Equilibrium isotherms Gibbs isotherm

The qualitative phase behaviour of the carbon dioxide-water-1-propanol system is discussed for the temperatures 303 and 333 K in Figure 1, where isothermal Gibbs phase diagrams are arranged in prisms to show the influence of pressure on phase equilibrium. 

Here we show that Eq. (8), together with the conditions of thermodynamic equilibrium for an isothermal adsorption system (equality of chemical potentials between the two phases), can be solved using the Gibbs ensemble Monte Carlo (GEMC) method in the modified form presented in the next section. 

The problem of these types of studies is the uncertainty about the potentialdetermining mechanism. Recall that the distinction is basically between relaxed and polarized interfaces (see the introduction to ch. 11.3). Relaxed interfaces are in thermodynamic equilibrium for these the isothermal Gibbs equation does not contain the applied field, but only terms. Polarized interfaces eu e not at... 

Equivalently (Figure 2.2), dS nw = 0 at equilibrium. The Gibbs function thus expresses the second law as dG < 0 for all possible processes in constant-temperature, constant-pressure systems. (Similarly, the Helmholtz free energy function. A, expresses the second law for isochoric, isothermal systems as dA < 0 for all possible processes.)... 

FIGURE 5.6 (a) Plot of the slope coefficient 5, vs. the surfactant (DDBS) concentration the points are the values of 5, for the curves in Figure 5.5 the fine is the theoretical curve obtained using the procedure described after Equation 5.85 (no adjustable parameters), (b) Plots of the relaxation time and the Gibbs elasticity vs. the DDBS concentration is computed from the equilibrium surface tension isotherm = n (S, /Eq) is calculated using the above values of 5,. 

Adsorption Isotherm The mathematical relationship between the equilibrium quantity of a material adsorbed and the composition of the bulk phase at constant temperature. See also Gibbs Isotherm, Langmuir Isotherm. 

Perhaps the most useful general approach is still the ideal adsorbed solution theory (lAST) developed many years ago by Myers and Prausnitz [26]. The spreading pressure for the pure components may be calculated as a function of equilibrium pressure (p,) by integration of the Gibbs isotherm ... 

Consider an adsorption isotherm which represents the dependence of surface concentration on the equilibrium pressure, p, of adsorbate, in the gas phase (Fig. 4.1.). The general relation between directly measurable quantities F and p and two-dimensional pressure is given by the Gibbs isotherm [75] ... 

The adsorbed mixture is treated as a two-dimensional phase. From the Gibbs isotherm, one can calculate a spreading pressure for each component based on its pure component isotherm. The basic assumption of the IAS theory is that the spreading pressures are equal for all components at equilibrium. 

The conventional picture of the interface of simple aqueous salt solutions is based on thermodynamic analysis of the equilibrium surface tension isotherm. Valuable sources for the equilibrium surface tension isotherm of a simple aqueous electrolyte solution are the papers of Jarvis and Scheiman, and P. Weissenborn and Robert J. Pugh (See Chap. 1, Fig. 8). In general, ions increase the surface tension in a specific manner. However, it is worth mentioning that certain combinations of ions decrease the surface tension or have a negligible effect on it. The thermodynamic analysis of the surface isotherm leads to the picture that the interfacial zone is depleted of ions. The surface deficiency is calculated using Gibbs equation as the derivative of the surface tension isotherm with a dividing plane chosen at a location that the surface excess of water vanishes. 

This is only an apparent contradiction to the conclusion drawn from the analysis of the equilibrium surface tension isotherm. Thermodynamics can accommodate several conflicting interfacial models provided that the integral excess or depletion is in accordance to Gibbs equation. Therefore, experiments are needed that yield direct insights in the interfacial architecture. These data can be obtained by optical techniques. 

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 ... 

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... 

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). 

Combining the IAS theory with the Gibbs equation for isothermal adsorption gives the relationship necessary for equilibrium calculations ... 

Thus far we have observed that the Gibbs and Planck functions provide the criteria of spontaneity and equilibrium in isothermal changes of state at constant pressure. If we extend our analysis to systems in which other constraints are placed on the system, and therefore work other than mechanical work can be performed, we find that the Gibbs and Helmholtz functions also supply a means for calculating the maximum magnitude of work obtainable from an isothermal change. 

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. 

When a surfactant is injected into the liquid beneath an insoluble monolayer, surfactant molecules may adsorb at the surface, penetrating between the monolayer molecules. However it is difficult to determine the extent of this penetration. In principle, equilibrium penetration is described by the Gibbs equation, but the practical application of this equation is complicated by the need to evaluate the dependence of the activity of monolayer substance on surface pressure. There have been several approaches to this problem. In this paper, previously published surface pressure-area Isotherms for cholesterol monolayers on solutions of hexadecy1-trimethyl-ammonium bromide have been analysed by three different methods and the results compared. For this system there is no significant difference between the adsorption calculated by the equation of Pethica and that from the procedure of Alexander and Barnes, but analysis by the method of Motomura, et al. gives results which differ considerably. These differences indicate that an independent experimental measurement of the adsorption should be capable of discriminating between the Motomura method and the other two. 

Cells are isothermal systems—they function at essentially constant temperature (they also function at constant pressure). Heat flow is not a source of energy for cells, because heat can do work only as it passes to a zone or object at a lower temperature. The energy that cells can and must use is free energy, described by the Gibbs free-energy function G, which allows prediction of the direction of chemical reactions, their exact equilibrium position, and the amount of work they can in theory perform at constant temperature and pressure. Heterotrophic cells acquire free energy from nutrient molecules, and photosynthetic cells acquire it from absorbed solar radiation. Both kinds of cells transform this... 

We wish to determine under isothermal and isobaric conditions the concentration of defects as a function of the solid solution composition (e.g NB in alloy (A, B)). Consider a vacancy, the formation Gibbs energy of which is now a function of NB. In ideal (A, B) solutions, we may safely assume that the local composition in the vicinity of the vacancy does not differ much from Ns and /VA in the undisturbed bulk. Therefore, we may write the vacancy formation Gibbs energy Gy(NE) (see Eqn. (2.50)) as a series expansion G%(NE) = Gv(0) + A Gv Ab+ higher order terms, so that AGv = Gv(Nb = l)-Gv(AfB = 0). It is still true (as was shown in Section 2.3) that the vacancy chemical potential /Uy in the homogeneous equilibrium alloy is zero. Thus, we have (see Eqn. (2.57))... 

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