Conductivity and molar conductivity, and how they are measured
Conductivity, κ, is the conductance of a unit cube of solution. Molar conductivity Λm = κ/c compares electrolytes mole for mole. Resistance is measured with AC in a conductivity cell.
Resistance, conductance and conductivity
The resistance R of a solution (in ohm, Ω) depends on its length l and area of cross-section A: R = ρ(l/A), where ρ is the resistivity (Ω m). Conductance G = 1/R, measured in siemens (1 S = 1 Ω⁻¹). Conductivity κ (kappa) = 1/ρ, in S m⁻¹. It is the conductance of a solution 1 m long with a cross-section of 1 m². The conductivity of an ionic solution depends on five things: the nature of the electrolyte, the size and solvation of its ions, the solvent and its viscosity, the concentration and the temperature. It rises as temperature rises.
Molar conductivity
Conductivity alone is not a fair way to compare electrolytes, because a more concentrated solution simply has more ions. Molar conductivity, Λm = κ/c, is the conductivity per unit concentration. Picture the conductance of a solution containing 1 mole of electrolyte, held between two large electrodes a unit distance apart. With κ in S cm⁻¹ and c in mol L⁻¹, use Λm = (κ × 1000) ÷ c to get S cm² mol⁻¹, because 1 L = 1000 cm³.
Measuring the conductivity of a solution
- 1Fill a conductivity cell with the solution. It has two platinum electrodes, coated with platinum black, fixed at a set distance apart.
- 2Connect the cell as one arm of a Wheatstone bridge, with an AC source and a detector such as a headphone.
- 3AC is used because a steady DC would cause electrolysis at the electrodes and change the solution.
- 4Adjust the variable resistance until no current flows through the detector. The resistance R of the cell then follows from the other three resistances.
- 5Find the cell constant by first filling the cell with a KCl solution of known κ: G* = R × κ.
- 6For the solution under test, **κ = G*/R**, and then Λm = κ/c.
How to draw the measuring set-up
- Draw a Wheatstone bridge: four resistances in a diamond shape.
- Make one arm the conductivity cell (a vessel with two Pt electrodes) and another arm a variable resistance.
- Join an AC source across one diagonal and a detector (headphone) across the other.
- Label: conductivity cell, Pt electrodes, variable resistance, fixed resistances, AC source, detector.
Worked example: cell constant and molar conductivity
A conductivity cell filled with 0.1 M KCl (κ = 0.0129 S cm⁻¹ at 298 K) has a resistance of 100 Ω. Filled with a 0.02 M solution of a salt, the same cell has a resistance of 645 Ω. Find (a) the cell constant, (b) κ of the salt solution and (c) its molar conductivity.
- (a) G* = R × κ = 100 Ω × 0.0129 S cm⁻¹ = 1.29 cm⁻¹.
- (b) κ = G*/R = 1.29 cm⁻¹ ÷ 645 Ω = 0.0020 S cm⁻¹ = 2.0 × 10⁻³ S cm⁻¹.
- (c) Λm = (κ × 1000) ÷ c = (2.0 × 10⁻³ × 1000) ÷ 0.02.
- Λm = 2.0 ÷ 0.02 = 100 S cm² mol⁻¹.
- In SI units: 100 × 10⁻⁴ = 1.0 × 10⁻² S m² mol⁻¹.
(a) 1.29 cm⁻¹; (b) 2.0 × 10⁻³ S cm⁻¹; (c) 100 S cm² mol⁻¹, or 1.0 × 10⁻² S m² mol⁻¹
Symbols and units
| Quantity | Symbol and formula | SI unit | Common unit |
|---|---|---|---|
| Resistance | R | Ω | Ω |
| Resistivity | ρ = R × A/l | Ω m | Ω cm |
| Conductance | G = 1/R | S | S |
| Conductivity | κ = 1/ρ | S m⁻¹ | S cm⁻¹ |
| Cell constant | G* = l/A | m⁻¹ | cm⁻¹ |
| Molar conductivity | Λm = κ/c | S m² mol⁻¹ | S cm² mol⁻¹ |
The TDS meter sold with home water purifiers in India is really a conductivity meter: dissolved salts raise κ, while very pure water conducts very poorly.
2 marks: “Define conductivity and molar conductivity, and give the SI unit of each.” Conductivity κ is the conductance of a unit cube of solution (1 m long, 1 m² cross-section); unit S m⁻¹. Molar conductivity Λm = κ/c; unit S m² mol⁻¹.
3 marks: “A 0.05 M solution of an electrolyte has a resistance of 250 Ω in a cell whose cell constant is 1.5 cm⁻¹. Calculate its conductivity and molar conductivity.” κ = G*/R = 1.5 ÷ 250 = 6.0 × 10⁻³ S cm⁻¹. Λm = (κ × 1000) ÷ c = 6.0 ÷ 0.05 = 120 S cm² mol⁻¹.
2 marks: “Why is alternating current used to measure the resistance of an electrolytic solution?” Direct current would cause electrolysis at the electrodes, which changes the concentration and so the resistance. AC prevents this.
Kappa is per cube, Lambda is per mole. Λm = κ × 1000 ÷ M gives S cm² mol⁻¹.