Someone argues: 'From the closed-association model,

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Amphiphiles and self-assembly

Problem

Someone argues: 'From the closed-association model, XN=N(X1eα)NX_N = N(X_1e^\alpha)^N. This is just a formula for X1X_1 raised to a power, so if I add more surfactant and the total concentration rises, X1X_1 must rise too, just more slowly once micelles start forming. So above the CMC, the free monomer concentration should keep climbing steadily, roughly in step with however much extra surfactant is added, only at a reduced rate.' For sodium dodecyl sulfate, α=8.8\alpha = 8.8 and N=60N=60; the free-monomer mole fraction is 0.79 e−α0.79\,e^{-\alpha} at the CMC, 0.81 e−α0.81\,e^{-\alpha} at twice the CMC and 0.83 e−α0.83\,e^{-\alpha} at five times the CMC. Find the error in the argument and state what actually happens to the free monomer concentration above the CMC.

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