§ C-33 Surface Adsorption & the Langmuir Isotherm: θ = K·P / (1 + K·P) (Grade 9–11 · Surface Chemistry)
Chapter XXXIII. Dynamic Adsorption–Desorption Equilibrium on Activated Carbon & Monolayer Saturation. «Every solid crystal surface possesses a definite number of elementary spaces capable of holding adsorbed molecules in a monomolecular layer.», Irving Langmuir, 1918
Theory Summary & Scientific Mechanism
Adsorption is the accumulation of gas or solute molecules (adsorbate) onto the surface of a solid (adsorbent, such as activated carbon or a metal catalyst). Because gas molecules lose translational freedom upon binding (Δ Sₐds < 0), spontaneous adsorption is universally exothermic (Δ Hₐds = -Qₐds < 0).
Langmuir’s monolayer model assumes a fixed number of equivalent surface sites. Equating the rate of adsorption on vacant sites kₐ P(1-θ) to the rate of desorption from occupied sites k_dθ yields the Langmuir isotherm: θ(P) = (K P)/(1 + K P), where K = kₐ / k_d.
At low pressures (KP ≪ 1), fractional coverage rises linearly (θ ≈ KP, Henry’s regime); at high pressures (KP ≫ 1), the surface saturates at a complete monolayer (θ → 1). Raising temperature T decreases K(T) and drives thermal desorption.
Everyday Analogy
An activated-carbon surface is like a parking lot with a fixed number of spaces. At low gas pressure cars park easily (θ ≈ KP), but once every spot is taken in a monolayer, coverage hits a ceiling (θ → 1)!
Interactive Question & Derivation
Why does the fractional surface coverage θ(P) in the Langmuir adsorption isotherm rise linearly at low gas pressures (θ ≈ KP) but approach a horizontal asymptote θ → 1 at high pressures, and how does heating affect physisorption?
- A: A solid surface has a finite number of active sites (θ → 1 when the monolayer is full); because adsorption is exothermic (Δ Hₐds < 0), raising temperature T decreases K(T) and promotes desorption
- B: Surface coverage θ grows linearly to infinity with pressure, and heating always increases gas adsorption
- C: At equilibrium, all gas molecules freeze permanently onto the surface with zero desorption rate
Study Hint: Surface sites saturate at a monolayer (θ → 1), and because adsorption is exothermic (Δ Hₐds < 0), heating promotes desorption.
Proof & Explanation: Irving Langmuir modeled a crystal or activated-carbon surface as an array of N₀ equivalent adsorption sites. Dynamic equilibrium is reached when the adsorption rate equals the desorption rate: kₐ P (1 - θ) = k_d θ, yielding the Langmuir Isotherm θ = (K P)/(1 + K P) where K = kₐ / k_d = K₀ e^Qₐds / (RT). Because gas adsorption reduces translational entropy (Δ Sₐds < 0), spontaneous adsorption is always exothermic (Δ Hₐds = -Qₐds < 0), so heating shifts equilibrium toward desorption.
Key Equations & 3D Simulation Parameters
rₐds = kₐ P(1-θ) = r_des = k_dθ ⇒ θ(P,T) = (K(T) P)/(1 + K(T) P), K(T) = K₀exp ≤ft((Qₐds)/(RT) )
- Adsorbate gas pressure P: 0.2 … 6 atm (default: 2 atm)
- Surface temperature T: 250 … 450 K (default: 300 K)
- Adsorption affinity constant K₀ (at 300 K): 0.3 … 3 atm⁻¹ (default: 1.2 atm⁻¹)
- Heat of adsorption Qₐds = −ΔHₐds: 15 … 45 kJ/mol (default: 25 kJ/mol)