Kinetics
Langmuir Adsorption Isotherm
θ = K p/(1 + K p) — surface coverage at equilibrium for non-interacting monolayer adsorption
The Langmuir adsorption isotherm θ = K p/(1 + K p) gives the fractional coverage of a uniform surface at equilibrium with a gas at partial pressure p, where K = kads/kdes. Irving Langmuir derived it between 1916 and 1918 from a kinetic equilibrium between adsorption and desorption, work that earned the 1932 Nobel Prize in Chemistry. The model assumes equivalent sites, one molecule per site, and no lateral interactions. Despite those approximations it is the foundation for heterogeneous catalysis kinetics (Langmuir-Hinshelwood), surface-area measurement (BET extension), and any setting where a finite number of binding sites equilibrates with a chemical potential — including pharmacological receptor binding.
- Equationθ = Kp/(1 + Kp)
- Kkads/kdes, units 1/pressure
- DerivedIrving Langmuir, 1916–1918
- Nobel PrizeChemistry 1932
- Linearized1/θ vs 1/p, slope 1/K
- BET extensionBrunauer-Emmett-Teller 1938 — multilayer
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Why the Langmuir isotherm matters
- First quantitative theory of adsorption. Before Langmuir's 1916 paper, adsorption was described by Freundlich's empirical power law. Langmuir put it on a kinetic-equilibrium foundation, derived saturation behavior, and won the 1932 Nobel Prize for his work on surface chemistry.
- Foundation of heterogeneous catalysis kinetics. The Langmuir-Hinshelwood mechanism (Cyril Hinshelwood, 1956 Nobel) treats two reactants as adsorbed in equilibrium, then reacting on the surface. It correctly predicts the volcano-shaped rate-vs-pressure curve seen in CO oxidation on Pt and N2 + 3H2 → 2NH3 on Fe — the Haber-Bosch reaction.
- Backbone of BET surface-area measurement. BET (Brunauer, Emmett, Teller 1938) extends Langmuir to multilayer adsorption of N2 at 77 K. Plotting p/(V(p* − p)) vs p/p* gives the monolayer capacity, which converts to specific surface area via the N2 cross-section of 0.162 nm2. Activated carbon: 100–1000 m2/g; MOF-210: 6240 m2/g.
- Saturation behavior. Langmuir predicts θ → 1 as p → ∞, the strict monolayer limit. The half-coverage pressure p1/2 = 1/K is the most useful single parameter — it sets the operating window for any process limited by surface coverage.
- Linearizable for parameter extraction. Rearranging gives 1/θ = 1 + 1/(Kp), so plotting 1/θ vs 1/p yields a straight line with slope 1/K and intercept 1. Modern fits use nonlinear regression but the linearized form remains the diagnostic plot — non-unit intercept signals model failure.
- Universal in pharmacology. Receptor binding follows the same form as Langmuir: fraction bound = [L]/(Kd + [L]), with Kd the dissociation constant. The Hill plot, Scatchard plot, and EC50 all descend from Langmuir's original equation applied to ligand-receptor equilibrium.
- Sets the catalyst design objective. Sabatier's principle — the optimum catalyst binds the substrate neither too weakly nor too strongly — is quantified by the Langmuir K. Volcano plots of activity vs ΔHads are the dominant catalyst-screening tool, descending directly from Langmuir-Hinshelwood kinetics.
Common misconceptions
- "Real surfaces obey Langmuir." Almost none do exactly. The four assumptions (equivalent sites, monolayer, no lateral interaction, dynamic equilibrium) all fail to some degree on real materials. Langmuir is a useful approximation over limited pressure ranges, not a microscopic truth.
- "K is just a fit parameter." K = kads/kdes is a real equilibrium constant; ln K vs 1/T gives the heat of adsorption ΔHads. Treating K as opaque hides the temperature dependence that distinguishes physisorption (ΔHads ≈ −20 kJ/mol) from chemisorption (ΔHads ≈ −80 to −400 kJ/mol).
- "Saturation means all sites are full." θ = 1 is asymptotic — at K p = 100 you have θ = 0.99, not exactly 1. The "monolayer capacity" extracted from BET is an extrapolation, not a directly observed plateau.
- "BET and Langmuir are independent models." BET is a Langmuir extension. Each layer above the first is treated as adsorbing onto the layer below according to a Langmuir form with the heat of liquefaction as ΔHads. The first layer keeps the original ΔHads of the substrate.
- "Langmuir-Hinshelwood is just two Langmuir isotherms." It's competitive Langmuir: θA = KApA/(1 + KApA + KBpB). The denominator's coupling is what produces the volcano-shaped rate vs pressure — independent isotherms would give monotonic rates.
- "BET works at any pressure." Valid only for p/p* between 0.05 and 0.30. Below 0.05 the data are dominated by micropore filling (use t-plot or DFT analysis); above 0.30 capillary condensation kicks in (interpret with BJH pore-size distribution). Reporting BET areas from outside this window is a common error in the materials literature.
Derivation
Consider a surface with a fixed total number of sites Ns. Let θ be the fraction occupied. The rate of adsorption is proportional to the rate of molecular impacts on empty sites: rads = kads p (1 − θ), where the impingement rate scales with pressure (Hertz-Knudsen) and the available fraction is 1 − θ. The rate of desorption is proportional to the occupied fraction: rdes = kdes θ. At equilibrium rads = rdes, so kads p (1 − θ) = kdes θ. Solve: θ = (kads/kdes) p/(1 + (kads/kdes) p) = K p/(1 + K p). K has units of inverse pressure (atm-1, Pa-1, or torr-1) and absorbs the kinetic prefactors.
The temperature dependence comes from kads/kdes. Using the van 't Hoff form, ln K = −ΔHads/(RT) + ΔSads/R. Adsorption is exothermic (ΔHads < 0) so K decreases with temperature — coverage drops at high T. Physisorption ΔHads ≈ −20 kJ/mol is similar to a heat of liquefaction; chemisorption ΔHads ≈ −80 to −400 kJ/mol involves chemical bond formation. The Langmuir form holds in both regimes, only K differs.
For competitive adsorption with two species A and B sharing the same sites, replace 1 − θ by 1 − θA − θB. Solving the two coupled equilibrium equations gives θA = KApA/(1 + KApA + KBpB). This is the kinetic core of the Langmuir-Hinshelwood mechanism: the surface reaction rate r = k θA θB has a maximum at intermediate pA/pB — the volcano. BET multilayer adsorption extends the same logic by treating each successive layer as a Langmuir adsorption on the layer below, and is the basis for surface-area measurement of porous materials.
Langmuir vs Freundlich vs BET vs Temkin vs Henry
| Isotherm | Form | Surface assumption | Coverage limit | Year / source |
|---|---|---|---|---|
| Henry | θ = K p | Dilute, no interaction | Linear (low p only) | William Henry 1803 (gas solubility analog) |
| Langmuir | θ = K p/(1 + K p) | Uniform sites, no lateral interaction | Saturates at θ = 1 | Irving Langmuir 1916–1918 |
| Freundlich | θ ∝ p1/n, n > 1 | Heterogeneous (log-distributed energies) | No saturation | Herbert Freundlich 1907 (empirical) |
| Temkin | θ = (RT/ΔQ) ln(K0 p) | Linearly decreasing ΔHads with coverage | Logarithmic | Mikhail Temkin 1940 |
| BET | (p/p*)/[V(1 − p/p*)] linearized | Multilayer, each layer Langmuir-like | No upper bound (n layers) | Brunauer-Emmett-Teller 1938 |
| Dubinin-Radushkevich | ln V = ln V0 − k ε2 | Micropore volume filling | Pore-volume saturation | Dubinin 1947 |
| Sips (Langmuir-Freundlich) | θ = (Kp)n/(1+(Kp)n) | Heterogeneous + saturation | θ = 1 | Robert Sips 1948 |
Physisorption vs chemisorption — Langmuir applies to both, K differs
| Property | Physisorption | Chemisorption |
|---|---|---|
| ΔHads | −5 to −40 kJ/mol | −40 to −400 kJ/mol |
| Bonding | Van der Waals / dispersion | Covalent or ionic surface bond |
| Specificity | Non-specific | Substrate-specific |
| Temperature range | Below 100 K (N2) to ambient | Up to several 100 °C |
| Activation energy | Near zero | Often non-zero (activated chemisorption) |
| Layers | Multilayer common — BET applies | Single layer only |
| Reversibility | Rapid, fully reversible | Slow desorption, often dissociative |
| Example | N2 on activated carbon at 77 K | H2 dissociated on Pt at 300 K |
Applications
- Ammonia synthesis (Haber-Bosch). N2 + 3H2 → 2NH3 over a fused Fe catalyst at 400–500 °C and 150–300 bar follows the Langmuir-Hinshelwood mechanism: N2 dissociatively adsorbs (rate-limiting), H2 dissociatively adsorbs in fast equilibrium, and the surface NHx intermediates combine. The volcano shape vs metal — Fe sits at the optimum N-binding energy of about −1 eV — was the original Sabatier-principle test case.
- BET surface area for catalyst screening. Modern catalyst datasheets quote BET specific surface area as the first specification: zeolite Y at 700 m2/g, MCM-41 mesoporous silica at 1000 m2/g, Pt/Al2O3 at 100 m2/g. The number sets the dispersion ceiling for active-metal loading.
- CO oxidation on Pt. 2CO + O2 → 2CO2 in catalytic converters runs through Langmuir-Hinshelwood; the reaction rate is volcano-shaped versus pCO/pO2. Strongly bound CO at high coverage poisons the surface — the Pt-Pd-Rh combination in three-way converters is engineered to avoid the high-CO branch.
- Activated-carbon water treatment. Granular activated carbon (BET surface 800–1200 m2/g) removes organic pollutants from drinking water by physisorption. Breakthrough capacity is calculated from a Langmuir or Freundlich fit to laboratory equilibrium data and scaled to bed volumes per regeneration cycle.
- Pharmacological receptor binding. The Hill-Langmuir equation y = [L]/(Kd + [L]) describes drug-receptor occupancy. EC50 = Kd for non-cooperative binding; the Hill coefficient n > 1 signals cooperativity, captured by the Sips form (Langmuir-Freundlich), which has been a textbook tool of pharmacology since Hill's 1910 paper on hemoglobin oxygen binding.
Frequently asked questions
What assumptions does the Langmuir isotherm make?
Four. (1) The surface has a finite, fixed number of equivalent adsorption sites. (2) Each site holds exactly one molecule — strict monolayer, no stacking. (3) Adsorbed molecules do not interact with their neighbors — no lateral attraction or repulsion, so the heat of adsorption is constant across coverage. (4) Adsorption-desorption is a dynamic equilibrium, not a kinetic snapshot. Real surfaces violate at least one of these in practice — heterogeneous sites, lateral interactions, or multilayer growth — but the Langmuir form often fits well over a limited pressure range, and its plate-like simplicity makes it the reference against which BET, Freundlich, and Temkin are framed.
How is the equilibrium constant K derived from rates?
Rate of adsorption equals kads p (1 − θ): proportional to gas pressure and to the fraction of empty sites. Rate of desorption equals kdes θ. At equilibrium they are equal, giving kads p (1 − θ) = kdes θ. Solving for θ yields θ = (kads/kdes) p / (1 + (kads/kdes) p) = K p/(1 + K p) with K = kads/kdes. K has units of 1/pressure and is itself temperature-dependent: K = K0 exp(−ΔHads/RT). Plotting 1/θ vs 1/p gives a line with slope 1/K and intercept 1, which is how K is extracted from data — the linearized Langmuir plot.
How does Langmuir compare to Freundlich and BET?
Freundlich (1907) is empirical: θ ∝ p1/n with n typically 1 to 10. It allows a heterogeneous surface where the heat of adsorption decreases logarithmically with coverage but never saturates — physically wrong at high pressure but often the best fit to soil and activated-carbon data. BET (Brunauer-Emmett-Teller 1938) extends Langmuir to multilayers by treating each adsorbed layer as a fresh Langmuir surface; the form θn = c p/((p* − p)(1 + (c − 1) p/p*)) where p* is saturation vapor pressure. BET is the standard for surface-area measurement: monolayer capacity → m2/g via the molecular footprint of N2 (16.2 Å2). Activated carbon gives BET surface areas of 100 to 1000 m2/g.
What is Langmuir-Hinshelwood kinetics?
A reaction model where two species A and B both adsorb onto the catalyst, then react on the surface. The rate is r = k θA θB with each coverage given by competitive Langmuir: θA = KA pA/(1 + KApA + KBpB). At low pressures r ∝ pA pB. At high pressure of one component the rate goes through a maximum and then decreases — the strongly adsorbed species crowds out its partner. This volcano shape is diagnostic of L-H and is observed for CO oxidation on Pt, hydrogenation on Ni, and ammonia synthesis on Fe. Contrast Eley-Rideal, where a gas-phase molecule reacts directly with an adsorbed one — rate is r = k θA pB, which never decreases with pB.
How is BET surface area measured in practice?
Outgas the sample at 200–300 °C under vacuum to remove water and impurities. Cool to 77 K (liquid N2) and dose with N2 in increments. At each pressure measure the gas uptake until equilibrium. Plot p/(V(p* − p)) vs p/p* — BET linearization — and extract the monolayer capacity Vm from the slope and intercept. Multiply Vm by Avogadro's number and the N2 cross-section (0.162 nm2 per molecule) to get specific surface area in m2/g. Activated carbon gives 100–1000 m2/g, zeolites 200–700 m2/g, MOFs 1000–7000 m2/g (record MOF-210 at 6240 m2/g). The valid p/p* range for BET fitting is 0.05 to 0.30 — outside that the assumptions fail.
When does the Langmuir model fail?
Three regimes. (1) High pressure where multilayer adsorption dominates — Langmuir saturates at θ = 1 but real measurements keep rising; switch to BET. (2) Strongly heterogeneous surfaces where ΔHads varies across sites — Langmuir predicts a single K, but data show coverage rising more gradually than the Langmuir curve; Freundlich or Temkin fit better. (3) Strong adsorbate-adsorbate interactions — repulsion makes coverage saturate below θ = 1 (e.g. H on Ni at 0.5 ML), attraction causes phase separation and stepped isotherms. Diagnostic: if the linear Langmuir plot 1/θ vs 1/p has a non-zero curvature or non-unit intercept, the model is failing.