Equations for the CMC Based on Theoretical Considerations

The decrease in the CMC/C20 ratio with increase in temperature (10-40°C) presumably occurs either because the size of the hydrophilic group decreases as a result of dehydration with this change or because the surface area of the micelle increases with this change. 

In general, then, ionic surfactants (both anionic and cationic) with a single straight-chain hydrophobic group, in distilled water against air at room temperature, show low CMC/C20 ratios of 3 or less, while POE nonionics under the same conditions show ratios of about 7 or more. Increase in the electrolyte content of the solution causes the CMC/C20 ratios of ionics to approach those of nonionics. Zwitterionics have CMC/C20 ratios intermediate between those of ionics and POE nonionics. 

Some investigators have assigned a CMC value to the range where a discontinuity appears in the plot of some property of the surfactant solution in nonaqueous media, even when the change is not sharp. Some values are listed in Table 3-6. 

Equations relating the CMC to the various factors that determine it have been derived from theoretical considerations by Hobbs (1951), Shinoda (1953), and Molyneux (1965). These equations are based on the fact that for nonionics the CMC 

AGmic can be broken into contributions from the component parts of the surfactant molecule, CH3(CH2)mVk, where W - the hydrophilic group, in the following fashion  

---