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BET ADSORPTION ISOTHERM• Brunauer–Emmett–Teller (BET) theory explains the physical adsorption of gas molecules on a solid surface.
• The BET theory applies to systems of multilayer adsorption.
• An important analysis technique for the measurement of the specific surface area of materials.
Stephen Brunauer Paul Emmett Edward Teller
( ) ( )1 1 1m
V cx
V x c x=
− + − ( )
( )00
11
m m
cP P
c V c V PV P P
−= +
−
ASSUMPTIONS OF BET THEORY
• The gas molecules undergo multilayer adsorption on solid surface.
• The principle of Langmuir theory can be applied to each layer.
• A dynamic equilibrium exists between successive layers. The rate of evaporation from a particular
layer is equal to the rate of condensation of the preceding layer.
• The enthalpy of adsorption of the first layer (E1) is a constant, whereas that of any layer above the
first layer is the energy of liquefaction of the adsorbate (EL)
• Condensation forces are the principal forces of attraction.
MODELLING OF ADSORPTION ACCORDING TO BET THEORY
S0, S0, S2, S3 ……Si represent the surface area of the adsorbent covered by 0, 1, 2, 3 …..i layers of adsorbates.
Rate of condensation on the bare surface = 1 0a PS
Rate of evaporation from the first layer = 1
1 1
ERTb S e
−
1 0 1 1
iERTa PS b S e
−
= According to assumption (iii) of BET
2
3
2 1 2 2
3 2 3 3
1
............................
i
ERT
ERT
ERT
i i i i
a PS b S e
a PS b S e
a PS b S e
−
−
−
−
=
=
=
Total surface area of the adsorbent 0
i
i
A S
=
=
Total volume of the adsorbed gas 0
0
i
i
V V iS
=
=
V0 is the volume of the gas adsorbed on 1 cm2 of the adsorbent surface monolayer coverage.
0
0
0
i
i
i
i
V iSV
AS
=
=
=
0
0
0
i
i
i
i
iSV
AVS
=
=
=
0
0
i
i
mi
i
iSV
VS
=
=
=
Vm is the volume of the gas for monolayer coverage.
Approximations by Stephen Brunauer, Paul Emmett and Edward Teller made a couple of approximations
2 3 4.......... i LE E E E E= = = =
32 4
2 3 4
......... i
i
b bb bg
a a a a= = = = =
1 0 1 1
iERTa PS b S e
−
=
2
3
2 1 2 2
3 2 3 3
1
............................
i
ERT
ERT
ERT
i i i i
a PS b S e
a PS b S e
a PS b S e
−
−
−
−
=
=
=
2 3 4.......... i LE E E E E= = = =
32 4
2 3 4
......... i
i
b bb bg
a a a a= = = = =
11
1 0
1
ERT
aS Pe S
b
=
1 0S yS=1
1
1
ERT
ay Pe
b
=
22
2 1
2
ERT
aS Pe S
b
=
2 1
LERT
PS e S
g
=
2 1S x S= LERT
Px e
g
=
2 0S x yS= Putting the value of S1 in the equation for S2,
3 2S x S=
2
3 1 1S x x S x S= = by putting 2 0S x yS=
2
3 0S x yS= by putting 1 0S yS=
3
4 0S x yS=
1
0
i
iS x yS−=
0
i
iS x c S= ( )
1
1
1
1 1
1
L
L
ERT
E ERT
ERT
aPe
b ayc ge
x bPe
g
−
= = =
0
0
i
i
mi
i
iSV
VS
=
=
=
0
i
iS x c S= 0 1
0
0 1
0i i
i i
mi i
i i
iS iSV
VS S S
= =
= =
+
= =
+
0 0
1 1
0 0 0 0
1 1
i i
i i
i im
i i
i x cS cS i xV
VS x cS S cS x
= =
= =
= =
+ +
1
1
1
i
i
im
i
c i xV
Vc x
=
=
=
+
Applying the expressions for summations( )
21 1
i
i
xi x
x
=
=−
( )1 1
i
i
xx
x
=
=−
and
( )
( )
( )( )
( )( )
21 1 1
1 1 11
1m
x cx cxc
x x xV
xV x cx c xc
x
− − −= = =
− + + −+
−
( ) ( )1 1 1m
V cx
V x c x=
− + −
( )
( )1 1
1 m
c xx
x V cV
+ − =−
( )( )11
1 m m
cxx
x V cV cV
−= +
−
( )( )
00
11
m m
cp p
cV cV pp p V
−= +
−
0
px
p=
p is the equilibrium pressure of the gas over the surface
p0 is the saturated vapour pressure of the gas at experimental conditions
Different forms of BET equation
BET equation and Langmuir equation:
( )( )( )
1
1
1 1
1 1 1
n n
n
m
n x nxV cx
V x c x cx
+
+
− + +=
− + − −
For Langmuir adsorption isotherm, n=1, then
( ) ( )
2
2
1 2
1 1 1m
V cx x x
V x c x cx
− +=
− + − −
( )( )
( ) ( )( )
( )( )
( ) ( )
2 2 2
2 2
1 1 1
1 1 1 1 11 1 1m
x x xV cx cx cx
V x x x x cx xc x cx cx x cx
− − −= = =
− − − − + − + − − + − −
( )( )
( )( )
21
1 1 1m
xV cx
V x x cx
−=
− − + ( )1m
V cx
V cx=
+
( )( )11
1 m m
cxx
x V cV cV
−= +
−
Explanations of Different isotherms by BET Theory:
i) When E1 >>> EL , the value of ‘c’ becomes very large.
The BET equation reduces to Langmuir isotherm for
monolayer adsorption. Hence Type I isotherm is
explained.
( )11
1
LE ERT
ac ge
b
− =
When E1 > EL , an intermediate value of ‘c’ is obtained.
This gives Type II adsorption isotherm indicating
multilayer adsorption.
( )11
1
LE ERT
ac ge
b
− =
When E1 < EL , small values of ‘c’ are obtained.
This results in Type III adsorption isotherm, where
multilayer adsorption begins even before the completion of
monolayer adsorption.
( )11
1
LE ERT
ac ge
b
− =
DETERMINATION OF SURFACE AREA
• Surface area of the adsorbent can be obtained by calculating the volume of monolayer
coverage (Vm) using BET and Langmuir isotherms.
Vm - The volume of the gas at STP required to cover the whole surface of adsorbent by monolayer adsorption.
Number of gas molecules in 22.414 dm3 of adsorbate gas at STP = N0 (Avogadro number)
Number of gas molecules in Vm dm3 of adsorbate gas at STP =0
22.414m
NV
‘S m2 - The area of single adsorbate gas molecule occupying the surface of adsorbent
Area occupied by number of particles = 0
22.414m
NV 20
22.414mN V S
m
Surface area of the adsorbent = 20
22.414mN V S
m
Nitrogen gas is generally used for finding out the surface area. The cross sectional area
of nitrogen is usually taken as 1.6210−19 m2
The value of Vm can be obtained from graphical methods.
From Langmuir Isotherm
1
bP
bP =
+
1 11
bP= +
11mV
V bP= +
m
V
V =
1 1 1
m mV bPV V= +
( ) ( )1 1 and V PGraph between
1
V
1
P
L an gm u ir Iso th erm
1Slope
1Inercept
m
m
bV
V
==
=
From BET Isotherm
( )( )
00
11
m m
cp p
cV cV pp p V
−= +
−Plot a graph between
( )0
P
V P P− and
0
P
P
V (P 0 -P )
P 0
B E T IS O T H E R M
P
P
( )1Slope
1Inercept
m
m
c
cV
cV
−=
=