Szyszkowski equation

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. 

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. 

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. 

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.,... 

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],... 

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... 

The corresponding equation from the variation of surface tension y with time is as follows (Langmuir-Szyszkowski equation) ... 

Since asm = 1016/TmN, this relation is similar to the one (2.30) obtained previously from the Langmuir and Szyszkowski equations. For surfactants whose asm values do not vary much,... 

Measurement of the high frequency modulus, c0, as a function of the equilibrium surface pressure, tt, should provide a sensitive criterion for interaction for monolayers that are quite soluble by normal standards, which involve much longer time spans than the inverse frequency of the compression/expansion experiment. A numerical example of the greater sensitivity of an e0 vs. tt plot, compared with that of the ir vs. log c relationship is shown in Figure 1 for a hypothetical case. The specific defini-nition of surface interactions used here to arrive at numerical values includes all mechanisms that produce deviations from Szyszkowski-Langmuir adsorption behavior. Ideal behavior, with zero surface interactions, then is represented by zero values of In fis in the equation of state ... 

This example emphasizes the danger of using the ideal Langmuir-Szyszkowski equation of state in converting surface tension-time data into adsorption—time data even for very dilute monolayers. Also it clearly shows that any conclusion about the existence or non-existence of an... 

II.2.2. Langmuir and Szyszkowski Equations. Accounting for the Adsorbed Molecules Own Size (Mutual Repulsion)... 

The Szyszkowski equation (II. 18) satisfies both limiting conditions it is consistent with the linear dependence of the surface tension on concentration within the Henry region, and agrees with eq. (11.17) at sufficiently high concentrations. At the same time, the criteria which define low and sufficiently high concentrations are set. Indeed, at low (as compared to a = IA) concentrations the logarithm can be expanded in series, yielding... 

Let us now analyze the general relationship between T and c values over the entire range of concentrations. The derivative of the Szyszkowski equation (11.18) with respect to concentration reads ... 

The comparison of the empirical Szyszkowski equation (II. 18) with the Gibbs equation (II.5) indicates that Langmuir adsorption isotherm (11.22) is well suited also for the description of adsorption at the air - surfactant solution interface. It is interesting to point out that at the gas - solid interface, for which eq. (11.22) was originally derived various deviations from Langmuirian behavior are often observed. 

The applicability of eq. (11.22) to a successful description of adsorption from a solution was established by Langmuir himself, when he compared his adsorption isotherm to the Gibbs equation and ended up with the Szyszkowski equation as a result. The transition from localized to non-localized adsorption (which can be viewed as the transition from fixed adsorption sites to moving ones) does not, therefore, change general trends in the adsorption in the cases described. One should also keep in mind that the liquid interface is more uniform in terms of energy than the solid interface, which contains active sites with different interaction potentials.4 The latter is probably the reason why... 

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