10.4 Surface Chemistry, Adsorption Isotherms, Colloids, and Macromolecular Solutions

10.4 Surface Chemistry, Adsorption Isotherms, Colloids, and Macromolecular Solutions

Quick Answer: Surface chemistry and colloids is a 10-item area of the Physical Chemistry TOS. It covers surface phenomena, how surface tension changes with concentration, adsorption equations, solutions of macromolecules, and colloids. Be ready to use the Young-Laplace, Kelvin, and Gibbs equations; to linearize Langmuir, Freundlich, and BET data; to apply the Schulze-Hardy rule; and to calculate number- and weight-average molar masses.

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Surface Tension and Curved Surfaces

Surface tension $\gamma$ is the work needed to create unit area of surface (J/m$^2$, equivalent to N/m). Molecules at the surface have fewer neighbours, so creating surface costs energy.

Surface tension falls as temperature rises and reaches zero at the critical point.

• Young-Laplace equation: the pressure inside a curved surface exceeds the outside by $\Delta p = 2\gamma/r$ for a droplet or cavity, or $4\gamma/r$ for a soap bubble with two surfaces.
• Capillary rise: $h = \frac{2\gamma\cos\theta}{\rho g r}$. Water ($\theta \approx 0$) rises about 14.7 cm in a capillary of radius 0.10 mm. Mercury ($\theta > 90$ degrees) is depressed.
• Kelvin equation: $\ln\frac{p}{p^*} = \frac{2\gamma V_m}{rRT}$. Small droplets have higher vapour pressure: a water droplet of radius 10 nm has about 11% more vapour pressure than a flat surface. This explains supersaturation, cloud seeding, and why small crystals dissolve while large ones grow (Ostwald ripening).

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Surface Tension and Concentration: the Gibbs Isotherm

For a dilute solute:

$\Gamma = -\frac{1}{RT}\frac{d\gamma}{d\ln c}$

• Surface-active solutes (alcohols, fatty acids, detergents) lower $\gamma$, so $\Gamma > 0$: they concentrate at the surface. Traube's rule says each added $CH_2$ group in a homologous series raises surface activity about threefold.
• Inorganic electrolytes slightly raise $\gamma$, so $\Gamma < 0$: they are depleted at the surface.
• Surfactants lower $\gamma$ until the critical micelle concentration (CMC), above which added molecules form micelles and $\gamma$ levels off. Sodium dodecyl sulfate has a CMC of about 8 mM at 25 C.

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Adsorption on Solids

Adsorption is spontaneous and lowers the gas's entropy, so it is almost always exothermic.

The Langmuir Isotherm

Assumptions: monolayer coverage, equivalent sites, and no interactions between adsorbed molecules.

$\theta = \frac{V}{V_m} = \frac{Kp}{1 + Kp} \qquad \Rightarrow \qquad \frac{p}{V} = \frac{p}{V_m} + \frac{1}{KV_m}$

A plot of $p/V$ against $p$ gives slope $1/V_m$. At low pressure coverage is proportional to $p$; at high pressure it approaches 1.

The Freundlich Isotherm

$\frac{x}{m} = kp^{1/n} \qquad \Rightarrow \qquad \log\frac{x}{m} = \log k + \frac{1}{n}\log p$

This empirical equation ($n > 1$) fits heterogeneous surfaces and adsorption from solution, such as activated carbon removing dyes.

The BET Isotherm and Surface Area

Brunauer, Emmett, and Teller extended Langmuir's model to multilayers:

$\frac{p}{V(p^* - p)} = \frac{1}{V_m c} + \frac{c - 1}{V_m c}\cdot\frac{p}{p^*}$

Nitrogen adsorption at 77 K gives the monolayer volume $V_m$. Each $N_2$ molecule occupies 0.162 nm$^2$. Example: $V_m = 25.0\text{ cm}^3$ (at 0 C, 1 atm) per gram corresponds to $\frac{25.0}{22{,}414} \times 6.022 \times 10^{23} \times 0.162 \times 10^{-18}\text{ m}^2 = 109\text{ m}^2/\text{g}$.

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Colloids

Colloidal particles are about 1 to 1000 nm across: large enough to scatter light (the Tyndall effect) but small enough to stay dispersed by Brownian motion.

Lyophilic colloids (proteins, starch) interact strongly with the solvent and are stable. Lyophobic colloids (gold, $As_2S_3$, $Fe(OH)_3$ sols) are stabilized only by surface charge.

Electrical Double Layer and Coagulation

Charged particles attract a diffuse layer of counter-ions. The potential at the shear plane is the zeta potential; magnitudes above about 30 mV usually indicate stability. DLVO theory balances van der Waals attraction against double-layer repulsion. Adding electrolyte compresses the double layer and causes coagulation.

Schulze-Hardy rule: coagulating power rises steeply with the charge of the counter-ion (DLVO theory predicts critical coagulation concentrations roughly proportional to $1/z^6$).

• Negative sols ($As_2S_3$, clay): $Al^{3+} > Ba^{2+} > Na^+$
• Positive sols ($Fe(OH)_3$): $PO_4^{3-} > SO_4^{2-} > Cl^-$

Alum clarifies drinking water this way. Protective colloids such as gelatin adsorb on lyophobic particles and prevent coagulation.

Emulsions

Emulsifiers have a hydrophilic-lipophilic balance (HLB): values of about 3-6 favour water-in-oil emulsions, and about 8-18 favour oil-in-water emulsions.

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Solutions of Macromolecules

Synthetic polymers contain chains of different lengths, so molar mass is an average:

$M_n = \frac{\sum N_iM_i}{\sum N_i}, \qquad M_w = \frac{\sum N_iM_i^2}{\sum N_iM_i}, \qquad \text{dispersity} = \frac{M_w}{M_n} \geq 1$

Example: Equal numbers of chains of 10,000 and 30,000 g/mol give $M_n = 20{,}000$ and $M_w = \frac{10{,}000^2 + 30{,}000^2}{40{,}000} = 25{,}000$, so the dispersity is 1.25.

The second virial coefficient $B$ measures polymer-solvent interaction: $B > 0$ in a good solvent and $B = 0$ in a theta solvent.