Adsorption in Water Purification — Freundlich and Langmuir Isotherms at Work
The Freundlich and Langmuir isotherms are usually taught as two curves to memorise for a surface chemistry question. They are also the working equations behind activated carbon in water treatment — the thing that decides how many kilograms of carbon a plant has to buy. Seeing the formula do a job makes it much harder to forget, and the linearisation step is a favourite in IIT-JAM, GATE and CSIR-NET numerical questions.
This article works the arithmetic all the way from a laboratory batch test to a carbon dose, and then states honestly where an equilibrium isotherm stops describing a real filter.
The three formulas you need
First, the quantity actually measured — the amount adsorbed per gram of adsorbent at equilibrium:
Then the two isotherms that describe how qe depends on Ce:
What each term means
| Symbol | Meaning | Unit |
|---|---|---|
| C0 | Starting concentration of the substance being removed | mg/L |
| Ce | Concentration left in the water once adsorption has reached equilibrium | mg/L |
| V | Volume of water in the test | L |
| m | Mass of adsorbent (activated carbon) used | g |
| qe | Loading — mass adsorbed per gram of carbon at equilibrium | mg/g |
| KF, 1/n | Freundlich constants: capacity term and a measure of how favourable the adsorption is. 1/n < 1 indicates favourable adsorption | KF mixed; 1/n dimensionless |
| qm | Langmuir monolayer capacity — the maximum loading the surface can reach | mg/g |
| b | Langmuir constant, related to how strongly the substance binds | L/mg |
Both curves rise steeply at low Ce and flatten off, but they differ in an important way. Langmuir has a genuine ceiling, qm: it is derived from the picture of a single layer of molecules on a uniform surface with a fixed number of identical sites. Freundlich has no ceiling at all — it is an empirical power law that fits a surface with a range of site energies, which is what a real activated carbon has.
Worked example 1 — reading a batch test
0.250 g of activated carbon is shaken with 0.500 L of water containing an organic contaminant at C0 = 50.0 mg/L. At equilibrium Ce = 8.0 mg/L.
Mass removed per litre = 50.0 − 8.0 = 42.0 mg/L
Total mass removed = 42.0 × 0.500 = 21.0 mg
qe = 21.0 mg ÷ 0.250 g = 84.0 mg/g
Removal efficiency = (42.0 ÷ 50.0) × 100 = 84.0%
Note that qe and the removal percentage answer different questions. The percentage tells you whether the water met the target; qe tells you how hard the carbon is working, and only qe lets you scale up.
Worked example 2 — fitting the Freundlich isotherm by linearisation
Taking logarithms turns the power law into a straight line, which is why this step appears in so many exam papers. Suppose two equilibrium points are measured:
Point A: Ce = 2.0 mg/L, qe = 30.0 mg/g
Point B: Ce = 20.0 mg/L, qe = 95.0 mg/g
Take logs of all four numbers:
log 2.0 = 0.30103, log 30.0 = 1.47712
log 20.0 = 1.30103, log 95.0 = 1.97772
Slope = (1.97772 − 1.47712) ÷ (1.30103 − 0.30103) = 0.50060 ÷ 1.00000 = 1/n = 0.501, so n = 1 ÷ 0.501 = 2.00
Intercept: log KF = 1.47712 − 0.501 × 0.30103 = 1.47712 − 0.15070 =
1.3264
KF = 101.3264 = 21.2
Now predict the loading at Ce = 10.0 mg/L:
qe = 21.2 × 10.00.501 = 21.2 × 3.167 = 67.1 mg/g
Because 1/n = 0.501 is comfortably below 1, this adsorption is favourable: the loading rises with concentration, but less than proportionally, which is the signature of a surface filling up.
With more than two points you would not use a pair of equations at all — you would fit the best straight line through all the (log Ce, log qe) points by least squares. That is exactly what a linear-regression tool does, and it is the honest way to handle real scattered data.
Worked example 3 — a Langmuir prediction
Suppose a fit gives qm = 120 mg/g and b = 0.15 L/mg. What is the loading at Ce = 8.0 mg/L?
Numerator: qm·b·Ce = 120 × 0.15 × 8.0 = 144.0
Denominator: 1 + b·Ce = 1 + 0.15 × 8.0 = 1 + 1.2 = 2.2
qe = 144.0 ÷ 2.2 = 65.5 mg/g
Sanity check on the model: as Ce grows very large, b·Ce dominates and qe → qm = 120 mg/g. At 65.5 mg/g the surface is a little over half saturated, which is consistent with b·Ce = 1.2 being close to 1.
Worked example 4 — from the isotherm to a carbon dose
Treat 1000 L of the same water from example 1: C0 = 50.0 mg/L down to a target of 8.0 mg/L, where the isotherm says the carbon carries qe = 84.0 mg/g.
Contaminant to be removed = (50.0 − 8.0) mg/L × 1000 L = 42 000 mg = 42.0 g
Carbon required = 42 000 mg ÷ 84.0 mg/g = 500 g
So 0.5 kg of carbon per 1000 L, i.e. a dose of 0.5 g/L — under ideal equilibrium conditions.
Where this is actually used
Adsorption on activated carbon is the standard step for substances that coagulation and filtration do not remove: taste and odour compounds, many synthetic organic chemicals, chlorination by-products and residual chlorine itself. Carbon is used in two physical forms — a powder dosed into the water and later separated, and granules packed into a bed that water flows through. Powder is flexible and suits an occasional problem; a granular bed gives long continuous service and can often be regenerated.
Related adsorption processes use different media for different targets: activated alumina and iron-based media are used for specific inorganic contaminants, and ion exchange resins remove charged species by a genuinely different mechanism — a stoichiometric swap of ions rather than accumulation on a surface. It is worth keeping that distinction clear, because exam questions sometimes test it.
Adsorption is also the sampling principle behind a great deal of environmental analysis: passing a large volume of water through a small solid bed concentrates trace organics so that an instrument can see them. Same equations, used to concentrate rather than to clean.
Where the simple formula stops being valid
- An isotherm is an equilibrium relation; a filter is a flow system. Example 4's 500 g assumes every gram of carbon reaches equilibrium with the final water concentration. In a real bed the water contacts the carbon for seconds to minutes, the front of the bed is saturated while the back is fresh, and performance is described by a breakthrough curve over time — not by a single qe. Batch isotherms give a best case; real doses are larger.
- Single-solute isotherms overstate capacity in real water. Natural water contains a large background of dissolved organic matter that competes for the same sites. A capacity measured on one pure compound in clean water will not be reached when that compound is a trace component of a real sample.
- Langmuir's assumptions do not describe activated carbon. The derivation needs a uniform surface, identical non-interacting sites and a single layer. Activated carbon is deliberately heterogeneous, with a wide pore-size and site-energy distribution. It often fits Langmuir well, but a good fit does not make the mechanism monolayer adsorption — this is a point examiners like.
- Freundlich has no ceiling, so do not extrapolate it. Push Ce high enough and the power law predicts loadings the surface cannot physically hold. Both isotherms are valid only over the concentration range actually measured.
- Temperature matters, and the word isotherm says so. KF, n, qm and b are all constants for one temperature. Physical adsorption is exothermic, so capacity generally falls as temperature rises.
- Adsorption does not destroy anything. It moves the contaminant from the water onto the solid. The spent carbon is then a concentrated waste to be regenerated or disposed of properly — a mass-balance point that a qe calculation quietly conceals.
- Units are the commonest arithmetic error. C in mg/L, V in L, m in g, qe in mg/g. Mixing in a µg/L figure without converting changes the answer by a factor of a thousand.
This article explains the chemistry for exam preparation. It is not a guide to making any water safe to drink; that requires testing by an accredited laboratory and treatment designed by a qualified engineer.
Where this appears in exams
| Exam | Typical question |
|---|---|
| IIT-JAM | Compute qe from batch data; identify KF and 1/n from a log–log plot |
| CUET-PG | Distinguish physisorption from chemisorption; recognise the Langmuir and Freundlich forms |
| GATE | Linearise either isotherm, extract constants from a fitted line, and predict a loading |
| CSIR-NET | Derivation of the Langmuir isotherm from rates of adsorption and desorption; surface coverage θ |
The fitting step is the one that costs marks. Both isotherms are solved by turning the data into a straight line and reading a slope and an intercept. The free linear regression tool fits y = mx + c to your (log Ce, log qe) or (Ce, Ce/qe) points so you can check the slope and intercept you obtained by hand.
Open the Linear Regression (Fit y = mx + c) Calculator →Preparing for IIT-JAM, GATE, CSIR-NET or CUET-PG? ABC Chemistry runs dedicated competitive-exam batches at its coaching centre and online for students across India — details at abcchemistry.in.