Dissolving a non-polar molecule in water incurs a free energy cost of about $10 kJ/(mol \cdot nm^2)$ (source: http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=101826)

The entropy gained from dissolving a solute is $k_B \ln(c_0/c)$ where $c$ is the number density of solutes in the water and $c_0$ is the number density of solutes when separated out from the water. Thus, I would expect the critical micelle concentration to be approximated by

$$c_{crit} \approx c_0 \exp\left(-\frac{10 kJ / (mol \cdot nm^2) SA}{k_B T}\right)$$ with $SA$ the surface area of the surfactant.

For SDS, using the information at https://en.wikipedia.org/wiki/Sodium_dodecyl_sulfate, I estimated that

$$c_0 \approx 1/(.5 nm^3)$$ $$SA \approx 4 nm^2$$ and $$k_B T \approx \frac{1}{40} eV$$

This gives me

$$c_{crit} \approx 2*10^{-7} mol \cdot L^{-1}$$

but the measured value is nearly $10^{-2} mol \cdot L^{-1}$.

Where are the missing five orders of magnitude coming from? Are there, perhaps, additional degrees of freedom gained when SDS dissolves besides just the location of the molecule?


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