 Critical micelle concentration from standard chemical potentials

With $\mu ={\mu }_1^0+k_{B}T\ln x_1$, large aggregates become abundant when monomer chemical potential approaches the favorable aggregate standard value ${\mu }_{\ast }^0$. This gives $x_{1,\ast }\simeq \exp [({\mu }_{\ast }^0-{\mu }_1^0)/(k_{B}T)]$ in the large-aggregation approximation. For one finite preferred size $n$, equal molecular populations $x_n=x_1$ instead give $\ln x_{1,\ast }=[n({\mu }_n^0-{\mu }_1^0)/(k_{B}T)-\ln n]/(n-1)$.