Why must q be less than qmax in the Langmuir model?

In the Langmuir model, qmax is the monolayer (maximum) adsorption capacity, so physically q should not exceed qmax. When solving for Ce or b using rearranged Langmuir equations, q<qmax is required to avoid division by zero and non-physical results.

What is the Langmuir linear form (Ce/q) used for?

The linearized Langmuir equation plots Ce/q versus Ce. If the system follows Langmuir behavior, the plot is approximately a straight line with slope 1/qmax and intercept 1/(qmax·b), which can be used to estimate parameters from experimental data.

How does mass balance relate to the Langmuir isotherm?

Mass balance computes adsorption capacity from measured concentrations and amounts using q=(C0−Ce)·V/m. It does not assume a Langmuir relationship by itself, but the resulting (Ce, q) points can be used to fit Langmuir parameters across multiple experiments.

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Langmuir Isotherm Calculator

Q: What does the Langmuir constant b mean?

The Langmuir constant b reflects the affinity (binding strength) between adsorbate and surface; higher b generally indicates stronger adsorption. Its units are inverse concentration, so concentration units must be consistent.

Q: What is the Langmuir linear form (Ce/q) used for?

The linearized Langmuir equation plots Ce/q versus Ce. If the system follows Langmuir behavior, the plot is approximately a straight line with slope 1/qmax and intercept 1/(qmax·b), which can be used to estimate parameters from experimental data.

Q: How does mass balance relate to the Langmuir isotherm?

Mass balance computes adsorption capacity from measured concentrations and amounts using q=(C0−Ce)·V/m. It does not assume a Langmuir relationship by itself, but the resulting (Ce, q) points can be used to fit Langmuir parameters across multiple experiments.

Calculate adsorption on a surface using the Langmuir isotherm. Solve for q (adsorbed amount), C_e (equilibrium concentration), or b (Langmuir constant), with optional step-by-step and a quick linearization check.

Background

The Langmuir model describes monolayer adsorption on a uniform surface with finite adsorption sites. It’s commonly used in chemistry, environmental engineering, and materials science to relate the amount adsorbed (q) to the equilibrium concentration in solution (C_e).

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What do you want to solve for?

Tip: Use Solve for q to predict adsorption at a measured equilibrium concentration.

Equilibrium concentration, C_e

Units can be mg/L, mol/L, etc. Keep units consistent.

Adsorbed amount, q

Typical units: mg/g, mmol/g, etc.

Monolayer capacity, q_max

Must be > 0.

Langmuir constant, b

Units are inverse concentration (e.g., L/mg).

Options:

Round results

Show step-by-step

Show Langmuir linear form (C_e/q)

Show saturation gauge + mini curve

Quick picks:

Chips prefill common adsorption–isotherm scenarios and run the calculation.

Result:

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How this calculator works

Langmuir isotherm: q = (qmax·b·Ce) / (1 + b·Ce)
Linear form: Ce/q = (1/(qmax·b)) + (Ce/qmax)
Mass-balance capacity: q = (C0 − Ce)·V / m

Keep units consistent (e.g., if concentration is mg/L and V is L, then q will be mg per mass unit).

Formula & Equation Used

Langmuir: q = (qmax·b·Ce) / (1 + b·Ce)

Solve for Ce: Ce = q / (b·(qmax − q)) (requires q < qmax)

Solve for b: b = q / (Ce·(qmax − q)) (requires q < qmax and Ce > 0)

Linear: Ce/q = 1/(qmax·b) + Ce/qmax

Mass balance: q = (C0 − Ce)·V / m

Example Problems & Step-by-Step Solutions

Example 1 — Predict q from Ce

Given C_e=10, q_max=8, b=0.15, find q.

Compute numerator: qmax·b·Ce = 8·0.15·10 = 12
Compute denominator: 1 + b·Ce = 1 + 0.15·10 = 2.5
q = 12 / 2.5 = 4.8

Example 2 — Solve Ce from q

Given q=4.2, q_max=8, b=0.15, find C_e.

Check: q < qmax → 4.2 < 8 ✓
Use: Ce = q / (b·(qmax − q))
Ce = 4.2 / (0.15·(8 − 4.2)) = 4.2 / (0.15·3.8) = 4.2 / 0.57 = 7.37

Example 3 — Mass-balance adsorption capacity

Given C0=50, Ce=12, V=0.25, m=1.5, find q.

ΔC = C0 − Ce = 50 − 12 = 38
Amount removed = ΔC·V = 38·0.25 = 9.5
q = 9.5 / 1.5 = 6.33

Frequently Asked Questions

Q: What does the Langmuir constant b mean?

b reflects how strongly the adsorbate binds to the surface (higher b → stronger affinity). Its units are inverse concentration (e.g., L/mg), so your concentration units must match.

Q: Why does the calculator warn that q must be less than q_max?

In the Langmuir model, q_max is the monolayer (maximum) capacity, so physically q should not exceed it. When solving for C_e or b, the formulas require q < q_max to avoid division by zero / non-physical results.

Q: What is the “linear form” (C_e/q) used for?

The linearized Langmuir equation plots C_e/q vs C_e. If data follow Langmuir behavior, the plot is roughly a straight line with slope 1/q_max and intercept 1/(q_maxb).

Q: How is the “mass balance” mode related to the Langmuir model?

Mass balance computes adsorption capacity from concentrations and amounts: q = (C0 − C_e)·V / m. It does not assume Langmuir by itself, but the resulting q values can be fit to Langmuir (or other isotherms) across multiple C_e.