The BET equation
Brunauer, Emmett and Teller (1938) extended Langmuir's monolayer model to multilayer adsorption. In the linear form used for calculation:
- n — amount adsorbed at relative pressure p/p₀ (often as volume V in cm³ STP/g)
- nm — monolayer capacity (Vm in the same units)
- C — BET constant, related exponentially to the energy of monolayer adsorption
Plotting the left-hand side against p/p₀ gives a straight line in the valid range. From the slope s and intercept i:
The specific surface area follows from the cross-sectional area σ of one adsorbed molecule:
With Vm in cm³ STP/g: SBET [m²/g] = Vm × NA × σ / 22414
| Adsorbate | σ (nm²) | SBET per cm³ STP/g of Vm |
|---|---|---|
| N₂ at 77 K | 0.162 | 4.353 m²/g |
| Ar at 87 K | 0.138 | 3.708 m²/g |
| Kr at 77 K | 0.202 | 5.427 m²/g |
Worked example
The isotherm points below are an illustrative nitrogen data set (generated from the BET equation with Vm = 50 cm³ STP/g and C = 100, so the answer is known).
| p/p₀ | V (cm³ STP/g) | V(1 − p/p₀) | (p/p₀) / [V(1 − p/p₀)] |
|---|---|---|---|
| 0.05 | 44.23 | 42.02 | 0.001190 |
| 0.08 | 48.74 | 44.84 | 0.001784 |
| 0.11 | 51.97 | 46.25 | 0.002378 |
| 0.14 | 54.77 | 47.10 | 0.002972 |
| 0.17 | 57.44 | 47.68 | 0.003566 |
| 0.20 | 60.10 | 48.08 | 0.004160 |
| 0.25 | 64.72 | 48.54 | 0.005150 |
| 0.30 | 69.80 | 48.86 | 0.006140 |
- Linear fit of the last column against p/p₀: slope s = 0.01980 g/cm³, intercept i = 0.000200 g/cm³.
- Monolayer capacity: Vm = 1/(0.01980 + 0.000200) = 50.0 cm³ STP/g.
- BET constant: C = 0.01980/0.000200 + 1 = 100.
- Surface area: S = 50.0 × 4.353 = 218 m²/g.
- Consistency check: 1/(√C + 1) = 1/11 = 0.091, which lies inside the 0.05–0.30 range used. V(1 − p/p₀) increases over every point.
Single-point BET estimates Vm ≈ V(1 − p/p₀) at one point near p/p₀ = 0.3, assuming C is large. Here that gives 48.9 cm³ STP/g and 213 m²/g, about 2 % low. The error grows as C gets smaller.
Multi-point BET calculator
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Choosing the range: the Rouquerol criteria
ISO 9277:2022 and the IUPAC 2015 report recommend checking the chosen points against these consistency criteria, especially for microporous materials:
- C must be positive. A negative intercept means the range is wrong.
- The BET range should only include points where n(1 − p/p₀) increases steadily with p/p₀.
- The p/p₀ value corresponding to nm should fall inside the selected range.
- The relative pressure predicted by 1/(√C + 1) should agree with that value to within about 10 %.
ISO 9277 also expects at least three, preferably five, points and a correlation coefficient of at least 0.999. For Type I isotherms these criteria usually push the range well below 0.05. The resulting number is then best described as a BET area (a fingerprint), not a literal geometric surface area.
Common sources of error
- Default range on microporous samples gives negative C or an inflated area.
- Incomplete degassing leaves water on the surface and lowers the measured area. See sample preparation & degassing.
- Too little total surface in the tube. Low-area samples (< ~1 m²/g) are better measured with krypton.
- Wrong σ. Argon at 87 K uses 0.138 nm² per ISO 9277; nitrogen 0.162 nm².
Related guides
- Gas adsorption (BET/BJH) method guide
- Total pore volume & average pore diameter (4V/A)
- t-plot micropore analysis
- IUPAC isotherm types
Frequently asked questions
What relative pressure range should I use for BET?
For Type II and Type IV(a) isotherms the classical range is p/p₀ ≈ 0.05–0.30. For microporous materials (Type I) the linear range lies at much lower relative pressure and should be selected with the Rouquerol criteria, often somewhere between 0.005 and 0.1.
What does a negative BET C constant mean?
A negative C (negative intercept) means the chosen points are not in a range where the BET model applies. It is common with microporous samples when the 0.05–0.30 default is used. Move the range to lower pressure until C is positive and the Rouquerol criteria are met; insufficient degassing can also distort the low-pressure data.
How do I convert monolayer volume to surface area?
For nitrogen (σ = 0.162 nm²), S_BET [m²/g] = 4.353 × V_m [cm³ STP/g]. For argon at 87 K (0.138 nm²) the factor is 3.708 and for krypton (0.202 nm²) 5.427.
Can I use BET with CO₂ at 273 K?
Not meaningfully. At 273 K and 1 bar CO₂ only reaches about p/p₀ ≈ 0.03, below the BET range, and the adsorption is mainly micropore filling. CO₂ data are evaluated with Dubinin–Radushkevich or DFT methods instead.