BET Isotherm and Multilayer Adsorption

Pour liquid nitrogen over a scoop of activated charcoal and it can adsorb a surprising volume of gas. Much more than a single monolayer could ever hold. The Langmuir model, built for one layer of adsorbed molecules, cannot explain this. Brunauer, Emmett, and Teller extended Langmuir's kinetic picture to stacks of physisorbed layers, and their isotherm is now the standard method for measuring surface area. The BET theory matters because surface area controls catalytic activity, gas storage capacity, and powder reactivity. Nearly every catalyst datasheet quotes a "BET surface area" measured from a nitrogen adsorption experiment. Understanding where that number comes from means understanding how multilayer physisorption differs from the single-layer chemisorption picture of Langmuir. This article builds the BET isotherm from a multilayer extension of Langmuir kinetics. You will see the derivation step by step. Then you will see the linearized BET plot used to extract surface area. Finally, two worked examples show how real nitrogen-adsorption data become a number in square meters.

Key takeaways

BET theory generalizes Langmuir's single-layer model to infinite stacked layers of physisorbed gas.
The isotherm introduces one new parameter, $c$ , related to the heat of adsorption of the first layer.
A linear BET plot of $p/[V(p^0-p)]$ against $p/p^0$ gives the monolayer volume $V_m$ from its slope and intercept.
$V_m$ combines with the molecular cross-sectional area to give the specific surface area, in m² g⁻¹.
The model is valid only over a limited relative pressure range, roughly $p/p^0 = 0.05$ to $0.35$ .
Key idea: treating adsorption as a dynamic equilibrium of stacked layers, each obeying Langmuir-like kinetics, replaces a single saturation limit with an unbounded uptake that diverges as $p \to p^0$ .

The core insight

BET theory keeps the Langmuir assumption of a dynamic equilibrium between adsorption and desorption on each site, but allows any number of gas layers to stack vertically. The first layer binds directly to the surface with its own equilibrium constant, while every layer above it condenses and evaporates like bulk liquid, so the whole tower of layers is described by just two numbers: the monolayer capacity $V_m$ and a single ratio of rate constants, $c$ .

The physical picture

In the Langmuir model, gas molecules can occupy surface sites, but once every site holds one molecule, adsorption stops. That produces a saturation curve that flattens out at high pressure. Real physisorption data on porous solids often do not flatten. Instead the amount adsorbed keeps rising, sometimes sharply, as the pressure approaches the saturation vapor pressure $p^0$ of the adsorbing gas at the experimental temperature. Brunauer, Emmett, and Teller (1938) explained this by allowing molecules to adsorb on top of already-adsorbed molecules. The bare surface sites adsorb gas with a characteristic binding energy $\Delta H_1$ , exactly as in Langmuir theory. But a molecule sitting on top of a first-layer molecule interacts with another molecule of the same kind, so its adsorption/desorption behavior resembles ordinary condensation of the bulk liquid, governed by the enthalpy of vaporization. In the limit $p \to p^0$ , an infinite number of layers can build up, mimicking bulk liquid condensation. This is exactly the physical limit enforced by the Kelvin equation for curved liquid surfaces, which is why BET and Kelvin-based capillary condensation analyses are typically applied together on the same isotherm. The result is an S-shaped ("Type II") isotherm for adsorption on a flat or macroporous surface: a knee at low pressure marking approximate monolayer completion, then a long linear rise, then an upward sweep as $p \to p^0$ and condensation floods the surface.

Figure

Building up layers: Langmuir versus BET adsorption

The core relations

V = \frac{V_m\, c\, x}{(1-x)\left[1 + (c-1)x\right]}, \qquad x \equiv \frac{p}{p^0}

where: $V$ is the volume of gas adsorbed at pressure $p$ , $V_m$ the monolayer volume, $p^0$ the saturation vapor pressure, and $c$ the BET constant.

c \approx \exp\!\left(\frac{\Delta H_1 - \Delta H_L}{RT}\right)

where: $\Delta H_1$ is the enthalpy of adsorption of the first layer, $\Delta H_L$ the enthalpy of liquefaction of the adsorbate, $R$ the gas constant, $T$ the temperature.

\frac{p}{V(p^0-p)} = \frac{1}{V_m c} + \frac{c-1}{V_m c}\cdot\frac{p}{p^0}

where: this is the linearized BET equation, plotted as $y = p/[V(p^0-p)]$ against $x = p/p^0$ .

A_{\text{s}} = \frac{V_m}{V_{\text{mol}}} N_A \sigma

where: $A_s$ is total surface area, $V_{\text{mol}}$ the molar volume of gas at STP, $N_A$ Avogadro's number, $\sigma$ the cross-sectional area of one adsorbate molecule.

Derivation AFrom Langmuir kinetics to the BET isotherm

Goal. Derive the BET isotherm by extending Langmuir's dynamic-equilibrium argument to an infinite ladder of adsorbed layers, then linearize the result into a plottable form.

Step 1: set up the layer populations. Let $\theta_0, \theta_1, \theta_2,\dots$ denote the fraction of surface sites bearing exactly $0, 1, 2,\dots$ adsorbed layers. Unlike Langmuir theory, sites are not capped at one molecule; a site can carry a whole stack. The total surface must be fully accounted for,

\sum_{i=0}^{\infty} \theta_i = 1.

The total volume adsorbed (relative to the monolayer volume $V_m$ , which corresponds to $\sum_i i\,\theta_i = 1$ if every site had exactly one layer) is

\frac{V}{V_m} = \sum_{i=0}^{\infty} i\,\theta_i.

Step 2: apply Langmuir-type kinetics to the first layer. Bare sites ( $\theta_0$ ) adsorb a first molecule with rate $a_1 p\,\theta_0$ and that first-layer molecule desorbs with rate $b_1 \theta_1$ . At equilibrium these rates balance,

a_1 p\,\theta_0 = b_1 \theta_1 \quad\Rightarrow\quad \theta_1 = \left(\frac{a_1 p}{b_1}\right)\theta_0 \equiv y\,\theta_0,

where $y = (a_1/b_1)p$ collects the first-layer rate constants. This step is identical in spirit to the ordinary Langmuir derivation for a single layer.

Step 3: treat higher layers as condensation of the bulk liquid. A molecule landing on top of an existing first-layer molecule to create a second layer, and every subsequent layer above it, behaves like ordinary vapor-liquid equilibrium rather than site-specific chemisorption. BET's key simplifying postulate is that the ratio of rate constants for every layer $i \ge 2$ equals the ratio for bulk liquefaction,

a_i/b_i = a_L/b_L \equiv g, \quad \text{for all } i\ge 2,

independent of layer index. Detailed balance between layer $i-1$ and layer $i$ (for $i \ge 2$ ) then gives

\theta_i = (g\,p)\,\theta_{i-1} \equiv x\,\theta_{i-1}, \qquad x \equiv g\,p.

At the saturation vapor pressure $p^0$ , bulk liquid and vapor coexist, so $x \to 1$ exactly when $p \to p^0$ ; this fixes $g = 1/p^0$ and identifies $x = p/p^0$ .

Step 4: unroll the recursion. Iterating Step 3 from $\theta_1$ gives $\theta_i = x^{i-1}\theta_1$ for $i \ge 1$ . Combining with Step 2, $\theta_1 = y\theta_0$ , and writing $c \equiv y/x$ (the ratio of first-layer to higher-layer equilibrium constants, evaluated at the same pressure) yields

\theta_i = c\,x^{i}\,\theta_0, \qquad i \ge 1.

The parameter $c$ is large when the surface binds the first layer much more strongly than molecules bind to each other in the liquid — physically, $c$ measures the "stickiness contrast" between chemisorption-like first-layer binding and ordinary liquefaction.

Step 5: normalize and sum the series. Substitute into the normalization condition from Step 1:

\theta_0 + c\,\theta_0\sum_{i=1}^{\infty} x^i = 1 \quad\Rightarrow\quad \theta_0\left[1 + \frac{c\,x}{1-x}\right] = 1 \quad\Rightarrow\quad \theta_0 = \frac{1-x}{1-x+cx}.

Here the geometric series $\sum_{i=1}^\infty x^i = x/(1-x)$ requires $x<1$ , i.e. $p<p^0$ , consistent with never exceeding the saturation vapor pressure.

Step 6: sum the adsorbed volume series. Now evaluate

\frac{V}{V_m} = \sum_{i=1}^{\infty} i\,\theta_i = c\,\theta_0 \sum_{i=1}^{\infty} i\,x^i = c\,\theta_0\cdot\frac{x}{(1-x)^2},

using the standard identity $\sum_{i=1}^\infty i x^i = x/(1-x)^2$ . Substituting $\theta_0$ from Step 5:

\frac{V}{V_m} = \frac{c\,x}{(1-x)^2}\cdot\frac{1-x}{1-x+cx} = \frac{c\,x}{(1-x)\left[1+(c-1)x\right]}.

Step 7: rearrange into the linear BET form. Multiply both sides by $(1-x)$ and divide by $x$ , then rearrange algebraically to isolate $x/[(1-x)]$ as a linear function of $x$ :

\frac{x}{(1-x)}\cdot\frac{V_m}{V} = \frac{1}{c} + \frac{c-1}{c}x.

Replacing $x = p/p^0$ and dividing through by $p^0$ gives the standard plotting form,

\boxed{\dfrac{p}{V(p^0-p)} = \dfrac{1}{V_m c} + \dfrac{c-1}{V_m c}\cdot\dfrac{p}{p^0}}

A plot of the left side against $p/p^0$ is a straight line with slope $(c-1)/(V_m c)$ and intercept $1/(V_m c)$ , from which both $V_m$ and $c$ can be extracted algebraically.

Worked examples

Example 1 — Extracting $V_m$ and $c$ from a BET plot (foundational). Nitrogen adsorption on a silica sample at 77 K gives the following two data points, already converted to the BET linear form $y = p/[V(p^0-p)]$ , with $V$ in cm³ (STP) g⁻¹: At $p/p^0 = 0.10$ : $y = 0.0230$ cm⁻³·g. At $p/p^0 = 0.30$ : $y = 0.0500$ cm⁻³·g.

Step 1 — slope and intercept. Treating this as two points on the line $y = \text{intercept} + \text{slope}\cdot x$ :

\text{slope} = \frac{0.0500 - 0.0230}{0.30 - 0.10} = \frac{0.0270}{0.20} = 0.1350 \text{ cm}^{-3}\text{g}

\text{intercept} = 0.0230 - 0.1350(0.10) = 0.0230 - 0.0135 = 0.00950 \text{ cm}^{-3}\text{g}

Step 2 — solve for $V_m$ and $c$ . From the theory, slope $=(c-1)/(V_mc)$ and intercept $=1/(V_mc)$ , so

\text{slope} + \text{intercept} = \frac{c-1}{V_mc} + \frac{1}{V_mc} = \frac{c}{V_mc} = \frac{1}{V_m}

V_m = \frac{1}{\text{slope}+\text{intercept}} = \frac{1}{0.1350+0.00950} = \frac{1}{0.1445} = 6.92 \text{ cm}^3\text{g}^{-1}

Then

c = 1 + \frac{\text{slope}}{\text{intercept}} = 1 + \frac{0.1350}{0.00950} = 1 + 14.2 = 15.2

Step 3 — surface area. At STP, one mole of gas occupies $V_{\text{mol}} = 22{,}400$ cm³. The number of moles of N₂ in the monolayer per gram of solid is

n_m = \frac{V_m}{V_{\text{mol}}} = \frac{6.92}{22400} = 3.09\times10^{-4} \text{ mol g}^{-1}

Each N₂ molecule occupies $\sigma = 0.162\ \text{nm}^2 = 1.62\times10^{-19}\ \text{m}^2$ (standard value from the liquid-nitrogen density). Total surface area per gram:

A_s = n_m N_A \sigma = (3.09\times10^{-4})(6.022\times10^{23})(1.62\times10^{-19}\ \text{m}^2)

A_s = (1.861\times10^{20})(1.62\times10^{-19}\ \text{m}^2) = 30.1\ \text{m}^2\text{g}^{-1}

Sanity check. $c \approx 15$ is a typical value for N₂ physisorption on oxide surfaces (moderate first-layer preference), and $30\ \text{m}^2\text{g}^{-1}$ is a reasonable specific surface area for a non-porous or mildly porous silica powder — consistent with literature values (tens of m² g⁻¹).

Example 2 — Comparing two catalysts and checking the pressure range (applied). A chemist runs full N₂ adsorption isotherms at 77 K on two catalyst supports, A and B, over $p/p^0 = 0.05$ to $0.35$ , and fits the linear BET equation to get: Support A: slope $= 0.0850$ g/cm³, intercept $= 0.00210$ g/cm³ (using $y$ in g cm⁻³ this time, with sample mass folded in — units cancel consistently within the ratio). Support B: slope $= 3.40$ g/cm³, intercept $= 0.0206$ g/cm³.

Step 1 — monolayer volumes.

V_{m,A} = \frac{1}{0.0850+0.00210} = \frac{1}{0.0871} = 11.5 \text{ cm}^3\text{g}^{-1}

V_{m,B} = \frac{1}{3.40+0.0206} = \frac{1}{3.42} = 0.292 \text{ cm}^3\text{g}^{-1}

Step 2 — BET constants.

c_A = 1+\frac{0.0850}{0.00210} = 1+40.5=41.5

c_B = 1+\frac{3.40}{0.0206}=1+165.0=166.0

Step 3 — surface areas (same $\sigma = 0.162\ \text{nm}^2$ for N₂):

n_{m,A} = \frac{11.5}{22400}=5.13\times10^{-4}\ \text{mol g}^{-1} \;\Rightarrow\; A_{s,A}=(5.13\times10^{-4})(6.022\times10^{23})(1.62\times10^{-19})=50.0\ \text{m}^2\text{g}^{-1}

n_{m,B} = \frac{0.292}{22400}=1.30\times10^{-5}\ \text{mol g}^{-1} \;\Rightarrow\; A_{s,B}=(1.30\times10^{-5})(6.022\times10^{23})(1.62\times10^{-19})=1.27\ \text{m}^2\text{g}^{-1}

Step 4 — interpretation. Support A has both a much larger surface area (50.0 vs 1.27 m² g⁻¹) and a moderate $c$ value (41.5), typical of a genuinely porous oxide with reasonably strong first-layer binding. Support B's very high $c$ (166) combined with tiny surface area suggests a nearly non-porous, low-area material where the few available sites bind nitrogen unusually strongly — worth flagging for further BJH pore-size analysis rather than accepting the number blindly.

Sanity check. Both areas fall within the physically sensible range for solid catalyst supports (roughly 1 to a few hundred m² g⁻¹), and $c$ values between 20 and 200 are typical for physisorption systems where BET theory is well-behaved; $c$ values far outside this range often signal microporosity effects that violate the theory's assumptions.

Limiting cases & common traps

Situation	What happens
$c \gg 1$ (strong first layer)	Sharp knee near monolayer completion; classic Type II isotherm; easy to read off $V_m$ visually
$c \approx 1$	First layer binds no differently from bulk liquid; isotherm looks more like Type III, BET plot less reliable
$x \to 0$	Isotherm reduces to the Langmuir form: $V/V_m \to cx$ , linear in $p$
$x \to 1$ ( $p \to p^0$ )	$V \to \infty$ : bulk condensation, unbounded uptake, correctly reproducing liquefaction
Microporous solids	BET plot curves; the model assumes infinite stacking room, which pores of molecular width do not have

The chemist behind it

Stephen Brunauer, Paul Emmett, and Edward Teller published the theory in 1938 while working at the U.S. Bureau of Mines and George Washington University, motivated by the very practical problem of measuring the surface area of industrial catalysts used in ammonia synthesis. Emmett had spent years studying iron catalysts for the Haber process and needed a reliable, general method to quantify catalyst surface area from a simple gas adsorption measurement — a need that the single-layer Langmuir isotherm, developed two decades earlier for chemisorption, could not meet for physisorbing gases like nitrogen or krypton.

Origins · 1 minPaul Hugh Emmett and how this was discovered

Paul Hugh Emmett

1900-1985 · American chemist who linked catalyst surface area to the BET method, transforming catalysis research from art to quantitative science

Ask the tutor: Want to see how the high-pressure branch of this same isotherm feeds into the Kelvin equation to extract a pore-size distribution?

How is the BET surface area different from the actual geometric surface area of a solid?

BET surface area is derived from the amount of gas needed to form one statistical monolayer, assuming a fixed molecular cross-section for the adsorbate. It captures all accessible surface, including internal pore walls, not just the outer geometric envelope of a particle. For rough or porous solids, BET area can be far larger than geometric area calculated from particle size and shape, which is exactly why BET analysis is essential for catalysts and powders.

Why is nitrogen at 77 K the standard gas for BET measurements?

Nitrogen is chemically inert toward almost all solid surfaces, has a well-characterized cross-sectional area (0.162 nm²), and its saturation vapor pressure at its own boiling point, 77 K, is conveniently near one atmosphere. This makes it easy to reach the required relative pressures experimentally. Other gases like krypton or argon are used for very low surface area materials where nitrogen gives too little measurable uptake.

What happens to the BET equation if c becomes very large?

As $c \to \infty$ , the isotherm sharpens into a near-vertical rise at low pressure followed by a plateau, essentially recovering Langmuir-like monolayer saturation before multilayer buildup dominates at higher pressure. The knee in the isotherm becomes very well defined, and $V_m$ can often be read almost directly from the graph without needing the full linear regression.

How does BET connect to the Kelvin equation and capillary condensation?

BET theory describes progressive layer-by-layer adsorption on open surfaces at low to moderate relative pressure. Once pores are narrow enough that opposing adsorbed layers meet, further uptake is governed by capillary condensation, where the curved liquid meniscus lowers the effective vapor pressure according to the Kelvin equation. Analyzing the high-pressure branch of the same isotherm with the Kelvin equation yields pore size distributions, a natural extension covered under capillary condensation and pore analysis.