CSIR-NET Surface Chemistry and Colloids — Isotherms, BET and Coagulation

By Aniket Bhardwaj · 11 September 2026 · CSIR-NET Chemistry

Surface chemistry is the part of physical chemistry that pays back straight-line thinking. Almost every quantitative question here is solved the same way: take a non-linear isotherm, rearrange it into y = mx + c, fit a straight line, and read the physical constants off the slope and intercept. This guide covers the four isotherms in the CSIR-NET syllabus, the surface-area calculation that follows from BET, the Gibbs treatment of surfactants, and the colloid stability rules — with the arithmetic done in full.

Physisorption or chemisorption? Decide first

Half of the descriptive marks in this chapter come from telling the two apart correctly.

Feature	Physisorption	Chemisorption
Forces	van der Waals	Genuine chemical bond
Enthalpy of adsorption	Small (comparable to condensation)	Large (comparable to bond formation)
Specificity	Non-specific	Highly specific to adsorbate and surface
Layers	Multilayer possible	Monolayer only
Reversibility	Readily reversible	Often irreversible
Activation energy	Essentially none	May be appreciable
Effect of raising T	Amount adsorbed falls	Rises, then falls

Both processes are exothermic, and the reason is thermodynamic rather than empirical: an adsorbed molecule has lost translational freedom, so ΔS is negative, and for ΔG to be negative ΔH must be negative too. Any question claiming endothermic adsorption at equilibrium is testing exactly that argument.

The Langmuir isotherm and its straight line

Langmuir's model assumes a fixed number of identical, independent sites, one molecule per site, and no interaction between adsorbed molecules. Equating rates of adsorption and desorption gives the fractional coverage θ.

θ = Kp / (1 + Kp) · linear form: p / (x/m) = 1/(K·x_m) + p / x_m

Here x/m is the amount adsorbed per gram of adsorbent, x_m is the monolayer capacity, and K is the adsorption equilibrium constant with units of (pressure)⁻¹. Plot p/(x/m) against p: the slope is 1/x_m and the intercept is 1/(K·x_m). Two limits are worth memorising — at low p, θ ≈ Kp (first order in pressure); at high p, θ → 1 (zero order, the surface is saturated).

Worked example 1 — extracting x_m and K from data. A gas adsorbs on charcoal as follows.

p / kPa	x/m (mg g⁻¹)	p ÷ (x/m) (kPa g mg⁻¹)
5	50.0	5 ÷ 50.0 = 0.100
10	66.7	10 ÷ 66.7 = 0.150
20	80.0	20 ÷ 80.0 = 0.250
40	88.9	40 ÷ 88.9 = 0.450
80	94.1	80 ÷ 94.1 = 0.850

The last column rises linearly with p. Take the first and last points:
slope = (0.850 − 0.100) / (80 − 5) = 0.750 / 75 = 0.0100 g mg⁻¹.
intercept = 0.100 − 0.0100 × 5 = 0.100 − 0.050 = 0.0500 kPa g mg⁻¹.

Therefore x_m = 1/slope = 1/0.0100 = 100 mg g⁻¹, and K = 1/(intercept × x_m) = 1/(0.0500 × 100) = 1/5.00 = 0.200 kPa⁻¹.

Check against a middle point: at p = 20, predicted p/(x/m) = 0.050 + 0.010 × 20 = 0.250, matching the table. A Langmuir fit that reproduces every point this cleanly is a sign the model is appropriate; real data would need a least-squares fit.

Freundlich — the empirical alternative

x/m = k·p^1/n · log(x/m) = log k + (1/n) log p, with n > 1

Worked example 2. x/m = 20 mg g⁻¹ at p = 10 kPa and 40 mg g⁻¹ at p = 40 kPa. Find n and k.

Divide the two equations: 40/20 = (40/10)^1/n, so 2 = 4^1/n.
Taking logs: log 2 = (1/n) log 4, so 1/n = 0.3010/0.6021 = 0.500 and n = 2.00.
Then k = 20 ÷ 10^0.5 = 20 ÷ 3.1623 = 6.32 mg g⁻¹ kPa⁻¹ᐟ².

Verify at p = 40: 6.32 × 40^0.5 = 6.32 × 6.3246 = 40.0 mg g⁻¹. Correct. Freundlich has no saturation limit built in, so it fails at high pressure — that is its known weakness and a favourite one-line answer.

BET and the measurement of surface area

Brunauer, Emmett and Teller extended Langmuir to multilayers by allowing molecules to stack on already-adsorbed molecules, with the second and higher layers having the enthalpy of liquefaction.

p / [ V(p₀ − p) ] = 1/(V_mc) + [ (c − 1)/(V_mc) ] × (p/p₀)

Plot the left side against the relative pressure p/p₀ over roughly 0.05–0.35 and you get a straight line. Adding slope and intercept gives 1/V_m directly, since slope + intercept = (c − 1 + 1)/(V_mc) = 1/V_m. From V_m, the surface area follows.

S = ( V_m / V_molar ) × N_A × a_m

Worked example 3 — specific surface area from N₂ adsorption. A BET plot gives a monolayer volume V_m = 0.850 cm³ of N₂ at STP per gram of solid. The cross-sectional area of an adsorbed N₂ molecule is a_m = 0.162 nm².

A units warning first. "STP" in BET work conventionally means 273.15 K and 1 atm, for which the molar volume is 22 414 cm³ mol⁻¹. Since 1982 IUPAC has defined standard pressure as 1 bar, giving 22 711 cm³ mol⁻¹ at the same temperature. Older textbooks and instrument software use 22 414; check which your source means, because the two differ by about 1.3%. We use 22 414 cm³ mol⁻¹ here.

Step 1 — moles in the monolayer: n = 0.850 ÷ 22414 = 3.792 × 10⁻⁵ mol g⁻¹.
Step 2 — molecules: 3.792 × 10⁻⁵ × 6.022 × 10²³ = 2.284 × 10¹⁹ g⁻¹.
Step 3 — convert the molecular area: 0.162 nm² = 0.162 × 10⁻¹⁸ m² = 1.62 × 10⁻¹⁹ m².
Step 4 — multiply: S = 2.284 × 10¹⁹ × 1.62 × 10⁻¹⁹ = 3.70 m² g⁻¹.

For perspective, activated charcoals and zeolites reach hundreds of square metres per gram; a value of a few m² g⁻¹ indicates a fairly non-porous powder.

Gibbs adsorption at a liquid surface

At a liquid–vapour interface there are no "sites", so the treatment is thermodynamic. The surface excess Γ is linked to the way surface tension γ changes with concentration.

Γ = − (1/RT) · (dγ / d ln c) · area per molecule = 1 / (Γ · N_A)

A solute that lowers γ has a negative dγ/d ln c and therefore a positive Γ — it accumulates at the surface. That is the definition of a surface-active substance. Inorganic salts do the opposite: they raise γ slightly and are negatively adsorbed.

Worked example 4 — headgroup area of a surfactant. For a non-ionic surfactant at 298 K, the plot of γ against ln c has a slope of −12.0 mN m⁻¹ just below the critical micelle concentration.

Γ = −(1/RT)(dγ/d ln c) = 12.0 × 10⁻³ ÷ (8.314 × 298) = 12.0 × 10⁻³ ÷ 2477.6 = 4.84 × 10⁻⁶ mol m⁻².
Area per molecule = 1 ÷ (4.84 × 10⁻⁶ × 6.022 × 10²³) = 1 ÷ (2.917 × 10¹⁸) = 3.43 × 10⁻¹⁹ m² = 0.343 nm², i.e. about 34 Å² per molecule.

Caution: this simple form applies to a non-ionic surfactant. For a 1:1 ionic surfactant with no added swamping electrolyte, both the surfactant ion and its counter-ion adsorb, and a factor of 2 appears: Γ = −(1/2RT)(dγ/d ln c). Forgetting that factor halves the answer.

Above the critical micelle concentration the surface is already saturated, so γ stops falling and the plot flattens — the standard way of locating the CMC experimentally. Conductivity, osmotic pressure and turbidity all show a break at the same concentration, which is why the CMC is regarded as a genuine physical property and not an artefact of one method.

Colloids — classification and stability

Basis	Categories	Note
Affinity for the medium	Lyophilic (solvent-loving) vs lyophobic	Lyophilic sols are thermodynamically stable and self-forming; lyophobic sols are not
Dispersed and dispersion phase	Sol, gel, emulsion, foam, aerosol	Both phases can be solid, liquid or gas except gas-in-gas
Particle nature	Multimolecular, macromolecular, associated (micelles)	Soaps and detergents are the associated type

What keeps a lyophobic sol from coagulating is charge. Ions adsorbed on the particle create an electrical double layer — a tightly held Stern layer plus a diffuse layer — and the potential at the shear plane is the zeta potential. Like-charged particles repel; adding electrolyte compresses the diffuse layer, the repulsion collapses, and the sol coagulates. That is the qualitative half of DLVO theory: net interaction = van der Waals attraction + electrical double-layer repulsion.

Schulze–Hardy rule: coagulating power rises steeply with the charge of the oppositely charged ion; the coagulation value scales roughly as z⁻⁶

Worked example 5 — relative coagulating power. For a negatively charged As₂S₃ sol, compare Na⁺, Mg²⁺ and Al³⁺ using the z⁻⁶ scaling.

Relative power ∝ z⁶: Na⁺ gives 1⁶ = 1; Mg²⁺ gives 2⁶ = 64; Al³⁺ gives 3⁶ = 729.
So the ratio is 1 : 64 : 729, and the coagulation values (the concentrations needed) go the other way, 729 : 11.4 : 1.

Two exam traps here. First, only the counter-ion charge matters — for a positively charged sol such as hydrated Fe₂O₃ you would compare Cl⁻, SO₄²⁻ and PO₄³⁻ instead. Second, z⁻⁶ is an approximate scaling from DLVO theory, not an exact law; quote it as a trend.

Surface catalysis — two rate laws to distinguish

For a bimolecular surface reaction, Langmuir–Hinshelwood assumes both reactants adsorb and then react, so the rate depends on the product of two coverages and passes through a maximum as one reactant's pressure rises (it crowds the other off the surface). Eley–Rideal assumes one reactant adsorbs and the other strikes it directly from the gas phase, so the rate rises monotonically with the gas-phase partner's pressure. Being asked to identify which mechanism fits a rate-versus-pressure curve is a standard CSIR-NET question, and the presence or absence of that maximum is the answer.

Common mistakes

Plotting the wrong linear form. Langmuir needs p/(x/m) against p; BET needs p/[V(p₀−p)] against p/p₀; Freundlich needs log–log. Mixing them gives a curve and a meaningless slope.
Forgetting that K in the Langmuir isotherm has units. If p is in kPa, K is in kPa⁻¹; an answer quoted as dimensionless is wrong.
Using 22.7 L mol⁻¹ in a BET calculation from an old textbook. State which standard state you are using; 22.414 L at 1 atm and 22.711 L at 1 bar are both correct, for different definitions.
Dropping the factor of 2 in the Gibbs isotherm for ionic surfactants.
Applying the Schulze–Hardy rule to the ion of the same charge as the sol. Only the counter-ion coagulates.
Calling every adsorption "chemisorption" because ΔH is negative. Both types are exothermic; the magnitude and the multilayer behaviour separate them.

Exam relevance

Exam	Typical demand from surface chemistry
IIT-JAM	Freundlich and Langmuir statements, colloid classification, Tyndall effect
GATE Chemistry	Isotherm linearisation, BET surface area, catalysis mechanisms
CSIR-NET	Gibbs adsorption and CMC, DLVO and zeta potential reasoning, Langmuir–Hinshelwood vs Eley–Rideal
CUET-PG	Definitions, emulsions and protective colloids

The CSIR-NET paper is set as a general aptitude Part A plus subject Parts B and C. For the number of questions, marks and negative marking, consult the current official notification rather than any remembered figure.

Every isotherm in this chapter ends in a straight-line fit. Once you have the linearised columns — p/(x/m) against p, or p/[V(p₀−p)] against p/p₀ — the Linear Regression tool returns the slope, the intercept and the correlation coefficient in one step, so you can check whether the model really fits before you convert the numbers into x_m, K or a surface area.

Open the Linear Regression Calculator →

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