Limiting Henry Isotherm

We will consider briefly the limiting behavior that one should expect when the adsorbate activity is low enough. According to Equation 4.4, the isotherm expression should be a function of adsorbate activity. This function can be expanded in Taylor series around = 0, yielding 

For sufficiently low values of a, all terms for powers greater than 1 vanish also, obviously it should be/(0) = 0, so that for one gets 

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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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. 

That is, if water sorption occurs on two types of sites, a polymer site and a pol3rmer-water site, the influence of the former will predominate as P and the amount of sorbed water simultaneously approach zero. A plot of vs P with a slope of k] (the inverse of Equation 2), as shown in Figure 1, is tangent to the experimental Isotherm at the origin. The analysis outlined allows a unique specification of x interaction parameter, through use of the limiting (Henry s Law) approximation of the Flory-Hugglns theory ( 3). That is... 

Make a plot of Eq. XVII-69 as 6 versus P, and, for comparison, one of a Langmuir adsorption isotherm of same limiting or Henry s law slope. Comment on the comparison. 

Thermodynamics requires that a linear limit be approached in the Henry s law region for all isotherm equations. 

The adsorption free energy and other parameters may be determined, provided that a proper adsorption isotherm is identified and is fitted to experimental data. However, it is usually difficult to unequivocally choose an appropriate isotherm an experimental isotherm may well be fitted to a multitude of theoretical isotherms having several adjustable parameters. If the adsorption isotherm at a very small surface coverage is accessible experimentally, the adsorption free energy can be determined from the limiting slope of the isotherm, as all isotherms reduce to Henry s law when 6 0 ... 

In other words, the model predicts a limiting form of a linear (Henry s Law) type of isotherm as the polymer concentration tends to zero. 

The simplest adsorption equation is Henry s Law, that is, the loading is directly proportional to the sorbate partial pressure. X= KP This linear isotherm equation adequately describes some adsorbents and, in the limit of low coverage, it actually describes most sorbents. For adsorption that is truly described by Henry s Hnear relationship, the loadings are low, the adsorption is bound to be essenhaUy isothermal and there are several published analytical solutions to describe both batch kinetics and column dynamic behavior for such systems. 

In the limit of low pressure the front end of the adsorption isotherm is approximated by the Henry regime which states that the number of adsorbed molecules per unit volume is proportional to the pressure and to the Henry coefficient, Kh. ... 

The adsorption isotherm —Equation (8) —associated with this surface equation of state is called the Henry law limit, in analogy with the equation that describes the vapor pressure of dilute solutions. The constant m, then, is the adsorption equivalent of the Henry law constant. When adsorption is described by the Henry law limit, the adsorbed state behaves like a two-dimensional ideal gas. 

It is interesting to look at this form of the isotherm in the limit of small values of 0 but still above Henry s limit. In this case the exponential term approaches unity, and the isotherm becomes... 

What is meant by the Henry law limit in the case of an adsorption isotherm ... 

For the sorption of hydrocarbons in type-A zeolites at ordinary temperatures, the region of linearity of the isotherm is limited to very low pressures, and Henry constants are usually obtained by extrapolation from measurements outside the linear region. 

The Henry constant J Cis a function of T but not P. (In some theoretical treatments, the Henry constant is the ratio of fugacity to quantity adsorbed, i.e., the inverse of the sense used here.) It is generally expected that adsorption will be governed by Henry s law at sufficiently low pressures. It is possible to construct theoretical models for adsorption in which an isotherm does not reduce to Henry s law, Equation (2.3), even in the limit P —> 0, but it is not clear that such situations obtain in practice and doubtful that they are important in noble gas geochemistry. 

At sufficiently low concentrations on a homogeneous surface the equilibrium isotherm for physical adsorption will always approach linearity (Henry s law). The limiting slope of the isotherm [limp o(dq/dp)T] is referred to as the Henry constant K . It is evident that the Henry con-... 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

On the other hand, kinetics of reactions occiuring on a solid surface, that is, catalysis or photocatalysis, must be significantly different. There may be two representative extreme cases. One is so-called a diffusion controlled process, in which siuface reactions and the following detachment process occur very rapidly to give a negligible surface concentration of adsorbed molecules, and the overall rate coincides with the rate of adsorption of substrate molecules. In this case, the overall rate is proportional to concentration of the substrate in a solution or gas phase (bulk), that is, first-order kinetics is observed IS). The other extreme case is so-called surface-reaction limited, in which surface adsorption is kept in equilibrium during the reaction amd the overall rate coincides with the rate of reaction occurring on the surface, that is, reaction of e and h+ with surface-adsorbed substrate (l9). Under these conditions, the overall rate is not proportional to concentration of the substrate in the bulk unless the adsorption isotherm obeys a Henry-type equation, in which the amount of adsorption is proportional to concentration in the bulk (20). In the former case, the rate... 

Figure 2. Simulated isotherms for methane-in carbon slit pores of varying widths are shown here. The number of adsorbed molecules per unit area of pore wall is plotted as a function of the pressure times the Henry s constant, which gives the single straight line shown for the limiting low pressme parts of the isotherms. Pore widths in A are shown on the Figure. From Ref. [22], Sep. Sci. and Tech. 27 (1992), 1837-1856.

Thus k is defined as tire limiting slope of an isotherm as P 0, and is known as Henry s constantfor adsorption. It is afunction of temperature only for a given adsorbentand adsorbate, and is characteristic of the specific interaction between a particular adsorbent and a particular adsorbate. 

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