Step 1: total pore volume (Gurvich rule)
Near saturation, mesopores and micropores are filled with condensed adsorbate. If that condensed phase is assumed to have the density of the bulk liquid (the Gurvich rule), the amount adsorbed at a high relative pressure converts directly into a pore volume:
| Adsorbate | M (g/mol) | ρL (g/cm³) | Factor (cm³ liquid per cm³ STP) |
|---|---|---|---|
| N₂, 77 K | 28.013 | 0.808 | 0.0015468 |
| Ar, 87 K | 39.948 | 1.394 | ≈ 0.00128 |
The value depends on where it is read. At p/p₀ = 0.95 nitrogen fills pores up to about 40 nm, and at 0.99 up to about 190 nm (Kelvin equation plus film thickness). On a Type I or Type IV isotherm with a clear plateau the choice hardly matters. On a Type II isotherm, or one with an H3 loop that keeps rising near p/p₀ = 1, it matters a lot, and the result is only meaningful together with the stated p/p₀.
Step 2: average pore diameter
If every pore is a cylinder of length L and diameter d, its volume is πd²L/4 and its wall area πdL. The ratio gives the hydraulic-diameter result:
w = 2 V / A (parallel-sided slits)
d [nm] = 4000 × V [cm³/g] / A [m²/g]
Worked example
- The isotherm shows 300 cm³ STP/g adsorbed at p/p₀ = 0.99.
- Total pore volume: 300 × 0.0015468 = 0.464 cm³/g.
- BET surface area from the same isotherm: 400 m²/g.
- Average pore diameter: 4000 × 0.464 / 400 = 4.6 nm.
Unit check: 4 × 0.464 × 10⁻⁶ m³/g ÷ 400 m²/g = 4.64 × 10⁻⁹ m.
When the average misleads
- Micropores present. Micropores add a lot of area but little volume, so 4V/A falls far below the size of the mesopores. Separate micro- and mesopore contributions with a t-plot or DFT.
- Broad or bimodal distributions. A single average hides two populations. Report the BJH or DFT distribution and its peak positions.
- Inconsistent windows. The BET area includes all accessible surface, but the volume read at 0.99 excludes pores above ~190 nm. Use the same basis when comparing samples.
The same idea in mercury porosimetry
MIP reports also give an “average pore diameter (4V/A)”. There V is the total intrusion volume and A the cumulative pore area from the pressure–volume integral (Rootare–Prenzlow). Because the two methods cover different size windows, MIP and nitrogen averages for the same sample are not expected to agree. See reading a MIP report.
Related quantities
- Porosity from pore volume: ε = Vp / (Vp + 1/ρs), with ρs the skeletal density (e.g. from helium pycnometry).
- Equivalent particle size from BET area (non-porous spheres): d [nm] = 6000 / (ρ [g/cm³] × S [m²/g]).
Related guides
Frequently asked questions
What is the formula for average pore diameter from BET?
For cylindrical pores, d = 4V/A, where V is the total pore volume and A the BET surface area. With V in cm³/g and A in m²/g, d [nm] = 4000 × V / A.
At what relative pressure should total pore volume be taken?
Usually at p/p₀ ≈ 0.95–0.99, on a plateau of the isotherm. At 0.99 the volume includes pores up to roughly 190 nm in diameter (N₂, 77 K). Always report the p/p₀ used.
How do I convert adsorbed volume (cm³ STP/g) to liquid pore volume?
For nitrogen at 77 K, multiply by 0.0015468 (= 28.013 / (0.808 × 22414)). For argon at 87 K the factor is about 0.00128.
Can I measure pore volume without BET?
Yes. Mercury intrusion gives the total intruded pore volume for pores from ~3 nm up to hundreds of micrometres. Helium pycnometry combined with envelope density gives total open porosity. Gas adsorption at high p/p₀ gives micro- plus mesopore volume.