Non-ideal mixture of homologues

The equation of state and adsorption isotherm for a non-ideal surface layer of two surfactants with the same molar areas are Eq. (3.30) and 

It can be shown that Eq. (3.41) can be used also for the approximate description of non-ideal mixtures of surfactants, n in Eq. (3.30) cein be expressed via the terms biCi using the isotherms (3.42). These products can in turn be expressed via the Oj values for individual solutions using Eqs. (3.1) and (3.2). This leads to a (dimensionless) equation of state for a mixed surface layer 

Mixtures of an ionic and a non-ionic surfactant are quite similar to those considered above, because the contribution of the DEL to the surface pressure for an ionic surfactant can be treated approximately as a non-ideality factor. If the solution of individual ionic surfactants and their mixtures are studied at a fixed ionic strength (which is usually accomplished by addition of an indifferent electrolyte), then the surface pressure of a mixed solution, similarly to nonionic surfactants, is calculated from the data obtained for individual solutions according to Eq. (3.41). 

Mixtures of ionic surfactants at any arbitrary ionic strength can be satisfactorily described by the Frumkin equation of state and isotherm if the electroneutral combination of the corresponding ions is considered. In a mixture of two anionic (or cationic) surfactants the counterion concentration is given by the sum of the concentrations of RiX, R2X and XY. For an ideal surface layer (a, = aj = a,2 = 0) it follows fi om Eqs. (3.32)-(3.34) that  

Combining Eqs. (3.44) and (3.45), one obtains the expression which describes the surface tension of a mixed solution 

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