The Langmuir adsorption isotherm

Result

The Langmuir adsorption isotherm

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Statement

Surface coverage as a function of pressure.

Why it matters

catalysis-activation-energy and heterogeneous-catalysis both describe how surface catalysis proceeds through an adsorb–react–desorb cycle, but neither, on its own, gives a quantitative expression for how much of a surface is actually covered by adsorbed reactant at a given gas pressure. The Langmuir isotherm supplies exactly this missing quantitative piece: a simple, physically motivated relation between surface coverage and pressure that underlies the rate law derived in heterogeneous-catalysis, and that remains the standard starting model for interpreting adsorption data across surface chemistry more broadly.

Hypotheses

The surface offers a fixed number of identical, energetically equivalent adsorption sites, each capable of holding at most one adsorbed molecule (monolayer adsorption).Real surfaces can have a distribution of site energies (a more strongly binding defect site versus a weaker terrace site, for instance) and, at sufficiently high pressure, can support adsorption in additional layers beyond the first (multilayer adsorption, treated by the more elaborate BET model); the Langmuir model's single-site-type, monolayer-only picture is a deliberate simplification valid over a more restricted pressure and coverage range. Adsorbed molecules do not interact with one another (no lateral, adsorbate–adsorbate interactions), so the energetics of adsorbing onto any given empty site is independent of how many neighbouring sites are already occupied.Real adsorbed layers frequently show measurable lateral interactions (either repulsive or attractive between neighbouring adsorbed molecules), which would make the effective binding constant \(K\) itself depend on coverage \(\theta\), a coverage-dependence entirely absent from the simple Langmuir model developed here.

Proof

\text{A(g)} + \text{site} \underset{k_d}{\overset{k_a}{\rightleftharpoons}} \text{A(ads)}

Adsorption and desorption are treated as a dynamic equilibrium: gas-phase molecules strike and stick to empty sites at a rate proportional to both the gas pressure and the fraction of sites still empty, while adsorbed molecules leave at a rate proportional simply to the fraction of sites currently occupied. A

\text{rate}_{\text{ads}} = k_aP(1-\theta), \qquad \text{rate}_{\text{des}} = k_d\theta

Writing the forward (adsorption) rate as proportional to pressure \(P\) and to the empty-site fraction \((1-\theta)\), and the reverse (desorption) rate as proportional to the occupied-site fraction \(\theta\) alone, follows directly from treating each empty or occupied site independently (Hypotheses, no lateral interactions). A

\text{At equilibrium: } k_aP(1-\theta) = k_d\theta \ \Longrightarrow\ \theta = \frac{k_aP}{k_d+k_aP} = \frac{KP}{1+KP}, \quad K\equiv\frac{k_a}{k_d}

Setting the forward and reverse rates of Step 2 equal (the defining condition of dynamic equilibrium) and solving algebraically for \(\theta\) gives the coverage directly as a function of pressure, with the two rate constants combining into a single equilibrium binding constant \(K\), the ratio of adsorption to desorption rate constants. A

KP\ll1:\ \theta\approx KP\ (\text{linear, low-pressure limit}); \qquad KP\gg1:\ \theta\to1\ (\text{saturated, high-pressure limit})

Expanding Step 3's result in the two limiting regimes shows coverage rising linearly with pressure at low pressure (few sites occupied, adsorption dominates unopposed) and saturating at complete monolayer coverage at high pressure (essentially every site occupied, no further increase possible) — the characteristic saturating shape of the isotherm. A

Result

\theta = \frac{KP}{1+KP}

Reading. Fractional surface coverage rises from zero, roughly linearly with pressure at low pressure, and asymptotically saturates toward complete monolayer coverage (\(\theta\to1\)) at high pressure, with the single parameter \(K\) (the ratio of adsorption to desorption rate constants) setting how quickly saturation is approached.

Scope. Valid for a single adsorbate on a uniform, single-site-type surface with no lateral interactions and no multilayer adsorption (Hypotheses); real surfaces with heterogeneous site energies, adsorbate–adsorbate interactions, or multilayer behaviour at high pressure require more elaborate isotherm models (such as the BET isotherm for multilayer adsorption) beyond this simple form.

Corollaries & converses

heterogeneous-catalysis's rate law, when the surface reaction step is rate-determining, is built directly on substituting this Result's \(\theta\) into a rate expression proportional to coverage — the Langmuir isotherm supplies the pressure-dependence machinery that catalytic kinetics needs as an input.
For a competitive system with two adsorbates \(A\) and \(B\) both binding the same finite pool of sites, the isotherm generalises directly to \(\theta_A = K_AP_A/(1+K_AP_A+K_BP_B)\), showing explicitly how a strongly binding second species (large \(K_B\)) can suppress \(A\)'s coverage even at fixed \(P_A\) — the quantitative basis for competitive catalyst poisoning.
Converse: measuring \(\theta\) (via, for instance, the volume of gas adsorbed) as a function of pressure and fitting to the Result's functional form is the standard experimental method for extracting a surface's binding constant \(K\) for a given adsorbate.

Fails without

Apply the single-monolayer hypothesis at pressures high enough for genuine multilayer adsorption to occur: the Result predicts \(\theta\) saturating smoothly at exactly \(1\) (Step 4), but a real adsorbate capable of forming additional layers on top of the first will show continued, unbounded uptake at high pressure that the simple Langmuir form cannot describe at all, requiring the BET isotherm instead.
Drop the no-lateral-interaction hypothesis, applying the Result to a system with strong adsorbate–adsorbate attraction or repulsion: the true binding constant then depends on coverage itself (adsorption becoming progressively easier or harder as neighbouring sites fill), so the single, coverage-independent \(K\) assumed throughout the Proof no longer accurately describes the system, and the fitted isotherm deviates systematically from the simple Result.

Common errors

Assuming coverage \(\theta\) rises linearly with pressure across the entire pressure range, rather than only in the low-pressure limit \(KP\ll1\) (Step 4); at higher pressure the relationship curves and saturates.
Applying the single-adsorbate Langmuir isotherm to a system where two or more species compete for the same sites, without using the competitive, multi-species generalisation (Corollaries).
Confusing the binding constant \(K\) (a ratio of rate constants, \(k_a/k_d\), governing the position of the adsorption/desorption equilibrium) with either rate constant individually.
Extracting \(K\) from adsorption data taken at pressures high enough that multilayer adsorption has already begun, without first confirming, from the shape of the isotherm itself, that the system remains in the genuinely monolayer, Langmuir-applicable regime.

Discussion

Irving Langmuir developed this isotherm in 1916–1918 as part of his broader programme of research into surface chemistry and adsorption, work recognised with the 1932 Nobel Prize in Chemistry; despite its deliberately simplified assumptions, the isotherm remains the standard first model taught and applied across surface chemistry, catalysis, and even, in modified form, in biochemistry (ligand–receptor binding follows a mathematically identical saturation curve).

The Brunauer–Emmett–Teller (BET) isotherm, developed in 1938 by extending Langmuir's single-layer picture to allow adsorption in multiple stacked layers, is the standard method for measuring a solid's total surface area from gas-adsorption data (via the volume of gas needed to complete a monolayer), a widely used practical technique in materials characterisation built directly on the Langmuir framework as its starting point.

Common misconception: that a low measured coverage at a given pressure necessarily means the adsorbate binds only weakly to the surface. Low coverage at a given pressure can equally well reflect operating in the low-pressure, linear regime of an isotherm with a genuinely large \(K\) (Step 4); the binding constant \(K\) itself, not the coverage at any single arbitrarily chosen pressure, is the correct measure of intrinsic binding strength.

Worked examples

K=0.500\,\text{atm}^{-1}; \quad P=1.00\,\text{atm}

Substituting directly into the Result: \(\theta = \dfrac{0.500\times1.00}{1+0.500\times1.00} = \dfrac{0.500}{1.500} = 0.333\), roughly one-third of the surface sites occupied at this pressure. A

\text{Same } K, \text{ at } P=10.0\,\text{atm}

\(\theta = \dfrac{0.500\times10.0}{1+0.500\times10.0} = \dfrac{5.00}{6.00} = 0.833\); increasing pressure tenfold raises coverage only from \(0.333\) to \(0.833\), not tenfold, directly illustrating the saturating, sub-linear shape of the isotherm at higher pressures. A

\theta(1\,\text{atm})=0.333; \qquad \theta(10\,\text{atm})=0.833

Reading. A tenfold increase in pressure produces a much smaller than tenfold increase in coverage once a substantial fraction of sites are already occupied, the direct experimental signature of approaching monolayer saturation.

Scope. The same formula, with any measured \(K\), predicts coverage at any pressure within the model's stated single-layer, non-interacting-site scope.

Problems

Given \(K=2.00\,\text{atm}^{-1}\), find the pressure at which the surface reaches exactly \(\theta=0.750\) coverage.
Solution
Rearranging the Result: \(\theta(1+KP)=KP \Rightarrow \theta = KP(1-\theta) \Rightarrow P = \dfrac{\theta}{K(1-\theta)} = \dfrac{0.750}{2.00\times0.250} = \dfrac{0.750}{0.500} = 1.50\,\text{atm}\).
Two adsorbates, \(A\) (\(K_A=1.00\,\text{atm}^{-1}\)) and \(B\) (\(K_B=5.00\,\text{atm}^{-1}\)), compete for the same sites, both present at \(P_A=P_B=1.00\,\text{atm}\). Using the competitive isotherm from the Corollaries, find \(\theta_A\).
Solution
\(\theta_A = \dfrac{K_AP_A}{1+K_AP_A+K_BP_B} = \dfrac{1.00\times1.00}{1+1.00\times1.00+5.00\times1.00} = \dfrac{1.00}{7.00} = 0.143\). Despite \(A\) and \(B\) being present at equal pressure, \(B\)'s much larger binding constant lets it dominate the available sites, suppressing \(A\)'s coverage well below what it would achieve alone (which, from Worked Example 1 at \(K=1.00\)... recomputing: alone, \(\theta_A=K_AP_A/(1+K_AP_A)=1.00/2.00=0.500\), substantially higher than the competitive value of \(0.143\)).
Explain, using Step 4 of the Proof, why measuring adsorption only at a single, arbitrarily chosen low pressure is insufficient to reliably determine the binding constant \(K\).
Solution
In the low-pressure limit, \(\theta\approx KP\) (Step 4), so \(\theta\) and \(K\) are not independently distinguishable from a single measurement alone: a low observed \(\theta\) could reflect either a small \(K\) at moderate pressure or a large \(K\) at very low pressure, and the linear relationship alone cannot separate the two without also knowing \(P\) precisely and checking consistency across pressure. Reliable determination of \(K\) requires measuring \(\theta\) across a range of pressures spanning from the low-pressure (linear) into the high-pressure (saturating) regime, and fitting the full functional form of the Result, rather than relying on any single data point.