Langmuir-Hinshelwood Mechanism: The Rate Law of Surface Catalysis

What It Is and Where It Applies

The Langmuir-Hinshelwood (LH) mechanism describes a heterogeneous catalytic reaction in which both reactants first adsorb onto the catalyst surface, migrate until they occupy adjacent sites, and then react as co-adsorbed species. The reaction A + B → products becomes a four-step cycle: adsorption of A, adsorption of B, the surface reaction between adsorbed A and B (usually rate-determining), and desorption of the product.

It is the default model for the majority of industrial gas-solid catalytic processes, including:

• CO oxidation (2 CO + O₂ → 2 CO₂) on Pt, Pd and Rh — the reaction inside every automotive three-way catalytic converter.
• Ammonia synthesis (N₂ + 3 H₂ → 2 NH₃) on Fe/K, where dissociated N and H atoms recombine on the surface.
• Hydrogenations of alkenes on Ni, Pd and Pt, where H atoms and the unsaturated molecule share the surface.

The LH picture matters because it predicts a maximum in rate at intermediate coverage — the surface must hold appreciable amounts of both partners at once, so neither species may dominate.

Deriving the Rate Law Step by Step

Start from the Langmuir adsorption isotherm. If A and B compete for identical single sites, the fractional coverages are:

• θ_A = K_A·p_A / (1 + K_A·p_A + K_B·p_B)
• θ_B = K_B·p_B / (1 + K_A·p_A + K_B·p_B)

where K_A and K_B are adsorption equilibrium constants (units of 1/pressure) and p_A, p_B are partial pressures. The shared denominator encodes site competition: adding B lowers θ_A.

Assume the rate-determining step is the bimolecular surface reaction between adjacent adsorbed A and B, so r = k·θ_A·θ_B. Substituting the isotherms gives the canonical LH rate law:

r = k·K_A·K_B·p_A·p_B / (1 + K_A·p_A + K_B·p_B)²

The denominator is squared because two sites are involved. Every symbol: k = surface rate constant, K = adsorption constants, p = partial pressures. If one reactant dissociates (e.g. O₂ → 2 O), its isotherm uses √(K·p) and the exponents change accordingly.

Key Quantities and a Worked Example

The most instructive limit is the volcano curve. Hold p_B fixed and vary p_A. At low p_A the rate rises roughly linearly (r ∝ p_A). At high p_A the strongly-adsorbing A saturates the surface, θ_B → 0, and r ∝ 1/p_A. The maximum occurs when K_A·p_A ≈ 1 + K_B·p_B.

Worked example (CO oxidation on Pt): take K_CO = 10 atm⁻¹ (CO adsorbs strongly), K_O ≈ 1 atm⁻¹, k = 100 s⁻¹, p_O₂ = 0.1 atm. At p_CO = 0.05 atm:

• Denominator = (1 + 10·0.05 + 1·0.1)² = (1.6)² = 2.56
• r = 100·10·1·0.05·0.1 / 2.56 = 5.0/2.56 ≈ 1.95 s⁻¹

Now raise p_CO to 0.5 atm: denominator = (1 + 5 + 0.1)² = 37.2, r = 100·10·1·0.5·0.1/37.2 = 50/37.2 ≈ 1.34 s⁻¹. Ten-fold more CO gives less reaction — the hallmark CO self-poisoning that plagues cold-start catalytic converters. Typical adsorption enthalpies run −100 to −150 kJ/mol for CO on Pt, versus −30 to −50 kJ/mol for physisorbed species.

How It's Measured and Used in Practice

LH kinetics are extracted from steady-state rate measurements in flow reactors: rates are recorded as functions of each partial pressure, and reaction orders are read off log-log plots. An LH system shows positive order in a reactant at low coverage that turns negative at high coverage — a diagnostic no power-law model reproduces.

• Temperature-programmed desorption (TPD) yields the adsorption energies that set K and their temperature dependence (K = K₀·e^(−ΔH_ads/RT)).
• Infrared / RAIRS spectroscopy confirms both reactants are present on the surface simultaneously (e.g. the ~2070 cm⁻¹ C-O stretch of adsorbed CO).
• Molecular-beam scattering and STM directly image co-adsorbed islands and reaction fronts.

Practically, engineers exploit the volcano: converters are run lean (excess O₂) to keep the surface from saturating with CO, and washcoat formulations balance metal loading against CO binding strength. The LH form also feeds directly into reactor-scale kinetic models used to size industrial beds.

Langmuir-Hinshelwood vs. Its Close Cousins

Two rival surface mechanisms share the stage:

• Eley-Rideal (ER): only one reactant adsorbs; the second strikes it directly from the gas phase. The rate law is r = k·K_A·p_A·p_B / (1 + K_A·p_A) — a single, not squared, denominator, and no self-inhibition volcano. Genuine ER cases are rare (e.g. H(ads) + H(gas) recombination); most reactions once assigned to ER proved to be LH on closer inspection.
• Mars-van Krevelen (MvK): the oxidant is lattice oxygen from the oxide itself; the surface is re-oxidised from the gas in a separate step. This governs selective oxidations and CO oxidation on RuO₂.

The distinguishing experiment is coverage dependence: LH requires two adsorbed partners and shows the classic rate maximum, while ER is monotonic. Both LH and ER descend from Langmuir's isotherm, but LH is a dual-site model — the squared denominator is its signature. Do not confuse LH with the gas-phase Arrhenius or the Michaelis-Menten enzyme rate law, which it superficially resembles algebraically but derives from entirely different assumptions.

Exceptions, Limits, and Famous Cases

The clean LH derivation assumes an ideal Langmuir surface: uniform sites, no adsorbate-adsorbate interactions, and a single rate-determining step. Real catalysts violate all three. Adsorbate islands, surface reconstruction, and coverage-dependent adsorption energies (Temkin/Freundlich behaviour) all bend the rate law.

• Ertl's CO oxidation on Pt(110): Gerhard Ertl won the 2007 Nobel Prize in Chemistry for showing that this LH reaction produces travelling waves and oscillations because CO adsorption drives a periodic surface reconstruction — dynamics no static LH law captures.
• Ammonia synthesis: here the rate-limiting step is N₂ dissociative adsorption, not the surface recombination, so the effective kinetics deviate from the textbook LH form (Ertl and Somorjai mapped this in detail).
• Strong-binding poisons: sulfur or excess CO can push the surface into the negative-order regime permanently, deactivating the catalyst.

Despite these caveats, LH remains the first model any catalytic chemist writes down. Its enduring value is qualitative as much as quantitative: it explains why more reactant can mean less product, and why optimal catalysts bind intermediates neither too weakly nor too strongly — the Sabatier principle in algebraic form.

Langmuir-Hinshelwood vs. Eley-Rideal vs. Mars-van Krevelen surface mechanisms

Feature	Langmuir-Hinshelwood	Eley-Rideal	Mars-van Krevelen
Reacting species	Both A and B adsorbed	One adsorbed, one from gas phase	Adsorbed A + lattice O
Rate law denominator	(1 + K_A p_A + K_B p_B)²	(1 + K_A p_A)	depends on lattice reoxidation
Self-inhibition	Yes — strong adsorber poisons	Weak / absent	Governed by O-vacancy refill
Rate vs. p_A shape	Volcano (max then decline)	Monotonic saturation	Saturating
Classic example	CO + O on Pt	H(ads) + H₂(gas) → H recombination	CO oxidation on RuO₂; selective oxidations
Site requirement	Two adjacent sites	One site	Redox lattice site + O