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Langmuir Isotherm Calculator

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θ = K·P / (1 + K·P).
Gas + Solution Adsorption.
5-Band Classification.
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No Data Stored.

How it Works

01Enter K_eq

Adsorption equilibrium constant — units of 1/[P]. Larger K = stronger binding to the surface

02Enter P

Adsorbate partial pressure (gas) or concentration (solution) — supports 8 unit options

03Apply Langmuir

θ = K·P / (1 + K·P) — the fractional surface coverage of a monolayer adsorption model

04Read Coverage

Get θ, % coverage, the K·P regime (linear / transitional / saturated), and 5-band classification

What is a Langmuir Isotherm Calculator?

The Langmuir adsorption isotherm is the foundational equation of surface chemistry — the model Irving Langmuir derived in 1918 that explained, for the first time, why a surface saturates with adsorbate as pressure increases instead of accumulating indefinitely. The equation is elegantly simple: θ = (Keq · P) / (1 + Keq · P), where θ is the fractional surface coverage (0 to 1), Keq is the equilibrium constant for adsorption (units of 1/), and P is the partial pressure (gas-phase) or concentration (solution-phase) of the adsorbate. Langmuir won the 1932 Nobel Prize in Chemistry for this work — the only Nobel awarded for an isotherm. Our Langmuir Isotherm Calculator implements his model with full unit flexibility, automatic gas-vs-solution detection, regime classification (linear, transitional, saturated), and a sample isotherm profile showing how θ evolves across multiple decades of P at your fixed K_eq.

Just enter Keq and P with consistent unit families (both gas-pressure: Pa, kPa, bar, atm, Torr — or both concentration: M, mM, μM). The calculator normalizes everything internally and computes the dimensionless product K·P, then forms θ = K·P / (1 + K·P). The result tells you what fraction of available surface sites is occupied by adsorbate at equilibrium under your conditions. The half-coverage point (θ = 0.5, where K·P = 1) corresponds to P = 1/K — a direct experimental measure of the adsorption affinity. Three regimes emerge naturally: K·P ≪ 1 gives the linear (Henry's-law) regime where θ ≈ K·P; K·P ≈ 1 gives the transitional region where the isotherm curves most strongly (best for fitting K from data); K·P ≫ 1 gives the saturation regime where the surface is nearly fully covered.

Designed for physical chemistry students learning surface chemistry, catalysis researchers studying CO/H₂/N₂ binding to metal surfaces, materials scientists characterizing porous adsorbents (zeolites, metal-organic frameworks, activated carbon), environmental engineers modeling pollutant removal by activated carbon filters, and biochemists working with enzyme-substrate or receptor-ligand binding (mathematically identical to Michaelis-Menten), the tool runs entirely in your browser — no data is stored or transmitted.

Pro Tip: Pair this with our Michaelis-Menten Equation Calculator — the enzyme-kinetics equation v = V_max· / (K_M + ) is mathematically equivalent to Langmuir's. Or use our Equilibrium Constant Calculator to relate K_eq to ΔG° via ΔG° = −RT·ln(K).

How to Use the Langmuir Isotherm Calculator?

Pick the Adsorbate Family: Both Keq and P must use units from the same family — either gas-pressure (Pa, kPa, bar, atm, Torr) for gas-phase adsorption studies, or concentration (M, mM, μM) for solution-phase adsorption. The calculator validates this automatically and flags mismatched families.
Enter Keq: The adsorption equilibrium constant in units of 1/. Larger K_eq means stronger binding to the surface. For typical gas adsorption studies K is in the range 10⁻⁵ to 10⁻² Pa⁻¹; for activated-carbon dye removal in water K is often 1-1000 M⁻¹.
Enter P: The adsorbate partial pressure (gas) or concentration (solution) at the conditions you want to evaluate θ. Choose any matching unit; conversion is automatic.
Press Calculate: The calculator forms the dimensionless K·P product, applies θ = K·P / (1 + K·P), and reports θ as a fraction (0-1) and as a percentage.
Read the Results: θ headline + 5-band classification (very-low / low / moderate / high / saturated); the K·P product (telling you which regime you're in); the empty-site fraction (1 − θ); the half-coverage pressure 1/K; a sample isotherm table showing θ at 9 different P values across 4 decades; and a full step-by-step calculation breakdown.

How is the Langmuir isotherm derived?

Langmuir's 1918 derivation set the standard for all later isotherm models. The genius was assuming a uniform monolayer with one molecule per site — and proving that this single assumption explains the universal saturation curve. Here's the complete derivation:

Langmuir treated adsorption as a chemical equilibrium between gas-phase adsorbate (A) and surface sites (S): A + S ⇌ A·S. He balanced the rates of adsorption and desorption to derive the now-famous expression for fractional coverage.

The Three Assumptions

  1. All sites are energetically equivalent. Every adsorption site has the same binding energy — no high-energy "preferred" sites, no low-energy "leftover" sites.
  2. One molecule per site. Adsorption forms a strict monolayer. Once a site is occupied, no more molecules can sit on top.
  3. No lateral interactions. Adsorbed molecules don't interact with each other (no attractive or repulsive forces between neighbors).

The Rate Argument

Let θ = fraction of sites occupied (so 1 − θ = fraction empty). At dynamic equilibrium:

  • Rate of adsorption = kads · P · (1 − θ). Adsorption is proportional to the partial pressure (collision rate) and to the empty fraction (available sites).
  • Rate of desorption = kdes · θ. Desorption is proportional to occupied fraction (only occupied sites can lose their molecule).

At equilibrium: kads · P · (1 − θ) = kdes · θ.

Solving for θ

Define Keq = kads / kdes (units 1/). Then:

Keq · P · (1 − θ) = θ

Keq · P − Keq · P · θ = θ

Keq · P = θ · (1 + Keq · P)

θ = (Keq · P) / (1 + Keq · P)

The Three Regimes

  • Linear (Henry's law) regime — K·P ≪ 1: denominator ≈ 1, so θ ≈ K · P. Surface coverage scales linearly with pressure; valid at low pressures where the surface is mostly empty.
  • Transitional regime — K·P ≈ 1: the curve bends. At K·P = 1, exactly half the sites are occupied (θ = 0.5). This is the most informative experimental regime for fitting K from data.
  • Saturation regime — K·P ≫ 1: denominator ≈ K·P, so θ → 1. Surface is essentially full; further pressure increases produce diminishing additional coverage. This saturation behavior was Langmuir's key prediction that distinguished his model from older "linear adsorption" assumptions.

Half-Coverage Pressure: P1/2 = 1/K

Setting θ = 0.5 gives K·P = 1, hence P1/2 = 1/K. This is a direct experimental measure of the adsorption affinity — the smaller P1/2, the stronger the binding. For chemisorption (covalent bonding), P1/2 is often in the µPa range; for physisorption, it's in the kPa to MPa range.

Linearized Forms (For Fitting K from Data)

Langmuir's equation can be linearized in two common ways:

  • 1/θ vs 1/P (Lineweaver-Burk-like): 1/θ = 1 + 1/(K·P) — slope 1/K, intercept 1.
  • P/θ vs P (more numerically stable): P/θ = 1/K + P — slope 1, intercept 1/K.

Modern practice prefers nonlinear least-squares fitting directly to the Langmuir form, but the linearized versions are still useful for visual data inspection.

Connection to Other Adsorption Models

  • BET (Brunauer-Emmett-Teller, 1938): extends Langmuir to multilayer adsorption; reduces to Langmuir for the first layer.
  • Freundlich (1909): empirical θ = K · P1/n; for heterogeneous surfaces. Special case of Langmuir-Freundlich for n = 1.
  • Temkin: θ = (RT/ΔH) · ln(K · P); accounts for linear binding-energy heterogeneity.
  • Toth: three-parameter generalization that handles both heterogeneity and saturation.

Langmuir is the simplest and most widely taught — the right starting point for any adsorption problem.

Real-World Example

Langmuir Isotherm Calculator – Worked Examples

Example 1 — N₂ on activated carbon at 77 K. Typical K_eq ≈ 5 × 10⁻⁵ Pa⁻¹. At P = 50 kPa = 50,000 Pa:
  • K · P = (5 × 10⁻⁵) × 50,000 = 2.5.
  • θ = 2.5 / (1 + 2.5) = 2.5 / 3.5 = 0.714 = 71.4%.
  • Surface is well into the high-coverage band, approaching saturation. Doubling P to 100 kPa would give K·P = 5, θ = 0.833 — only +12 percentage points more coverage. Diminishing returns at high P is the Langmuir signature.
  • Half-coverage pressure: P1/2 = 1/K = 1/(5×10⁻⁵) = 20,000 Pa = 20 kPa. ✓

Example 2 — Methylene blue dye onto activated carbon (water treatment). Typical K_eq ≈ 100 M⁻¹. Initial dye concentration P = 0.01 M (10 mM):

  • K · P = 100 × 0.01 = 1.0. Right at the half-coverage point.
  • θ = 1.0 / (1 + 1.0) = 0.500 = 50%. Surface is exactly half-occupied.
  • This is the most informative regime for fitting K from experimental data — the isotherm curve has its maximum slope here.
  • To increase coverage to 90%: solve 0.9 = K·P/(1+K·P) → K·P = 9 → P = 0.09 M (9× more dye). Diminishing returns again.

Example 3 — H₂ on platinum at room temperature. K_eq ≈ 0.01 atm⁻¹ (chemisorption is strong). At P = 0.001 atm = 1 mTorr:

  • K · P = 0.01 × 0.001 = 10⁻⁵ ≪ 1. Deep in the linear regime.
  • θ = 10⁻⁵ / (1 + 10⁻⁵) ≈ 10⁻⁵ = 0.001%. Essentially empty surface.
  • In this regime θ ≈ K·P (Henry's-law approximation). To raise θ to 50%, need P = 1/K = 100 atm — much higher than the lab conditions allow without a pressure vessel.

Example 4 — Strong binding (low P at half-coverage). CO on Ni(100) chemisorption at room T: K_eq ≈ 10⁶ Pa⁻¹ (very strong). At P = 10⁻⁶ Pa (ultra-high vacuum, adsorbate dosing range):

  • K · P = 10⁶ × 10⁻⁶ = 1.0 → θ = 0.5 (50%).
  • P1/2 = 10⁻⁶ Pa. CO binds so strongly to Ni that even a millionth of a pascal — barely measurable — gives half-coverage of the surface.
  • This explains why ultra-high vacuum (P < 10⁻⁹ Pa) is required for clean-surface studies in catalysis: even traces of CO will cover the surface.

Who Should Use the Langmuir Isotherm Calculator?

1
Physical Chemistry Students: Solve textbook adsorption problems and visualize how θ depends on K and P; compute the half-coverage pressure 1/K from binding-energy data.
2
Catalysis Researchers: Predict CO, H₂, N₂, O₂ surface coverage on metal catalysts (Pt, Pd, Ni, Cu) — coverage controls reaction selectivity and rate.
3
Materials Scientists: Characterize porous adsorbents (zeolites, MOFs, activated carbon) — fit K_eq from experimental N₂ or Ar isotherms at 77 K.
4
Environmental Engineers: Design activated-carbon filters for water treatment (dye removal, heavy metals, organic pollutants) — predict adsorption capacity at design concentrations.
5
Biochemists: The Michaelis-Menten enzyme kinetics equation v = V_max· / (K_M + ) is mathematically equivalent to Langmuir — same θ = K·P / (1 + K·P) form, with K = 1/K_M and θ = v/V_max.
6
Pharmaceutical Scientists: Receptor-ligand binding (drug-target interactions) follows the same isotherm — θ becomes the fraction of receptors occupied; EC₅₀ = 1/K.

Technical Reference

Langmuir's Original Paper: Irving Langmuir, "The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum," J. Am. Chem. Soc. 40, 1361-1403 (1918). 43 pages of dense theory + experimental data on ~50 gas-surface systems. Langmuir derived the isotherm from a kinetic equilibrium argument (his "principle of independent adsorption sites") and tested it against unprecedented quality data from his GE Research Lab. The 1932 Nobel Prize in Chemistry recognized this and his related work on monomolecular films at the air-water interface (Langmuir-Blodgett films).

Generalized to Multiple Adsorbates (Competitive Langmuir). When two species A and B compete for the same surface sites: θ_A = K_A · P_A / (1 + K_A · P_A + K_B · P_B); θ_B has analogous form. The competitive denominator is what underlies enzyme inhibition kinetics in biochemistry (Michaelis-Menten with competitive inhibitor: v = V_max· / (K_M(1 + /K_I) + )).

Connection to Thermodynamics. The Langmuir K_eq relates to the standard adsorption free energy via ΔG°_ads = −RT · ln(K_eq · P°), where P° is the standard reference pressure (1 bar for gases, 1 M for solutes). For a half-coverage condition (K·P = 1): ΔG_ads = 0 — surface and gas-phase have equal chemical potential. Stronger adsorbents have more negative ΔG°_ads and consequently higher K.

Typical K_eq Ranges:

  • Physisorption (van der Waals): K = 10⁻⁶ – 10⁻³ Pa⁻¹; P1/2 = kPa to MPa. Examples: N₂, Ar at 77 K on activated carbon.
  • Weak chemisorption: K = 10⁻³ – 10⁰ Pa⁻¹; P1/2 = Pa to kPa. Examples: H₂ on Cu, NH₃ on metal oxides.
  • Strong chemisorption: K = 10⁰ – 10⁶ Pa⁻¹; P1/2 = µPa to Pa. Examples: CO on Ni, O₂ on Pt at high T.
  • Solution-phase adsorption (water treatment): K = 10⁰ – 10⁴ M⁻¹; P1/2 = 0.1 – 10000 mM. Examples: dyes, organic pollutants on activated carbon.
  • Receptor-ligand (pharmacology): K = 10⁶ – 10¹² M⁻¹; P1/2 = pM to µM. Drug-target binding affinities (Kd = 1/K).

When Langmuir Fails.

  • Heterogeneous surfaces: different sites have different K_eq. Use the Freundlich isotherm θ = K·P1/n (n > 1 for heterogeneous surfaces).
  • Multilayer adsorption: when multiple layers form (relative pressure P/P° > 0.3 for vapors), use the BET equation; the BET surface area of a porous solid is the standard characterization metric.
  • Strong adsorbate-adsorbate interactions: at high coverage, neighboring molecules attract or repel each other. Use the Frumkin isotherm: ln(K·P) = ln(θ/(1−θ)) − a·θ, where a quantifies the lateral interaction.
  • Site dissociation: if molecules dissociate on adsorption (H₂ → 2 H_ads), use the dissociative Langmuir form: θ = √(K·P) / (1 + √(K·P)).

Connection to Michaelis-Menten Enzyme Kinetics. The Michaelis-Menten equation v = V_max · / (K_M + ) has identical form to Langmuir if you map: v ↔ θ · V_max, ↔ P, K_M ↔ 1/K. The K_M (Michaelis constant) is the substrate concentration at which enzyme is half-saturated — exactly P1/2 = 1/K in Langmuir language. This deep connection means anything you learn about Langmuir transfers directly to enzyme kinetics, drug binding, and biological signaling.

Key Takeaways

The Langmuir isotherm θ = (Keq · P) / (1 + Keq · P) is the foundational equation of surface adsorption: simple, universal, and underpinning almost every later model (BET, Freundlich, Temkin, Michaelis-Menten). Three regimes emerge from one equation: linear at K·P ≪ 1 (θ ≈ K·P, Henry's law), transitional at K·P ≈ 1 (θ = 0.5 at K·P = 1; best regime for fitting K), and saturated at K·P ≫ 1 (θ → 1, monolayer complete). The half-coverage pressure P1/2 = 1/K is a direct experimental measure of binding affinity. Use the ToolsACE Langmuir Isotherm Calculator with 8 unit options for both gas-phase and solution-phase systems, automatic family validation, regime classification, and a sample isotherm profile. Bookmark it for surface-chemistry homework, catalysis surface-coverage estimates, water-treatment filter design, enzyme-kinetics analogies, and any adsorption problem where one molecule binds one site.

Frequently Asked Questions

What is the Langmuir Isotherm Calculator?
It computes the fractional surface coverage θ = (Keq · P) / (1 + Keq · P) using Irving Langmuir's 1918 monolayer adsorption model — the foundational equation of surface chemistry. Inputs: K_eq (adsorption equilibrium constant, units 1/) and P (partial pressure for gases, concentration for solutes). Supports 8 unit options across two families (gas-pressure: Pa, kPa, bar, atm, Torr; concentration: M, mM, μM) with automatic family validation.

Output: θ (fractional coverage, 0-1 or %); the dimensionless K·P product (which classifies the regime — linear, transitional, or saturated); empty-site fraction (1 − θ); half-coverage pressure 1/K (a direct measure of binding affinity); 5-band classification (very-low → saturated); a sample isotherm profile showing θ at 9 different P values; and full step-by-step calculation breakdown. Designed for physical chemistry students, catalysis researchers, materials scientists, and biochemists working with adsorption or binding equilibria.

Pro Tip: The Michaelis-Menten enzyme kinetics equation has identical form — try our Michaelis-Menten Calculator with K_M = 1/K.

What's the formula for the Langmuir isotherm?
θ = (Keq · P) / (1 + Keq · P), where θ is the fractional surface coverage (0 to 1), Keq is the adsorption equilibrium constant (units of 1/), and P is the partial pressure (gas-phase) or concentration (solution-phase) of the adsorbate. Derived by balancing rates of adsorption (kads·P·(1−θ)) and desorption (kdes·θ) at equilibrium, with Keq = kads/kdes.
What does θ = 0.5 mean?
θ = 0.5 means exactly half of the available surface sites are occupied by adsorbate. This occurs when K · P = 1, i.e., at the pressure (or concentration) P1/2 = 1/K. The half-coverage point is a direct experimental measure of binding affinity: smaller P1/2 means stronger binding. It's also the most informative experimental regime for fitting K from data because the isotherm has its maximum slope here.
What are the three regimes of the Langmuir isotherm?
(1) Linear (Henry's law) regime — K·P ≪ 1: denominator ≈ 1, so θ ≈ K · P. Coverage scales linearly with pressure. Valid at low P / dilute conditions where the surface is mostly empty.
(2) Transitional regime — K·P ≈ 1: the curve bends. At K·P = 1, exactly half the sites are occupied (θ = 0.5).
(3) Saturation regime — K·P ≫ 1: denominator ≈ K·P, so θ → 1. The surface is essentially full; further pressure increases produce diminishing additional coverage. This saturation behavior is Langmuir's defining prediction.
What units should K_eq and P use?
Keq must have units of 1/ — they're reciprocals so K·P is always dimensionless. The calculator supports 8 P units across two families: gas-pressure (Pa, kPa, bar, atm, Torr) for gas-phase adsorption studies, and concentration (M, mM, μM) for solution-phase adsorption. K and P must come from the same family; the calculator validates this and flags mismatches.
What are Langmuir's three assumptions?
(1) All sites are energetically equivalent — every adsorption site has the same binding energy. Real surfaces often violate this (use Freundlich isotherm for heterogeneous surfaces). (2) One molecule per site (monolayer) — adsorbed molecules cannot stack. For multilayer adsorption use the BET equation. (3) No lateral interactions — adsorbed molecules don't attract or repel each other. For interacting adsorbates use the Frumkin isotherm.
How is Langmuir related to Michaelis-Menten enzyme kinetics?
Identical form, different variables. Michaelis-Menten: v = V_max · / (K_M + ). Langmuir: θ = K·P / (1 + K·P), or equivalently θ = P / (1/K + P). Map ↔ P, K_M ↔ 1/K, v/V_max ↔ θ. K_M is the substrate concentration at half-maximal velocity — exactly the half-coverage point P1/2 = 1/K in Langmuir language. The deep connection means insights from one transfer directly to the other; receptor-ligand binding in pharmacology follows the same form (EC₅₀ = K_d = 1/K).
Can the Langmuir isotherm describe water-treatment activated carbon?
Yes — for solution-phase adsorption of dyes, organic pollutants, or heavy metals onto activated carbon, the Langmuir isotherm is the standard starting model. Replace P with concentration C (in M, mM, or μM); K has units of 1/C. Typical K values for organic pollutants on activated carbon are 10-1000 M⁻¹, giving P1/2 ranges of 0.001-0.1 M. Real activated carbon often shows energetic heterogeneity, so the Freundlich isotherm (θ = K·P1/n) sometimes fits better at high concentrations.
What's the difference between Langmuir and BET isotherms?
Langmuir assumes monolayer adsorption only — once a site is occupied, no more adsorbate can stack on top. Saturates at θ = 1. BET (Brunauer-Emmett-Teller, 1938) extends Langmuir to multilayer adsorption, allowing molecules to stack on already-adsorbed molecules. BET diverges as P → P° (the saturation vapor pressure), reproducing capillary condensation. The first layer in BET still follows Langmuir-like form. BET surface area (computed from N₂ adsorption at 77 K) is the standard metric for porous-material characterization.
How do I fit K_eq from experimental data?
Best practice: nonlinear least-squares fitting directly to θ = K·P / (1 + K·P), using software like Python (scipy.optimize.curve_fit), R (nls), or Excel Solver. Quick visual checks: plot 1/θ vs 1/P (slope = 1/K, intercept = 1) or P/θ vs P (slope = 1, intercept = 1/K). The latter is more numerically stable. For best precision, collect data points spanning K·P from ~0.1 to ~10 (the curved region) — too low and you only see linear behavior; too high and you only see saturation.
What does saturation (θ → 1) mean physically?
θ → 1 means the surface monolayer is essentially complete — every available adsorption site is occupied. Further increases in pressure produce vanishingly small additional coverage because there are no empty sites left. This is the experimental signature that distinguishes Langmuir from older 'linear adsorption' theories: no matter how much you increase P, you can't fit more than one monolayer. To go beyond saturation, multilayer adsorption must begin (BET regime), capillary condensation occurs in pores, or new chemisorption sites must be created (e.g., by surface restructuring at high coverage).

Author Spotlight

The ToolsACE Team - ToolsACE.io Team

The ToolsACE Team

Our chemistry tools team implements the Langmuir adsorption isotherm — Irving Langmuir's 1918 derivation that founded modern surface chemistry and earned him the 1932 Nobel Prize. The calculator computes the fractional surface coverage θ = K·P / (1 + K·P) for both gas-phase adsorption (with P as partial pressure in Pa, kPa, bar, atm, or Torr) and solution-phase adsorption (with P as concentration in M, mM, or μM). The K_eq input takes matching reciprocal units. Output includes θ as both a fraction (0-1) and percent, the dimensionless K·P product (which classifies the regime — linear when K·P ≪ 1, saturated when K·P ≫ 1), the empty-site fraction (1 − θ), the half-coverage pressure 1/K (a direct measure of binding affinity), a 5-band coverage classification (very-low → saturated), a sample isotherm profile showing how θ evolves across multiple decades of P at your fixed K_eq, and a complete step-by-step calculation breakdown.

Surface ChemistryAdsorption ThermodynamicsSoftware Engineering Team

Disclaimer

The Langmuir model assumes (1) all surface sites are energetically equivalent, (2) one molecule per site (monolayer only), (3) no lateral interactions between adsorbed species. Real surfaces often violate these assumptions: use Freundlich for heterogeneous surfaces, BET for multilayer adsorption, Frumkin for strongly interacting adsorbates, dissociative Langmuir for H₂-on-metal-style systems. The model is most accurate at low-to-moderate coverage; near saturation 5-15% deviations from real data are common.