11.4 Diffusion-controlled reactions

Diffusion-controlled reactions are fascinating processes where reaction rates are limited by how quickly molecules can move through a medium. These reactions are crucial in many systems, from biological processes to industrial applications, and understanding them is key to predicting and controlling reaction speeds.

The Smoluchowski equation is the cornerstone of diffusion-controlled reaction theory. It links reaction rates to factors like particle size, medium viscosity, and temperature. By grasping these concepts, we can better understand and manipulate reaction rates in various settings.

Diffusion-Controlled Reactions

Derivation of Smoluchowski equation

• Describes rate of diffusion-controlled reactions assumes reaction limited by diffusion rate of reactants considers diffusion of one reactant species (A) towards another (B)
• Reaction rate given by: $k = 4\pi DR$
  • $k$ rate constant
  • $D$ sum of diffusion coefficients of reactants ($D_A + D_B$)
  • $R$ sum of radii of reactants ($R_A + R_B$)
• Diffusion coefficient related to Stokes-Einstein equation: $D = \frac{k_BT}{6\pi\eta r}$
  • $k_B$ Boltzmann constant
  • $T$ absolute temperature
  • $\eta$ viscosity of medium
  • $r$ radius of diffusing particle
• Equation assumes spherical reactants, no intermolecular forces, and diffusion occurs in three dimensions
• Provides upper limit for reaction rate in diffusion-controlled systems (aqueous solutions, biological systems)

Effects on diffusion-controlled reactions

• Viscosity of medium affects diffusion rate
  • Higher viscosity leads to slower diffusion and lower reaction rate (glycerol, honey)
  • Lower viscosity results in faster diffusion and higher reaction rate (water, ethanol)
• Particle size influences diffusion coefficient and reaction rate
  • Larger particles have smaller diffusion coefficient and slower reaction rate (proteins, colloids)
  • Smaller particles have larger diffusion coefficient and faster reaction rate (ions, small molecules)
• Stokes-Einstein equation relates particle size and viscosity to diffusion coefficient
  • $D$ inversely proportional to radius of particle ($r$) and viscosity ($\eta$)
• Temperature affects diffusion rate and reaction rate
  • Higher temperature increases diffusion coefficient and reaction rate (boiling water)
  • Lower temperature decreases diffusion coefficient and reaction rate (ice)

Encounter complex in reactions

• Intermediate state formed when reactants come into close proximity occurs due to diffusion of reactants towards each other represents transient, non-covalently bound complex
• Formation of encounter complex prerequisite for reaction to proceed complex must overcome activation barrier to form final products
• Stability and lifetime of encounter complex can influence reaction rate
  • More stable encounter complex may have longer lifetime, increasing probability of successful collisions and reactions (enzyme-substrate complexes)
  • Less stable encounter complex may have shorter lifetime, decreasing probability of successful collisions and reactions (weakly interacting molecules)
• Encounter complex formation affected by electrostatic interactions, hydrophobic effects, and shape complementarity
• Concept applies to bimolecular reactions in solution phase (protein-ligand binding, electron transfer reactions)

Activation energy from Stokes-Einstein equation

• Activation energy for diffusion-controlled reactions related to diffusion coefficient higher activation energy leads to slower diffusion rate and lower reaction rate
• Stokes-Einstein equation can estimate activation energy: $D = \frac{k_BT}{6\pi\eta r}e^{-\frac{E_a}{RT}}$
  • $E_a$ activation energy
  • $R$ gas constant
• Rearranging equation, activation energy can be determined: $E_a = -RT \ln(\frac{D}{D_0})$
  • $D_0 = \frac{k_BT}{6\pi\eta r}$ pre-exponential factor
• Experimentally, activation energy found by measuring diffusion coefficient at different temperatures and plotting $\ln(D)$ vs. $\frac{1}{T}$
  • Slope of resulting line equal to $-\frac{E_a}{R}$
• Activation energy provides insight into energy barrier for diffusion process and temperature dependence of reaction rate
• Lower activation energy indicates faster diffusion and higher reaction rate at given temperature (ion transport in solution)
• Higher activation energy indicates slower diffusion and lower reaction rate at given temperature (protein unfolding)