Equilibria, Gibbs’ theory

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... 

When the kinetics are unknown, still-useful information can be obtained by finding equilibrium compositions at fixed temperature or adiabatically, or at some specified approach to the adiabatic temperature, say within 25°C (45°F) of it. Such calculations require only an input of the components of the feed and produc ts and their thermodynamic properties, not their stoichiometric relations, and are based on Gibbs energy minimization. Computer programs appear, for instance, in Smith and Missen Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982), but the problem often is laborious enough to warrant use of one of the several available commercial services and their data banks. Several simpler cases with specified stoichiometries are solved by Walas Phase Equilibiia in Chemical Engineering, Butterworths, 1985). 

The central focus of Gibbs theory is the equilibrium state S, a quiescent limiting condition of a sufficiently large ( macroscopic ) physical system that exhibits characteristically simple responses to attempted changes of the control variables that specify the state. 

While the equilibrium kinetic theory permits us to develop in fairly simple manner the properties of a dilute hard sphere gas, it becomes progressively more complicated and difficult to apply to both dense systems and systems in which there are forces acting between particles. To deal with such systems, we shall here outline briefly the very powerful statistical methods of Gibbs. ... 

At equilibrium. Tf = T, and the equilibrium version of the Adam-Gibbs theory, discussed earlier, is recovered. For a glass that is not able to relax, T/ is constant, and then follows an Arrhenius temperature-dependence, which accords well with experimental measurements on deeply quenched glasses. 

The difference is explained by Gibbs theory of equilibria and was studied experimentally by Donnan the overall process is known as the Gibbs-Donnan equilibrium. 

As the basis of his theory of equilibria, Gibbs adopted the laws of thermodynamics, a method of treatment which had first been employed by Horstmann. In deducing the law of equilibrium, Gibbs regarded a system as possessing only three independently variable factors —temperature, pressure, and the concentration of the components of the system—and he enunciated the general theorem now usually known as the Phase Rule, by which he defined the conditions of equilibrium as a relationship between the number of what arc called the phases and the components of the system. 

In the transition from thermodynamics to dynamics of adsorption transport and energetic aspects have to be distinguished. The main question on energetic aspects belongs to the validity of Gibbs equation under non-equilibrium conditions. Defay et al. (1966) demonstrated the outcome of non-equilibrium thermodynamics for this topic. This direction deserves further attention. Comparison of experimental data with theory enables us to determine the range of applicability of Gibbs theory to non-equilibrium systems. 

The decrease in the internal radius, simultaneously with increasing segregation, causes the shift of the equilibrium boundary concentration of B inside IMC at the curved interface between B and IMC. Let us compare these two equilibrium concentrations. According to Gibbs theory, at curved interfaces, the mentioned shift is equal to... 

However, some subjects neglected in most textbooks are treated here. An example is thermodynamic stability theory, which plays an important role both at equilibrium and away from equilibrium. Thermodynamic theory of stability and fluctuation, which originated in the work of Gibbs, is the subject of Chapters 12-14. We begin with the classical theory of stability, as Gibbs formulated it. 

Two major variables which effect E are the film thickness and surface concentration. The concentration of surface-active material influences the concentration gradients. For a freshly produced foam to survive, then surface tension gradients are necessary. However, the main deficiency in the early studies on Gibbs elasticity was that it applies to thin films and the diffusion effects from bulk solution were neglected. In fact, the Gibbs theory only applies to a hypothetical equilibrium state (i.e. it is assumed that there is insufficient surfactant in the film to diffuse to the surface and lower the surface tension). 

The essence of the Adam and Gibbs theory is that n can be ealeulated on thermodynamie equilibrium grounds as follows ... 

A more general, and for the moment, less detailed description of the progress of chemical reactions, was developed in the transition state theory of kinetics. This approach considers tire reacting molecules at the point of collision to form a complex intermediate molecule before the final products are formed. This molecular species is assumed to be in thermodynamic equilibrium with the reactant species. An equilibrium constant can therefore be described for the activation process, and this, in turn, can be related to a Gibbs energy of activation ... 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

The structure of hydrogels that do not contain ionic moieties can be analyzed by the Flory Rehner theory (Flory and Rehner 1943a). This combination of thermodynamic and elasticity theories states that a cross-linked polymer gel which is immersed in a fluid and allowed to reach equilibrium with its surroundings is subject only to two opposing forces, the thermodynamic force of mixing and the retractive force of the polymer chains. At equilibrium, these two forces are equal. Equation (1) describes the physical situation in terms of the Gibbs free energy. 

The effect of pressure on chemical equilibria and rates of reactions can be described by the well-known equations resulting from the pressure dependence of the Gibbs enthalpy of reaction and activation, respectively, shown in Scheme 1. The volume of reaction (AV) corresponds to the difference between the partial molar volumes of reactants and products. Within the scope of transition state theory the volume of activation can be, accordingly, considered to be a measure of the partial molar volume of the transition state (TS) with respect to the partial molar volumes of the reactants. Volumes of reaction can be determined in three ways (a) from the pressure dependence of the equilibrium constant (from the plot of In K vs p) (b) from the measurement of partial molar volumes of all reactants and products derived from the densities, d, of the solution of each individual component measured at various concentrations, c, and extrapolation of the apparent molar volume 4>... 

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