Szyszkowski-Langmuir equation

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. 

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

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... 

Fig. 2.9 Equilibrium surface pressure for BHBCi solutions symbols - experimental data from [13,25], curves -theoretical calculations curve 1 - Langmuir-Szyszkowski equation curves 2 - 5 - reorientation model for 2, 3, 7 and over 50 adsorption states of the BHBCie molecule, respectively (n j =2.52 lO mVmol,...

If the surface tension is known as a function of concentration of surfactant, then F can be evaluated. One possibility is to use the Langmuir-Szyszkowski equation (Wang and Yoon, 2008) ... 

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. 

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,... 

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

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... 

Many adsorption experiments on long chain fatty acids and other amphiphiles at the liquid/air interface and the close agreement with the von Szyszkowski equation is logically one proof of the validity of Langmuir s adsorption isotherm for the interpretation of y - log c -plots of typical surfactants in aqueous solutions (cf. Appendix 5D). This evidence is also justification for use of the kinetic adsorption/desorption mechanism based on the Langmuir model for interpreting the kinetics and dynamics of surface active molecules. 

For most of the conventional amphiphiles it was demonstrated by Rosen [141] that at a surface pressure H = 20 mN/m the surface excess concentration reaches 84-100 % of its saturation value. Then, the (l/c)n=2o value can be related to the change in free energy of adsorption at infinite dilution AG , the saturation adsorption F and temperature T using the Langmuir and von Szyszkowski equations. The negative logarithm of the amphiphile concentration in the bulk phase required for a 20 mN/m reduction in the surface or interfacial tension can be used as a measure of the efficiency of the adsorbed surfactant ... 

The general principles of the adsorption of surfactants at liquid/gas interfaces and the structure of adsorption layers are well described in detail for example in [10] as well as in Chapter 2. The main effect of adsorption is the substantial change of the interfacial pressure n of a given interface. This property is described in a first approximation by the Langmuir- von Szyszkowski equation (2.16)... 

Thus, for any time t the value of a can be calculated from Eqs (71) and (72). If we assume a Langmuir-Szyszkowski adsorption isotherm and interfacial tension equations, the parameters % and can be expressed via the values of the dynamic and equilibrium interfacial pressures, Yl(t) and... 

The quantities determined from surface tension measurements at 25°C are summarized in Table 11.1. The data are compared with those previously reported for TX-lOO in water [21]. In addition to CMC and surface tension, we are able to determine the area per molecule, a, at the air/solution interface when it is satmated with surfactant molecules. It is calculated from a linear fit to the data below the CMC by using the Langmuir-Szyszkowski adsorption equation. We can notice that both ILs display very high CMC valnes. Indeed, there is a factor of 1000 between these valnes and the CMC measured in water. 

Derivation of the dependency of surface tension of solutions with the concentration if a specific theory of adsorption is assumed, e.g. it can be shown that the Langmuir theory (discussed in Chapter 7) yields the Szyszkowski equation. 

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.,... 

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 ... 

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 ... 

The thermodynamics and dynamics of interfacial layers have gained large interest in interfacial research. An accurate description of the thermodynamics of adsorption layers at liquid interfaces is the vital prerequisite for a quantitative understandings of the equilibrium or any non-equilibrium processes going on at the surface of liquids or at the interface between two liquids. The thermodynamic analysis of adsorption layers at liquid/fluid interfaces can provide the equation of state which expresses the surface pressure as the function of surface layer composition, and the adsorption isotherm, which determines the dependence of the adsorption of each dissolved component on their bulk concentrations. From these equations, the surface tension (pressure) isotherm can also be calculated and compared with experimental data. The description of experimental data by the Langmuir adsorption isotherm or the corresponding von Szyszkowski surface tension equation often shows significant deviations. These equations can be derived for a surface layer model where the molecules of the surfactant and the solvent from which the molecules adsorb obey two conditions ... 

Finally, for an ideal surface layer of a n-component ideal bulk solution, Eqs. (2.26) and (2.27) transform into a generalised von Szyszkowski-Langmuir equation of state... 

It is seen from the von Szyszkowski-Langmuir surface tension isotherm, Eq. (2.41), that at a given temperature the shape of the surface tension isotherm is determined by only one parameter cOg =cO =cd. The other parameter b enters this equation as a dimensionless variable be, in combination with the concentration. Therefore, the value of b does not affect the shape of surface tension isotherm, and only scales this curve with respect to the concentration axis. It should be noted that this dependence on b is characteristic to all the equations presented above. The dependence of the surface pressure isotherm on the molar area co is illustrated by Fig. 2.1. It is seen, that the lower ro is, hence the higher the limiting adsorption T = 1/co, the steeper is the slope of the n(c)-curve. 

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