Electrochemistry: Conductance, Kohlrausch’s Law, Debye–Hückel Theory and Transport Number

8.1Conductance, cell constant, specific and molar conductance

A solution of an electrolyte conducts by the movement of ions, not electrons. Its resistance is measured like that of a metal wire, but with alternating current (about 1 kHz) and platinised platinum electrodes, so that electrolysis and polarisation at the electrodes do not change the reading. The cell is placed in one arm of a Wheatstone bridge.

• Conductance G = 1/R, unit siemens (S = Ω⁻¹).
• Specific conductance (conductivity) κ is the conductance of a 1 cm cube of solution (or 1 m cube in SI). Units: S cm⁻¹ or S m⁻¹; 1 S m⁻¹ = 0.01 S cm⁻¹.
• Cell constant G* = l/A (cm⁻¹ or m⁻¹) depends only on the geometry of the cell. It is never measured with a ruler: the cell is filled with a standard KCl solution of known κ, and G* = κ(KCl) × R(KCl). For any other solution in the same cell, κ = G*/R. So “cell constant divided by resistance” is the specific conductance.
• Two solutions in the same cell: κ₂ = κ₁R₁/R₂. The cell constant cancels.

Molar and equivalent conductance

Molar conductance Λ_m is the conductance of all the ions from one mole of electrolyte, placed between electrodes 1 cm apart. Equivalent conductance Λ_eq is the same for one gram-equivalent.

• In the 1000κ/c form, κ is in S cm⁻¹ and c in mol L⁻¹. In SI, Λ_m = κ/c with κ in S m⁻¹ and c in mol m⁻³ (1 mol L⁻¹ = 1000 mol m⁻³), giving S m² mol⁻¹. 1 S cm² mol⁻¹ = 10⁻⁴ S m² mol⁻¹.
• For a salt with ν₊ cations of charge z₊: N = cν₊z₊, so Λ_eq = Λ_m/(ν₊z₊). For NaCl they are equal; for BaCl₂, Λ_eq = Λ_m/2.

What conductance depends on

• Nature of the electrolyte: strong electrolytes (salts, HCl, NaOH) are fully ionised and conduct well; weak ones (CH₃COOH, NH₄OH, HCOOH) are partly ionised. At the same concentration a weak acid has the lowest κ.
• Size and solvation of ions: a heavily hydrated ion is effectively bigger and moves slower.
• Viscosity of the solvent: a more viscous solvent slows the ions.
• Temperature: conductance of a solution rises by about 2 % per degree, because viscosity falls and weak electrolytes ionise more. (Metallic conductance falls with temperature.)
• Concentration, as below.

Effect of dilution

• κ falls on dilution: there are fewer ions in each cm³.
• Λ_m and Λ_eq rise on dilution: Λ counts the ions of one mole, and on dilution these ions move more freely. For a strong electrolyte the reason is weaker attraction between ions (thinner ionic atmosphere, 8.4); for a weak electrolyte it is mainly a larger degree of ionisation α.
• Question setters sometimes write “conductivity” in a question about molar conductivity. Read the stem to see which quantity is meant.

• Strong electrolytes follow Kohlrausch’s empirical law, a straight line in √c: Λ_m = Λ°_m − A√c. Extrapolating the line to c = 0 gives Λ°_m. Λ_m changes only a little on dilution (about 1.2 to 1.5 times).
• Weak electrolytes have a low Λ_m that shoots up only at very low concentration (often 20 times or more), because α keeps increasing. The curve cannot be extrapolated; Λ°_m of a weak electrolyte is found from Kohlrausch’s law of independent migration (8.2).

8.2Kohlrausch’s law, ionic conductance and ionic mobility

Kohlrausch’s law of independent migration of ions: at infinite dilution every ion moves independently of its partner, and contributes a fixed share, its ionic conductance λ°, to the limiting molar conductance of the electrolyte.

• Evidence: the difference between a potassium and a sodium salt of the same anion is the same for every anion: Λ°(KCl) − Λ°(NaCl) = 149.9 − 126.4 = 23.5 and Λ°(KNO₃) − Λ°(NaNO₃) = 145.0 − 121.5 = 23.5 S cm² mol⁻¹. The difference is λ°(K⁺) − λ°(Na⁺).
• For BaCl₂: Λ°_m = λ°(Ba²⁺) + 2λ°(Cl⁻) = 127.2 + 2(76.3) = 279.8 S cm² mol⁻¹ (= 279.8 × 10⁻⁴ S m² mol⁻¹). Tables may list λ° for “½Ba²⁺” (63.6); in molar terms double it.

Values at 25 °C in water. Papers usually give the data; these are for checking orders and for rough answers.

Λ° of a weak electrolyte by combination

Write the wanted electrolyte as a sum and difference of strong electrolytes whose Λ° can be measured, so that the unwanted ions cancel:

• With stoichiometry: Λ°(NH₄OH) = Λ°(NH₄Cl) + ½Λ°(Ba(OH)₂) − ½Λ°(BaCl₂); Λ°(CaI₂) = Λ°((CH₃COO)₂Ca) + 2Λ°(NaI) − 2Λ°(CH₃COONa).
• The same trick gives Λ° of any salt, strong or weak, from three others.

Ionic mobility

Ionic mobility u is the speed of an ion under a potential gradient of 1 V cm⁻¹ (unit cm² V⁻¹ s⁻¹). Ionic conductance and mobility are proportional:

• Hydration decides the order of alkali-metal ions. The smaller Li⁺ holds the largest hydration shell and moves slowest: mobility Li⁺ < Na⁺ < K⁺ < Rb⁺, so Λ°: LiCl < NaCl < KCl. Na⁺ moves slowly because of extensive hydration.
• Halide ions: F⁻ < Cl⁻ < Br⁻ ≈ I⁻ for the same reason. In molar terms ½SO₄²⁻ (80.0) is ahead of Br⁻ (78.1): SO₄²⁻ > Br⁻ > Cl⁻ > F⁻.
• H⁺ and OH⁻ are abnormally fast (349.8 and 198.6). Charge is passed by a proton jump along the hydrogen-bonded chain of water molecules (Grotthuss mechanism); no single ion has to travel the whole way. The amide ion NH₂⁻ in liquid ammonia is fast for the same reason.
• Mobility is speed divided by the potential gradient: at a fixed field it goes up with speed; written as u = v/(dV/dx) it is inversely proportional to the gradient.

Walden’s rule

For the same ions in different solvents, Λ°η° ≈ constant (η° = viscosity of the solvent). A more viscous solvent slows the ions in proportion. The rule holds only for large ions that are hardly solvated, such as tetraalkylammonium (R₄N⁺) and picrate, whose effective size is the same in every solvent. It fails for small ions, whose solvation shell, and so effective size, changes from solvent to solvent.

8.3Equivalent and molar conductance at infinite dilution

• Λ°_m (or Λ_∞) is the limiting value of Λ_m as c → 0. Here the ions are so far apart that they do not affect each other, and a weak electrolyte is fully ionised.
• Λ°_eq = Λ°_m/(ν₊z₊), as for any concentration.
• Strong electrolyte: extrapolate the straight Λ_m–√c line to zero (Fig. 8.1). Weak electrolyte: use Kohlrausch’s law (8.2); extrapolation fails because the curve rises almost vertically near c = 0.
• Answer options often differ only in units: 279 S cm² mol⁻¹ appears as 279 × 10⁻⁴ S m² mol⁻¹.