Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section D / 25 / c

The amphiphile packing parameter is $p=v/(a\ell )$: hydrocarbon volume divided by the product of effective head-group area and maximal tail length. For a spherical core of radius $R$, $v/a=R/3$; requiring $R\le \ell$ gives spheres for $p\lesssim 1/3$. Cylindrical cores instead have $v/a=R/2$, favouring cylinders or rods for roughly $1/3<p\lesssim 1/2$. Lower-curvature bilayer structures are favoured towards $p\simeq 1$; intermediate ranges can include flexible or closed bilayer morphologies.

The charged sulfate head groups make the effective area sensitive to electrostatic repulsion, salt and counterion binding. Screening can reduce that area, increasing $p$ and favouring growth from spheres towards elongated micelles. These are geometric selection rules, not exact phase boundaries or a claim that every morphology is stable at any concentration.

For a dodecyl tail, take $n=12$. The given relations yield $v=0.3502$ nm${}^3$ and $\ell =1.672$ nm. The spherical-core condition is $v/(a\ell )\le 1/3$, henceEquality corresponds to a maximally extended tail reaching the core centre. A smaller head-group area would require a spherical radius larger than the available chain length. The estimate assumes all twelve tail carbons contribute to the hydrocarbon core; partial penetration or hydrated head-group geometry would modify it.

At the limiting area, the spherical core radius is $R=\ell =1.672$ nm. Its aggregation number isAbove the critical micelle concentration, free monomers remain approximately at 8 mM, so 42 mM of surfactant units are in aggregates. ThusThe concentration is of whole aggregates, not the concentration of surfactant molecules inside them. The estimate neglects micelle polydispersity and concentration-dependent growth.