Szyszkowski

We have considered the surface tension behavior of several types of systems, and now it is desirable to discuss in slightly more detail the very important case of aqueous mixtures. If the surface tensions of the separate pure liquids differ appreciably, as in the case of alcohol-water mixtures, then the addition of small amounts of the second component generally results in a marked decrease in surface tension from that of the pure water. The case of ethanol and water is shown in Fig. III-9c. As seen in Section III-5, this effect may be accounted for in terms of selective adsorption of the alcohol at the interface. Dilute aqueous solutions of organic substances can be treated with a semiempirical equation attributed to von Szyszkowski [89,90]... 

Bianco and Marmur [143] have developed a means to measure the surface elasticity of soap bubbles. Their results are well modeled by the von Szyszkowski equation (Eq. III-57) and Eq. Ill-118. They find that the elasticity increases with the size of the bubble for small bubbles but that it may go through a maximum for larger bubbles. Li and Neumann [144] have shown the effects of surface elasticity on wetting and capillary rise phenomena, with important implications for measurement of surface tension. 

There are three forms of the Langmuir-Szyszkowski equation, Eq. III-57, Eq. Ill-107, and a third form that expresses ir as a function of F. (n) Derive Eq. III-57 from Eq. Ill-107 and (b) derive the third form. 

Very careful measurements by the method of capillary rise have been carried out by Volkmann, v. Szyszkowski, Richards and Harkins. 

Similar conclusions as to the attainment of a finite maximum value of r as pointed out by Langmuir J.A.G.S. xxxix. 1883, 1917) can be obtained from an empiric equation put forward by V. Szyszkowski Zeit Phys. Qhem. LXiv. 385,1908) in the following form ... 

In the following table are given the limiting values of A calculated from Milner s and v. Szyszkowski s equations by Langmuir and Harkins. 

The data of v. Szyszkowski have been employed for determining FA. ... 

The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

These two equations represent the generalized Szyszkowski-Langmuir adsorption model. 

Finally, assuming the ideality of both the enthalpic and entropic mixing gives p = 0 and Eqs. 17 and 18 simplify to the well-known Szyszkowski-Langmuir equation given by... 

Equations 21 and 22 present the useful extension of the Szyszkowski-Langmuir model to the adsorption with two orientational states at the interface. If the molecular interactions are considered, a similar simphfied model with P = 2 = P and b = b2 = b can be obtained from Eqs. 10 and 11, giving... 

Equations 27 and 28 present the extension of the Szyszkowski-Langmuir model to the adsorption of one-surfactant systems with aggregation at the interface. For the formation of dimmers on the surface, n = 2 and Eqs. 27 and 28 can be expanded to obtain the Frumkin equation of adsorption state. In general, the surface aggregation model described by Eqs. 27 and 28 contains four free parameters, including coi, n, b and Fc, which can be obtained by regression analysis of the data for surface tension versus surfactant concentration in the solution. 

The Frumkin theory with Eqs. 17-18 presents the first improvement of the Szyszkowski-Langmuir theory and is shown in Pig. 1 by the thick line. The Frumkin theory requires input for the surfactant interaction parameter... 

Fig.l Surface tension versus solution concentration of nonionic surfactant CnEs as measured at r = 298.15 K (data points) [45], and as predicted by the Szyszkowski-Langmuir adsorption model (thin line) described by Eq. 20 and by the Frumkin adsorption model (thick line) described by Eqs. 17-18... 

To resolve the problem of negative /3 values obtained with the Frumkin theory, the improved Szyszkowski-Langmuir models which consider surfactant orientational states and aggregation at the interface have been considered [17]. For one-surfactant system with two orientational states at the interface, we have two balances, i.e., Ft = Fi + F2 and Ftco = Ficoi + F2C02, which can be used in conjunction with Eq. 24 to derive two important equations for determining the total surface excess and averaged molecular area required in the calculation of surface tension, i.e.,... 

Fig. 3 Comparison of the surface tension for nonionic surfactant CnEg as measured at T = 298.15 K, data points [45], with improved models considering orientational states of surfactant molecules at the surface. The data shown are obtained by regression analysis minimizing the revised chi-square The calculation with fi = 0 represents the best fit of the improved Szyszkowski-Langmuir model described by Eqs. 21 and 22. The other calculated curve with =- 3.921 shows the best fit of the improved Frumkin adsorption model described by Eqs. 23 and 24...

Where T is the excess surface concentration and R and T have their usual meanings. In order to evaluate the slopes, dyint/dCp, the experimental data of dyint versus Qc and Cpeo can be adjusted to the empirical equation of Szyszkowski [30],... 

Von Szyszkowski, B. (1908) Experimentelle Studien Uber kapillare Eigenscchaften derWasseriyen Ldsungen van Fettsauren. Z. Phys. Chem., 64, 385-414. 

Since c- -ac c, the number of adsorbed mols in the surface is per cm.2, or iVTco molecules per cm.2. At T=293° K., (Xq=72 5 for water, and hence the surface covered by one molecule (which is equal to 1 jFtoN) is 12 8xlO i /j3. Szyszkowski found ji3=0411, hence the surface is 31 x 10 i cm., approximately the square of the molecular diameter. Langmuir also derived an inteipretation of the constant a. Tamamushii used an approximate form of Szyszkowski s equation ... 

Equation [4.3.13] is the Szyszkowski equation, after B. von Szyszkowski who proposed... 

The Szyszkowski equation has proved useful for describing the relatively simple adsorptions under consideration. However, its foundation is at issue because the Langmuir equation was not derived for mobile but for localized adsorbates the translational entropy is not properly accounted for. We return to this issue below, in the mean time accepting [4.3.13] as a useful empirical expression. 

An analogical - but empirical - equation was given earlier by Szyszkowski (1908). When the surface tension of both the components does not differ much, Eq. (6.34) leads to a simple additivity law for surface tensions. If ai and a2 are sufficiently close, then the exponentials can be expanded into a MacLaurin series giving... 

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