Gas Adsorption BET & BJH

BET Equation (Linear Form)

1/[W(P₀/P - 1)] = 1/(Wₘ·C) + (C-1)/(Wₘ·C) · (P/P₀)

From Monolayer to Surface Area

S_BET = (Wₘ × N_A × A_m) / M

• N_A = Avogadro's number (6.022 × 10²³)
• A_m = molecular cross-sectional area
  - N₂ at 77 K: 16.2 ų
  - Ar at 87 K: 13.8 ų (or 14.2 ų on heterogeneous surfaces)
  - Kr at 77 K: 20.2 ų
• M = molecular weight of adsorbate

⚠️ Critical Validation:

The C-constant must be positive (typically 50-300). Negative or very low C values (<20) indicate invalid BET application. Very high C values (>1000) suggest strong microporosity where BET may overestimate surface area.

1. Initial slope (P/P₀ < 0.05): Steep = micropores; gradual = weak interaction
2. Knee sharpness: Sharp = strong adsorbate-surface interaction (high C-constant)
3. Middle section (0.1-0.8): Linear = monolayer-multilayer; curved = capillary condensation
4. High P/P₀ behavior: Plateau = limited pore volume; continuous rise = macropores/interparticle voids
5. Hysteresis presence: Indicates mesopores > 4 nm with network effects

For nitrogen at 77 K, cavitation occurs at P/P₀ ≈ 0.42-0.45, causing forced closure of hysteresis loops. This represents the tensile strength limit of liquid nitrogen, not a pore size effect.

Implications:
• Pores with necks < 5-6 nm show cavitation-controlled desorption
• True pore size cannot be determined from desorption branch
• Argon at 87 K shows cavitation at P/P₀ ≈ 0.38
• Consider using argon or alternative methods for narrow-necked pores

Modified Kelvin Equation

r_k = -2γV_m cosθ / RT ln(P/P₀)

Where r_k is the Kelvin radius (core radius), to which the statistical thickness t must be added:
r_p = r_k + t

⚠️ Known BJH Limitations:

• • Underestimates pore size by 20-30% for pores < 10 nm
• • Assumes cylindrical pore geometry
• • Neglects fluid-wall interactions in small pores
• • Statistical thickness models may be inaccurate
• • Tensile strength effects below P/P₀ = 0.42

The t-plot compares sample adsorption to a reference non-porous material with similar surface chemistry. Deviations from linearity indicate microporosity.

log W = log W₀ - (B/2.303β²) × [log(P₀/P)]²

• W = amount adsorbed at P/P₀
• W₀ = micropore capacity
• B = structural constant related to pore size
• β = affinity coefficient (N₂ = 0.33, Ar = 0.29, CO₂ = 0.36)
• Valid range: P/P₀ = 10⁻⁵ to 0.02

Superior to t-plot as it uses reduced adsorption (αs = n/n₀.₄), eliminating surface chemistry effects.

• • Plot V_ads vs. αs
• • Reference: non-porous material at P/P₀ = 0.4
• • More robust for heterogeneous surfaces
• • Standard references available for carbons, silicas, aluminas

Latest ML-Enhanced DFT Capabilities:

• Automated kernel selection: ML algorithms analyze isotherm shape to recommend optimal DFT kernel
• Custom kernel generation: Neural networks generate material-specific kernels from limited reference data
• Real-time fitting: 100x faster PSD calculation through ML-accelerated numerical methods
• Multi-probe correlation: Combines N₂, Ar, and CO₂ data for comprehensive pore analysis

1. C-constant must be positive and typically > 50
2. V(1 - P/P₀) must increase continuously with P/P₀
3. P/P₀ at monolayer should be ≈ 1/(√C + 1)
4. Linear correlation R² > 0.9999 (4 nines minimum)
5. Calculated monolayer capacity must be within selected range

⚠️ Common Pitfalls to Avoid:

• • Using default 0.05-0.30 range without validation
• • Ignoring negative intercept (indicates micropores)
• • Applying BET to Type I isotherms above P/P₀ = 0.1
• • Not checking if n_m falls within selected range
• • Using wrong cross-sectional area for adsorbate
• • Insufficient equilibration time during measurement

Optimal sample mass depends on expected surface area:

• • > 1000 m²/g: 30-50 mg
• • 100-1000 m²/g: 50-100 mg
• • 10-100 m²/g: 100-300 mg
• • 1-10 m²/g: 300-500 mg
• • < 1 m²/g: 1-2 g (use Kr)

Target: Total surface area in tube > 10 m² for accurate measurement