Gibbs equilibrium isotherm

Adsorbed-Solution Theoiy The common thennodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is... 

This is the important Gibbs adsorption isotherm. (Note that for concentrated solutions the activity should be used in this equation.) Experimental measurements of y over a range of concentrations allows us to plot y against Inci and hence obtain Ti, the adsorption density at the surface. The validity of this fundamental equation of adsorption has been proven by comparison with direct adsorption measurements. The method is best applied to liquid/vapour and liquid/liquid interfaces, where surface energies can easily be measured. However, care must be taken to allow equilibrium adsorption of the solute (which may be slow) during measurement. 

In equilibrium the adsorbed vapor has the same chemical potential as in the gas phase. Inserting Eq. (14) into Eq. (I3) yields the Gibbs adsorption isotherm... 

The thermodynamics of 2D Meads overlayers on ideally polarizable foreign substrates can be relatively simply described following the interphase concept proposed by Guggenheim [3.212, 3.213] and later applied on Me UPD systems by Schmidt [3.54] as shown in Section 8.2. A phase scheme of the electrode-electrolyte interface is given in Fig. 8.1. Thermodynamically, the chemical potential of Meads is given by eq. (8.14) as a result of a formal equilibrium between Meads and its ionized form Me in the interphase (IP). The interphase equilibrium is quantitatively described by the Gibbs adsorption isotherm, eq. (8.18). In the presence of an excess of supporting electrolyte KX, i.e., c , the chemical potential is constant and... 

An excellent application of LEED not requiring detailed understanding of diffracted intensities is the study of diffusion of adsorbed atoms into the substrate bulk. Conversely, segregation at a surface of atoms previously dissolved in the substrate lattice can be observed too. These processes, bulk solution and precipitation at a surface, are ultimately controlled by the Gibbs adsorption isotherm which determines equilibrium concentration at the interface. Atoms from the gas phase can contribute to such surface equilibria also. Distribution of foreign atoms between surface and bulk may be important in catalysis (393). 

The tv o-dimensional spreading pressure jr is an intensive property of the adsorbed phase. Considering the adsorption of just a single (superscript 0) component i and respecting the Gibbs-Duhem equation provides for constant temperature and pressure and equilibrium conditions the vell-kno vn Gibbs adsorption isotherm (Myers and Prausnitz, 1965) ... 

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

The selection of the reactor depends on the data available. We can use an equilibrium reactor and include the equilibrium constants or a Gibbs equilibrium reaction. For this particular case, we select a Gibbs reactor and we leave the use of the equilibrium reactor as exercise for the reader. We select isothermal operation at the inlet temperature and the reaction will take place in vapor or mixed phase at 50 bar. We can select the presence of inerts in the third tab of the dialogue box in Figure 8.43. 

We have considered the equilibrium distribution of a nonelectrolytic surfactant solute between the hulk liquid phase and the interfacial phase in a gas-Uquid system via relation (3.3.106). We have thereby iUustrated the appUca-tion of the Gibbs adsorption isotherm (3.3.40a) to a single nonionic surface-active solute. Chemical reactions can influence such adsorption isotherms in a number of ways. If the surface-active solutes are ionic, the adsorption equilibria are affected. In other cases, the solute to be removed (the colligend) is not surface active but it reacts with or is... 

Gibbs adsorption isotherm is valid in thermodynamic equilibrium. In equilibrium, the surface tension should not depend on the total surface area if new surface is produced, surfactant from the bulk diffuses to the surface and the same surface tension is estabhshed as before. If, however, the system is not given enough time to equilibrate, the local surface tension changes with an expansion or shrinkage of the geometric surface area A This is characterized with the surface elasticity E, also called surface dilatational modulus [684]. The surface elasticity is defined as... 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

Determination of the equilibrium spreading pressure generally requires measurement and integration of the adsorption isotherm for the adhesive vapors on the adherend from zero coverage to saturation, in accord with the Gibbs adsorption equation [20] ... 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

Isothermal crystallization was carried out at some range of degree of supercooling (AT = 3.3-14 K). AT was defined by AT = T - Tc, where Tj is the equilibrium melting temperature and Tc is the crystallization temperature. T s was estimated by applying the Gibbs-Thomson equation. It was confirmed that the crystals were isolated from each other by means of a polarizing optical microscope (POM). 

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