Catalysis & Adsorption Isotherms | Physical Chemistry PDF Download

CATALYSIS

A catalyst is a substance that participates in chemical reaction by increasing the rate of reaction, yet the catalyst itself remains intact after the reaction is complete.
The mechanism describing a catalytic process is as follows:

Where S represents the reactant; C is catalyst and P is the product. The reactant or substrate-catalyst complex is represented by SC and is an intermediate.
The rate expression for product formation is

              …(1)

Because SC is an intermediate than apply S.S.A. on the formation of SC.

                               …(2)
= composite constant 

then                              …(3)

The relationship between initial concentration and the concentration of all species present after the reaction is:
          [S]₀ = [S] + [SC] + [P]
          [C]₀ = [C] + [SC]
then   [S] = [S]₀ – [SC] – [P]               …(4)
&        [C] = [C]₀ – [SC]                             …(5)
Substituting these values in equation (2), we get

K_m[SC] = ([S]₀ – [SC] – [P]) ([C]₀ – [SC])

then rate of the reaction becomes

Case I.             

[C]₀ <<  [S]₀

i.e. much more substrate is present in comparison to catalyst. Then

if                           k_m < [S]₀

Then

i.e. zero order reaction with respect to substrate.

&  

 when concentration of substrate [S]₀ >> km then reaction rate

R₀ = k₂[C]₀ = R_max

i.e. the rate of reaction will reach a limiting value where the rate becomes zero order in substrate concentration.
 

Case II.

[C]₀ >>  [S]₀

i.e. the reaction rate is first order in [S]₀, but can be first or zero order in [C]₀ depending on the size of [C]₀ relative to km.

Homogeneous and Heterogeneous Catalysis.

A homogeneous catalyst is a catalyst that exist in the same phase as the species involved in the reaction, and heterogeneous catalysts exist in a different phase. Enzymes surve as an example of a homogeneous catalyst, they exist in solution and catalyze reactions that occur in solution.
In heterogeneous catalysis reaction, an important step in reactions involving solid catalysis is the absorption of one or more of the reactants to the solid surface. The particles absorb to the surface without changing their internal bonding. An equilibrium exists between the free and surfaceabsorbed species or adsorbate and surface adsorption and deadsorption can be obtained.
A critical parameter in evaluating surface adsorption is the fractional coverage, θ, defined as

The variation of θ with pressure at fixed temperature is called adsorption isotherm.

(a)  The Langmuir Isotherm : The simplest kinetic model describing the adsorption process is known as the Langmuir model, where adsorption is described by the following mechanism

R is reagent, M is an unoccupied absorption site of catalyst and RM is an occupied adsorption site. k_a and k_d is the rate constant for adsorption and deadsorption.
Three approximations are employed in the Langmuir model:

(1)  Adsorption is complete once monolayer coverage has been reached.

(2)  All adsorption site are equivalent and the surface is uniform

(3)  Adsorption and deadsorption are uncooperative processes. The occupancy state of the adsorption site will not affect the probability of adsorption or deadsorption for adjacent site.

The rate of change in θ will depends on the rate constant for adsorption k_a, reagent pressure P and the number of vacant site which is equal to the total number of adsorption sites, N, times the fraction of sites that are open (1 – θ)

The change in θ due to deadsorption is 

At equilibrium, the change in θ with time is zero i.e

. (k_a PN + k_dN) θ = k_aPN

where k is the equilibrium constant defined as  

This equation is the equation for the Langmuir isotherm.
In many instances adsorption is accompanied by dissociation of the adsorbate, a process that is described by the following mechanism:

&  

The condition for no net change leads to the isotherm.

i.e. the surface coverage now depends more weakly on pressure than for non-dissociative adsorption.

(b)  The BET isotherm.
If initial adsorbed layer can be act as a substance for further  adsorption, then, instead of the isotherm leveling off the some saturated value at high pressure, it can be expected to rise 

 indefinitely. The most widely used isotherm dealing with multiplayer adsorption was derived by Brunauer, Emmett and Teller, and is called the BET isotherm.

P° = vapour pressure above a layer of absorbate
V_mono = volume of monolayer coverage.
C = constant The Langmuir-Hinshelwood mechanism for adsorption and catalysis.

(1)  Unimolecular surface Reaction.
A is reactant and S is the vacant site on surface.
If r is the rate of the reaction, then according to the Langnuir-H. hypothesis,

 r ∝ θ

r = k₂θ

Apply S.S.A. for the formation of [AS].

If Cs is the total concentration of act ive site on surface, then the concentration [S] of the vacant sites on the surface is equal to the product of C_s and (1 – θ). Thus

[S] = C_s (1 - θ)

Also, the concentration of AS on the surface is,

 [AS] = C_sθ

 then   

or     

or   

or     

                   …(1)
thus, 

                      ...(2)

The concentration is expressed in terms of partial pressure
Then

or                      ...(3)

Two limiting cases 

Case I.

∴         r = k₁P_A it is first order w.r.t. A

Case II. k₂ << k₁P_A + k_–1

 

∵ 

Two situations arise depending upon the pressure:

(1)  At low pressure θ → 0 and keq P_A << 1 so that

It is first order w.r.t. P_A or [A].

(a) At high pressure; θ → 1 and k_eqP_A >> 1 so that 

 r = k₂
it is zero order with respect to P_A or [A].

(2)  Bimolecular surface reaction

then it follow the rate law