λ₋°. This gives Λm° for weak electrolytes, which cannot be extrapolated, by combination: Λm°(CH₃COOH) = Λm°(CH₃COONa) + Λm°(HCl) − Λm°(NaCl). For a weak electrolyte the degree of dissociation is α = Λm/Λm° and Ostwald’s dilution law gives K_a = cα²/(1 − α).

The ionic mobility u is the drift speed per unit field (m² V⁻¹ s⁻¹), related to the ionic conductivity by λ = zuF, and the transport number t₊ = λ₊/(λ₊ + λ₋) is the fraction of current carried by the cation. H⁺ and OH⁻ have exceptionally high mobilities (λ° about 350 and 198 S cm² mol⁻¹, against about 50–76 for ordinary ions) because they move by proton hopping along hydrogen bonds (the Grotthuss mechanism). This is what shapes conductometric titrations: titrating a strong acid with a strong base replaces fast H⁺ by slow Na⁺, so conductance falls to a sharp minimum at equivalence and then rises as OH⁻ accumulates — a V-shaped curve; with a weak acid the conductance rises gently from a low start and more steeply after the end point.

Debye–Hückel theory explains why ions are less "active" than their concentration: each ion is surrounded by an ionic atmosphere of opposite charge that lowers its chemical potential. The limiting law is log γ± = −A|z₊z₋|√I, with A = 0.509 for water at 298 K and ionic strength I = ½Σcᵢzᵢ²; it holds for I up to about 0.01 M, and extended forms add a denominator 1 + Ba√I. The same atmosphere slows ions in a field through the relaxation and electrophoretic effects, and the Debye–Hückel–Onsager equation Λm = Λm° − (A + BΛm°)√c gives Kohlrausch’s empirical K from first principles.