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

−=

=