Engineering & Science
Brunauer-Emmett-Teller (BET) Multilayer Adsorption & Specific Surface Area
Calculate multilayer nitrogen physisorption uptake v = (v_m * c * x) / ((1 - x) * (1 + (c - 1) * x)) (cm^3 STP/g) across relative pressure x = P / P_0 and specific BET catalyst surface area S_BET (m^2/g).
Turn 77 K nitrogen physisorption monolayers into catalyst m^2/g surface area!
Langmuir’s monolayer isotherm saturates at v_m—failing above P / P_0 ~ 0.05 because physisorbed N2 molecules at 77 K stack into second, third, and infinite liquid-like layers on porous zeolites, activated carbons, and MOFs! Stephen Brunauer, Paul Hugh Emmett, and Edward Teller (JACS 1938) generalized Langmuir kinetics to multilayer condensation via the BET Isotherm: v = (v_m * c * x) / ((1 - x) * (1 + (c - 1) * x)), where x = P / P_0 (standard linear BET range 0.05 <= x <= 0.30), v_m is monolayer STP volume (cm^3/g), and c = EXP((E_1 - E_L) / (R * T)) measures first-layer adsorption strength!
Once monolayer capacity v_m (cm^3 STP/g) is known, multiplying by Avogadro’s number and N2 molecular cross-section sigma_N2 = 0.162 nm^2 divided by molar STP volume (22,414 cm^3/mol) gives specific surface area S_BET = 4.3525 * v_m (m^2/g)! For mesoporous alumina with v_m = 125.0 cm^3 STP/g (S_BET = 544.1 m^2/g) and c = 80: at x = 0.05, uptake is 106.33 cm^3/g, rising across multilayer buildup at x = 0.20 to 148.81 cm^3/g!
| Relative Pressure x = P / P0 (vm = 125 cm^3/g, c = 80) | BET Adsorbed Gas v (cm^3 STP/g) | Coverage Ratio v / vm | Adsorption Regime |
|---|---|---|---|
| Low BET Window (x = 0.05) | 106.33 cm^3/g | 0.851 Monolayers | Sub-Monolayer Filling |
| Mid BET Window (x = 0.20) | 148.81 cm^3/g | 1.190 Monolayers | Multilayer Stacking Active |
| Saturation Singularity (x >= 1.0) | Capillary Condensation Guard | "Invalid Relative Pressure (0-1)" | Requires 0 <= P/P0 < 1 |
BET Multilayer Adsorbed Volume v (cm^3 STP/g) at x = P / P_0
Evaluates (v_m * c * x) / ((1 - x) * (1 + (c - 1) * x)) across valid relative pressures 0 <= x < 1.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 >= 1), "Invalid Relative Pressure (0-1)", ROUND((B2 * B3 * B4) / ((1 - B4) * (1 + (B3 - 1) * B4)), 2))Specific Catalyst BET Surface Area S_BET (m^2/g) for 77 K N2 (0.162 nm^2)
Converts N2 STP monolayer volume B2 (cm^3/g) into m^2/g using ((N_A * 0.162e-18) / 22414) = 4.3525 m^2/cm^3.
=IF(B2 <= 0, "Invalid Monolayer Volume", ROUND(4.3525 * B2, 1))Try the example
Use the original values or change them to test your understanding.