Molar Conductivity and the Factors Affecting It – Conductivity of Solutions (Senior 5 Chemistry) | Ganzaa

Conductivity of Solutions

Lessons

Senior 5 · Chemistry

Conductivity of Solutions

4 lessons

Lesson 2 of 4

Molar Conductivity and the Factors Affecting It

25 min

A fair comparison between solutions

A 0.1 mol dm⁻³ solution of potassium chloride conducts about nine times better than a 0.01 mol dm⁻³ solution. That is no surprise: it holds ten times as many ions. To compare how well each mole of an electrolyte conducts, chemists divide the conductivity by the concentration. The result, the molar conductivity, behaves very differently for strong and weak electrolytes, and that difference is a window into how far a substance is ionised.

Definition

Molar conductivity, $\Lambda_m$ : the conductivity of a solution divided by its concentration, $\Lambda_m = \frac{\kappa}{c}$ . With $\kappa$ in S cm⁻¹ and $c$ in mol dm⁻³:

$\Lambda_m = \frac{1000\,\kappa}{c} \quad (\mathrm{S\,cm^{2}\,mol^{-1}})$

The factor 1000 converts $c$ from mol dm⁻³ to mol cm⁻³ (1 dm³ = 1000 cm³). Example: 0.0100 mol dm⁻³ KCl, $\kappa = 1.413 \times 10^{-3}\,\mathrm{S\,cm^{-1}}$ , so $\Lambda_m = 141.3\,\mathrm{S\,cm^{2}\,mol^{-1}}$ .

Limiting molar conductivity, $\Lambda^{\circ}_m$ : the molar conductivity at infinite dilution ( $c \to 0$ ), where ions are so far apart that they no longer affect each other.

Strong electrolyte: fully ionised in solution (NaCl, HCl, KOH). Weak electrolyte: only partly ionised (ethanoic acid, ammonia).

How molar conductivity changes with concentration

$c$ / mol dm⁻³	KCl: $\kappa$ / S cm⁻¹	KCl: $\Lambda_m$ / S cm² mol⁻¹	ethanoic acid: $\Lambda_m$ / S cm² mol⁻¹
0.001	$1.47 \times 10^{-4}$	146.9	48.4
0.010	$1.41 \times 10^{-3}$	141.3	16.0
0.100	$1.29 \times 10^{-2}$	129.0	5.1

Table 12.1: KCl values measured (CRC Handbook); ethanoic acid values calculated from $K_a = 1.75 \times 10^{-5}$ and $\Lambda^{\circ}_m = 390.6$ , which agree with measured values to within about 2 %.

Two trends stand out. First, on dilution, $\kappa$ falls (fewer ions in each cm³) but $\Lambda_m$ rises (each mole conducts better). Second, the rise is small for KCl but enormous for ethanoic acid.

Figure 12.2: $\Lambda_m$ plotted against $\sqrt{c}$ at 25 °C. KCl data from the CRC Handbook; the ethanoic acid curve is calculated.

Strong electrolytes are fully ionised at every concentration. In more concentrated solutions each ion is surrounded by an "ionic atmosphere" of oppositely charged ions that drags on it as it moves, so $\Lambda_m$ falls a little. Kohlrausch found that for dilute solutions the graph against $\sqrt{c}$ is a straight line: $\Lambda_m = \Lambda^{\circ}_m - K\sqrt{c}$ Extrapolating the line to $c = 0$ gives $\Lambda^{\circ}_m$ .
Weak electrolytes ionise more and more as they are diluted, so the number of ions per mole, and with it $\Lambda_m$ , rises steeply at low concentration. The curve is so steep near $c = 0$ that $\Lambda^{\circ}_m$ cannot be found by extrapolation; Lesson 3 shows how it is found instead.

Factors affecting molar conductivity

Factor	Effect on $\Lambda_m$
Type of electrolyte	strong electrolytes have much higher $\Lambda_m$ than weak ones at the same concentration
Concentration	$\Lambda_m$ increases on dilution: slightly (strong) or steeply (weak)
Charge and size of ions	higher charge carries more current; smaller hydrated size moves faster; $\mathrm{H^{+}}$ and $\mathrm{OH^{-}}$ are exceptionally fast
Temperature	increases $\Lambda_m$ (lower viscosity, faster ions; for weak electrolytes, often more ionisation)
Solvent	a more viscous or less polar solvent lowers $\Lambda_m$

Worked example

Problem. The conductivity of 0.0500 mol dm⁻³ hydrochloric acid at 25 °C is $1.996 \times 10^{-2}\,\mathrm{S\,cm^{-1}}$ . Calculate its molar conductivity, and compare it with KCl at a similar concentration (133.4 S cm² mol⁻¹ at 0.0500 mol dm⁻³).

Solution.

$\Lambda_m = \frac{1000 \times 1.996 \times 10^{-2}}{0.0500} = 399\,\mathrm{S\,cm^{2}\,mol^{-1}}$

Answer: 399 S cm² mol⁻¹, three times that of KCl. Both are fully ionised and share the $\mathrm{Cl^{-}}$ ion, so the difference comes from $\mathrm{H^{+}}$ , which moves far faster than $\mathrm{K^{+}}$ .

Worked example

Problem. For KCl at 25 °C, $\Lambda_m = 146.95$ S cm² mol⁻¹ at 0.00100 mol dm⁻³ and 141.27 S cm² mol⁻¹ at 0.0100 mol dm⁻³. Assuming $\Lambda_m = \Lambda^{\circ}_m - K\sqrt{c}$ , find $K$ and $\Lambda^{\circ}_m$ .

Solution. $\sqrt{0.00100} = 0.03162$ and $\sqrt{0.0100} = 0.1000$ .

$K = \frac{146.95 - 141.27}{0.1000 - 0.03162} = \frac{5.68}{0.06838} = 83.1$

$\Lambda^{\circ}_m = 146.95 + 83.1 \times 0.03162 = 149.6\,\mathrm{S\,cm^{2}\,mol^{-1}}$

Answer: $K = 83.1$ , $\Lambda^{\circ}_m = 149.6$ S cm² mol⁻¹, within 0.2 % of the accepted 149.9. Using only two points limits the accuracy; a straight line through many points is better.

Warning

Do not say "molar conductivity falls on dilution". Conductivity $\kappa$ falls on dilution; molar conductivity $\Lambda_m$ rises. Keep the two quantities, and their units, separate: $\kappa$ in S cm⁻¹, $\Lambda_m$ in S cm² mol⁻¹.

Tip

To get $\Lambda_m$ in S cm² mol⁻¹, remember "1000 κ over c", with $\kappa$ in S cm⁻¹ and $c$ in mol dm⁻³. If $\kappa$ is given in S m⁻¹, first divide by 100.

The Organic Chemistry Tutor: Identifying Strong Electrolytes, Weak Electrolytes and Nonelectrolytes. Watch which acids, bases and salts are strong, weak or non-electrolytes; this decides the shape of their molar conductivity curves.

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Practice

Calculate $\Lambda_m$ for a 0.0200 mol dm⁻³ solution with $\kappa = 2.77 \times 10^{-3}\,\mathrm{S\,cm^{-1}}$ .
The molar conductivity of 0.00500 mol dm⁻³ NaCl is 120.6 S cm² mol⁻¹. Calculate its conductivity.
On dilution, the conductivity of a solution decreases but its molar conductivity increases. Explain both changes.
Sketch, on the same axes, $\Lambda_m$ against $\sqrt{c}$ for NaOH and for aqueous ammonia, and explain the shapes.
For NaCl, $\Lambda_m = 123.74$ at 0.00100 and 118.51 at 0.0100 mol dm⁻³ (S cm² mol⁻¹). Estimate $\Lambda^{\circ}_m$ .
Why can $\Lambda^{\circ}_m$ for ethanoic acid not be found by extrapolating its graph?
A student says the molar conductivity of HCl is higher than that of NaCl because HCl is "more ionised". Evaluate this explanation.

Show answersHide answers

$\frac{1000 \times 2.77 \times 10^{-3}}{0.0200} = 138.5\,\mathrm{S\,cm^{2}\,mol^{-1}}$ .
$\kappa = \frac{120.6 \times 0.00500}{1000} = 6.03 \times 10^{-4}\,\mathrm{S\,cm^{-1}}$ .
$\kappa$ depends on the number of ions per unit volume, which falls on dilution. $\Lambda_m$ is per mole: on dilution, interionic attractions weaken (strong electrolytes) and ionisation increases (weak electrolytes), so each mole conducts better.
NaOH: high values, a nearly straight, gently falling line (strong, fully ionised). Ammonia: low values falling steeply as $\sqrt{c}$ rises, curving up sharply near $c = 0$ (weak, ionisation increases on dilution).
$K = \frac{123.74 - 118.51}{0.1000 - 0.03162} = 76.5$ ; $\Lambda^{\circ}_m = 123.74 + 76.5 \times 0.03162 = 126.2\,\mathrm{S\,cm^{2}\,mol^{-1}}$ , close to the accepted 126.4.
Its $\Lambda_m$ rises very steeply as $c \to 0$ , so the curve cannot be extended to the axis reliably, and the solution would have to be impossibly dilute to measure.
Wrong: both are strong electrolytes, fully ionised. The difference is the much greater mobility of $\mathrm{H^{+}}$ (349.7) compared with $\mathrm{Na^{+}}$ (50.1 S cm² mol⁻¹).

Sources and credits

Molar conductivities of KCl, NaCl and HCl, and the acid dissociation constant of ethanoic acid: CRC Handbook of Chemistry and Physics (equivalent conductivity of electrolytes in aqueous solution; dissociation constants of organic acids).
Video: Identifying Strong Electrolytes, Weak Electrolytes, and Nonelectrolytes, The Organic Chemistry Tutor, embedded from YouTube.
The text, Figure 12.2, the tables, worked examples and Practice items were written and drawn by Ganzaa.

1A fair comparison between solutions

2How molar conductivity changes with concentration

3Factors affecting molar conductivity

A fair comparison between solutions

How molar conductivity changes with concentration

Factors affecting molar conductivity