Gibbs adsorption isotherm determined using

Equation (4.3.37) can be used to determine the function = T1(c1), which is the adsorption isotherm for the given surface-active substance. Substitution for c1 in the Gibbs adsorption isotherm and integration of the differential equation obtained yields the equation of state for a monomole-cular film = T jt). 

The problem has been treated theoretically by the use of the Gibbs adsorption isotherm, which has been used with success in treating the interfaces between liquids and gases (30). One of the most easily measurable properties of a liquid is its surface tension, and changes in this quantity can be determined with great accuracy. The surface tension of a liquid is numerically equal to its surface energy, as also are changes in these quantities. 

If there is still a significant proportion of the amphiphile dissolved in the liquid we talk about Gibbs monolayers. Solubility in water is increased by using molecules with short alkyl chain or a high polarity of the headgroup. In this case T is determined from the reduction of the surface tension according to the Gibbs adsorption isotherm (Eq. 3.52). 

Plot y versus In m and determine the surface excess of acetic acid using the Gibbs adsorption isotherm. (Note We can use the molality, m, in the isotherm instead of C2, the molarity.) b) At 25 °C, the surface tensions of propionic acid solutions in water are... 

The MPB approximation has been used recently, in 1991, by Bhuiyan et al. [25] for the surface tension problem. The excess surface tension was determined by numerical integration of the Gibbs adsorption isotherm (the Gibbs equation), with the electrolyte activity obtained from the bulk MPB approximation. A simple model of the interface, used previously in another study [16], was adopted. In an earlier study [26], this version of the MPB equation had been reported to be... 

Fig. 2. Adsorption isotherms in all four model systems. F is the Gibbs excess adsoiption. The pressures corresponding to the three configurations shown in Figure 3 are marked with arrows. The pressure is plotted relative to the vapor pressure of the model fluid, as determined by independent Gibbs Ensemble Monte Carlo simulations. Chemical potentials were converted to pressures using a virial equation of state.

Under electrochemical conditions and T, P = constant, adsorption isotherms can be derived using standard statistical considerations to calculate the Gibbs energy of the adsorbate in the interphase and the equilibrium condition for the electrochemical potentials of the adsorbed species i in the electrolyte and in the adsorbed state (eq. (8.15) in Section 8.2). A model for the statistical considerations consists of a 2D lattice of arbitrary geometry with Ns adsorption sites per unit area. In the case of a 1/1 adsorption, each adsorbed particle can occupy only one adsorption site so that the maximal number of adsorbed particles per unit area in the compact monolayer is determined by A ax = Ng. Then, this model corresponds to the simple Ising model. The number of adsorbed particles, A ads< and the number of unoccupied adsorption sites, No, per unit area are given by... 

A threedimensional diagram, expressing the amount adsorbed as a function of sorbate equilibrium concentration and total pressure, is drawn using both direct experimental data and computed data for C /N- mixtures on K-clinoptilolite at room temperature. The real adsorption isotherms follow the Freundlich law for Nj and Henry s law for C>2 Freundlich and henry constants are computed and correlated to sorbate equilibrium concentration. Selectivity coefficients,, equilibrium constants, K, and the changes in the standard Gibbs free energy, aG°, are determined. Selectivity towards N. is satisfactory and K-clinoptilolite could be used for industrial air enrichment. 

The Gibbs equation contains three independent variables T, a, and p (defined either via concentration or pressure, c or p, respectively), and is a typical thermodynamic relationship. Therefore, it is not possible to retrieve any particular (quantitative) data without having additional information. In order to establish a direct relationship between any two of these three variables, it is necessary to have an independent expression relating them. The latter may be in a form of an empirical relationship, based on experimental studies of the interfacial phenomena (or the experimental data themselves). In such cases the Gibbs equation allows one to establish the dependencies that are difficult to obtain from experiments by using other experimentally determined relationships. For example, the surface tension is relatively easy to measure at mobile interfaces, such as liquid - gas and liquid - liquid ones (see Chapter I). For water soluble surfactants these measurements yield the surface tension as a function of concentration (i.e., the surface tension isotherm). The Gibbs equation allows one then to convert the surface tension isotherm to the adsorption isotherm, T (c), which is difficult to obtain experimentally. 

This is the spreading pressure of the mixture. Knowing this analytical expression for z, we can determine the adsorption isotherm for each component in the mixture by using the Gibbs equation (5.5-8) ... 

Pore size distribution of the micropore system determined by the water adsorption isotherm using the Gibbs-Kelvin equation determined for the macropore system by mercury penetration and mercury or helium displacement, an example is given in Fig. 4-8. 

For fluid interfaces, the Gibbs equation is often used to establish adsorbed amounts of i from the experimentally determined dependency of y on X . This approach is especially useful when little surface area is available so that F cannot be established analytically. In this way, the functionality r,(X ), the adsorption isotherm, can be derived. 

The adsorption of polymers at the liquid/liquid interface is somewhat different from that at the solid/liquid interface as the polymer can penetrate both phases, x determines the adsorption behavior of polymers at liquid/liquid interfaces. The presence of the polymer at the interface between the two immiscible liquids lowers the surface tension. Determination of the adsorption isotherm (see Section IX.B) is more straightforward compared to particulate dispersions as surface tension measurements, interpreted using the Gibbs equation, can be used to give accurate adsorbed amounts. 

Though this interpretation of adsorption kinetics and pressure isotherms is very helpful in deriving structure-functionality relationships, the results are limited to the validity of the theoretical models or approaches. This concerns both the three-step concept of adsorption kinetics and the commonly used simplification of the original Gibbs equation [95], on which the determination of the surface area... 

The adsorption at the liquid/gas interface can be determined by using a microtome or molecular tracers however, the most precise and universal methods of determining r(c) at mobile interfaces, liquid-gas and liquid-liquid, are indirect methods based on the simultaneous use of the Gibbs equation and the surface (or interfacial) tension isotherm, a(c) (Figure 2.8). 

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