Equilibrium constant Gibbs adsorption equation

At constant temperature, the interfacial tension y of a water oil system containing a single surfactant solute can be calculated at thermodynamic equilibrium starting with the Gibbs adsorption equation... 

The Gibbs monolayer is continuously in equilibrium with the adjacent solution. Hence, for an ideal solution (where C is proportional with X ) at constant temperature, combining the Gibbs adsorption equation (3.88) with (7.1) yields... 

Even in the primitive versions of the van dcr Waals theory with m independent of p, that coefficient may still depend on the temperature T (or, equivalently, on the chemical potential fi) at which the phases are in equilibrium. While in Chapter 5 we shall see some examples, or limiting idealized cases, in which m is a fixed constant, independent of T, and is determined by the intermolecular forces alone, as in (1.38), it will, more generally, depend on T and in that event, as we shall see in 3.4, the connection between this theory and the Gibbs adsorption equation (2.31) is not entirely straightforward and requires discussion. 

Hydrophilic surfactants adsorb best on aqueous phases, whereas hydrophobic surfactants adsorb best on lipophilic surfaces (oils). Data on adsorption at constant temperature are usually plotted as a function of the surfactant equilibrium concentration plots for solid substrates are termed Langmuir isotherms. From such isotherms the maximum surfactant concentration at the interface (Fmax) can be derived and the maximum area occupied by the surfactant at the interface ( max) can be calculated. In addition, the Gibbs adsorption equation can be extracted. 

Turning now to adsorption equilibrium, let us apply algebraic methods to a two component 1,2 phase system. From the phase rule there will be two degrees of freedom, but we shall reduce this to one by maintaining the temperature constant. Then for the total system there exists a Gibbs-Duhem equation... 

The value a, being an adsorption equilibrium constant, is related to the standard Gibbs energy of adsorption, AG °, by the equation... 

Gibbs [2] derived a thermodynamic relationship between the surface or interfacial tension y and the surface excess F (adsorption per unit area). The starting point of this equation is the Gibbs-Duhem equation, as given above [see Eq. (5.1)]. At equilibrium, where the rate of adsorption is equal to the rate of desorption, dG = 0. Hence, at a constant temperature, but in the presence of adsorption. 

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) ... 

Adsorption from solutions was fully studied by G. C. Schmidt. He first showed that the adsorbed amount reaches a maximum, when the surface is saturated, and does not then increase if the concentration of the solution is increased (1910). He proposed an adsorption formula (1911) taking this into account, which he later modified (1916). Extensive researches carried out from 1906 by Freundlich showed that a thermodynamic theory given by J. W. Gibbs (1877, see p. 742) could be used as a guide. A modification of the adsorption equation (5), viz. xlm=kc f (6), applies to solutions, where adsorbed amount, m=mass of adsorbent, equilibrium concentration of solution, k and n are constants (i/n varies from o i to o-8). It was apparently first used by C. H. D. Bodeker, then by W. Biltz, and Freundlich. 

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... 

Adsorption isotherms represent a relationship between the adsorbed amount at an interface and the equilibrium activity of an adsorbed particle (also the concentration of a dissolved substance or partial gas pressure) at a constant temperature. The analysis of adsorption isotherms can yield thermodynamic data for the given adsorption system. Theoretical adsorption isotherms derived from statistical and kinetic data, and using the described assumptions (see 3.1), are known only for the gas-solid interface or for dilute solutions of surfactants (Gibbs). Those for the system gas-solid are of a few basic types that can be thermodynamically predicted81. From temperature relations it is possible to calculate adsorption and activation energies or rate constants for individual isotherms. Since there are no theoretically founded equations of adsorption isotherms for dissolved surfactants on solids, the adsorption of gases on solides can be used as a starting point for an interpretation. 

Equation (2-80) expresses the retention of an ionizable basic analyte as a function of pH and three different constants ionization constant adsorption constant of ionic form of the analyte (7 bh+) and adsorption constant of the neutral form of the analyte (T b)- These three constants describe three different equilibrium processes, and they have their own relationships with the system temperature and Gibbs free energy with respect to the particular analyte form. 

Hall proposed an experimental procedure, in which changes in the area of the monolayer in equilibrium with the soluble surfactants should be performed to keep constant or control the chemical potential of the first component when the activity of the second component is varied. To summarise, the rigorous thermodynamic analysis of the penetration equilibrium, based on the Gibbs equation, can neither provide an equation of state of the monolayer, nor an adsorption isotherm for the soluble component. This analysis only enables one to formulate the conditions for a penetration experiment, which are, however, very difficult to implement. Therefore, to describe the thermodynamic behaviour of real mixed monolayers, at present one should use some approximate theoretical models. 

This argument ignores the fact that by considering the displacement equilibrium at constant surface area, terms in ir can be eliminated, and also that in any case use of the Gibbs equation enables ir to be calculated. This distinction between alternative formulations is, of course, the essence of the difference between adsorption theory and solution theory of the adsorption of vapours as developed by HilP° many years ago. 

InFig. 9, aplotofEq. (114) results in predicted adsorption as a function of pH with total cation and surface-site mole fraction held constant. For the Gibbs plot in Fig. 10 of equation (115), adsorption is plotted as the areal surface concentration, Tg, as a function of increasing total cation mole fraction [or equilibrium solution mole fraction with Eq. (102)] at constant pH and total surface-site concentration. 

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