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(6.13) Substituting in equation (6.14):

where X has a numerical value varying from 1 (for ionic surfactants in dilute solution or in the presence of excess electrolyte) to 2 (in concentrated solution).

Application of the Gibbs equation to surfactant solutions

The slope of the surface tension against log concentration plot reaches a constant value at concentrations just below the cmc before becoming approximately zero at higher concentrations (Fig. 6.2). The surfactant molecules are closely packed in the surface over this narrow concentration range and we can calculate the area A that each molecule occupies at the surface from

where r2 is the value of surface excess concentration calculated from the Gibbs equation using the value of dy/d(ln c) just below the cmc and NA is the Avogadro constant. The calculation is shown in Example 6.1 using surface tension data for the surface-active drug diphenhydramine.

EXAMPLE 6.1 Calculation of area per molecule using the Gibbs equation

The slope of a plot of y against log c just below the cmc for the antihistamine diphenhydramine hydrochloride (see Box 6.3) is -0.0115 N m 1 at 30°C. Calculate the area per molecule of this drug at the air/solution inter-

A = 1/(6.023 x 1023 x 1.982 x 10-6) = 83.8 x 10-20 m2 per molecule

The area per molecule of diphenhydramine = 0.84 nm2.

An interesting effect arises when the surfactant is contaminated with surface-active impurities. A pronounced minimum in the surface tension-log c plot is observed at the cmc, which would seem to be an apparent violation of the Gibbs equation, suggesting a desorption (positive dy/d[log c] value) in the vicinity of the cmc. The minimum in fact arises because of the release below the cmc of the surface-active impurities on the breakup of the surfactant micelles in which they were solubilised.

6.2.4 The influence of the surfactant structure on surface activity

The surface activity of a particular surfactant depends on the balance between its hydro-philic and hydrophobic properties. For the simplest case of a homologous series of surfactants, an increase in the length of the hydrocarbon chain as the series is ascended results in increased surface activity. Conversely, an increase in the hydrophilicity, which for polyoxyethylated nonionic surfactants may be effected by increasing the length of the ethylene oxide chain, results in a decreased surface activity. This latter effect is demonstrated by Fig. 6.4, from which it is noted that lengthening of the hydrophilic

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