VKC Notes · Electrochemistry · Civil Services P-I · IFoS P-I · Geo-Scientist P-II and Prelims

8.3 Electrolytes: Conductance, Kohlrausch, Debye–Hückel and Transport Numbers

★★★ 2021–2025: 14 Mains parts, 124 marks (Civil Services 2 / 20 m · IFoS 6 / 64 m · Geo-Scientist 6 / 40 m), plus 44 Geo-Scientist Prelims questions (2021–2026): about seven a year in Chemistry Paper II, more than any other physical chemistry topic.

Syllabus lines. Civil Services and IFoS: “Debye-Huckel theory of strong electrolytes and Debye-Huckel limiting law for various equilibrium and transport properties.” Geo-Scientist Mains: “Activity and activity coefficients of electrolytes / ions in solution. Debye-Hückel limiting law.” Conductance, Kohlrausch’s law and transport numbers are asked heavily in Geo-Scientist Prelims. Cells, emf and the Nernst equation are in the 8.2 notes.

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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes

8.3.1 Conductance, cell constant and molar conductivity ★★★ Prelims every year

(8.3.1)

G^* = l/A is the cell constant, found with standard KCl: G^* = κ(KCl) × R(KCl). Then any solution in the same cell: κ = G^*/R. 0.1 M KCl, κ = 1.12 S m⁻¹, R = 55 Ω: G^* = 61.6 m⁻¹ = 0.616 cm⁻¹ Geo Pre 2021 Q70 Geo Pre 2024 Q66 Geo Pre 2025 Q72 Geo Pre 2026 Q70 Geo Pre 2022 Q71.SN
Units of Λ_m: S m² mol⁻¹ with c in mol m⁻³ (1 mol L⁻¹ = 1000 mol m⁻³), or S cm² mol⁻¹ with Λ_m = 1000 κ(S cm⁻¹)/c(mol L⁻¹). 1 S cm² mol⁻¹ = 10⁻⁴ S m² mol⁻¹.SN
Temperature: electrolytic conductance rises with temperature (about 2 % per K): the solvent's viscosity falls, ions move faster, and weak electrolytes dissociate more. Metallic conductance falls with temperature: stronger lattice vibrations scatter the electrons (ρ ∝ T) Geo 2021 Q11(c) 5m.SN

8.3.2 Molar conductivity against concentration ★★★

(8.3.2)

κ falls on dilution (fewer ions per cm³), but Λ_m rises: per mole, the ions interfere less (strong electrolytes) or more molecules dissociate (weak) Geo Pre 2022 Q78 Geo Pre 2025 Q79.SN
Strong (KCl, HCl): a straight line in √c; extrapolate to c = 0 for Λ°_m. Weak (CH₃COOH): Λ_m is low and rises steeply only at very low c; it cannot be extrapolated, so Λ°_m comes from Kohlrausch's law (8.3.3) Geo Pre 2021 Q74 Geo Pre 2024 Q72 Geo Pre 2023 Q73.SN
Why strong electrolytes fall at all: interionic attraction, explained by Debye–Hückel–Onsager (8.3.5) IFoS 2021 Q4(a) 15m.

Fig. 8.3.1 Λm against √c at 25 °C (measured values).

Draw it in 60 s: axes √c and Λm; a gently falling straight line from 150 (KCl, dashed to the axis); a curve hugging the axis that shoots up near √c = 0 (acetic acid).

8.3.3 Kohlrausch's law of independent migration and its uses ★★★

(8.3.3)

At infinite dilution each ion moves independently and contributes a fixed λ°. For BaCl₂: 127 + 2 × 76 = 279 S cm² mol⁻¹ = 279 × 10⁻⁴ S m² mol⁻¹ Geo Pre 2023 Q67.SN

Use	How	Prelims
Λ° of a weak electrolyte	Λ°(CH₃COONa) = Λ°(CH₃COOH) + Λ°(NaNO₃) − Λ°(HNO₃) = 391.0 + 121.5 − 421.2 = 91.3	Geo Pre 2024 Q67 Geo Pre 2025 Q73 Geo Pre 2026 Q71
Degree of dissociation, K_a	α = Λ_m/Λ°_m; K_a = cα²/(1 − α) (Ostwald). HCOOH, 0.02 M, Λ_m 40: α = 40/400 = 0.10, K_a = 2.2 × 10⁻⁴	Geo Pre 2022 Q76 Geo Pre 2023 Q75 Geo Pre 2024 Q75 Geo Pre 2025 Q79
Solubility, K_sp	s = κ/Λ°_m (κ of water subtracted). BaSO₄: 3 × 10⁻⁶/150 = 2 × 10⁻⁸ mol cm⁻³ = 2 × 10⁻⁵ M, K_sp = s² = 4 × 10⁻¹⁰; CaF₂: K_sp = 4s³ = 3.2 × 10⁻¹¹	Geo Pre 2022 Q77 Geo Pre 2023 Q74 Geo Pre 2024 Q76 Geo Pre 2026 Q73
Ionic product of water	κ(pure water) = 5.5 × 10⁻⁶ S m⁻¹; [H⁺] = κ/(λ°_H⁺ + λ°_OH⁻) = 5.5 × 10⁻⁶/0.0548 = 1.0 × 10⁻⁴ mol m⁻³ = 1.0 × 10⁻⁷ M	–

Solved PYQs CSE 2025 Q4(c) 10m IFoS 2022 Q1(d) 8m

0.1 M Ca(NO₃)₂, κ = 26.2 mS cm⁻¹ = 2.62 S m⁻¹; c = 100 mol m⁻³: Λ_m = 2.62/100 = 26.2 mS m² mol⁻¹.
Kohlrausch: Λ_m = λ(Ca²⁺) + 2λ(NO₃⁻); λ(Ca²⁺) = 26.2 − 2 × 7.14 = 11.9 mS m² mol⁻¹.
t(Ca²⁺) = 11.92/26.2 = 0.455; t(NO₃⁻) = 14.28/26.2 = 0.545; the two nitrate ions carry more of the current.SN
Pure water at 37 °C: van't Hoff, ln(K_w,2/10⁻¹⁴) = −(55 820/8.314)(1/310.15 − 1/298.15) = 0.871, K_w = 2.39 × 10⁻¹⁴; pH = ½pK_w = 6.81, still neutral since [H⁺] = [OH⁻].

8.3.4 Ionic mobility and transport numbers ★★★ Prelims

(8.3.4)

The transport number is the fraction of the current carried by an ion; it is proportional to its speed Geo Pre 2026 Q78. Hittorf's method measures concentration changes near the electrodes; the moving-boundary method times a visible boundary.SN
Moving boundary: t₊ = z c V F/Q (V = area × distance swept, Q = It). 0.01 M AX, V = 3.0 × 2.0 cm³, Q = 3.0 mA × 2700 s = 8.1 C: t = 10⁻⁵ × 6 × 96 485/8.1 = 0.71 Geo Pre 2023 Q79; AgNO₃ (1 M, 0.1 cm³, 27 C): 0.36 Geo Pre 2024 Q79 Geo Pre 2022 Q80 Geo Pre 2026 Q80.SN
Indicator electrolyte: shares the anion, has a slower cation and a denser solution placed below, so the boundary stays sharp: Cd(NO₃)₂ for AgNO₃, CdCl₂ for HCl Geo Pre 2021 Q80 Geo Pre 2023 Q78 Geo Pre 2024 Q80.SN
Abnormal mobilities: H⁺ (349.8) and OH⁻ (198.6) far exceed other ions (≈ 50–80): protons hop along hydrogen bonds (Grotthuss mechanism); NH₂⁻ in liquid NH₃ moves the same way Geo Pre 2025 Q74 Geo Pre 2023 Q72. Hydration: Li⁺ is smallest but most hydrated, so slowest: Λ°: KCl > NaCl > LiCl Geo Pre 2024 Q68.SN
Negative transport number: in concentrated CdI₂, Cd forms CdI₃⁻, CdI₄²⁻, so cadmium travels to the anode and its apparent t(Cd²⁺) becomes negative Geo Pre 2024 Q78.

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