Lesson 5 · 30 min

Fast reactions: relaxation and the diffusion limit

Neutralisation is over before mixing is — so how is it measured? Eigen nudged systems already at equilibrium and watched them return: pure water relaxes in about 36 μs. Then compute diffusion coefficients from ionic conductivities and see why H⁺ and OH⁻ combine at 1.4 × 10¹¹ L mol⁻¹ s⁻¹.

In this lesson

Explain how relaxation methods measure reactions faster than mixing, and obtain rate constants from 1/τ = k₊([A] + [B]) + k₋
Estimate the upper limit set by diffusion with the Smoluchowski equation, and explain the effect of ionic charges
Compute diffusion coefficients from ionic conductivities and compare the neutralisation rate of H⁺ and OH⁻ with Eigen’s measurement

Pour acid into base and neutralisation is over “instantly” — but how fast is that? Mixing two solutions takes at least a millisecond or so, and neutralisation is finished long before. In the first half of the twentieth century chemists could only call it “immeasurably fast”.

Around 1954 the German physical chemist Manfred Eigen tried another route: rather than mixing, give a system already at equilibrium a sudden small push — a temperature jump of a degree or two, or a pulse of strong electric field — and watch it return. The time it takes, the relaxation time, is fixed directly by the forward and reverse rate constants and can be as short as nanoseconds. Eigen shared the 1967 Nobel Prize in Chemistry with Norrish and Porter; his lecture was titled “Immeasurably fast reactions”.

Predict first

In pure water [H⁺] = [OH⁻] = 1.0 × 10⁻⁷ mol/L. A pulse of electric field slightly upsets the equilibrium H₂O ⇌ H⁺ + OH⁻. Roughly how long does it take to recover?

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

About 1 ns About 40 μs About 1 s About 1 h

Relaxation: nudge an equilibrium

For A + B ⇌ C, suppose the equilibrium is pushed away by a small amount x. The forward rate k₊[A][B] and the reverse rate k₋[C] no longer match, and the difference pulls the system back. As long as x is small the difference is proportional to x, and x decays exponentially: x = x₀ e^(−t/τ). The relaxation time depends only on the rate constants and the equilibrium concentrations.

So measure τ at a few different concentrations, plot 1/τ against ([A] + [B]), and the slope is k₊ and the intercept k₋ — reactions a million times faster than mixing, measured without mixing anything.

\frac{1}{\tau} = k_+\left([\mathrm{A}]_\mathrm{eq} + [\mathrm{B}]_\mathrm{eq}\right) + k_-

For the ionisation of water, H₂O ⇌ H⁺ + OH⁻, the water concentration barely changes and 1/τ = k₊([H⁺] + [OH⁻]).

$\tau$: relaxation time
$k_+$: forward (association) rate constant
$k_-$: reverse (dissociation) rate constant

H₂O ⇌ H⁺ + OH⁻, k = 1.4 × 10¹¹ L mol⁻¹ s⁻¹ (Eigen) · 25 °C

Chemical relaxation: nudge an equilibrium

Solution pH = 7.0

Conductivity departure from equilibrium after the pulse

Relaxation time against pH

1/τ = k([H⁺] + [OH⁻]): at pH 7.0, [H⁺] + [OH⁻] = 2.0 × 10⁻⁷ mol/L and τ = 35.7 μs.

Drag the pH: pure water (pH 7) relaxes slowest, about 36 μs; add a little acid to reach pH 4 and [H⁺] rises a thousandfold while τ drops to about 71 ns — the more ions, the more frequent the encounters.

The diffusion limit: how fast can partners meet?

If two molecules react on every encounter, the rate is set by how fast they find each other in solution — diffusion control. In 1917 Smoluchowski showed that a “reaction sphere” of radius R absorbing molecules that diffuse towards it gives a rate constant k_D = 4πN_A R(D_A + D_B). For neutral molecules of equal size, substituting the Stokes–Einstein relation makes R cancel: k_D = 8RT/(3η), about 7 × 10⁹ L mol⁻¹ s⁻¹ in water at 25 °C, whatever the molecules are.

Ions also feel Coulomb forces. Debye (1942) supplied the correction factor: opposite charges attract and meet faster, like charges repel and meet more slowly. H⁺ and OH⁻ carry opposite charges and also diffuse unusually fast, because a proton can relay along chains of hydrogen bonds (their diffusion coefficients follow from their very large molar conductivities) — so their neutralisation is among the fastest bimolecular reactions in water.

k_\mathrm{D} = 4\pi N_\mathrm{A} R\,(D_\mathrm{A} + D_\mathrm{B})\,f,\qquad f = \frac{\delta}{e^{\delta} - 1},\quad \delta = \frac{z_\mathrm{A} z_\mathrm{B}\, l_\mathrm{B}}{R}

l_B = e²/(4πε₀εᵣkT) is the Bjerrum length, about 0.71 nm in water at 25 °C; for neutral molecules f = 1.

$R$: encounter (reaction) distance
$D$: diffusion coefficient
$f$: Debye electrostatic factor
$l_\mathrm{B}$: Bjerrum length

D from ionic conductivities · Smoluchowski–Debye model · compared with Eigen’s measurement

How fast can a reaction be? The diffusion limit

Encounter distance R = 0.50 nm

Diffusion-limited rate constant of H⁺ + OH⁻

D(H⁺) = 9.31 × 10⁻⁹ m²/s and D(OH⁻) = 5.27 × 10⁻⁹ m²/s (from λ°); the attraction between opposite charges speeds up encounters 1.88-fold; k_D = 1.0 × 10¹¹ L mol⁻¹ s⁻¹.

For comparison, two neutral molecules of equal size in water at 25 °C have k_D = 8RT/(3η) = 7.4 × 10⁹ L mol⁻¹ s⁻¹.

From λ°, D(H⁺) = 9.31 × 10⁻⁹ m²/s and D(OH⁻) = 5.27 × 10⁻⁹ m²/s. Pull the encounter distance out to about 0.87 nm and the model meets Eigen’s 1.4 × 10¹¹ — larger than the ions themselves, a sign that the proton jumps across a few water molecules along a hydrogen-bond chain.

Worked example

Relaxation after acidifying

HCl is added to pure water to make [H⁺] = 1.0 × 10⁻⁴ mol/L. What is the relaxation time of H₂O ⇌ H⁺ + OH⁻? (k₊ = 1.4 × 10¹¹ L mol⁻¹ s⁻¹, K_w = 1.0 × 10⁻¹⁴)

Concept check

Why is the neutralisation of H⁺ and OH⁻ about 20 times faster than the diffusion limit for two neutral molecules (about 7 × 10⁹)?

Both diffuse unusually fast (the proton relays along hydrogen bonds), and their opposite charges attract The neutralisation has a negative activation energy The viscosity of water does not affect ions They combine before they meet

Try the calculation

Use k_D = 8RT/(3η) to estimate the diffusion-limited rate constant for two neutral molecules in water at 25 °C (η = 0.890 mPa s), in units of 10⁹ L mol⁻¹ s⁻¹.

Key takeaways

Relaxation methods disturb a system at equilibrium and time its return; for A + B ⇌ C, 1/τ = k₊([A] + [B]) + k₋, and pure water relaxes in about 36 μs.
The diffusion limit is k_D = 4πN_A R(D_A + D_B) f; equal-sized neutral molecules in water reach about 7 × 10⁹ L mol⁻¹ s⁻¹, and oppositely charged ions go faster thanks to their attraction.
The neutralisation H⁺ + OH⁻ has k = 1.4 × 10¹¹ L mol⁻¹ s⁻¹ (Eigen): diffusion coefficients from λ° plus the Debye factor reproduce it with an encounter distance of about 0.87 nm.

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Data and references

M. Eigen, “Immeasurably fast reactions”, Nobel Lecture, 11 December 1967, in Nobel Lectures, Chemistry 1963–1970, Elsevier (1972), pp. 170–203.
M. Eigen, L. De Maeyer, “Untersuchungen über die Kinetik der Neutralisation. I”, Z. Elektrochem. 59, 986–993 (1955).
M. von Smoluchowski, “Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen”, Z. Phys. Chem. 92, 129–168 (1917).
P. Debye, “Reaction rates in ionic solutions”, Trans. Electrochem. Soc. 82, 265–272 (1942).
D. P. Fernández et al., “A formulation for the static permittivity of water and steam”, J. Phys. Chem. Ref. Data 26, 1125–1166 (1997).