Exercise 11
ADSORPTION KINETICS
11.1. Introduction
11.1.1. General terms. Definition of adsorption systems
Adsorption is a surface phenomenon. On the border between two contacting phases a
substance is divided into bulk and surface phases. In each phase components are in different
quantitative ratios. Density of the surface phase (surface layer covering the area which
remains in the field of adsorption forces) is elevated as compared to the bulk phase which
gives an effect of absorption and storage of substances.
Adsorption processes are usually classified according to the types of phases which form
interfacial surface and according to types of forces that act on these surfaces. Because of the
phase type, we consider adsorption in liquid-liquid, liquid-gas, solid-liquid and solid-gas
systems. Physical and chemical adsorption is distinguished due to the type of intermolecular
interactions.
Of greatest practical importance is the physical adsorption in the solid-liquid and solid-gas
system. Porous solid with highly developed surface is called the adsorbent. The real
adsorbent surface, which is involved in mass transfer, per mass or volume unit determines its
specific surface area. An adsorbed substance in the gas phase is referred to as the adsorptive,
and after the transition into adsorbed state it is the adsorbate.
11.1.2. Physical and chemical adsorption
Physical adsorption is induced by intermolecular forces (van der Waals forces, hydrogen
bridge bonding). These forces are revealed when the distance between molecules is very
small, of the order of nanometers. The field of internal forces inside the adsorbent is
balanced, while molecules near the interface are subject to unbalanced forces. The system
tends to saturate these forces which causes attraction of molecules from the bottom of the
bulk phase and the deposition of a layer of molecules on the adsorbent surface.
Intramolecular forces are weak in this case and the process is reversible. The molecules are
not adsorbed on the entire surface but only in the so called active centers, i.e. in places where
the forces are intensified (cavities, pores, etc.). This is why the molecules are not stably
placed on the adsorbent surface, but being in the sphere of these interactions they can move
along it.
A characteristic feature of the physical adsorption is the formation of multimolecular layers.
Molecules adsorbed on the surface, subject to cohesion forces from deep inside the phase,
attract next molecules of a liquid or gas from the environment, partly balancing these forces.
In this way on the first adsorbed layer the next one is formed.
Chemical adsorption (the so called chemisorption) is connected with the transition of
electrons between the adsorbent and a substance being adsorbed. As a result, a chemical bond
is formed between active groups on the adsorbent surface (solid body) and liquid molecules
(gas or liquid). Forces determining this type of bonds are covalent forces involving ionic
strength. Chemisorption is localized, irreversible, while formation of chemical bonds destroys
individuality of adsorbate and adsorbent molecules which now form a uniform system.
The most important criteria which help to distinguish the two types of adsorption are as
follows:
1) thermal effect of the process – lower at physical adsorption (heat of adsorption is equal to
the heat of condensation or crystallization 4-80 kJ/mol);
2) reversibility of the process – in the case of physical adsorption the adsorbed substance can
be easily removed from the surface (e.g. by heating up the adsorption system), in the case
of chemical adsorption it is very difficult;
3) thickness of the adsorption layer – in physical adsorption the layers are formed with
thickness corresponding to several molecule diameters; during chemisorptions
monomolecular layers are formed;
4) activation energy for physical adsorption is relatively low, for chemical adsorption it is
high, similar to the values for chemical reactions.
11.1.3. Adsorbents
The most often used adsorbent is activated carbon. This is the product of carbonization and
activation of a raw carbon which includes charcoal, anthracite, peat semi-coke, xylites,
lignite, coal, sawdust, nutshells, stones, bones, etc. Due to thermal processing in the absence
of air, without vapor-gas or chemical activation, a complex porous structure is formed with
different size of pores penetrating into the particle. A result is a well developed specific
surface area (good activated carbon has a sorption area equal to 800-1800 m2/g), which
reflects adsorption forces and along with pore size distribution, mechanical strength,
adsorption capacity and particle size determines the applicability of sorbents in various
technological processes.
Activated carbon is used in food industry to purify solutions (e.g. in sugar industry), in water
and sewage technology to remove pollutants present at small concentrations. Activated
carbon is also very important in gas purification, hermetization of transshipment processes,
storage and distribution of liquid fuels, separation of multicomponent gas mixtures and
recovery of organic solvents.
The next frequently applied adsorbent is silica gel. This is an amorphous solid of the general
chemical formula SiO2·n H2O. Due to an adequate chemical and physical processing a porous
structure is obtained which ensures specific surface area in the range of 100 to 500 m2/g.
Silica gels are usually used in drying of gases and in selective adsorption of hydrocarbons.
In water and sewage technology very important are adsorbents called the ion exchangers.
These are natural aluminosilicates (zeolites, bentonites) or synthetically obtained products
(sulphonated carbon, ion-exchange resins). Ion exchangers have active acidic (cationexchange resins) and alkaline groups (anion exchangers), capable of chemical reactions with
ions of the solution. As a result of this reaction the ion from the solution is exchanged to the
ion of an active group. Ion exchangers are used mainly to soften and decarbonize water.
Very important are molecular sieves. These are crystalline metal aluminosilicates (zeolites)
and a new type of carbon adsorbents (carbon sieves). A characteristic feature of these
adsorbents is their high selectivity related to regular network spatial structure of a molecule.
11.1.4. Adsorption equilibrium
On the solid-liquid interface the adsorbate is divided into bulk (gas or liquid) and surface
phases. This process proceeds until reaching an equilibrium which can be presented by the
general equation:
f(a,p,T) = 0 (for gas)
or
f(a,c,T) = 0 (for liquid)
(11.1)
where: a – absorption capacity of the adsorbent, i.e. the amount of substance adsorbed by
 kg 
the unit of adsorbent mass   ,
 kg 
p – partial pressure of the adsorptive in gas [Pa],
 kg 
c – adsorptive concentration in liquid in the equilibrium state  3  ,
m 
T – temperature [K].
In the case when the process proceeds at a constant temperature, equilibrium is written down
as an adsorption isotherm. The adsorption isotherm is dependent on temperature and
adsorption system properties. Since adsorption is an exothermal process, at an elevated
temperature absorption capacity of the adsorbent is lower. The equilibrium adsorption
capacity depends, among other things, on the size of specific surface area of the adsorbent, its
porous structure, the type and size of molecules of the absorbed substance (bigger molecules
are absorbed stronger).
The simplest equation which describes the adsorption isotherm in the case of low partial
pressures or adsorbate concentrations is Henry’s equation:
a
H
p
RT
or
a  H c
where: H – Henry’s constant for the adsorption system and temperature,
 kJ 
R – universal gas constant 
.
 kmol  K 
(11.2)
In the description of adsorption equilibrium very important is the Langmuir isotherm
equation derived theoretically under the assumption that adsorption occurs only in active sites
on the surface and that adsorption surface layer is monomolecular:
a
a max  b  p
1 b  p
a
or
amax  b  c
1 b  c
(11.3)
where: amax – maximum adsorption capacity corresponding to the total coverage of the
 kg 
adsorbent surface by the adsorbed substance   ,
 kg 
b–
constant for a given adsorption system and temperature.
The Langmuir equation well describes many real adsorption systems, both liquid and
gaseous. In the region of very low pressures or concentrations equation (11.3) becomes an
analytical notation of Henry’s law.
Another very often used equation is BET derived by Brunauer, Emmet and Teller for multilayer adsorption assuming that the Langmuir equation is used for every subsequent layer:
a max  (b  1) 
a
a max  b  1 
p
pn

p  
p 
1 
  1  b 

p
p
n  
n 

or
a

c
1 
 cn
c
cn
 
c
  1  b 
cn
 



(11.4)
where: pa – saturated vapor pressure [Pa],
 kg 
cn – saturation concentration  3  .
m 
For adsorption from the liquid phase equation (11.4) can be used for a substance with limited
solubility (cn << ∞).
In the description of an isotherm the empirical Freundlich equation is also used
a  k  cn
(11.5)
where: k,n – empirical constants for a given adsorption system.
The equation has a suitable mathematical form and in many cases gives good agreement with
empirical results.
Figure 1 shows a comparison of the above discussed adsorption isotherms.
Fig. 11.1. Comparison of adsorption isotherms
For low concentrations all three basic types of isotherms are similar, for higher
concentrations there are significant differences between them. The Langmuir isotherm has a
horizontal asymptote (for c →∞, a→amax), the Freundlich isotherm is monotonous (for for c
→∞, a→∞), the BET isotherm has a vertical asymptote (for c→cn, a→∞).
11.1.5. Adsorption kinetics
For practical use of adsorption important is the rate at which the system reaches the state of
equilibrium, i.e. the knowledge of adsorption kinetics. Considering the process as mass
transfer by diffusion in one direction, the following stages can be distinguished:
-
external diffusion covering diffusion or convection in the main bulk of the liquid and
mass transfer from the liquid core to the external surface of the solid body,
-
mass transport through the interface,
-
internal diffusion which covers diffusion in the liquid contained inside the adsorbent
pores, specific adsorption in active centers and diffusion in the surface layer of the
solid body, i.e. migration of molecules adsorbed from the sites with a higher
concentration to the sites of lower concentration.
The effect of subsequent stages on the process kinetics is not constant. It depends on the
adsorption system properties, hydrodynamic conditions, and additionally it changes while
approaching the state of equilibrium. It was found that the stages which determine the rate of
the whole process are the slowest processes of diffusion because mass transport through the
interface and adsorption in active centers takes place immediately.
In the adsorption process from a constant volume, at good mixing of the bulk phase, a stage
limiting the mass transfer is internal diffusion. Its rate depends on the type of diffusing
substance and on the capillary-porous structure of the adsorbent. The diffusing molecules
penetrate the porous medium not through its entire surface but only through the inlet surface
of pores and that is why the active diffusion surface is smaller than the surface of adsorbent
granule. Because of the tortuosity of pores and capillaries in the granule, the actual diffusion
path is longer than the apparent path, that is, the radius of the adsorbent granule. Moreover, in
addition to diffusion in the liquid filling the capillaries, a parallel process of surface diffusion
proceeds.
Because including all parameters that affect the internal diffusion is very difficult, in practice,
to describe the rate of the process, a simplified equation, analogous to the Fick equation for
the rate of diffusion in solutions is used:
dN
dc
 De 
A  d
dx
(11.6)
where: N – the amount of diffusing substance [mol] or [g],
A – apparent surface of diffusion, i.e. surface of the sphere of radius r [cm2],
τ – time [s],
 cm 2 
De – effective diffusion coefficient for the adsorption system 
,
 s 
dc
g
 mol  

– concentration gradient in the adsorbent granule  3
or  3
.

dx
 cm  cm   cm  cm 
The effective diffusion coefficient takes into account the effect of both types of internal
diffusion and a reduction of the active surface and elongation of the actual path of diffusion
in the pores. For low adsorbate concentrations in a narrow range of concentrations, the
adsorption equilibrium can be described by a straight-line isotherm and constant values of
Henry’s coefficient and diffusion coefficients can be assumed. The dependence of the
effective diffusion coefficient on diffusion coefficients in a liquid filling the pores DL and
surface diffusion DS is then defined by the formula:
De 
w
  D L  H  DS 
2
(11.7)
where: H – Henry constant,
εw – internal porosity, i.e. the ratio of pore volume to the volume of adsorbent granule,
μ – tortuosity factor, i.e. the ratio of actual pore length to adsorbent granule radius.
In practically applied adsorbents internal porosity is 0.2 to 0.5 and the tortuosity factor is 2 to
4.
One of the methods to determine the effective diffusion coefficient on the basis of
experimental data is the technique developed by Eagle and Scott. Integrating equation (11.6)
they obtained an analytical solution for the linear and homogeneous isotherm of spherical
adsorbent particle. The solution gives the relationship between an instantaneous composition
of the bulk phase and time:
E
X 0  X
6  1
 1  2   2  exp  n 2  k  
X0  Xr
 n1 n

where: E – degree of equilibrium,
Xo – initial concentration of adsorbate in the solution,
Xt – concentration after time τ,

(11.8)
Xr – equilibrium (final) concentration,
k – adsorption rate constant [s-1].
The relationship of the adsorption rate constant with effective diffusion coefficient is
determined by the formula:
k
2
r2
 De
(11.9)
where: r – radius of the adsorbent granule [cm].
In calculations the relation E = f(k·τ) in the form of a diagram or table is used.
11.2. Aim of the exercise
The aim of the exercise is to measure the rate of adsorption of the mixture of hydrocarbons
on silica gel and to determine the effective diffusion coefficient. In the measurement nheptane and toluene are used.
11.3. Apparatus
The experimental set-up is shown in Figure 11.2. It is equipped with a water ultrathermostat
(4), thermostated refractometer (5) and measuring flask with a ground-glass stopper (2)
placed in a holder (3). Water temperature in the thermostat is controlled using thermometer
(1).
Fig. 11.2. Schematic of the experimental set-up: 1 – thermometer, 2 – measuring flask, 3 –
holder to place the flask, 4 – thermostat, 5 – Abbe refractometer
11.4. Method of measurement
11.4.1. Preparation of silica gel
Before measurement silica gel used as an adsorbent should be dried in a dryer at a
temperature of 100°C for 5-6 hours. Due to hygroscopic properties silica gel should be stored
in a closed desiccator – in an open vessel it absorbs water vapor and loses its adsorption
properties.
11.4.2. Measurement of the concentration of hydrocarbon mixture
The concentration of toluene in the mixture with n-heptane is given in volume fractions and
is measured by the refractometric method. To do this one should:
1) using a pipette take a small amount of solution (about 0.2 ml) and transfer it to the prism
of the refractometer;
2) looking into the eyepiece, with the control knob set the prism of the refractometer so that
the line separating light from dark field passes through the center of the cross;
3) read on the scale the value of refractive index (nD) with an accuracy to the fourth decimal
place;
4) from the refractometer scaling curve at the experimental set-up, for an adequate
temperature, read off the concentration of toluene.
The value of toluene concentration can be calculated from the formulas:
X  9,4967  nD  1,3853
for
t  25C
(11.10)
X  0,9206  nD  1,3848
for
t  30C
where: X – volume fraction of toluene in the mixture with n-heptane.
11.4.3. The exercise
1. In the flask prepare volume V (40-60 ml) of the mixture of toluene and n-heptane at the
concentration given by the tutor’s assistant (10-40% vol. of toluene). Close the flask
tightly with the stopper.
2. Turn on the thermostat. Check if the refractometer is connected to water circulation in the
thermostat. Using the contact thermometer set temperature T given by the assistant (25 or
30°C). Wait until reaching the desired temperature (read it on the thermometer placed in
the thermostat).
3. Put the flask with the mixture into the thermostat and after about 5 minutes measure initial
concentration Xo (see section 11.4.2).
4. Weigh mass m of the silica gel (30-70 g) taken from the desiccators and quickly pour into
the flask with the mixture of hydrocarbons. From the moment of pouring start to measure
time τ of the process duration.
5. Periodically, take with a pipette a small amount of the solution from the flask and measure
the instantaneous concentration Xt (before sampling mix the contents of the flask by
shaking). Set the time of measurement at 1, 2, 3, 4, 5, 7, 10 minutes and then every 5 to 10
minutes until reaching identical concentrations in three subsequent trials. The state of
equilibrium determined by the constant concentration of toluene Xr should be attained
after about 40-70 minutes.
6. After reaching the state of equilibrium, turn the thermostat off, take the flask out from the
thermostat, pour the solution into the cylinder and pour the gel into the cylinder with waste
gel.
Write results of the measurements in Tables 11.1 and 11.2.
11.5. Safety note
Be careful when pouring and pipetting liquid, volatile hydrocarbons, and execute quickly and
efficiently all activities connected with it. Store solutions of hydrocarbons in tightly closed
containers. Never pour waste hydrocarbons into the sink, but to specially prepared vessels.
Pay special attention when using glass equipment.
11.6. Description of results
1. Draw a diagram of changes in toluene concentration in time (on the basis of results of the
measurements given in Table 11.1).
2. Calculate for each measurement point the instantaneous adsorption capacity according to
the following equation taking into account change in the volume of solution:
a
V X 0  X

m 1 X
(11.11)
where: a – instantaneous adsorption capacity [cm toluene/g gel],
V – volume of the mixture taken for measurement [cm3],
m – mass of silica gel [g].
Write down results of the calculations of instantaneous adsorption capacity in relevant
columns of Table 11.1. Draw a diagram of adsorption capacity as a function of time. Write
down in Table 11.2 the value of equilibrium adsorption capacity ar (corresponding to
concentration Xr).
3. For each measurement point calculate the degree of conversion E according to the formula:
E
X 0  X
X0  Xr
(1.12)
From Table 11.3 of the function E = f(k·τ) (or a diagram prepared on the basis of data in
this Table) for each value of E read off the value of product k·τ. Write down the results in
relevant columns of Table 11.1. Draw a diagram of product k·τ as a function of time. The
relation should be linear. Read from the diagram the rate constant k (the slope of straight
line in the diagram). In the case of a wide range of experimental points the value of k
should be determined by the least squares method. Write down the value of rate constant k
in Table 11.2.
4. Determine the radius of gel granule r as a mean value from the measurement of 10 to 15
granules; write down the results in Table 11.2. Calculate effective diffusion coefficient D e
from equation (11.10). Write down the value of diffusion coefficient De in Table 11.2.
Table 11.1
Results of measurements and calculations
No.
τ
nD
Xt
a
E
k·τ
[min]
[-]
[% vol.]
[cm3/g]
[-]
[-]
Table 11.2
Results of measurements and calculations
T
Xo
M
V
Xr
ar
r
k
De
[°C]
[% vol.]
[g]
[cm3]
[% vol.]
[cm3/g]
[cm]
[s-1]
[cm2/s]
Table 11.3
1
Values of function E  1  2   2  exp  n 2  k   
 n 1 n
6

k·τ
0.0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
E1
0.001
0.226
0.310
0.421
0.499
0.560
0.610
0.652
k·τ1
0.71
0.81
0.9
1.0
1.2
1.4
1.6
1.8
0.689
0.721
0.749
0.774
0.816
0.850
0.877
0.899
2.01
2.51
3.01
3.51
4.01
4.51
5.01
0.981
0.950
0.970
0.982
0.989
0.993
0.996
E1
k·τ1
E1