BET surface area calculator with the consistency (Rouquerol) criteria

$$n=\frac{n_{\mathrm{m}}\cdot C\cdot x}{(1-x)(1-x+C\cdot x)}$$$$\frac{x}{n(1-x)}=\frac{1}{n_{\mathrm{m}}C}+\frac{C-1}{n_{\mathrm{m}}C}\cdot x$$$$S=n_{\mathrm{m}}\cdot N_{\mathrm{A}}\cdot \sigma$$

BET plot. With $x=p/p_0$ and $n$ the amount adsorbed, the BET model has the form above, and its linearised form, the BET plot of $x/[n(1-x)]=1/[n((p_0/p)-1)]$ against $x$, gives the monolayer capacity $n_{\mathrm{m}}$ and the BET constant $C$ from a straight line (Shimizu & Matubayasi, Langmuir 2022, doi:10.1021/acs.langmuir.2c00753). The calculator fits the line by least squares over the chosen range; with slope $s$ and intercept $i$, $n_{\mathrm{m}}=1/(s+i)$ and $C=1+s/i$. The BET plot is linear only over a limited part of the isotherm, usually not above $p/p_0\approx 0.3$, and the range is typically chosen between 0.05 and 0.30 (same paper, quoting IUPAC).

Surface area. The BET surface area is the monolayer capacity multiplied by the molecular cross-sectional area $\sigma$ of the adsorbate (Shimizu & Matubayasi 2022). With $n_{\mathrm{m}}$ in mmol/g and $\sigma$ in nm², $S({\mathrm{m}}^2/\mathrm{g})=n_{\mathrm{m}}\times 602.214\times \sigma$, where 602.214 is Avogadro’s number ($6.02214\times {10}^{23}{\mathrm{mol}}^{-1}$) $\times {10}^{-18}{\mathrm{m}}^2/{\mathrm{nm}}^2\times {10}^{-3}\mathrm{mol}/\mathrm{mmol}$; $\sigma =0.162{\mathrm{nm}}^2$ is often used for N₂ at 77 K and 0.142 nm² for Ar at 87 K (Manz & Delgass, ACS Omega 2026, doi:10.1021/acsomega.5c13463). Amounts in cm³(STP)/g are divided by 22.414 cm³/mmol (ideal-gas molar volume at 273.15 K and 101.325 kPa) and amounts in mg/g by the molar mass (28.014 g/mol for N₂, 39.948 g/mol for Ar). Argon at 87.3 K is preferred over N₂ at 77 K for pores smaller than about 0.7 nm (Heliyon 2023, doi:10.1016/j.heliyon.2023.e23840).

Consistency criteria. Because the linear region is hard to identify and some ranges give unphysical parameters, four consistency criteria are used (Shimizu & Matubayasi 2022): A, the BET constant $C$ must be positive; B, the BET equation is applied only where $n(1-p/p_0)$ increases continuously with $p/p_0$; C, the monolayer capacity should correspond to a relative pressure inside the selected range; D, the relative pressure of the monolayer calculated from BET theory should equal the one found in C, within 20 %. Setting $n=n_{\mathrm{m}}$ in the BET equation gives that relative pressure, $(p/p_0{)}_{\mathrm{m}}=1/(\sqrt{C}+1)$. The calculator reads the $p/p_0$ at which the measured isotherm reaches $n_{\mathrm{m}}$ by linear interpolation between points, and checks B point by point on the Rouquerol plot of $n(1-p/p_0)$. A $C$ of at least about 80 means the knee of the isotherm is sharp; below about 100 the monolayer capacity is less reliable (IUPAC and NIST, quoted by Shimizu & Matubayasi 2022).

Auto range. Different research groups have extracted different BET areas from the same data by choosing different pressure ranges: 61 laboratories given the same 18 isotherms reported a wide range of values (Osterrieth et al., Adv Mater 2022, doi:10.1002/adma.202201502), and their BETSI software expands on the Rouquerol criteria to assign the range systematically. This calculator uses a simpler rule: it fits every run of three or more neighbouring points, keeps those that pass A, B and C (and D if you ask), and takes the one with the most points, then the highest R². On a microporous carbon, an instrument’s fixed 0.05–0.30 window gave a negative C despite a high R², while a range chosen with the Rouquerol criteria ($p/p_0$ 0.0051–0.0597) gave 944 m²/g instead of the 777.5 m²/g reported (Morgan et al., Materials 2026, doi:10.3390/ma19173629).

Limits. Even when all the criteria are met, the BET calculation can overestimate the true monolayer loading of MOFs that combine mesopores and large micropores, because pore filling overlaps with monolayer formation (Gómez-Gualdrón et al., J Am Chem Soc 2016, doi:10.1021/jacs.5b10266); for NU-1103 the overestimate was about 16 % (Wang et al., J Am Chem Soc 2015, doi:10.1021/ja512973b). Report the range, the criteria it passes and σ with the area.

The built-in example is an illustrative type I N₂ isotherm at 77 K, not a measured one: 24 points from $p/p_0=0.0005$ to 0.9, with 131.1 to 669.0 cm³(STP)/g adsorbed, most of it below $p/p_0=0.01$, as in a microporous MOF or carbon.

The usual range fails. Over $p/p_0$ 0.05–0.30 (7 points) the BET plot is straight enough (R² = 0.9940), but the intercept is negative (−0.001469 g/mmol, slope 0.07307 g/mmol), so $C=1+0.07307/(-0.001469)=\mathbf{-}\mathbf{48.7}$. Criterion A fails, $n(1-p/p_0)$ falls over that range (criterion B fails), and the 1363 m²/g it would give is not a valid BET area.

Auto range. Of 253 ranges of three or more neighbouring points, 10 pass criteria A–C; the longest is $p/p_0$ 0.0005–0.04 (11 points):

1. Slope $s=0.05305\mathrm{g}/\mathrm{mmol}$, intercept $i=5.606\times {10}^{-5}\mathrm{g}/\mathrm{mmol}$, R² = 0.99996.
2. $n_{\mathrm{m}}=1/(s+i)=\mathbf{18.83}\mathbf{mmol}\mathbf{/}\mathbf{g}$ (422.1 cm³(STP)/g).
3. $C=1+s/i=\mathbf{947}$.
4. The isotherm reaches $n_{\mathrm{m}}$ at $p/p_0=0.0307$, inside the range (criterion C); $1/(\sqrt{947}+1)=0.0315$, 2.5 % away (criterion D passes).
5. $S=18.83\times 602.214\times 0.162$ $=\mathbf{1837}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{g}$.

A check with exact numbers. An ideal BET isotherm with $n_{\mathrm{m}}=5\mathrm{mmol}/\mathrm{g}$ and $C=100$ gives back exactly those values over 0.05–0.30, $p/p_0$ at the monolayer $1/(\sqrt{100}+1)=0.0909$, and $S=5\times 602.214\times 0.162=487.8{\mathrm{m}}^2/\mathrm{g}$ with N₂ (427.6 m²/g with $\sigma =0.142{\mathrm{nm}}^2$).