Conductance, Conductivity & Molar Conductivity

CHEConductance & Kohlrausch's Law

This subtopic defines conductance, conductivity (specific conductance) and molar conductivity with their SI units, and how they vary with dilution. Numericals use the cell constant from resistance measurements and convert conductivity to molar conductivity, so getting units and the dilution trends right is essential.

Conductance and conductivity

G = \dfrac{1}{R} \qquad \kappa = G\cdot\dfrac{l}{A} = \dfrac{1}{R}\cdot\dfrac{l}{A}

G

in siemens

S

;

\kappa

(conductivity) in

S\,\text{cm}^{-1}

;

l/A

= cell constant (

\text{cm}^{-1}

Cell constant

\text{cell constant} = \dfrac{l}{A} = \kappa\, R

l

= electrode separation,

A

= electrode area; found using a solution of known

\kappa

(e.g. KCl).

Molar conductivity

\Lambda_m = \dfrac{\kappa \times 1000}{c}

\kappa

S\,\text{cm}^{-1}

c

\text{mol L}^{-1}

; gives

\Lambda_m

S\,\text{cm}^2\,\text{mol}^{-1}

; the

1000

converts

\text{cm}^3\to\text{L}

Debye-Huckel-Onsager (strong electrolyte)

\Lambda_m = \Lambda^\circ_m - A\sqrt{c}

linear in

\sqrt{c}

; intercept at

c\to0

gives the limiting molar conductivity

\Lambda^\circ_m

Conductance $G = \dfrac{1}{R}$ , unit siemens $S$ ( $\Omega^{-1}$ ); conductivity (specific conductance) $\kappa = G \times \dfrac{l}{A} = \dfrac{\text{cell constant}}{R}$ , unit $S\,\text{cm}^{-1}$ (SI $S\,\text{m}^{-1}$ ).
Cell constant $= \dfrac{l}{A}$ (unit $\text{cm}^{-1}$ ); from a resistance reading $\kappa = \dfrac{1}{R}\cdot\dfrac{l}{A}$ .
Conductivity $\kappa$ is the conductance of a unit cube of solution ( $1\,\text{cm}^3$ between electrodes $1\,\text{cm}^2$ apart, $1\,\text{cm}$ apart); it depends on the number of ions per unit volume.
Molar conductivity $\Lambda_m = \dfrac{\kappa \times 1000}{c}$ (with $\kappa$ in $S\,\text{cm}^{-1}$ and $c$ in $\text{mol L}^{-1}$ ), giving units $S\,\text{cm}^2\,\text{mol}^{-1}$ ; the factor $1000$ converts $\text{cm}^3$ to litres.
On dilution, molar conductivity $\Lambda_m$ increases (the same mole of ions is dispersed in more solution, and for weak electrolytes the degree of ionisation rises), but conductivity $\kappa$ decreases because the number of ions per unit volume falls.
Strong electrolytes: $\Lambda_m$ increases only slightly with dilution and follows $\Lambda_m = \Lambda^\circ_m - A\sqrt{c}$ (Debye-Huckel-Onsager); $\Lambda^\circ_m$ is found by extrapolating the straight $\Lambda_m$ vs $\sqrt{c}$ line to $c \to 0$ .
Weak electrolytes: $\Lambda_m$ rises steeply near infinite dilution and the $\Lambda_m$ vs $\sqrt{c}$ curve is not linear, so $\Lambda^\circ_m$ cannot be obtained by extrapolation — it is found instead using Kohlrausch's law.
Conductivity falls with dilution while molar conductivity rises — distinguish carefully: $\kappa$ is per unit volume, $\Lambda_m$ is per mole.
Conductance increases with temperature (ionic mobility rises), unlike metallic conductance which decreases with temperature; electrolytic conduction is by movement of ions, metallic by free electrons.
Worked check (KCl, $c=0.20\,\text{M}$ , $\kappa=2.48\times10^{-2}\,S\,\text{cm}^{-1}$ ): $\Lambda_m = \dfrac{2.48\times10^{-2}\times1000}{0.20} = 124\,S\,\text{cm}^2\,\text{mol}^{-1}$ .
Worked check (cell constant): for $l=0.5\,\text{cm}$ , $A=1.5\,\text{cm}^2$ , cell constant $=\dfrac{0.5}{1.5}=0.333\,\text{cm}^{-1}$ ; with $R=50\,\Omega$ , $\kappa = \dfrac{0.333}{50} = 6.67\times10^{-3}\,S\,\text{cm}^{-1}$ .
SI conversions: $1\,S\,\text{cm}^{-1} = 100\,S\,\text{m}^{-1}$ and $1\,S\,\text{cm}^2\,\text{mol}^{-1} = 10^{-4}\,S\,\text{m}^2\,\text{mol}^{-1}$ — watch units in problems quoting SI.

Where the marks go

Mixing up the two dilution trends: $\Lambda_m$ INCREASES on dilution while $\kappa$ DECREASES — stating both move the same way loses marks.
Forgetting the factor of $1000$ in $\Lambda_m = \dfrac{\kappa\times1000}{c}$ , or using $c$ in the wrong units (must be $\text{mol L}^{-1}$ when $\kappa$ is in $S\,\text{cm}^{-1}$ ).
Confusing cell constant ( $l/A$ , units $\text{cm}^{-1}$ ) with conductivity ( $\kappa$ , units $S\,\text{cm}^{-1}$ ).
Attempting to find $\Lambda^\circ_m$ of a weak electrolyte by extrapolating its $\Lambda_m$ vs $\sqrt{c}$ plot — the curve is non-linear and shoots up near $c\to0$ .
Reversing the temperature trend: electrolytic conductance rises with temperature; do not apply the metallic (decreasing) behaviour to electrolytes.

How the board asks it

Numericalcell constant, resistance and molar conductivity
The resistance of a $0.20\,M$ $KCl$ solution in a conductivity cell is $50\,\Omega$ . If the cell constant is $0.333\,\text{cm}^{-1}$ , calculate the conductivity $\kappa$ and the molar conductivity $\Lambda_m$ of the solution.
Give reasonsopposite dilution trends of conductivity and molar conductivity
Account for the fact that on dilution the molar conductivity $\Lambda_m$ of an electrolyte increases whereas its conductivity (specific conductance) $\kappa$ decreases.
Distinguishstrong vs weak electrolyte variation with $\sqrt{c}$
Distinguish between the variation of molar conductivity with $\sqrt{c}$ for a strong electrolyte and for a weak electrolyte, and state how $\Lambda^\circ_m$ is obtained in each case.
Define / statedefinitions and SI units
Define molar conductivity and conductivity (specific conductance), and give the SI unit of each.
Give reasonstemperature dependence of electrolytic vs metallic conduction
Explain why the conductance of an electrolytic solution increases with rise in temperature, whereas that of a metallic conductor decreases.
NumericalSI unit conversion of conductivity
A solution has conductivity $\kappa = 6.67\times10^{-3}\,S\,\text{cm}^{-1}$ . Express this value in the SI unit $S\,\text{m}^{-1}$ .

Practise this topic

PreviousΔG, ΔG° and Equilibrium Constant NextKohlrausch's Law, Degree of Dissociation & K_a

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.