Table 4.

Diffusion rate equations in the 0th and first order

	Langmuir–Hinshelwood	First order
Step	Diffusion (ao → oa)
Rate	$k_{diff}{\theta }_a{\theta }_o$	$k_{diff}{\theta }_{ao}$
Ensemble-specific stoichiometric coefficients
$d{\theta }_a/dt$	0	0
$d{\theta }_o/dt$	0	0
$d{\theta }_{aa}/dt$	No explicit rate equations. ${\theta }_{ij}={\theta }_i{\theta }_j$	$6\left({\theta }_{ao}/{\theta }_o-{\theta }_{aa}/{\theta }_a\right)$
$d{\theta }_{ao}/dt$		$3\left({\theta }_{aa}-{\theta }_{ao}\right)/{\theta }_a+3\left({\theta }_{oo}-{\theta }_{ao}\right)/{\theta }_o$
$d{\theta }_{oo}/dt$		$6\left({\theta }_{ao}/{\theta }_a-{\theta }_{oo}/{\theta }_o\right)$

The rate equation for surface diffusion with respect to each site configuration is the product of the corresponding entry in the row labeled “Rate” and the corresponding entry in the ensemble-labeled row. For example, (dθ_aa/dt)_diff = 6(θ_ao/θ_o − θ_aa/θ_a) × k_diffθ_ao in the first order. The Langmuir–Hinshelwood model is unable to describe dynamics of multisite ensembles, and therefore entries for aa, ao, and oo are absent.