Glossary

11.4 Diffusion-controlled reactions

3 min read•july 22, 2024

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 is the cornerstone of 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

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  • 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=4πDRk = 4\pi DR
    • kk rate constant
    • DD sum of diffusion coefficients of reactants (DA+DBD_A + D_B)
    • RR sum of radii of reactants (RA+RBR_A + R_B)
  • related to Stokes-Einstein equation: D=kBT6πηrD = \frac{k_BT}{6\pi\eta r}
    • kBk_B Boltzmann constant
    • TT absolute temperature
    • η\eta viscosity of medium
    • rr 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
    • DD inversely proportional to radius of particle (rr) and viscosity (η\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

  • 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=kBT6πηre−EaRTD = \frac{k_BT}{6\pi\eta r}e^{-\frac{E_a}{RT}}
    • EaE_a activation energy
    • RR gas constant
  • Rearranging equation, activation energy can be determined: Ea=−RTln⁡(DD0)E_a = -RT \ln(\frac{D}{D_0})
    • D0=kBT6πηrD_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)\ln(D) vs. 1T\frac{1}{T}
    • Slope of resulting line equal to −EaR-\frac{E_a}{R}
  • Activation energy provides insight into energy barrier for diffusion process and 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)

Key Terms to Review (13)

Activation Energy: Activation energy is the minimum amount of energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to be transformed into products, linking the concepts of kinetics and thermodynamics in the context of chemical reactions.
Collisional Frequency: Collisional frequency refers to the rate at which particles collide with one another in a given volume over a specific time period. This concept is crucial for understanding reaction rates, particularly in diffusion-controlled reactions, where the frequency of collisions between reactants can dictate how quickly a reaction occurs. Higher collisional frequencies generally lead to increased reaction rates, as particles are more likely to encounter each other and react.
Concentration Gradient: A concentration gradient refers to the gradual change in the concentration of solutes in a solution as a function of distance through a solution. It plays a crucial role in various processes, such as the movement of molecules across membranes and the rate of chemical reactions. Understanding how concentration gradients influence molecular movement helps in grasping concepts like diffusion and reaction rates, which are central to many chemical and biological phenomena.
Diffusion Coefficient: The diffusion coefficient is a numerical value that quantifies how quickly a substance spreads through another medium due to molecular motion. It provides insights into the rate at which reactants diffuse to one another in reactions, impacting the overall speed of the reaction, particularly in diffusion-controlled scenarios. A higher diffusion coefficient indicates faster diffusion and can significantly affect the kinetics of reactions where diffusion is a limiting factor.
Diffusion-controlled reaction: A diffusion-controlled reaction is a type of chemical reaction where the rate of reaction is limited by the rate at which reactants diffuse together rather than by the intrinsic properties of the reactants or the reaction itself. In these reactions, reactants must come into close proximity to react, making the diffusion process the rate-limiting step. This concept emphasizes the importance of molecular motion and concentration gradients in determining how quickly a reaction can proceed.
Fick's Laws of Diffusion: Fick's Laws of Diffusion describe the process by which particles move from an area of higher concentration to an area of lower concentration, driven by concentration gradients. The first law states that the diffusion flux is proportional to the negative gradient of concentration, while the second law provides a way to calculate how diffusion causes the concentration of a substance to change over time. These laws are essential in understanding diffusion-controlled reactions, where the rate of reaction is limited by the speed at which reactants diffuse together.
Reactant concentration: Reactant concentration refers to the amount of a substance present in a given volume of solution or reaction mixture, which significantly influences the rate at which chemical reactions occur. Higher reactant concentrations typically lead to more frequent collisions between molecules, thus increasing the rate of reaction. Understanding how concentration affects reaction rates is crucial for predicting the behavior of chemical systems and is foundational in analyzing rate laws and reaction orders.
Reaction-limited process: A reaction-limited process refers to a scenario in which the rate of a chemical reaction is slower than the rate at which reactants can diffuse to the reaction site. This means that the overall speed of the process is dictated by how fast the reactants can come together rather than how quickly they can react once they are in proximity. In this context, these processes often occur when the reactants are widely spaced or when conditions limit their ability to collide effectively, emphasizing the importance of diffusion in determining the reaction kinetics.
Single-step vs multi-step processes: Single-step processes refer to reactions that occur in one straightforward step, resulting in the formation of products from reactants without intermediate species. In contrast, multi-step processes involve a series of individual steps or reactions, often including one or more intermediates, which must be completed before the final products are formed. Understanding the difference is crucial in chemical kinetics, especially when analyzing reaction mechanisms and their rates.
Smoluchowski Equation: The Smoluchowski equation is a mathematical expression that describes the rate of diffusion-controlled reactions, particularly in systems where the reactants are small and mobile. This equation connects the concepts of diffusion and reaction kinetics, providing insights into how the concentration of reactants changes over time as they diffuse and react with each other. It's crucial for understanding how particles interact in solutions, especially in cases where diffusion is the primary limiting factor in reaction rates.
Surface Area: Surface area refers to the total area that the surface of an object occupies. It plays a crucial role in chemical reactions, particularly in determining the rate at which reactants can collide and interact, influencing both the Arrhenius equation and the kinetics of diffusion-controlled reactions. A larger surface area allows for more collisions between reactants, thereby enhancing reaction rates and impacting the overall efficiency of chemical processes.
Temperature Dependence: Temperature dependence refers to how the rate of a chemical reaction changes with varying temperatures. Generally, an increase in temperature tends to accelerate reaction rates due to enhanced molecular motion and increased frequency of collisions between reactants, thus influencing various chemical processes.
Unimolecular vs Bimolecular Reactions: Unimolecular reactions involve a single reactant species undergoing a transformation, while bimolecular reactions involve two reactant species that collide and react. The distinction between these types of reactions is essential for understanding the kinetics and mechanisms of chemical processes, particularly in how they relate to molecular interactions and the factors that influence reaction rates.
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