Equilibrium Aspects Thermodynamics of Micellisation

Various approaches have been employed to tackle the problem of micelle formation. The most simple approach treats micelles as a single phase, and this is referred to as the phase-separation model. In this model, micelle formation is considered as a phase-separation phenomenon, and the cmc is then taken as the saturation concentration of the amphiphile in the monomeric state, whereas the micelles constitute the separated pseudophase. Above the cmc, a phase equilibrium exists with a constant activity of the surfactant in the micellar phase. The Krafft point is viewed as the temperature at which a solid-hydrated surfactant, the micelles, and a solution saturated with undissociated surfactant molecules are in equiUbrium at a given pressure. 

Consider an anionic surfactant, in which n surfactant anions, S and n counterions M associate to form a micelle that is  

The micelle is simply a charged aggregate of surfactant ions plus an equivalent number of counterions in the surrounding atmosphere, and is treated as a separate phase. At any given temperature the chemical potential of the surfactant in the micellar state is assumed to be constant, and this may be adopted as the standard chemical potential, by analogy to a pure liquid or a pure solid. Considering the equilibrium between micelles and monomer, then 

In Equation (3.10), the cmc is expressed as a mole fraction, which is equal to C/(55.5 + C), where C is the concentration of surfactant in moldm , that is 

It must be stated that A G° should be calculated using the cmc expressed as a mole fraction, as indicated by Equation (3.12). However, most cmc values quoted in the literature are given in mol dm and, in many cases, AG°-values have been quoted when the cmc was simply expressed in mol dm . Strictly speaking, this is incorrect, as AG° should be based on Xj rather than on C. The value of AG°, when the cmc is expressed in moldm is substantially different from the AG° value when the cmc is expressed in mole fraction. For example, for dodecyl hexaoxyethylene glycol the quoted cmc value is 8.7X10 moldm at 25 °C. Therefore, 

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