Henry isotherm

Henry Isotherm In the simplest case, the degree of surface coverage is proportional to the bulk concentration ... 

An analogous law was established in 1803 by W. Henry for the solubilities of gases in water hence, this expression is called the Henry isotherm. The adsorption coefficient B (units dmVmol) depends on the heat of adsorption B = B° e,xp(q RT). The Henry isotherm is valid for low surface coverages (e.g., at 9 < 0.1). 

For small pressures the Langmuir isotherm becomes the Henry isotherm describing the domain of linear adsorption... 

At different types of adsorption isotherms plotted for adsorption of donor particles on oxides (see section 1.5) expressions (1.112) - (1.115) provide the rise in and decrease in with the growth of partial pressure of gas P, the functions themselves being different. Thus, in case of applicability of the Henry isotherm at small P we have the function oi - exp const-P becoming a power function <7s P with the rise in P which is often observed in experiments [154, 155, 169]. 

In case of applicability of the Henry isotherm, i.e. in range of small H2 pressures the expression (2.70) acquires the shape... 

As a first example, the transient case with Henry isotherm can be considered. Expressions developed in Section 2.3 apply with D replacing Dm,ct m replacing cM (including the substitution of c v M by < M and cfsM by c f ) and Ku (defined as r/cM(r0,t) in both cases, i.e. with or without the presence of L) by AT i / (1 + Kc ). Other cases with analytical solutions arise from the steady-state situation. The supply flux under semi-infinite steady-state diffusion is [57] ... 

In principle, the FIAM does not imply that the measured flux. / s should be linear with the metal ion concentration. The linear relationship holds under submodels assuming a linear (Henry) isotherm and first-order internalisation kinetics [2,5,66], but other nonlinear functional dependencies with for adsorption (e.g. Langmuir isotherm [11,52,79]) and internalisation (e.g. second-order kinetics) are compatible with the fact that the resulting uptake is a function (not necessarily linear) of the bulk free ion concentration cjjjj, as long as these functional dependencies do not include parameters corresponding with the speciation of the medium (such as or K [11]). 

To avoid misunderstanding, we note that Equation (9) does not represent the Henry isotherm, as might seem at first sight, since the quantity e,+ in (9), as will be seen from the following (see Sec. V,A), is itself a function of N. 

If adsorption is fast but not sufficiently strong to justify the assumption C0 0, then C0 will, at any instant, be determined by the adsorption isotherm (2.103). This boundary condition leads to mathematical problems the integral equation resulting from (2.110) then becomes a Volterra equation. This has been solved for only some very simple isotherms. Delahay and Trachtenberg [203] solved it for the Henry isotherm (2.104), the solution being... 

Two examples are given here, to be followed up in Chap. 10. Lovric and Komorsky-Lovric [371] expressed (2.115) for the simple Henry isotherm (2.104), as... 

In this chapter, it is shown how to simulate the adsorption of a substance, not taking into account any electrochemical reactions the substance may undergo. That is, only the adsorption itself is dealt with here. In Chap. 2, Sect. 2.5, some theory is presented, laying the groundwork for the simulation. It is noted there that adsorption may be controlled by transport and the adsorption isotherm, in which case there is equilibrium at all times between the solution and surface phases or that the adsorption step itself may limit the rate of adsorption. In this latter case, there are rate constants whose values must be known. In both cases, for isotherms more complicated than the Henry isotherm (2.104), nonlinear terms will enter the equations to be solved in a simulation. 

Of these shapes, long linear isotherms are uncommon for adsorption on solids (Just like type c in fig. 2.8) and occur only if penetration into the solid takes place, leading to a Nemst-type distribution law as in liquid-liquid partitioning, see (1.2.20.1]. Linearity is also found for the (relatively short) initial parts of all isotherms on homogeneous surfaces. We shall call them, as before, Henry isotherms or the Henry parts of curved Isotherms. 

As a rough approximation we assume that in the case of unsaturated steam p[Pg.183]

Of course, the assumption of the validity of a Henry isotherm is questionable from the theoretical point of view but the results at this stage of investigation do not indicate a need for a more detailed model with more unknown parameters which have to be estimated additionally. 

Solving Equation 5.61 together with some of the adsorption isotherms Ej = Ei(cij) in Table 5.2, we can in principle determine the two unknown functions T,(t) and Cj/O- Because the relation r,(c,j) is nonlinear (except for the Henry isotherm), this problem, or its equivalent formulations, can be solved either numerically or by employing appropriate approximations. ... 

In adsorption equilibrium, when the adsorption flux equals the desorption flux, the Langmuir and Henry isotherms results. 

As the most simple form of an isotherm for surfactant mixtures, a generalised Henry isotherm can be used,... 

The analytical solutions presented above are most of all derived on the basis of the very simple Henry isotherm or the more physically sensible Langmuir isotherm. Beside these analytical solutions a direct integration of the initial and boundary value problem of the diffusion-controlled model is possible. To do so differentials are replaced by differences. This approximation leads to linear equation systems for each time step which have to be solved. As... 

Analytical solutions as presented above are based on the very simple Henry isotherm, while for the frequently applied Langmuir isotherm an approximate solution as a power series can be obtained. For any other, more sophisticated isotherm, an analytical solution does not exist. Thus, a direct integration of the initial and boundary value problem of the diffusion-controlled model is required. Using a difference scheme [63] numerical results can be obtained for any type of an adsorption isotherm. The following models rely on such numerical methods. 

The eommon simulation hypotheses are ID simulation of transport between eomponents and food no interaction between substances no transfer of food eonstituents into the paekaging no reaction Henry isotherms constant diffusion and partition coefficients during the same stage uniform diffusion coefficients the simulations may be ehained, inherited, branched etc. with arbitrary complexity. 

The deviation of the adsorption isotherm at a uniform electrode surface from the linear behavior (Henri isotherm) is related to the interaction between the adsorbed species. A substantiated derivation of its form can only be made on the basis of an analysis of the statistical-mechanical properties of the whole ensemble of adsorbed ions, which, in turn, requires the knowledge of the interaction potential between the ions, U, as a function of their distance, R, along the surface. This quantity is defined as a difference between the energies of the system, when these two ions are fixed at distance R or are very far from each other. 

Table 1 lists the six most popular surfactant adsorption isotherms, i.e., those of Henry, Freundlich, Langmuir, Volmer (10), Frumkin (11), and van der Waals (9). For cj— 0 all other isotherms (except that of Freundlich) reduce to the Henry isotherm. The physical difference between the Langmuir and Volmer isotherms is that the former corre-... 

We now see that except for the case of Henry isotherm, all other isotherms exhibit an increase in the thermodynamic correction factor versus loading. 

This is a linear relationship between activity and amount sorbed, having the same form as Henry s law in solution chemistry, so that it is often termed Henry isotherm, and K, is the Henry constant. In principle, it is expected that isotherm equations should approximate to the Henry isotherm for low sorbate activities. 

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