Colloids and surfactants — micelles, CMC, emulsions

1. Colloids — the stage

Dispersed	Medium	Name	Example
liquid	gas	aerosol	fog
solid	liquid	sol	ink
liquid	liquid	emulsion	milk
gas	liquid	foam	shaving foam

WHY 1–1000 nm? Below 1 nm it's a true solution (molecules dissolved). Above 1000 nm gravity wins and it settles (suspension). In between, particles are big enough to scatter light (Tyndall effect) but light enough that random thermal kicks (Brownian motion) keep them suspended.

2. Surfactants and surface tension

3. Micelles and the CMC — derive it

WHY does a sharp CMC exist? (Derivation from chemical potential)

Treat micelle formation as an equilibrium between $N$ monomers $S$ and one micelle $M_N$ : $N\,S \rightleftharpoons M_N$

At equilibrium the chemical potentials balance: $\mu_{M_N} = N\,\mu_S$

Write each as standard part + activity (use concentrations $\approx$ activities): $\mu^\circ_{M} + RT\ln\!\frac{[M_N]}{N} = N\big(\mu^\circ_S + RT\ln[S]\big)$

Why this step? Each species' chemical potential is $\mu = \mu^\circ + RT\ln a$ ; the micelle's "concentration per molecule" is $[M_N]/N$ .

Solving for the fraction in micelles vs monomer and using the equilibrium constant $K$ : $[M_N] = K\,[S]^N, \qquad \ln K = \frac{N\mu^\circ_S - \mu^\circ_M}{RT}$

Formula The cooperativity that creates a sharp CMC

Because $N$ is large (typically 50–100), $[M_N]\propto[S]^N$ is an extremely steep function. Below a threshold monomer concentration $[S]$ , $[S]^N\approx 0$ (no micelles); just past it, $[S]^N$ explodes → micelles suddenly appear. $\boxed{\;[S]_{\text{free}} \approx \text{CMC} = \text{const for } [S]>\text{CMC}\;}$

The free-energy of micellisation per mole of monomer (for ionic surfactant, simplified): $\Delta G_{\text{mic}}^\circ \approx RT\ln(\text{CMC})$ Derivation: for large $N$ , dividing $\mu_M^\circ - N\mu_S^\circ = -RT\ln K$ by $N$ and noting at CMC monomer ≈ CMC gives $\Delta G_{\text{mic}}^\circ = \tfrac{1}{N}(\mu_M^\circ) - \mu_S^\circ \approx RT\ln(\text{CMC})$ (CMC in mole-fraction units). The negative value (CMC ≪ 1) means micellisation is spontaneous — driven mostly by the hydrophobic effect (entropy gain of released "caged" water).

How CMC shows up experimentally

Figure — Colloids and surfactants — micelles, CMC, emulsions

Many properties show a kink at the CMC because micelles behave totally differently from monomers:

Surface tension: drops as monomers fill the surface, then flattens at CMC (surface saturated; extra surfactant goes to micelles, not surface).
Molar conductivity (ionic): drops more steeply after CMC (micelles carry many charges but move slowly + bind counterions).
Osmotic pressure / turbidity / detergency: all bend at CMC.

4. Worked examples

Worked example Example 1 — Estimating

$\Delta G_{\text{mic}}$ SDS has CMC $= 8.0\times10^{-3}$ mol/L at 298 K. Estimate $\Delta G_{\text{mic}}^\circ$ (nonionic approximation, mole-fraction scale, water = 55.5 mol/L).

Step 1. Convert CMC to mole fraction: $x = \dfrac{8.0\times10^{-3}}{55.5} = 1.44\times10^{-4}$ . Why? $\Delta G^\circ$ depends on standard state; mole-fraction is the conventional one.

Step 2. $\Delta G_{\text{mic}}^\circ = RT\ln x = (8.314)(298)\ln(1.44\times10^{-4})$ . Why? Direct from the derived formula.

Step 3. $= 2477 \times (-8.846) = -2.19\times10^{4}\ \text{J/mol} = \mathbf{-21.9\ kJ/mol}$ . Negative ⇒ spontaneous. ✓

5. Emulsions

HOW does soap clean? Grease has no head, so it won't dissolve in water. Soap micelles swallow grease into their oily core (solubilisation), and the charged outer heads make the whole package water-soluble → washes away. This only works above CMC (need micelles!).

Common mistake Steel-manned misconceptions

"More surfactant ⇒ lower surface tension, always." Why it feels right: below CMC, adding surfactant really does lower $\gamma$ . Fix: once the surface is saturated (at CMC), extra surfactant builds micelles in the bulk, not at the surface — $\gamma$ plateaus. That plateau is literally how we measure CMC.

"Micelles form one molecule at a time, gradually." Why it feels right: most equilibria are gradual. Fix: the $[M_N]\propto[S]^N$ cooperativity with large $N$ makes onset essentially sudden — hence a sharp CMC, not a smooth ramp.

"Colloids are just dirty solutions." Fix: colloids are thermodynamically distinct — particle size 1–1000 nm gives Tyndall scattering and Brownian stability, unlike true solutions (no scattering) or suspensions (settle out).

" $\Delta G_{\text{mic}}<0$ because micelles release heat." Why it feels right: spontaneity ↔ exothermic, usually. Fix: micellisation is mostly entropy-driven (the hydrophobic effect releases ordered water); $\Delta H$ is often near zero or even positive.

Flashcards

What size range defines colloidal particles?

1 nm to 1000 nm (between true solutions and suspensions).

What is a surfactant/amphiphile?

A molecule with a hydrophilic head and a hydrophobic tail (e.g. SDS).

Define CMC.

Critical Micelle Concentration — the surfactant concentration above which micelles begin to form.

Why is there a sharp CMC rather than a gradual onset?

Micelle formation is cooperative:

[M_N]\propto[S]^N

with large aggregation number

N

, so micelles appear suddenly once monomer concentration crosses threshold.

What happens to free monomer concentration above the CMC?

It stays roughly constant; all added surfactant goes into micelles.

What is the main thermodynamic driving force of micellisation?

The hydrophobic effect — releasing ordered "caged" water increases entropy (

T\Delta S

), making

\Delta G<0

Formula linking CMC and free energy of micellisation.

\Delta G_{\text{mic}}^\circ \approx RT\ln(\text{CMC})

(CMC in mole fraction).

How does surface tension vs concentration locate the CMC?

\gamma

falls as monomers adsorb, then flattens (plateau) at CMC because the surface is saturated.

In a micelle, which part points inward?

The hydrophobic tails (core); hydrophilic heads face outward into water.

How does soap remove grease?

Micelle cores solubilise grease; charged heads make the package water-soluble — works only above CMC.

State Bancroft's rule for emulsions.

The phase in which the emulsifier is more soluble becomes the continuous phase.

Difference between O/W and W/O emulsion (example each)?

O/W = oil drops in water (milk); W/O = water drops in oil (butter).

What is the Tyndall effect and why do colloids show it?

Scattering of light by colloidal particles; their size (1–1000 nm) is comparable to light wavelength so they scatter visible light.

Recall Feynman: explain to a 12-year-old

Imagine tiny soldiers with waterproof boots and a head that loves water. Drop them in a glass of water — they hate getting their boots wet, so first they line up along the top surface with boots in the air. Once the surface is packed full, the rest of them grab each other and make little balls with all the boots tucked inside and heads facing out — that ball is a micelle. The exact crowd size where balls suddenly start forming is the CMC. The cool trick: grease can hide inside those balls' boot-room, so soap drags dirt away in water. That's washing, in one idea!

Connections

Surface Tension and Adsorption
Gibbs Free Energy and Spontaneity — $\Delta G_{\text{mic}}=RT\ln(\text{CMC})$
Hydrophobic Effect and Entropy
Cell Membranes and Lipid Bilayers — same amphiphile logic
Tyndall Effect and Light Scattering
Detergents and Cleaning Action
Le Chatelier and Equilibrium — micelle equilibrium $NS\rightleftharpoons M_N$

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, surfactant ek aisi molecule hai jiska "do dimaag" hota hai — ek sira (head) ko paani bahut pasand hai (hydrophilic), aur doosra sira (tail) paani se nafrat karta hai (hydrophobic), kyunki wo greasy/oily hota hai. Ab problem yeh hai ki dono ends ek saath khush nahi reh sakte. Isliye pehle ye molecules paani ki surface par line lagati hain, jisse surface tension ghat jaata hai. Jab surface poori bhar jaati hai, tab extra molecules aapas mein clump banakar ek ball banati hain — usko micelle kehte hain, jisme saare tails andar chhup jaate hain aur heads bahar paani ki taraf.

Jis concentration par ye micelles banna shuru hote hain, usko CMC (Critical Micelle Concentration) kehte hain. Important baat: CMC ek "sharp" point hota hai, dheere-dheere nahi badalta. Iska reason cooperativity hai — micelle banta hai $N$ molecules se ek saath ( $[M_N]\propto[S]^N$ ), aur $N$ bada (50–100) hota hai, isliye threshold cross karte hi micelles ekdam se ban jaate hain. CMC ke neeche free monomers, CMC ke upar saara extra surfactant micelle mein chala jaata hai, monomer concentration constant reh jaati hai.

Driving force kya hai? Mostly hydrophobic effect — jab greasy tail paani mein akela hota hai to uske aas-paas paani ordered "cage" banata hai (low entropy). Tail ko micelle ke andar chupa do to wo ordered paani free ho jaata hai → entropy badhti hai → $\Delta G = \Delta H - T\Delta S$ negative ho jaata hai. Yaani micelle banना spontaneous, lekin enthalpy se nahi, entropy se! Formula yaad rakho: $\Delta G_{\text{mic}}^\circ \approx RT\ln(\text{CMC})$ .

Practical importance: soap isi liye kaam karta hai — grease micelle ke oily core mein ghus jaata hai (solubilisation) aur charged heads usko paani mein dho dete hain. Yeh sab sirf CMC ke upar hota hai. Aur emulsion (jaise doodh, mayonnaise) bhi surfactant se stabilise hota hai — Bancroft rule: jis phase mein emulsifier zyada ghulta hai, wahi continuous phase banta hai.

Test yourself — Physical Chemistry (Advanced)

Connections