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An amphiphile contains a solvent-compatible head and a solvent-avoiding tail. In water, aggregation hides hydrocarbon tails while leaving heads exposed. This reduces the free-energy cost of tail–water contact, competing with loss of translational entropy, headgroup repulsion and chain packing. A micelle is a finite equilibrium aggregate, not necessarily a permanently bound particle.

Equality of molecular chemical potentials controls aggregation. For a large preferred aggregation number, ${\mu }_1={\mu }_1^0+k_{B}T\ln x_1$ approaches an aggregate value ${\mu }_{\ast }^0$, givingAbove the critical micelle concentration, added surfactant mainly increases aggregate population while monomer activity changes relatively little. Finite aggregation numbers give a smooth crossover rather than an exact bulk phase transition. The amphiphile packing parameter $p=v/(a_0{l}_c)$ compares tail volume, preferred head area and chain length; small $p$ favours spherical micelles, while larger values favour less curved aggregates. These geometric criteria are approximate, not universal shape phase boundaries.^[ref]

Surface-tension or conductivity changes with concentration locate an operational CMC; scattering probes size and aggregation number. temperature, impurities and salt must be controlled, and independent methods help distinguish aggregation from precipitation. In an ionic surfactant, salt screens headgroup repulsion and can change both the CMC and aggregate shape. This discussion concerns generic synthetic surfactants only.

Interpret the stated free energy as per lattice site of volume $v_0$. Flory–Huggins solution theory counts polymer chains, giving the entropy term $(\phi /N)\ln \phi$, and solvent molecules, giving $(1-\phi )\ln (1-\phi )$. The interaction term penalizes unlike contacts when $\chi >0$; a simple contact model has $\chi =z[{ϵ}_{ps}-({ϵ}_{pp}+{ϵ}_{ss})/2]/(k_{B}T)$. Connectivity reduces polymer translational entropy relative to independent monomers. This is an incompressible mean-field model with composition-independent $\chi$ for the calculations below.^[ref]

Expanding the dilute osmotic pressure identifies the effective second virial coefficient. A good solvent has $\chi <1/2$ and positive excluded volume, promoting swollen chains. A poor solvent has $\chi >1/2$, favouring contraction and potentially phase separation. The theta-solvent condition is $\chi =1/2$ in this long-chain mean-field description: the second virial term vanishes, but higher terms remain. For finite $N$, this condition is not the exact critical point, which instead has ${\chi }_c=(1+N^{-1/2}{)}^2/2$.

Let $g=f_{\mathrm{mix}}/(k_{B}T)$. The osmotic pressure in Flory-Huggins solution theory isThus $\Pi =(k_{B}T/v_0)[\phi /N+(1/2-\chi ){\phi }^2+{\phi }^3/3+\cdots ]$. For mass concentration $c={\rho }_p\phi$ and chain molar mass $M_n=N{\rho }_p{v}_0{N}_A$,osmometry uses a membrane permeable to solvent but retaining every polymer chain, at known uniform temperature. Extrapolate $\Pi /c$ to zero concentration to remove interactions; the intercept gives the number-average molar mass. Aggregation, leakage of small chains, ions with a Donnan contribution, or a poor-solvent two-phase sample invalidate the simple molecular-weight interpretation.

The measured ratios give an exactly linear reduced-pressure fit:The reduced osmotic-pressure intercept is $185\mathrm{Pa}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$. The data determine $M_n/T=0.04494\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$; the measurement temperature is needed to convert this into an absolute molar mass. For an explicitly assumed room temperature $T=300\mathrm{K}$, $M_n\simeq 13.5\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}=1.35\times 10^4\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The boxed expression is the general answer for an arbitrary experimental temperature.