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Lesson 5 · 25 min

Conductometry: how fast ions move

Ultrapure water conducts only 0.055 µS/cm while seawater reaches 50 mS/cm. Ionic molar conductivities, Kohlrausch’s law and the proton relay of H⁺ explain why a conductometric titration curve is V-shaped.

In this lesson

  • Distinguish conductance, conductivity and molar conductivity, and convert with the cell constant
  • Use Kohlrausch’s law to build Λ° from ionic limiting conductivities, and explain why H⁺ and OH⁻ are so fast
  • Extract the degree of dissociation and dissociation constant of a weak electrolyte from conductivity
  • Predict and explain conductometric titration curves of strong acids, weak acids and mixtures

The ultrapure water of a chip fab must reach 18.2 MΩ·cm, a conductivity of just 0.055 µS/cm; seawater conducts about 50 mS/cm, nearly a million times more. Remarkably, that tiny conductivity of ultrapure water is not due to impurities but to the water’s own 10−710^{-7} M of HX+\ce{H+} and OHX−\ce{OH-} — their ionic conductivities give exactly 0.055 µS/cm. Conductivity meters are cheap, robust and fast, and they underpin water-quality monitoring, ion-chromatography detectors and conductometric titrations.

Predict first

Dilute hydrochloric acid is titrated with NaOH while the conductivity is monitored. How does the conductivity change before the equivalence point?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

Conductivity and molar conductivity

Solutions conduct because ions drift in an electric field. A conductivity meter measures the conductance G=1/RG = 1/R between two electrodes (in siemens, S) and multiplies it by the cell constant Kcell=l/AK_\text{cell} = l/A (electrode spacing over area, calibrated with standard KCl) to give the conductivity κ — the conductance of a 1 cm cube of solution. To avoid polarising the electrodes or electrolysing the sample, it uses alternating current near 1 kHz.

κ grows with concentration, so to compare electrolytes we divide by concentration and get the molar conductivity Λ=κ/c\Lambda = \kappa/c. If ions ignored one another, Λ would not depend on concentration; in fact Λ of a strong electrolyte falls slightly as c rises, because the “ionic atmosphere” around each ion holds it back. In the 1870s Kohlrausch found that in dilute solution Λ is a straight line in c\sqrt{c}, extrapolating at c→0c \to 0 to the limiting molar conductivity Λ° — and that Λ° splits into separate contributions from the cation and the anion: the law of independent migration of ions.

κ=G Kcell=∑iciλiΛ=κc=Λ∘−KcΛ∘=ν+λ+∘+ν−λ−∘\kappa = G\,K_\text{cell} = \sum_i c_i\lambda_i\qquad \Lambda = \frac{\kappa}{c} = \Lambda^\circ - K\sqrt{c}\qquad \Lambda^\circ = \nu_+\lambda_+^\circ + \nu_-\lambda_-^\circ
With κ in S/cm and c in mol/cm³, Λ is in S cm² mol⁻¹. For 1:1 electrolytes Onsager’s theory gives K = A + BΛ° (A = 60.20 and B = 0.229 at 25 °C).
κ\kappa
conductivity
KcellK_\text{cell}
cell constant l/A
λi\lambda_i
molar conductivity of ion i
Λ∘\Lambda^\circ
limiting molar conductivity at infinite dilution
Strong electrolytes extrapolate linearly to Λ°; weak ones get Λ° from the ions

Molar conductivity and Kohlrausch’s law

00.050.10.150.20.250.3√c / (mol/L)^½0100200300400Λ / (S cm² mol⁻¹)
  • HCl
  • KCl
  • NaCl
  • CH₃COOH
  • measured (CRC)
Λ° (sum of ions)Λ at c (model)Conductivity κ = Λ·c
HCl425.96410.194100 µS/cm
KCl149.79140.341400 µS/cm
NaCl126.39117.481170 µS/cm
CH₃COOH390.5516.27163 µS/cm

Acetic acid: α = Λ/Λ° = 0.0420

Independent migration: Λ°(KCl) − Λ°(NaCl) = 23.40 and Λ°(KNO₃) − Λ°(NaNO₃) = 23.40 S cm² mol⁻¹ — the difference ignores the anion.

Limiting ionic conductivities λ° at 25 °C

  • H⁺ 349.6
  • OH⁻ 198.0
  • K⁺ 73.5
  • NH₄⁺ 73.5
  • Cl⁻ 76.3
  • NO₃⁻ 71.4
  • Na⁺ 50.1
  • Li⁺ 38.7
  • CH₃COO⁻ 40.9
  • ½SO₄²⁻ 80.0

H⁺ and OH⁻ are exceptional: rather than whole ions moving, protons are relayed along the hydrogen-bond network (the Grotthuss mechanism).

Ionic λ° and the measured Λ values at each concentration come from the CRC Handbook (25 °C). Strong-electrolyte lines are the Debye–Hückel–Onsager limiting law Λ = Λ° − (A + BΛ°)√c (A = 60.20, B = 0.229), within 1 % of experiment below 0.01 M and too low above it; the acetic-acid line takes the exact degree of dissociation α from Ka (pKa 4.756), applies the Onsager correction to the ionised part and a Debye–Hückel activity correction — it stays within 3 % of the measurements.
The three strong electrolytes fall almost linearly with √c (open circles are CRC measurements); acetic acid lies flat along the bottom at high concentration and shoots up on dilution — it cannot be extrapolated to Λ° at all, which instead comes from λ°(H⁺) + λ°(CH₃COO⁻) = 390.6. Look at the bars below: H⁺ and OH⁻ are far ahead of every other ion.

Why are HX+\ce{H+} (349.65) and OHX−\ce{OH-} (198) several times faster than other ions? They do not push through the water as whole ions; charge is relayed along the hydrogen-bond network — one water molecule takes a proton while handing another on to its neighbour. This is the Grotthuss mechanism: the charge passes along like falling dominoes while hardly any atom moves far.

For a weak electrolyte only the ionised part conducts. Arrhenius took the degree of dissociation as α=Λ/Λ∘\alpha = \Lambda/\Lambda^\circ, and Ostwald combined it with the law of mass action in his dilution law, Ka=α2c/(1−α)K_\mathrm{a} = \alpha^2c/(1-\alpha) — historically the first way to measure a weak acid’s dissociation constant.

Worked example

Ka of acetic acid from conductivity

At 25 °C, 0.0100 M acetic acid has a conductivity of 1.63×10−41.63\times10^{-4} S/cm. With λ∘(HX+)=349.65\lambda^\circ(\ce{H+}) = 349.65 and λ∘(CHX3COOX−)=40.9\lambda^\circ(\ce{CH3COO-}) = 40.9 S cm² mol⁻¹, find α and KaK_\mathrm{a}.

Conductometric titrations

A conductometric titration looks for straight lines instead of a jump: on each side of the end point the ionic composition changes linearly, the conductivity plots as two (or more) straight segments, and they intersect at the end point. It suits very dilute or very weak systems where the potential break is poor:

  • strong acid + strong base: a V — fast HX+\ce{H+} is swapped for slow NaX+\ce{Na+}, then surplus OHX−\ce{OH-} arrives;
  • weak acid + strong base: hardly conducting at first, the growing CHX3COOX−\ce{CH3COO-} and NaX+\ce{Na+} raise the conductivity slowly, and after equivalence OHX−\ce{OH-} steepens the slope;
  • mixed acids: hydrochloric acid is neutralised first (falling), then acetic acid (rising slowly) — two breaks, two acids in one run;
  • weak acid + weak base (ammonia): after equivalence the surplus ammonia barely ionises, the curve goes flat, and the break is sharpest of all.

The titrant should be at least ten times more concentrated than the sample to limit dilution, or the readings can be multiplied by (V0+V)/V0(V_0 + V)/V_0.

κ = Σ cᵢλᵢ · composition from the exact acid–base equilibrium

Conductometric titration: watching ions being swapped

0246810Titrant volume V / mL0123Conductivity κ / (mS/cm)
  • H⁺
  • Na⁺
  • Cl⁻
  • CH₃COO⁻
  • OH⁻
  • Fitted lines and intersection

Intersection (end point) 1: 2.51 mL (Equivalence 2.50 mL)Intersection (end point) 2: 4.95 mL (Equivalence 5.00 mL)

At V = 2.00 mL: pH 2.98, κ = 0.927 mS/cm

Ionc / mMλ / (S cm² mol⁻¹)Contribution to κ
H⁺1.04341.939.9 %
Na⁺3.8547.220.4 %
Cl⁻4.8173.039.4 %
CH₃COO⁻0.079638.10.3 %
At every volume the composition (H⁺, OH⁻, Na⁺ or NH₄⁺, Cl⁻, CH₃COO⁻) comes from the exact charge balance of the acid–base kernel, and κ = Σ cᵢλᵢ with λ° from the CRC Handbook; with the Onsager correction each monovalent ion uses λᵢ = λᵢ° − (A/2 + Bλᵢ°)√I, reliable near 0.01 M and over-corrected above that. The dilution correction scales the reading back to the starting volume so the branches become straight; the end point is the intersection of straight lines fitted to the branches (skipping points within ±20 % of the break). The stacking order of the coloured areas only shows each ion’s share.
The hydrochloric–acetic mixture: drag the handle to see which ions carry the current at each stage and in what share. Turn off the dilution correction and the lines bend slightly and the intersections shift; turn off the Onsager correction to see how much interionic effects matter. Then try “acetic acid ← ammonia”.

Concept check

Titrating dilute acetic acid with NaOH, why does the conductivity slope rise sharply after the equivalence point?

Try the calculation

0.0100 M KCl has a molar conductivity of 141.27 S cm² mol⁻¹ at 25 °C. What is its conductivity in µS/cm? (This is the most common calibration standard for conductivity meters.)

µS/cm

Key takeaways

  • κ = G·K_cell = Σcᵢλᵢ; the molar conductivity Λ = κ/c falls linearly with √c in dilute solution (Kohlrausch).
  • Λ° = Σνλ° (independent migration); H⁺ and OH⁻ are relayed by the Grotthuss mechanism and run several times faster than other ions.
  • Weak electrolytes: α=Λ/Λ∘\alpha = \Lambda/\Lambda^\circ and Ka=α2c/(1−α)K_\mathrm{a} = \alpha^2c/(1-\alpha) (Ostwald’s dilution law).
  • A conductometric end point is the intersection of two straight lines, suited to very dilute, very weak or mixed acid–base systems.
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Data and references

  1. “Ionic conductivity and diffusion at infinite dilution” and “Equivalent conductivity of electrolytes in aqueous solution”, CRC Handbook of Chemistry and Physics, 97th ed. (2016), section 5.
  2. P. Atkins, J. de Paula, Atkins’ Physical Chemistry (Oxford University Press), chapter on molecular motion in liquids: ion mobility, Kohlrausch’s law and the Debye–Hückel–Onsager theory.
  3. J. Mendham, R. C. Denney, J. D. Barnes, M. Thomas, Vogel’s Textbook of Quantitative Chemical Analysis, 6th ed. (Prentice Hall, 2000), conductometric titrations.
  4. R. A. Robinson, R. H. Stokes, Electrolyte Solutions, 2nd ed. (Butterworths, 1959), Ch. 6–7.
  5. N. Agmon, “The Grotthuss mechanism”, Chem. Phys. Lett. 244, 456–462 (1995).