Langmuir Isotherm Calculator
Estimate equilibrium surface coverage from the Langmuir monolayer adsorption model.
Langmuir model parameters
Surface coverage
Calculation details
| Quantity | Expression | Value | Interpretation |
|---|---|---|---|
| Dimensionless loading | K × P | 2.0000 | Relative strength of adsorption at the current state |
| Occupied fraction | (K × P) / (1 + K × P) | 0.6667 | Fraction of identical sites occupied |
| Occupied percent | θ × 100 | 66.67% | Occupied sites as a percentage |
| Vacant percent | (1 – θ) × 100 | 33.33% | Sites still available for adsorption |
How to use the Langmuir isotherm calculator
What this calculator does
This calculator estimates the fraction of a uniform adsorbent surface occupied by one adsorbate at equilibrium. It applies the single-site Langmuir relation, θ = KP/(1 + KP), where θ is the occupied surface fraction, K is an adsorption equilibrium constant, and P is either gas partial pressure or dissolved-species equilibrium concentration. The result describes ideal monolayer coverage; it does not fit experimental parameters, predict multilayer adsorption, or prove that a real surface satisfies the model assumptions.
When to use it
Use the calculator to check a surface-coverage value during physical chemistry work, compare adsorption states at different pressures or concentrations, verify a hand calculation before fitting laboratory data, or identify the pressure or concentration associated with half saturation. The Chemistry LibreTexts treatment of one-adsorbate Langmuir adsorption gives the kinetic interpretation and core assumptions behind the equation.
How to calculate
- The calculator opens with a complete demonstration: Keq = 2.5 reciprocal units and P = 0.8 matching units. Its result and a validated Excel workbook are immediately available.
- Replace Equilibrium constant (Keq) with a positive decimal. Enter ordinary decimal notation such as 2.5 or 0.012; commas and scientific notation are intentionally rejected to prevent ambiguous interpretation.
- Replace Partial pressure or equilibrium concentration (P) with a nonnegative decimal in the unit paired with K. Results update as you type.
- Read Surface percent first, then use the secondary values and calculation table to inspect the dimensionless product KP, fractional coverage, vacant percentage, and half-saturation point.
- Select Download Excel to export the current validated inputs and canonical results. Select Reset to clear the demonstration and calculated state; export remains unavailable until both required fields again form a complete valid model.
Input guide
Equilibrium constant (Keq) is required, must be finite and greater than zero, and carries the reciprocal unit of P. A realistic example is 2.5 L/mol when P is expressed in mol/L. Increasing K at fixed P raises KP and therefore raises predicted coverage. A common mistake is to combine a constant reported in one reciprocal unit with a pressure or concentration reported in another.
Partial pressure or equilibrium concentration (P) is required, must be finite and zero or greater, and may represent gas pressure or a solution equilibrium concentration. A realistic example is 0.8 mol/L paired with K = 2.5 L/mol. Increasing P at fixed K increases coverage but with diminishing gains as θ approaches 1. A common mistake is to use an initial concentration rather than the equilibrium concentration required by an equilibrium isotherm.
Output guide
Surface percent is the primary estimate, shown from 0% toward 100%. Zero means no predicted occupancy at P = 0; values near 100% indicate approach to monolayer saturation, not more than one layer. Surface fraction (θ) is the same occupancy as a number from 0 to less than 1. Vacant surface is 1 – θ expressed as a percentage. Dimensionless loading (K × P) is the unit-canceling product that drives the equation; KP = 1 corresponds exactly to 50% coverage. Half-saturation P equals 1/K in the current P unit and identifies the pressure or concentration that would yield θ = 0.5. The calculation table repeats these identities with their current values. These are model estimates, except the algebraic relationships themselves, which are exact identities within the Langmuir model.
Worked example
For the startup values K = 2.5 and P = 0.8, first calculate KP = 2.5 × 0.8 = 2. Then θ = 2/(1 + 2) = 2/3 = 0.6667 at four displayed decimals. Multiplying by 100 gives 66.67% occupied surface, leaving 33.33% vacant. Half saturation occurs at P = 1/K = 0.4 in the matching pressure or concentration unit. These values match the first-open result cards, detail table, and exported workbook.
Model assumptions and interpretation
The Langmuir model assumes a fixed number of equivalent adsorption sites, one adsorbate per site, monolayer occupancy, and no lateral interaction that changes site energy. The original surface-science framework was introduced in Irving Langmuir's 1918 paper on gas adsorption on plane surfaces; the Journal of the American Chemical Society record for the original paper provides the historical source.
At low KP, the denominator is close to 1, so coverage is approximately proportional to pressure or concentration. At high KP, the occupied fraction approaches 1 asymptotically because the finite site population becomes saturated. This saturating response is why a doubling of P does not generally double θ. A modern discussion of adsorption-isotherm fitting and model limitations is available in the peer-reviewed article on nonlinear regression for adsorption isotherm data.