MTD 07_Ads25(1)

7/24/2019 MTD 07_Ads25(1)

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Molecular Thermodynamics (CH3141)

N.A.M. (Klaas) Besseling

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Adsorption; the Langmuir Model

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Ad sorption;

a surface/interface attracts molecules

from an adjoining gas phase or solution

This is a very important phenomenon e.g.

- heterogeneous catalysis

- purification (removing a certain component)

- separation (e.g. chromatography)

-

modifying surfaces- detergency

- emulsification

- …

Adsorption from a liquid solution is similar to what we describe

here for adsorption from a gas phase

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Adsorption;the Langmuir model

Irving Langmuir

(1881-1957)

relatively simple model:

• molecules bind to ‘sites’

• one molecule can bind to one site

•

molecules do not interact

hence sites are independent: ‘divide and rule’

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sites on acrystal surface

solid surface in x- y plane; z is normal to surface

variation of

potential

energy

x

y

z

z ! x, y!

It is reasonable to assume that an adsorbed atom is caught in a

3D harmonic potential well.

We assume that all sites are the same (same potential well)

!m

z = z m

z

m

top view side view

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- equilibrium between gas phase and adsorbed layer .

- use a common zero of energy!

- choose the potential energy far from the surface as 0

- !m is the potential energy at the bottom of the

potential-energy well associated with the adsorption sites.

! = ! m + !

l x

+ ! l y

+ ! l z

l x, l y and l z are vibrational quantum numbers

The energy of an adsorbed atom is:

Vibrations for the x, the y, and the z direction are independent.

For the moment we assume molecules are mono-atomic

(or that intramolecular vibrations, rotations not influenced by adsorpti

Adsorption is (often) an equilibrium ‘reaction’

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partition function for single atom adsorbed at a specific site:

qad = exp !

! m + !

l x

+ ! l y

+ ! l z

kT

"

# $$

%

& ' ' l

z

(l y

(l x

(

= exp ! !

m

kT

"

# $%

& ' exp !

! l x

kT

"

# $

%

& ' exp !

! l y

kT

"

# $$

%

& ' '

exp ! !

l z

kT

"

# $

%

& '

l z

(l y

(l x

(

qad = exp ! ! m

kT

" # $

% & ' q xq yq z

or

qvib, x qvib, zqvib, y

Note the factor where !m is the potential

energy at the bottom of the well

exp !" m

kT ( )

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(recall that for the vibrational ground state ,

vibrational temperature was defined as )

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low-T :

qvib, x ! exp "! x,0

kT

# $ %

& ' ( = exp "

h" x

2kT

# $ %

& ' ( = exp "

)vib, x

2T

# $ %

& ' (

qad = exp ! ! m

kT

" # $

% & ' exp !

! x,0

kT

" # $

% & ' exp !

! y,0

kT

" # $

% & ' exp !

! z,0

kT

" # $

% & '

= exp !! m + ! x,0 + ! y,0 + ! z,0

kT " # $

% & '

= exp !! ad

kT

" # $

% & ' defining ! ad ( ! m + ! x,0 + ! y,0 + ! z,0

! 0 =

1

2h"

!vib

= "! k = h" k

only vibrational ground state relevantkT << !! T <<"

vib

! !"

= effective binding energy = ‘adsorption energy’; pos or neg?

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M surface sites

N adsorbed atoms

If the number of adsorbed atoms N is the same

as the number of sites M , then

the total partition function of the adsorbed layer would be:

qad,1qad,2 . . . qad, N =

qad

N

if N < M there are many different configurations

(ways to arrange the N atoms on the M sites)

Call this number of different ways !conf

hence Qad

=!conf

qad

N

However,

with

Partition function for

entire adsorbed layer

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The number of ways to arrange N atoms at M sites

can be calculated as follows.

Imagine that we place the atoms at the lattice of sites one by one.

( N ! M )

for the 1st atom there are M possibilities

for the 2nd atom there are possibilities

for the N th atom there are possibilities

M ! 1

M ! ( N ! 1) = M ! N +1

M M ! 1( ) M ! 2( ) . . . M ! N +1( ) = M !

M ! N ( )!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(check this for yourself)

The number of ways to place N distinguishable atoms on M sites

is hence

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the result is !conf = M !

N ! M " N ( )!

(this is a binomial coeficient ) M

N

! "

# $

The number of ways to arrange N distinguishable

atoms on N sites is N !

The number of different ways to arrange N distinguishable atoms is

a factor of N ! larger than

the number of different ways to arrange N indistinguishable atoms

When the adsorbed atoms are all of the same kind we have to divide

by N !

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nr. of ways to arrange N indistinguishable particles on M sites,and leave M - N sites vacant.

nr. of ways to arrange

M distinguishable particles on M sites

nr. of ways to arrange

N distinguishable particles

on N sites

nr. of ways to arrange N indistinguishable particles on M sites,

and M - N other indistinguishable particles on the remaining sites.

!conf = M !

N ! M " N ( )! nr. of ways to arrange

N distinguishable particles on

M sites

is also equal to:

if then N =

M !

conf =

M !

M !=1

is the number of ways to arrange M indistinguishable particles on

M sites

M !

N ! M ! N ( )!

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The Helmholtz energy of the adsorbed layer is

Aad

= !kT lnQad

= !kT ln "conf qad

N ( )

= !kT ln"conf ! kTN lnqad

In the low-T limit !kTN lnqad

= N ! ad

(check this)

Elaborate the term and try to find an compact expression

in terms of (hint: use the Stirling formula).

Does this term contribute to the total mean energy ? to the entropy?

!kT ln"conf

! = N M

E

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! ln"conf = ! ln M !

N ! M ! N ( )!= ! ln M !+ ln N !+ ln M ! N ( )!

= ! M ln M + M + N ln N ! N + M ! N ( )ln M ! N ( )! M ! N ( )

= ! M ln M + N ln N + M ! N ( )ln M ! N ( )

= ! N ln M ! M ! N ( )ln M + N ln N + M ! N ( )ln M ! N ( )

= N ln

N

M + M ! N ( )ln

M ! N

M

= M ! ln! + 1"! ( )ln 1"! ( )( )

Elaborate the term and try to find an expression

in terms of (hint: use the Sterling formula).

Does this term contribute to the total energy ? to the entropy?

Did you expect this beforehand?

!kT ln"conf

! = N M

E

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Aad = kTM ! ln! + 1!! ( ) ln 1!! ( )( )! kTN lnqad

= kTM ! ln! + 1!! ( ) ln 1!! ( )( )+ N ! ad

We obtain for the Helmholtz energy:

!kT ln"conf = kTM # ln# + 1!# ( )ln 1!# ( )( )

The contribution to is

This contributes to the entropy (not to the energy)

S conf = k ln!conf = "kM # ln# + 1"# ( )ln 1"# ( )( )

low T limit

Aad

= !kT lnQad

= !kT ln "conf qad

N ( )

(can be checked using Gibbs-Helmholtz relation)

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At equilibrium the chemical potential in the gas phase is

equal to that of the of the adsorbed layer.

Hence, need to find expression for µad

A relation between number density (or the pressure

) in the gas phase and the “surface density”

can be found from condition:

p = ! kT

µ gas( ! ) = µ

ad (" )

! = N gas

V

! = N ad M

µ ad=

dAad

dN ad

! " #

$ % & M ,T

= 'kT d lnQ

ad

dN ad

! " #

$ % & M ,T

(we already have expression for µgas )

Agas Aad equilibrium

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lnQad= M ln M ! N ln N ! M ! N ( ) ln M ! N ( )+ N lnqad

!!"!#$

!"#$

"

# $%

& ' # %$

= (!"" + !" # ("( )+ !"%#$

= (!" "

# ("+ !"%

#$

= (!" !

1(! + !"%

#$

µ ad= !kT

d lnQad

dN ad

" # $

% & ' M ,T

for differentiation of

with respect to N it is convenient to write

A( N , M ,T )

N

M =! ,

M ! N

M =1!!

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µ ad= kT ln

!

1!! ! lnq

ad" # $

% & '

µ !"#

= !" !" ! !#( ) (as derived earlier on)

µ !"= µ

#!$! %&

!

1!!

1

!"#

!

" #$

% & = %& ! '3( )

!

1!! = p

"3qad

kT

!

1!! = " "3

qad

!

1!! = " K "

!

1!! = pK

p the Langmuir

adsorption equation

K p = K ! kT Note that

adsorption equilibrium:

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we did not bother about intramolecular contributions

check that the result is the same for adsorption of molecules,

when the intramolecular degrees of freedom are not influenced

by attachment to the surface

µ !"

= µ #!$

! %& !

1"! " %&!

!"" %&!

'&( = %& ! #

3( )" %&!'&(

both and get same additional term µ ad

µ gas

what would change if intramolecular degrees of freedom are

influenced by attachment to the surface?

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!

1!! = " K "

! = " K "

1+ " K "

if then ! K ! <<1 ! = K " "

if then ! K ! ! " ! "1

Henri equation

surface gets full

or

Langmuir

Henri

K ! ! !

!"

!"

in e.g.

mole/m2

or in

mole/gr

or in

gr/gr

! or p!

if then ! K ! =1 ! = 1

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Langmuir equation not only applicable for gas adsorption,

also for adsorption from ideal-dilute solutions.

Expressions for K " then more complicated (often unknown)

Handled as an adjustable parameter in experimental studies

! = " K "

1+ " K "

! ! = !

!"#$ =

%"#$

"&

where qad = partition function of an adsorbed atom

q g = partition function of an atom in the gas

more general (molecules, even in liquid solutions) ! ! =

"# $%

# &

where q AD = partition function of an adsorbed molecule

q f = partition function of molecule in the fluid

qg!V and q

f !V ( )

for adsorption from monoatomic ideal gas:

?does K depend on V ?

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Surface pressure according to the Langmuir model

Analogous to the pressure in a 3D system,

the surface pressure is given by

pad

= ! " A

ad

" O

#

$ % &

' ( N ,T

where is the surface area

with o as the area per site

O = Mo

pad

= ! ! A

ad

! Mo

" # $

% & ' N ,T

= ! 1

o

! Aad

! M

" # $

% & ' N ,T

=

kT

o

! lnQad

! M

" # $

% & ' N ,T

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lnQad= M ln M ! N ln N ! M ! N ( ) ln M ! N ( )+ N lnqad

We knew already

!

! M

"

# $

%

& ' N ,T

ln M +1 ! ln M ! N ( )!1

! !"!

! !

!

" #$

% & " ##

= !" !

! '"= !"

$

%$'! &= '!"%$'! &

0 0

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!!"=

"#

$

! #$%!"

! &

! " #

$ % & ' %#

if # << 1 then

pad= !

kT

oln 1!" ( )

pad

= !kT

oln 1!" ( ) # kT

"

o+ ...= kT $ + ...

~ ideal gas law

if then! "1 pad!"

ln 1+ x( ) = x

!

1

2 x

2 + 1

3 x

3...!

!=

"

#!=

"

$! "

which is more realistic than the ideal-gas result!

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The surface pressure of an adsorbed layer yields a reduction of

the surface tension (interfacial tension) :

! (" ) = ! 0 ! pad(" )

where $ is the surface tension,

and $0 the surface tension of the “clean” surface

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An important relation in interfacial thermodynamics is

the Gibbs adsorption law.

For a one-component system it can be written as:

!"

! µ

# $ %

& ' (

T

= )*

Check that our expressions for and are

thermodynamically consistent according the Gibbs adsorption law.

pad(! ) µ (! )

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Check that our expressions for and are

thermodynamically consistent according the Gibbs adsorption law

pad(! ) µ (! )

! = !

0 " pad

= ! 0+kT

oln 1"# ( ) = !

0 "

kT

oln $ +1( )

µ = kT ln

!

1!! ! lnqad"

# $ %

& '

µ ! µ (1 2)

kT = ln

"

1!" µ

(1 2)! "kT lnq

ad

exp µ ! µ

(1 2)

kT

" # $

% & ' ( ! =

"

1!" ) 1!" =

1

! !1) " =

!

1+ !

where

first write ! as a function of µ

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! = exp µ ! µ

(1 2)

kT

"

# $%

& ' with

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! = ! 0 !

kT

oln ! +1( )

!

! µ

" # $

% & ' T

!

kT

o

1

! +1!

1

kT

!"

! µ

# $ %

& ' ( T

= )1

o

*

* +1= )

+

o= ),

Q.E.D.

(use the “chain rule”)

MTD 07_Ads25(1)

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