Langmuir Isotherm - Monolayer Coverage From K_eq and P

The langmuir isotherm is a physical-chemistry model that gives the fraction of an adsorbent surface covered by an adsorbate as a function of the equilibrium constant K_eq and the partial pressure of a gas (or the molarity of a dissolved species) at constant temperature. The model assumes a single monolayer of adsorbate on energetically identical sites and turns the kinetic balance between sticking and leaving into one closed-form fraction.

• • Gas adsorption on a solid: Predict how much of a metal or carbon surface is covered by a gas such as CO, H2, or CH4 at a given partial pressure.
• • Dissolved-species adsorption from solution: Estimate dye, ion, or surfactant uptake onto activated carbon or a polymer resin at a given molarity.
• • Lab adsorption isotherm fitting: Convert a measured coverage-versus-pressure dataset into the equilibrium constant K_eq for a single-component monolayer.

Once you know K_eq for a given adsorbate-adsorbent pair at a given temperature, the coverage fraction is set by the partial pressure of the gas (or the molarity of the dissolved species) around the surface.

The same equation underpins many surface-area techniques and gas-storage screening, which is why it is part of standard physical-chemistry coursework.

The fraction form of this isotherm is mathematically the same shape as the Michaelis-Menten equation calculator, where theta plays the role of fractional saturation and K_eq * P plays the role of substrate concentration over K_M.

The equation comes from balancing the rate at which adsorbate particles stick to empty surface sites with the rate at which they leave occupied sites, then solving for the steady-state coverage.

• theta: Fraction of the surface covered, between 0 (empty) and 1 (full monolayer).
• K_eq: Equilibrium constant of adsorption. Units are the inverse of the units of P (atm^-1 for gas, M^-1 for solution). Higher K_eq means stronger binding.
• P: Partial pressure of the gas in atm, or molarity of the dissolved species in M, depending on the input mode.

The same fraction covers both gases (use partial pressure in atm) and dissolved species (use molarity in M), so the only thing that changes between the two cases is the unit of P and the matching unit of K_eq.

According to Wikipedia, the model gives the surface coverage fraction as theta = (K_eq * P) / (1 + K_eq * P) and assumes a single monolayer of adsorbate on energetically identical sites with no lateral interactions. As published by LibreTexts Chemistry, the same equation reduces to theta approximately K_eq * P in the low-pressure linear limit and approaches theta = 1 at high pressure.

According to Wikipedia, The langmuir isotherm gives the surface coverage fraction as theta = (K_eq * P) / (1 + K_eq * P) and assumes a single monolayer of adsorbate on energetically identical sites with no lateral interactions between adsorbed particles.

As published by LibreTexts, The langmuir isotherm reduces to theta approximately K_eq * P at low pressure (linear regime) and approaches theta = 1 at high pressure (saturation plateau).

If you later need to know the pH of the solution the adsorbate is dissolved in, the buffer pH calculator runs the same Henderson-Hasselbalch algebra on the conjugate acid-base pair at that temperature.

Four ideas make the equation useful, and four assumptions decide whether the model fits your experiment.

The same math is the same shape as Michaelis-Menten enzyme kinetics and as the Hill-Langmuir binding equation, so the rearrange-for-K_eq step is the same algebra you see in biochemistry, surface chemistry, and gas-storage screening.

Because the equilibrium constant K_eq depends on temperature through the adsorption enthalpy, the Arrhenius equation calculator gives the k-versus-T shape you use to scale K_eq to a different operating temperature.

Pick the variable you want to solve for, choose the gas-or-solution input mode, and enter the two known quantities. The calculator returns the third variable plus the coverage fraction and percent.

1. 1 Open Solve For and pick a target: Choose Surface coverage theta when you already know K_eq and P, Equilibrium constant K_eq when you have measured coverage and P, or Partial pressure / molarity P when you know the coverage you want and K_eq.
2. 2 Set the P Units toggle: Pick Partial pressure (atm) for a gas-phase adsorption or Molar concentration (M) for a dissolved species. The unit label on the P field and the K_eq output will update to match.
3. 3 Enter the two known quantities: Fill in K_eq and P for a coverage solve, theta and P for a K_eq solve, or theta and K_eq for a P solve. Leave the third numeric input at its default; the calculator will overwrite it with the solved value.
4. 4 Read the coverage and solved variable: The primary output is theta on a 0 to 1 scale, with the same number scaled to 0 to 100 as percent covered. When you asked to solve for K_eq or P, that variable is shown in the matched unit.
5. 5 Check the saturation gap: The saturation gap (1 - theta) tells you how much of the monolayer is still empty, useful for sizing gas-storage or sensor dynamic range. Press Reset to clear any validation message and start a new case.

For CO on activated carbon at P = 0.5 atm with K_eq = 4 atm^-1, set Solve For to theta and P Units to pressure. The calculator returns theta = 0.6667 (66.67% covered) and a saturation gap of 0.3333, matching what a Langmuir linearization gives for a monolayer-fitted system.

If the input mode is Molar concentration (M) and your raw data is in mass per litre, the concentration calculator converts mass and volume into the molarity the isotherm expects as the P input.

The equation is short, but the rearrangement step is easy to get wrong by hand. This calculator handles the algebra and the unit bookkeeping in one place.

• • Rearrange in one click: Switch the Solve For menu to go from coverage to K_eq to P without rewriting the equation each time, the most common source of algebra mistakes in homework.
• • Two input modes, one equation: The pressure-or-molarity toggle switches the unit label on P and the matching inverse unit on K_eq, so the same calculator covers gas and solution adsorption.
• • Three useful outputs per case: The calculator returns theta, the percent coverage, and the saturation gap (1 - theta), the form most surface-area and gas-storage lab notes use.
• • Honest edge-case handling: Asking for K_eq or P at theta = 1 (full monolayer) returns a validation message instead of an infinite value, so the result is always meaningful for downstream analysis.

The calculator is intentionally narrow: it solves a single-component, single-monolayer adsorption balance. For multi-component competitive adsorption, BET multilayer fits, or Freundlich power-law fits, you will need a more general model.

When the gas partial pressure in this calculator is reported as a total system pressure that needs to be converted to a partial pressure first, the ideal gas calculator handles the P times y_i step that gives the partial pressure the isotherm actually needs.

What the result actually tells you depends on a few assumptions and a few numerical choices.

• • The model assumes a single monolayer, identical sites, and no lateral interactions, so it cannot describe multilayer physical adsorption (BET is the standard extension) or cooperative binding (the Hill equation handles that).
• • The rearrangement for K_eq or P needs theta strictly between 0 and 1. At theta = 1 the math says K_eq or P is infinite, so the calculator returns a validation message instead of a misleading finite number.

According to Omni Calculator, the equation gives the surface coverage fraction as (K_eq * P) / (1 + K_eq * P) and is the standard tool for single-component monolayer adsorption. For real catalysts and porous adsorbents, BET multilayer fits are usually needed on top of the single-component K_eq reported here.

According to Omni Calculator, The langmuir isotherm gives the surface coverage fraction as theta = (K_eq * P) / (1 + K_eq * P) and is the standard model for single-component monolayer adsorption on energetically identical sites.

When the K_eq value you measured at one pH needs to be checked against the expected protonation state of the surface, the pH pOH calculator returns the pH and pOH that the adsorption experiment actually ran at.