Page 74 - Adsorbents fundamentals and applications

HORV ´ ATH–KAWAZOE APPROACH 59

This implies that a gas molecule has a continuous distribution of positions in the
filled micropore, which it is free to occupy. In reality, however, the gas molecule
has only fixed positions at which it can exist when the pore is filled to capacity,
especially when the pore width is only a few multiples of the adsorbate molecule
diameter. Besides, the Boltzmann law of distribution of energy requires that
the above average be instead computed by weighing the energy terms with the
probability of a molecule to possess that energy, namely e −ε/kT .Inother words
the molecule is expected to occupy a position at which the energy potential is
minimum. This aspect was not considered in the original work. In the present
work, a discretized average, which attempts to incorporate the above described
features in the model, is proposed.
Finally, the average potential energy that is calculated is related to the free
energy change upon adsorption, RT ln(P/P 0 ). The resulting slit-pore HK model
is as follows:
P N S A S + N A A A

RT ln = N Av · ε(z) = N Av 4
P 0 σ (L − 2d 0 )
σ σ σ σ
4 10 4 10
× − − + (4.12)
3(L − d 0 ) 3 9(L − d 0 ) 9 3d 3 9d 9
0 0
As can be seen from Eq. 4.12, the interaction energy of a molecule is indetermi-
nate when L = 2d 0 , that is, when the free space of the pore (L − d S ) equals the
diameter of an adsorbate molecule (d A ). However, a finite value for the potential
exists for a molecule of this size, as is obvious from Eq. 4.9. This anomaly is
overcome in the modified model.
An implicit assumption in the original derivation by Horv´ ath and Kawazoe
(1983) was that the isotherm followed Henry’s law. It is well known that most
isotherm data show a considerable deviation from Henry’s law at pressures near
the saturation pressure of the adsorbate, and the equilibrium loading usually fol-
lows a Langmuir-type behavior. Cheng and Yang (1994) introduced a correction
term to incorporate the Langmuir isotherm in the model and the corrected model
(referred to as the HK–CY model) was given as:

P 1
RT ln + RT 1 + ln(1 − θ) = N Av · ε(z)
P 0 θ
4 10 4 10
N S A S + N A A A σ σ σ σ
= N Av 4 3 − 9 − 3 + 9 (4.13)
σ (L − 2d 0 ) 3(L − d 0 ) 9(L − d 0 ) 3d 9d
0 0
The Langmuir isotherm is typically represented as follows:

P
K
P 0
θ = (4.14)
P
1 + K
P 0