 Solution

ID: 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/iii/solution

At the limiting area, the spherical core radius is

R = ℓ = 1.672

N_{agg} = \frac{4 π R ^{2}}{a} = \frac{4 π R ^{3}}{3 v} ≃ 55.9.

Above the critical micelle concentration, free monomers remain approximately at 8 mM, so 42 mM of surfactant units are in aggregates. Thus

c_{micelle} ≃ \frac{50 - 8}{55.9} = 0.751 mM .

The concentration is of whole aggregates, not the concentration of surfactant molecules inside them. The estimate neglects micelle polydispersity and concentration-dependent growth.

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