The Eley-Rideal mechanism: a gas molecule strikes an adsorbed one

In the Eley-Rideal mechanism (Eley and Rideal, 1940) only reactant A adsorbs. B reacts with adsorbed A straight from the gas phase, so the rate is \(k\theta_AP_B\) (er_rate()). A and B never compete for sites, so raising \(P_A\) can only help: the rate rises and levels off, and it stays first order in \(P_B\). In the dual-site Langmuir-Hinshelwood mechanism (lh_rate_dual_site()), too much A pushes B off the surface and the rate turns over. That difference is the standard kinetic test for telling the two mechanisms apart.

import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.surface.systems.eley_rideal import er_rate
from chemistrykit.surface.systems.langmuir_hinshelwood import lh_rate_dual_site

k, K_A, K_B, P_B = 4.0, 2.0, 5.0, 0.2
P_A = np.linspace(0.001, 20.0, 400)
rate_er = er_rate(k, K_A, P_A, P_B)
rate_lh = lh_rate_dual_site(k, K_A, P_A, K_B, P_B)
print(f"Eley-Rideal rate is monotonic in P_A: {bool(np.all(np.diff(rate_er) > 0))}")
print(f"Eley-Rideal saturates at k*P_B = {k * P_B:.2f}; at P_A=20: {rate_er[-1]:.3f}")
print(f"Langmuir-Hinshelwood peaks at P_A = {P_A[np.argmax(rate_lh)]:.3f}, then falls")

Eley-Rideal rate is monotonic in P_A: True
Eley-Rideal saturates at k*P_B = 0.80; at P_A=20: 0.780
Langmuir-Hinshelwood peaks at P_A = 1.003, then falls

P_B_scan = np.linspace(0.0, 5.0, 200)
er_vs_B = er_rate(k, K_A, 1.0, P_B_scan)
lh_vs_B = lh_rate_dual_site(k, K_A, 1.0, K_B, P_B_scan)

fig, axes = plt.subplots(1, 2, figsize=(11, 4))
axes[0].plot(P_A, rate_er / rate_er.max(), label="Eley-Rideal")
axes[0].plot(P_A, rate_lh / rate_lh.max(), label="Langmuir-Hinshelwood")
axes[0].set_xlabel("$P_A$")
axes[0].set_ylabel("rate (normalized)")
axes[0].set_title("Rate vs. pressure of adsorbed reactant A")
axes[0].legend()
axes[1].plot(P_B_scan, er_vs_B / er_vs_B.max(), label="Eley-Rideal: first order in $P_B$")
axes[1].plot(P_B_scan, lh_vs_B / lh_vs_B.max(), label="Langmuir-Hinshelwood")
axes[1].set_xlabel("$P_B$")
axes[1].set_ylabel("rate (normalized)")
axes[1].set_title("Rate vs. pressure of gas-phase reactant B")
axes[1].legend()
plt.tight_layout()
plt.show()