The Langmuir isotherm calculator above works in both directions. Give it a monolayer capacity and an affinity constant and it returns the adsorbed amount and the fractional surface coverage at any equilibrium concentration. Give it a set of measured concentration and loading pairs instead and it fits the Langmuir model to them by linear regression, returning the capacity, the constant and a coefficient of determination that tells you whether the model was appropriate in the first place.
Arb Digital publishes free calculators that make the assumptions visible. The Langmuir equation is short enough that everyone trusts it, and it rests on four conditions that real adsorbents routinely violate. This page computes the numbers and then spends most of its length on when those numbers mean something, because a beautifully fitted isotherm on an unsuitable system is worse than no isotherm at all.
What This Langmuir Isotherm Calculator Does
Adsorption isotherms describe how much of a substance sticks to a surface as a function of how much is left in the surrounding phase, at constant temperature. The Langmuir model is the simplest one with a physical derivation behind it, and it dominates water treatment, catalysis, soil chemistry and biosensor work for that reason. Adsorptive media beds are a mainstream drinking water technology, as the EPA's overview of drinking water treatment technologies sets out, and isotherm fitting is how the media used in them are compared.
In prediction mode this page evaluates the isotherm directly and reports four supporting values: the fractional coverage, the capacity and constant you supplied, and the dimensionless separation factor. In fit mode it linearises your data, performs a least squares regression, recovers the capacity and constant from the slope and intercept, and reports R² alongside them. The bar breakdown shows predicted coverage across a spread of concentrations so you can see where on the curve your working point sits.
Two boundaries with adjacent tools are worth stating. The surface area calculator works out geometric areas of shapes; it has nothing to do with the specific surface area of a porous adsorbent, which is measured rather than computed. And the equilibrium constant calculator handles homogeneous solution equilibria, whereas the Langmuir constant describes a surface equilibrium between occupied and vacant sites. They share algebra and nothing else.
How to Use It
- Choose the mode. Prediction if you already have the parameters, fitting if you have bench data and need them.
- Keep the units consistent. If concentration is in mg/L then K is in L/mg. If it is in mol/L then K is in L/mol. The equation does not check for you.
- In fitting mode, paste pairs of equilibrium concentration and adsorbed amount, one per line. Blank lines and headers are skipped.
- Read R² before you read anything else. A poor fit means the Langmuir model does not describe your system, and the returned capacity is meaningless.
- Check the coverage. If θ is below about 0.1 across your whole data set, you are on the linear part of the curve and cannot resolve the capacity at all.
The Formula and How It Is Calculated
The Langmuir isotherm in its coverage form is θ = KC / (1 + KC), where θ is the fraction of sites occupied. Multiplying by the monolayer capacity gives the working form used in solution chemistry: qe = qm K Ce / (1 + K Ce), with qe the amount adsorbed per gram of adsorbent and Ce the equilibrium concentration. The derivation treats adsorption and desorption as opposing rates that balance at equilibrium, which is set out clearly in the LibreTexts derivation of the Langmuir isotherm from equilibrium considerations.
Fitting uses the double reciprocal linearisation, 1/qe = 1/qm + 1/(qm K Ce). Plotting 1/qe against 1/Ce gives an intercept of 1/qm and a slope of 1/(qmK), so the capacity is the reciprocal of the intercept and the constant is the intercept divided by the slope.
Worked through with the default data: five pairs generated from qm = 80 mg/g and K = 0.1 L/mg give equilibrium loadings of 13.33, 26.67, 40.00, 53.33 and 66.67 mg/g at 2, 5, 10, 20 and 50 mg/L. Regressing the reciprocals returns an intercept of 0.0125 and a slope of 0.125, so qm = 1/0.0125 = 80 mg/g and K = 0.0125/0.125 = 0.1 L/mg, recovering the inputs exactly with R² = 1. In prediction mode, the same parameters at Ce = 10 mg/L give KC = 1, so θ = 1/2 and qe = 40 mg/g. The concentration at which half the sites are filled is always 1/K.
The Four Assumptions, and What Breaks Them
The Langmuir derivation requires four things to be true. First, the surface has a fixed number of identical adsorption sites. Second, each site holds at most one molecule, so the maximum loading is a single complete layer. Third, adsorbed molecules do not interact with their neighbours. Fourth, the enthalpy of adsorption is the same for every site and does not change as coverage builds.
Real materials break these routinely. Activated carbon has a distribution of pore sizes and surface chemistries, so its sites are emphatically not identical; that heterogeneity is exactly what the competing Freundlich model was built to describe empirically. Physical adsorption of gases at low temperature builds multiple layers rather than stopping at one, which is why nitrogen adsorption for surface area measurement uses the BET model instead. Charged adsorbates on a charged surface repel one another as loading rises, which makes the effective affinity fall with coverage rather than staying constant.
The consequence is not that Langmuir is useless. It is that a good fit is weak evidence for the mechanism. Plenty of heterogeneous systems produce a convincingly straight double reciprocal plot over a narrow concentration range, and the fitted qm then behaves as a fitting parameter rather than as a real monolayer capacity. Treat it as an empirical description unless you have independent evidence for the site model.
Why the Linearisation Distorts Your Fit
This is the most technically important section on the page and the one most often skipped. Linearising by taking reciprocals is convenient but it does not preserve the error structure of the data. The 1/qe against 1/Ce form compresses the high concentration points into a tight cluster near the origin and stretches the low concentration points out along the axis, so a least squares fit is dominated by exactly the points that were measured least precisely.
A small absolute error on a small qe becomes an enormous error on 1/qe. The practical result is a systematic bias: the double reciprocal form tends to underestimate the capacity and overestimate the affinity, and the R² it reports flatters the fit because the transformation itself creates correlation.
Three other linearisations exist and each biases differently. The Hanes form plots Ce/qe against Ce and is generally the best behaved of the linear options. The Eadie–Hofstee and Scatchard forms put the dependent variable on both axes, which is statistically improper. The honest answer is that nonlinear least squares on the untransformed equation is the correct method and every linearisation is a legacy of the era before that was easy. This page reports the double reciprocal fit because it is the form quoted in the overwhelming majority of the published literature, so it is what your result will be compared against. If your capacity matters, refit nonlinearly and report both.
The Separation Factor and What Counts as Favourable
The dimensionless separation factor RL = 1 / (1 + K C0), computed from the Langmuir constant and the initial concentration, is a compact way to describe the shape of the isotherm. A value between zero and one indicates a favourable isotherm, one that bends over towards saturation. A value of exactly one is a linear isotherm; above one is unfavourable, meaning adsorption becomes easier as loading rises, which normally signals cooperative behaviour the Langmuir model cannot represent. A value approaching zero means adsorption is effectively irreversible over the range studied.
The trap is that RL depends on the initial concentration, so the same adsorbent reports different values in different experiments. It is a description of a particular run, not a property of the material. Quoting an RL without the C0 that produced it conveys nothing, and comparing RL values between papers that used different starting concentrations is meaningless.
Temperature, and Why It Is Called an Isotherm
Every parameter here is defined at one temperature. Adsorption is almost always exothermic, so raising the temperature reduces the affinity constant and shifts the whole curve down; a capacity measured at 25 °C does not apply at 45 °C. Repeating the fit at several temperatures and plotting ln K against reciprocal temperature yields the enthalpy of adsorption from the slope, which is the same van 't Hoff construction used elsewhere in thermodynamics. Our activation energy calculator handles the closely related Arrhenius analysis for rate constants, and the Gibbs free energy calculator converts an equilibrium constant into a free energy change.
Time is the other variable an isotherm cannot see. Every point on the curve is an equilibrium point, which means the experiment must have run long enough for the system to stop changing. Sampling a slow adsorbent after two hours when it needs twenty-four produces a set of pairs that fit beautifully and describe kinetics rather than equilibrium. Establishing the contact time by a separate kinetic experiment is a prerequisite, not an optional refinement.
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Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Using the initial concentration as Ce — the isotherm is defined against what remains in solution at equilibrium, and substituting C0 inflates the apparent capacity.
- Mixing units between C and K — K carries the reciprocal of the concentration unit, so switching from mg/L to mol/L changes its numerical value.
- Reporting qm from data that never approaches saturation — if coverage stays below about 0.1 the extrapolation to a monolayer is unconstrained.
- Treating a high R² as proof of monolayer adsorption — the reciprocal transformation inflates R², and heterogeneous surfaces often fit well over a narrow range.
- Comparing isotherms measured at different temperatures — both qm and K are temperature dependent, which is the whole point of the word isotherm.
Related Free Tools From Arb Digital
Prepare the solutions with the molarity calculator and the solution dilution calculator, and convert between concentration units with the concentration converter. The equilibrium constant calculator covers solution phase equilibria, the Gibbs free energy calculator turns a constant into a free energy, and the percent error calculator compares a fitted value against a reference. The full free online tools hub lists everything else.
Frequently Asked Questions
It is a model describing how much of a substance adsorbs onto a surface as a function of the concentration remaining in the surrounding phase at constant temperature. It assumes a fixed number of identical single-occupancy sites and saturates at a complete monolayer.
qm is the monolayer capacity, the adsorbed amount when every site is filled, usually in mg per gram of adsorbent. K is the affinity constant, whose reciprocal is the concentration at which half the sites are occupied. K carries the reciprocal units of the concentration used.
Plot the reciprocal of adsorbed amount against the reciprocal of equilibrium concentration. The intercept is one over the capacity and the slope is one over the capacity times the constant, so the capacity is the reciprocal intercept and the constant is intercept divided by slope.
Taking reciprocals compresses high concentration points and stretches low concentration points, so the regression is dominated by the least precise measurements. The double reciprocal form tends to underestimate capacity and overestimate affinity, and inflates R squared.
Langmuir is derived from a physical site model and saturates at a monolayer. Freundlich is an empirical power law with no saturation limit, built to describe heterogeneous surfaces where sites differ in energy. Neither being a better fit proves a mechanism.
Between zero and one it indicates a favourable isotherm, one means linear, above one is unfavourable and near zero suggests effectively irreversible adsorption. It depends on the initial concentration, so it describes a particular experiment rather than the material itself.
Yes, which is why the curve is called an isotherm. Adsorption is usually exothermic, so raising temperature lowers the affinity constant. Fitting at several temperatures and plotting the log of K against reciprocal temperature gives the enthalpy of adsorption.
Only where a single layer forms. Physical adsorption of gases at low temperature builds multiple layers, which is why surface area measurement uses the BET model instead. Chemisorption, which stops at one layer by definition, is the better Langmuir case.
This calculator is provided for education and general reference. It describes how the Langmuir isotherm is evaluated and fitted and is not laboratory, treatment design or safety guidance; follow the methods and risk assessments issued by your own institution.
