9.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Adsorption | 3/6 | https://en.wikipedia.org/wiki/Adsorption | reference | science, encyclopedia | 2026-05-05T10:45:47.677105+00:00 | kb-cron |
As SD is dictated by factors that are taken into account by the Langmuir model, SD can be assumed to be the adsorption rate constant. However, the rate constant for the Kisliuk model (R') is different from that of the Langmuir model, as R' is used to represent the impact of diffusion on monolayer formation and is proportional to the square root of the system's diffusion coefficient. The Kisliuk adsorption isotherm is written as follows, where θ(t) is fractional coverage of the adsorbent with adsorbate, and t is immersion time:
d
θ
(
t
)
d
t
=
R
′
(
1
−
θ
)
(
1
+
k
E
θ
)
.
{\displaystyle {\frac {d\theta _{(t)}}{dt}}=R'(1-\theta )(1+k_{\text{E}}\theta ).}
Solving for θ(t) yields:
θ
(
t
)
=
1
−
e
−
R
′
(
1
+
k
E
)
t
1
+
k
E
e
−
R
′
(
1
+
k
E
)
t
.
{\displaystyle \theta _{(t)}={\frac {1-e^{-R'(1+k_{\text{E}})t}}{1+k_{\text{E}}e^{-R'(1+k_{\text{E}})t}}}.}
=== Adsorption enthalpy === Adsorption constants are equilibrium constants, therefore they obey the Van 't Hoff equation:
(
∂
ln
K
∂
1
T
)
θ
=
−
Δ
H
R
.
{\displaystyle \left({\frac {\partial \ln K}{\partial {\frac {1}{T}}}}\right)_{\theta }=-{\frac {\Delta H}{R}}.}
As can be seen in the formula, the variation of K must be isosteric, that is, at constant coverage. If we start from the BET isotherm and assume that the entropy change is the same for liquefaction and adsorption, we obtain
Δ
H
ads
=
Δ
H
liq
−
R
T
ln
c
,
{\displaystyle \Delta H_{\text{ads}}=\Delta H_{\text{liq}}-RT\ln c,}
that is to say, adsorption is more exothermic than liquefaction.
=== Single-molecule explanation === The adsorption of ensemble molecules on a surface or interface can be divided into two processes: adsorption and desorption. If the adsorption rate wins the desorption rate, the molecules will accumulate over time giving the adsorption curve over time. If the desorption rate is larger, the number of molecules on the surface will decrease over time. The adsorption rate is dependent on the temperature, the diffusion rate of the solute (related to mean free path for pure gas), and the energy barrier between the molecule and the surface. The diffusion and key elements of the adsorption rate can be calculated using Fick's laws of diffusion and the Einstein relation (kinetic theory). Under ideal conditions, when there is no energy barrier and all molecules that diffuse and collide with the surface get adsorbed, the number of molecules adsorbed
Γ
{\displaystyle \Gamma }
at a surface of area
A
{\displaystyle A}
on an infinite area surface can be directly integrated from Fick's second law differential equation to be:
Γ
=
2
A
C
D
t
π
{\displaystyle \Gamma =2AC{\sqrt {\frac {Dt}{\pi }}}}
where
A
{\displaystyle A}
is the surface area (unit m2),
C
{\displaystyle C}
is the number concentration of the molecule in the bulk solution (unit #/m3),
D
{\displaystyle D}
is the diffusion constant (unit m2/s), and
t
{\displaystyle t}
is time (unit s). Further simulations and analysis of this equation show that the square root dependence on the time is originated from the decrease of the concentrations near the surface under ideal adsorption conditions. Also, this equation only works for the beginning of the adsorption when a well-behaved concentration gradient forms near the surface. Correction on the reduction of the adsorption area and slowing down of the concentration gradient evolution have to be considered over a longer time. Under real experimental conditions, the flow and the small adsorption area always make the adsorption rate faster than what this equation predicted, and the energy barrier will either accelerate this rate by surface attraction or slow it down by surface repulsion. Thus, the prediction from this equation is often a few to several orders of magnitude away from the experimental results. Under special cases, such as a very small adsorption area on a large surface, and under chemical equilibrium when there is no concentration gradience near the surface, this equation becomes useful to predict the adsorption rate with debatable special care to determine a specific value of
t
{\displaystyle t}
in a particular measurement. The desorption of a molecule from the surface depends on the binding energy of the molecule to the surface and the temperature. The typical overall adsorption rate is thus often a combined result of the adsorption and desorption.
== Quantum mechanical – thermodynamic modelling for surface area and porosity == Since 1980 two theories were worked on to explain adsorption and obtain equations that work. These two are referred to as the chi hypothesis, the quantum mechanical derivation, and excess surface work (ESW). Both these theories yield the same equation for flat surfaces:
θ
=
(
χ
−
χ
c
)
U
(
χ
−
χ
c
)
{\displaystyle \theta =(\chi -\chi _{c})U(\chi -\chi _{c})}
where U is the unit step function. The definitions of the other symbols is as follows:
θ
:=
n
ads
/
n
m
,
χ
:=
−
ln
(
−
ln
(
P
/
P
vap
)
)
{\displaystyle \theta :=n_{\text{ads}}/n_{m},\quad \chi :=-\ln {\bigl (}-\ln {\bigl (}P/P_{\text{vap}}{\bigr )}{\bigr )}}
where "ads" stands for "adsorbed", "m" stands for "monolayer equivalence" and "vap" is reference to the vapor pressure of the liquid adsorptive at the same temperature as the solid sample. The unit function creates the definition of the molar energy of adsorption for the first adsorbed molecule by:
χ
c
=:
−
ln
(
−
E
a
/
R
T
)
{\displaystyle \chi _{c}=:-\ln {\bigl (}-E_{a}/RT{\bigr )}}