Mass-action model of micellization ($a_q=K_q{a}_1^q$)

For equilibrium association of $q$ isolated amphiphiles into one micelle, equality of chemical potentials gives the displayed relation between dimensionless thermodynamic activities. Large aggregation number makes aggregate abundance rise rapidly with monomer activity. A finite-aggregate model gives a concentration crossover, rather than an exactly discontinuous thermodynamic transition.

Here $\mu$ is the common equilibrium chemical potential per amphiphile and ${\mu }_{\alpha }^0$ the standard free energy per amphiphile in an aggregate of size $\alpha$. A minimum of the aggregate free-energy balance at a finite size produces a micellar population around that size, in addition to monomers. Translational entropy leaves a small aggregate population below the nominal critical micelle concentration; a preferred finite size is not supplied by the mass-action equation alone.

If $x_{\alpha }$ counts amphiphile molecules belonging to aggregates of size $\alpha$, the corresponding dimensionless aggregate count is $n_{\alpha }=x_{\alpha }/\alpha$. Its ideal translational chemical potential contains $\ln n_{\alpha }$. Confusing these two distributions drops a factor of $\alpha$ and changes both normalization and the chemical-potential interpretation.