11.3 Surface reactions and adsorption kinetics

Langmuir Adsorption Isotherm and Surface Reaction Kinetics

The Langmuir adsorption isotherm describes how gas molecules adhere to a solid surface, forming the foundation for understanding heterogeneous catalysis. Since most industrial catalytic reactions happen on surfaces, knowing how molecules adsorb, react, and desorb is essential for predicting reaction rates and designing better catalysts.

Langmuir Adsorption Isotherm

The Langmuir isotherm relates surface coverage ($\theta$, the fraction of occupied sites) to the gas-phase pressure ($P$) at equilibrium:

$\theta = \frac{KP}{1 + KP}$

Here, $K$ is the adsorption equilibrium constant, which reflects how strongly molecules bind to the surface. A large $K$ means strong adsorption.

This model rests on four key assumptions:

• Adsorption is limited to a monolayer (no stacking of molecules on top of each other)
• All adsorption sites are equivalent and independent
• There are no lateral interactions between adsorbed molecules
• At equilibrium, the rate of adsorption equals the rate of desorption

These assumptions are idealized. Real surfaces have defects, varying site energies, and adsorbate-adsorbate interactions. Still, the Langmuir isotherm works surprisingly well as a first approximation and appears frequently in rate expressions for catalytic reactions.

Langmuir-Hinshelwood Mechanism

The Langmuir-Hinshelwood (LH) mechanism is the most common model for bimolecular surface reactions. It assumes that both reactants must adsorb onto the catalyst surface before they can react with each other. The mechanism has three steps:

1. Adsorption of reactants A and B onto vacant surface sites ($*$): $A_{(g)} + * \rightleftharpoons A^*$ $B_{(g)} + * \rightleftharpoons B^*$

2. Surface reaction between the two adsorbed species: $A^* + B^* \rightarrow AB^* + *$

3. Desorption of the product back into the gas phase: $AB^* \rightleftharpoons AB_{(g)} + *$

If the surface reaction (step 2) is the rate-determining step, the overall rate expression becomes:

$r = k\theta_A\theta_B = \frac{kK_AK_BP_AP_B}{(1 + K_AP_A + K_BP_B)^2}$

where $k$ is the surface reaction rate constant, and $K_A$, $K_B$ are the adsorption equilibrium constants for A and B. The squared denominator arises because both A and B compete for the same pool of surface sites.

Notice the denominator includes terms for both species. This means that if one reactant adsorbs too strongly (very large $K$), it can dominate the surface and block the other reactant from adsorbing. This self-inhibition effect is a distinctive prediction of the LH model: increasing the pressure of one reactant beyond a certain point can actually decrease the overall rate.

Surface Coverage in Catalytic Reactions

The Langmuir isotherm predicts two limiting regimes that directly affect reaction kinetics:

• Low pressure ($KP \ll 1$): Coverage is approximately $\theta \approx KP$, so it increases linearly with pressure. The reaction rate also increases linearly because more molecules are available on the surface.
• High pressure ($KP \gg 1$): The surface becomes saturated ($\theta \approx 1$). Adding more gas doesn't increase coverage, so the reaction rate plateaus and becomes zero-order in that reactant. The rate is now limited entirely by the surface reaction step.

Temperature has a competing effect. Raising the temperature increases the rate constant $k$ for the surface reaction (Arrhenius behavior), but it also promotes desorption, which decreases surface coverage. There's typically an optimal temperature that balances these two effects to maximize the overall rate.

Turnover Frequency of Heterogeneous Catalysts

Turnover frequency (TOF) quantifies how productive each active site on a catalyst is:

$TOF = \frac{\text{Reaction rate}}{\text{Number of active sites}}$

It's reported as molecules converted per site per second. TOF is the standard metric for comparing the intrinsic activity of different catalysts, because it normalizes out differences in surface area or catalyst loading. A high TOF means each active site is efficiently converting reactants.

TOF depends on the nature of the active site (metal identity, oxidation state, presence of promoters) and on reaction conditions (temperature, pressure, reactant concentrations).

Factors Influencing Surface Reactions and Catalytic Activity

Several properties of the catalyst and the reaction environment determine how well a surface reaction performs:

Surface area and porosity. More surface area means more active sites. This is why catalysts are often prepared as nanoparticles or deposited on high-surface-area porous supports (like alumina or silica). Nanostructured catalysts can have dramatically higher activity per gram of material.

Catalyst composition and surface structure. The identity of the metal or oxide determines how strongly reactants bind and how easily bonds break. Beyond composition, the specific crystal facets exposed on the surface matter too. Steps, edges, and defect sites often have different adsorption energies and can be more catalytically active than flat terraces.

Support effects. The support material isn't always inert. Strong metal-support interactions (SMSI) can modify the electronic properties of the active metal, changing how it binds reactants. For example, a metal nanoparticle on a reducible oxide support (like $TiO_2$) can behave very differently than the same metal on an inert support (like $SiO_2$).

Catalyst deactivation. Catalysts lose activity over time through several mechanisms:

• Poisoning: Strong adsorption of impurities (e.g., sulfur on metal catalysts) that block active sites
• Sintering: Agglomeration of small nanoparticles into larger ones at high temperatures, reducing total surface area
• Coking: Deposition of carbonaceous residues that physically cover active sites

Regeneration procedures (such as oxidation to burn off coke, or reduction to restore the metal state) can often recover lost activity.

Structure sensitivity. Some reactions show TOFs that vary with particle size or crystal face, meaning the reaction is structure-sensitive. Ammonia synthesis on Fe is a classic example: certain crystal planes are far more active than others. In contrast, structure-insensitive reactions (like some hydrogenation reactions) show roughly constant TOF regardless of surface geometry.