11.4 Diffusion-controlled reactions

Diffusion-Controlled Reactions

In many reactions, the chemistry itself is nearly instantaneous. The bottleneck isn't breaking or forming bonds; it's getting the reactant molecules close enough to react. These are diffusion-controlled reactions, and their rates depend on how fast molecules move through the surrounding medium.

Understanding these reactions matters because they set the upper speed limit for bimolecular reactions in solution. Any reaction in a viscous environment or involving large molecules (think enzyme-substrate binding or protein-ligand interactions) may be governed by diffusion rather than by the intrinsic chemical step.

Derivation of the Smoluchowski Equation

The Smoluchowski equation predicts the rate constant for a reaction that is entirely limited by diffusion. The setup: picture one reactant species (B) sitting stationary while the other (A) diffuses toward it through solution. Once A reaches a critical distance (the sum of the two molecular radii), reaction occurs instantly.

The resulting rate constant is:

$k = 4\pi D R$

• $k$ is the bimolecular rate constant
• $D = D_A + D_B$, the sum of the diffusion coefficients of both reactants
• $R = R_A + R_B$, the sum of the radii of both reactants (the "encounter distance")

Each reactant's diffusion coefficient comes from the Stokes-Einstein equation:

$D = \frac{k_B T}{6\pi \eta r}$

• $k_B$ is Boltzmann's constant
• $T$ is absolute temperature (K)
• $\eta$ is the viscosity of the medium
• $r$ is the hydrodynamic radius of the diffusing particle

Key assumptions built into this derivation:

• Reactants are spherical
• No long-range intermolecular forces (no electrostatic attraction or repulsion)
• Diffusion is three-dimensional and occurs in a continuous medium

Because the model ignores any activation barrier for the chemical step, the Smoluchowski equation gives an upper limit for the rate constant. For small molecules in water at 25 °C, this limit is typically on the order of $10^9$ to $10^{10}$ $\text{M}^{-1}\text{s}^{-1}$.

Factors Affecting Diffusion-Controlled Reactions

Three variables in the Stokes-Einstein equation control how fast molecules diffuse, and therefore how fast diffusion-limited reactions proceed.

Viscosity ($\eta$) acts as a brake on molecular motion. $D$ is inversely proportional to $\eta$, so doubling the viscosity cuts the diffusion coefficient in half.

• In low-viscosity solvents like water ($\eta \approx 0.001$ Pa·s) or ethanol, molecules diffuse quickly and diffusion-controlled rates are high.
• In high-viscosity media like glycerol ($\eta \approx 1.5$ Pa·s) or the crowded interior of a cell, diffusion slows dramatically and reaction rates drop accordingly.

Particle size ($r$) also appears in the denominator of the Stokes-Einstein equation. Larger particles diffuse more slowly.

• Small species like ions or dissolved $O_2$ have large diffusion coefficients and reach encounter distance quickly.
• Large species like proteins or colloidal particles diffuse much more slowly, which is one reason macromolecular association rates are often diffusion-limited.

Temperature ($T$) appears in the numerator. Raising the temperature increases thermal energy, which speeds up diffusion. This is why reactions in solution generally speed up when heated, even for diffusion-controlled processes. Note, though, that temperature also affects viscosity (most liquids become less viscous when heated), so the real effect of raising $T$ is often larger than the Stokes-Einstein numerator alone would suggest.

The Encounter Complex

Before reactants can undergo a chemical transformation, they must first find each other through diffusion. The transient, non-covalently bound state that forms when two molecules arrive within encounter distance is called the encounter complex (sometimes called a "cage pair" in solution).

Formation of the encounter complex is a necessary first step, but it doesn't guarantee reaction. The complex must still overcome any activation barrier for the chemical step to form products. Two outcomes are possible:

1. The encounter complex proceeds over the activation barrier and reacts.
2. The molecules diffuse apart before reacting, and no product forms.

The stability and lifetime of the encounter complex influence which outcome dominates:

• A longer-lived encounter complex gives the reactants more attempts to cross the activation barrier. Enzyme-substrate complexes are a good example: shape complementarity, electrostatic interactions, and hydrophobic effects all stabilize the encounter, increasing the probability of reaction.
• A shorter-lived encounter complex (weakly interacting molecules with poor shape or charge complementarity) means the partners separate quickly, lowering the overall rate.

This concept is most relevant for bimolecular reactions in the solution phase, including protein-ligand binding, electron transfer reactions, and radical recombination.

Activation Energy from Diffusion Data

Even diffusion-controlled reactions have a temperature dependence that can be characterized by an effective activation energy. This $E_a$ reflects the energy barrier for a molecule to push past its solvent neighbors, not the barrier for the chemical step itself.

When the diffusion process has an Arrhenius-type temperature dependence, you can write:

$D = D_0 \, e^{-E_a / RT}$

where $D_0 = \frac{k_B T}{6\pi \eta r}$ is the pre-exponential factor and $R$ is the gas constant.

Rearranging gives:

$E_a = -RT \ln\!\left(\frac{D}{D_0}\right)$

How to determine $E_a$ experimentally:

1. Measure the diffusion coefficient $D$ at several different temperatures.
2. Plot $\ln(D)$ versus $1/T$.
3. Fit a straight line to the data. The slope equals $-E_a / R$.
4. Multiply the magnitude of the slope by $R$ to get $E_a$.

What the result tells you:

• A low $E_a$ (e.g., ion transport in dilute aqueous solution) means diffusion is relatively easy and the rate responds weakly to temperature changes.
• A high $E_a$ (e.g., diffusion of large molecules through viscous or structured media) means the process is more sensitive to temperature, and cooling the system will slow the reaction significantly.

Typical activation energies for diffusion in water are in the range of 15–20 kJ/mol, which is noticeably smaller than activation energies for most bond-breaking chemical steps (often 40–100+ kJ/mol). This difference is one way to diagnose whether a reaction is diffusion-controlled or activation-controlled: if the measured $E_a$ is unusually low, diffusion is likely the rate-limiting step.