= Solution
The <critical micelle concentration> is the approximate <surfactant> <concentration> above which added <amphiphile> predominantly joins <micelles>. Isolated <amphiphiles> expose their tails to the surrounding <solvent>; aggregation can reduce the unfavorable exposure, while losing translational <entropy> and incurring head-group and chain-packing costs. The favored aggregate minimizes the total <free energy>.
A <mass-action model of micellization> makes the concentration dependence explicit. For aggregation of $q$ monomers into one <micelle>, equilibrium requires
$$
q\mu_1=\mu_q,\qquad
\mu_1=\mu_1^\circ+k_BT\ln a_1,\qquad
\mu_q=\mu_q^\circ+k_BT\ln a_q.
$$
The dimensionless <thermodynamic activities> therefore obey
$$
\boxed{a_q=K_q a_1^q,\qquad
K_q=\exp\!\left[-\frac{\mu_q^\circ-q\mu_1^\circ}{k_BT}\right].}
$$
For a large <aggregation number> $q$, a modest increase in monomer activity produces a steep increase in <micelle> abundance. In an ideal dilute approximation with the same reference <concentration> $c^\circ$ for both species, the amphiphile balance is
$$
\frac{c_{\rm tot}}{c^\circ}=a_1+qK_qa_1^q.
$$
Well below the crossover almost every <amphiphile> is isolated. Above it, most additional <amphiphile> enters <micelles>, so the free-monomer <concentration> rises only slowly. It is not exactly constant for a finite $q$, and a small aggregate population can exist below the nominal <critical micelle concentration>. \b[The experimentally sharp change is an association crossover, not an assertion that aggregation is absolutely absent below one concentration.]
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2016/paper-1-micelle-crossover.png]
{title=A finite-aggregation <mass-action model of micellization> produces a concentration crossover}
In the original illustrative model, $q=50$ and $qK_q=1$ after choosing the arbitrary reference <concentration>. The normalization is a teaching choice, not a measured <critical micelle concentration>. The monomer curve flattens relative to total <concentration>, while the aggregated fraction rises rapidly.
An experimental <critical micelle concentration> can be estimated from a change of slope in <surface tension> against the logarithm of total <surfactant> <concentration>, as interfacial coverage approaches saturation and bulk aggregation becomes important. Changes in <electrical conductivity> for ionic <surfactants>, or in <light scattering>, supply other probes. Different observables can give slightly different operational values because the crossover has finite width. Molecular structure, <temperature> and ionic conditions affect the balance of <chemical potentials>; screening charged head groups can reduce repulsion and favor aggregation, but no universal sign of the temperature dependence should be assumed.
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