We demonstrate that the ubiquitously-used Langmuir-Hinshelwood formalism is an incomplete kinetic description and is inappropriate even for catalytic reactions as simple as A + A → A₂. The Langmuir-Hinshelwood description is predicated on the Hinshelwood assumption – that all adsorbates are randomly-distributed on the surface. The utility of this approximation is in reducing coverages of multi-site ensembles (e.g. A^*–A^*) to a product of coverages of constituent single-site ensembles (e.g. A^*). For example, the rate of surface reaction between two A^* species is rigorously r = k_rθ_A_{*– A*} where θ_A_*–A* is the fraction of pairs of sites which contain two A^* species. The Langmuir-Hinshelwood formalism simplifies the rate to r = k_rθ_A_*² by application of the Hinshelwood approximation – mathematically, that the mean-field metric, μ_ij = θ_ij(θ_iθ_j)^-1, is unity for all site pairs ij, giving θ_A_*–A* = θ_A_*².

The Hinshelwood approximation, however, crucially neglects the propensity of multi-site steps (e.g. two-site A^*–A^* → *–* + A₂) to engender (i) aggregation (μ_ij >> 1) of slowly-consumed ensembles and (ii) partitioning (μ_ij << 1) of rapidly-consumed ensembles. For example, Scheme 1 illustrates derivation of the ensemble-specific rate of surface reaction with respect to A^*–A^* pairs, dθ_A_*–A*/dt, which is characterized by a coverage-dependent stoichiometric coefficient (1 + 6θ_A_*–A*/θ_A_*) that captures spatial coupling unique to the A^*–A^* ensemble. Per the [A^*–A^*]-specific stoichiometric coefficient, as many as seven A^*–A^* pairs are annihilated per surface reaction – engendering partitioning of A^*–A^* pairs (i.e. μ_A_*–A* << 1).

Configurationally non-ideal phenomena such as A^*–A^* partitioning are neglected by the Langmuir-Hinshelwood formalism and result in a ‘poorly-mixed’ surface. Poor ‘surface-mixing’ effects profound changes in rates, selectivities, and identities of rate-determining steps incognizable in the Langmuir-Hinshelwood formalism – as we demonstrate in the context of (i) the A + A → A₂ reaction and (ii) industrially-relevant catalytic ammonia synthesis.

Aditya Bhan

Professor, Univ of Minnesota

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