© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
ADSORPTION:
physical consistency and Henry constant
General Integral Equation / GL (Generalized Langmuir) / All equations (preview)
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)
Physical consistency:
(for gas and dilute solute adsorption)
This condition is in fact equivalent to:
lim_{p→0}(φ) = 1
or
lim_{c→0}(φ) = 1
where the so-called φ-function is:
φ = δ log(a)/
δ log(p)
or
φ = δ log(a)/
δ log(c)
In a similar manner a Langmuir-like behaviour (see below) at p → ∞ may also be obtained.
Langmuir constant K_{L} may be defined as:
or
or
A detailed discussion and mathematical formulation for Henry and (above formulated) Langmuir constants may be found in the Appendix of the paper:
Physical consistency condition
It is usually believed, that the very existence of such a limit (i.e. isotherm tends to Henry behavior at very low adsorptions) is a consequence of the existing maximum of adsorption energy and is sometimes called a physical consistency condition. Though the maximum energy value always produces Henry behavior, the opposite is not necassarily true. It is easy to prove that it is enough that the energy distribution function defined in the infinite energy range (-∞ , +∞) behaves in a special way and such a limit is obtained (e.g. for Toth or RP isotherm equations).
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: 1^{st} depends only on the average adsorption energy, 2^{nd} is the integral that depends only on the shape and width of the energy distribution:
K_{H} = I_{1}(E_{avg}) I_{2}(σ_{E})
So, e.g. for:
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, K_{L}), 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:
Calculability - conditions for finite values of Henry and Langmuir constants
Generally, finite values of the above 2^{nd} integral term are obtained if:
General Integral Equation / GL (Generalized Langmuir) / All equations (preview)
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