A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

physical consistency and Henry constant

Henry Isotherm | Physical consistency | Definitions ( Henry constant | Langmuir constant ) References | Calculation: ( Henry constant | Langmuir constant | Calculability )

Henry isotherm:

Many experimental isotherms display a behavior corresponding to the simplest isotherm, the Henry isotherm:
a = K c     (for solute adsorption)
or
a = K p     (for gas or vapor adsorption)

This adsorption law is formally identical with the well known Henry law for gas absorption in liquids (which in turn is a consequence of a more general Nernst partition law):
vabs = K p (Henry law for gas absorption)

For adsorption isotherms measured in most systems this behavior may be observed for relatively high temperatures and/or very low pressures (or concentrations). However, as explained below it results from the so-called physical consistency requirement for adsorption isotherms and is a boundary condition for low relative adsorptions (coverages).

Physical consistency:
(for gas and dilute solute adsorption)

Existence and calculation of Henry and Langmuir constants:
• Calculation of Henry constant for theoretical isotherms
By using the General Integral Equation and after putting p→0 (or c→0) one easily obtains a formula being a product of 2 terms: 1st depends only on the average adsorption energy, 2nd is the integral that depends only on the shape and width of the energy distribution:
KH = I1(Eavg) I2E)
So, e.g. for:

• "true" Gauss energy distribution, where there is no energy limit on both sides, one obtains:
KH = Ko exp(Eavg) exp(0.5 σE2)
• Langmuir-Freundlich (LF), Generalized Freundlich aka. Sips (GF) (and GL where m ≠ 1) or Rudzinski (R) equations: the 2nd term goes to +∞ and accordingly these equations do not display Henry behaviour.
• Toth (T) or "Square" (Sq) aka. UNILAN equations: the 2nd term is finite and accordingly these equations do display Henry behaviour.
• One may expect that the Henry constants KH are always in the range:
KH ⊂ (Kavg , Kmax)
where:
Kavg = Ko exp(Eavg)   and   Kmax = Ko exp(Emax)

• Calculation of Langmuir constant for theoretical isotherms
By using the same approach one may find an approximation of an isotherm close to monolayer filling. If such an equation reduces to Langmuir isotherm for p→∞, then an analogue of Henry constant (let's call it Langmuir equilibrium constant, KL), may be calculated. In the calculations below it was assumed that no surface-screening effect is observed, i.e. Langmuir model and not Flory-Huggins (used e.g. for polymer adsorption) describes adsorption on homogeneous surface.
E.g. for:

• "true" Gauss energy distribution, where there is no energy limit on both sides, one obtains:
KL = Ko exp(Eavg) exp(-0.5 σE2)
• Langmuir-Freundlich (LF) or Tóth (T) (and GL where n ≠ 1) equations: the 2nd term goes to -∞ and accordingly these equations do not display Langmuir behaviour.
• Generalized Freundlich / Sips (GF) and "Square" (Sq) aka. UNILAN equations: the 2nd term is finite and accordingly these equations do display Langmuir behaviour.
• One may expect (provided, that lateral interactions and multilayer formation are absent) that the Langmuir constants KL are always in the range:
KL ⊂ (Kmin , Kavg)
where:
Kmin = Ko exp(Emin)   and   Kavg = Ko exp(Eavg)
• If lateral interactions of the Fowler-Guggenheim (non-specific) or Kiselev (associative, here simplified model) must be considered (and multilayer effects are separated or non-existant), one may expect (provided, that lateral interactions are absent) that the Langmuir constants KL are in the range:
KL ⊂ (Kmin , Kavg)
where:
Kmin = Ko exp(Emin) exp(α)   (α - FG interaction factor)
Kmin = Ko exp(Emin) (1 + Kn)   (Kn - Kiselev association constant)
and
Kavg = Ko exp(Eavg)

• Calculability - conditions for finite values of Henry and Langmuir constants
Generally, finite values of the above 2nd integral term are obtained if:

• for p→0 (or c→0) (Henry range):
if for E → +∞ (or Emax) the energy distribution function decreases faster than exp(-E)
It means that the energy distribution function with behavior described in this range by the function exp(-kE) and with 0<k<1 will not produce finite integral. However, if k>1 then the integral will be finite.
E.g. for Tóth isotherm equation and energy distribution we have this coefficient k=(1+n)>1 (m=1, 0<n≤1 where m and n are heterogeneity coefficients)
• for p→∞ (or c→∞) ("Langmuir" range):
if for E → -∞ (or Emin) the energy distribution function decreases faster than exp(E) (here "decrease" if you go from average energy towards -∞ or Emin)
It means that the energy distribution function with behavior described in this range by the function exp(kE) and with 0<k<1 will not produce finite integral. However if k>1 then the integral will be finite.

Henry Isotherm | Physical consistency | Definitions ( Henry constant | Langmuir constant ) References | Calculation: ( Henry constant | Langmuir constant | Calculability )

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