Chem 162 – Spring 2005 Kinetics IV Page 1 of 8

Kinetics IV

Reading: SAB Chapter 24.1 – 24.3 and 24.7 – 24.8 VI. Reactions at Surfaces

Preliminaries: System – gas molecules adsorbed at solid surface Adsorption – accumulation of particles at a solid surface Absorption – incorporation of particles within a bulk substance Adsorbate – particles undergoing adsorption process Adsorbent – solid surface Desorption – removing particles from surface General reaction: There is a dynamic equilibrium between free and adsorbed particles

( ) ( ) adsorption

desorptionAdsorbate g Adsorbent s Covered Surface→+ ←

Class of adsorption: Physisorption – Physical adsorption – van der Waals forces are forces of adsorption.

• Longer range interactions that are generally weak. • Molecules retain their identity, although arrangements might be altered • No appreciable activation energy: ~ 20 /physH kJ mol∆ − • Multi-layer adsorption may occour

Chemisorption – Chemical absorption – covalent bonds are the forces of adsorption.

• Strong forces • Molecules may fragment • Adsorption occurs at ‘active sites’ • Only a monolayer is formed, then surface becomes saturated • ~ 200 /chemH kJ mol∆ −

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Adsorption Isotherm: Extent of surface coverage, which depends on the gas pressure for a fixed T and a fixed system of solid and gas We define the fraction of occupied sites as θ , so the fraction of open sites is ( )1 θ− Our goal is to determine the adsorption isotherm in terms of kinetic parameters We will primarily consider chemisorption processes.

Langmuir Isotherm (ideal chemisorption): Assumptions:

• Uniform surface with a limited number of equivalent sites • Gas molecules form a monolayer only • Gas molecules do not react with one another.

The rate of adsorption is equal to the rate of surface coverage:

( )[ ] 1a adrate k Adtθ θ= = −

This rate law is ‘bimolecular’ in terms of the concentration (pressure) of gas particles and the ‘concentration’ of free sites, with an adsorption rate constant of ak . The corresponding desorption process depends only on the ‘concentration’ of occupied sites (since unoccupied sites will obviously not exhibit desorption):

( )1d d

drate k

dtθ

θ−

= =

Since there is a dynamic equilibrium between the adsorption and desorption processes, these two rates are equal:

( )[ ] 1a dk A kθ θ− =

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We can define an equilibrium constant between adsorption and desorption, then rearrange the above expression to obtain surface coverage as a function of [A]:

( )[ ] 1a

d

kKk A

θθ

= =−

so [ ]1 [ ]

K AK A

θ =+

Consider this graphically:

To analyze adsorption data – a plot of 1/θ vs. 1/[ ]A gives a slope equal to 1/ K and an intercept of unity. Dissociation upon absorption: In some instances, after absorption of a substance, dissociation occurs on the surface. One example of this is when H2(g) adsorbs, then dissociates into hydrogen atoms. If we think about this qualitatively, we expect that the rate of adsorption will change because we now need 2 sites for the molecule to adsorb. The rate of desorption will also be affected by the new stoichiometry of desorbing particles (i.e.: we do not desorb an H2, we desorb two hydrogen atoms to retain the original number of free sites.) The new rates of adsorption and desorption become:

( )22[ ] 1a a

drate k Adtθ θ= = − and ( ) 21

d d

drate k

dtθ

θ−

= =

At equilibrium:

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( )2 22[ ] 1a dk A kθ θ− = so

( ) ( )1/ 2

1/ 22 2[ ] [ ]

1a

d

k A K Ak

θθ

= = −

Finally, this equation can be rewritten as:

1/ 2 1/ 22

1/ 2 1/ 22

[ ]1 [ ]

K AK A

θ =+

Consider this graphically:

To analyze this type of adsorption data – a plot of 1/θ vs. 1/ 21/[ ]A gives a slope equal to

1/ 21/ K with a slope of unity. In general, if you have adsorption followed by dissociation into n particles, the adsorption isotherm becomes:

1/ 1/2

1/ 1/2

[ ]1 [ ]

n n

n n

K AK A

θ =+

Competitive adsorption: Now consider the adsorption isotherm when two distinct species (A and B) are competing for the same sites on the surface. There are two sets of parallel reactions:

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In this case, Aθ is the fraction of sites occupied by A, Bθ is the fraction of sites occupied by B and ( )1 A Bθ θ− − is the fraction of unoccupied sites available for A or B. The rates of adsorption and desorption for each species simply take into account these new fractions. Species A:

The rate of adsorption for A is: ( ), , [ ] 1A a A a A Brate k A θ θ= − − While the rate of desorption is: , ,A d A d Arate k θ= (note that the desorption rate is not different than the non-competitive case)

These rates are equivalent at equilibrium, and , ,/A A a A dK k k= , so:

[ ]1

AA

A B

K A θθ θ

=− −

[1]

Species B:

The rate of adsorption for B is: ( ), , [ ] 1B a B a A Brate k B θ θ= − − While the rate of desorption is: , ,B d B d Brate k θ=

These rates are equivalent at equilibrium, and , ,/B B a B dK k k= , so:

[ ]1

BB

A B

K B θθ θ

=− −

[2]

Between equations [1] and [2], we have two unknowns. Solving this system of equations yields the isotherms for each component:

[ ]1 [ ] [ ]

AA

A B

K AK A K B

θ =+ +

and [ ]1 [ ] [ ]

BB

A B

K BK A K B

θ =+ +

To analyze this type of adsorption data – a double plot of 1/θ vs. 1/[ ]X (where X is an arbitrary component) gives two lines with slopes equal to 1/ XK and a common intercept of unity.

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Reactions at surfaces: The first consideration we need to address surface reactions is the molecularity of the reaction, i.e.: the number of particles (adsorbed or not) that undergo a reaction at a solid surface. Additional vacant sites needed for adsorption of products are not counted. Unimolecular reactions at surfaces: In unimolecular reactions, the reactant is adsorbed to the surface, then undergoes a reaction (typically a dissociation): The key assumption is that the first step (absorption) forms a fast equilibrium, so the total rate of the mechanism is only dependent on the rate of the reaction step. Furthermore, the reaction is an irreversible unimolecular reaction. The rate of the reaction is expected to be first order in [A], but the reaction will only occur when A is adsorbed to the surface, so:

rate kθ= We already know the fraction of sites occupied by A during the (fast, equilibrium) absorption process, so we can express the rate as:

[ ] [ ]1 [ ]

d A kK Aratedt K A

= =+

Note that this result is equivalent to the Mechaelis-Menten mechanism of enzyme catalysis. Consider this result graphically:

At low [A] (when K[A] << 1), the reaction rate appears first order in [A]. At high [A] (when K[A]>>1), the reaction appears zero-order in [A] because the concentration of A is effectively constant at high surface coverage (similar to a pseudo-zero order reaction!)

Unimolecular reactions with competitive absorption:

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If there is an inhibitor present on the surface (competing for sites), the reaction rate is expected to decrease as Aθ decreases. This is actually easy to take into account by changing the θ term in the reaction rate to Aθ :

[ ]1 [ ] [ ]

AA

A I

kK Arate kK A K I

θ= =+ +

where [I] is the concentration of inhibitor. From this expression, qualitatively, we see that for low [I], this expression reduces to the ideal case above, while at high [I] it reduces to:

[ ][ ]

A

I

kK ArateK I

=

which is first order in [A], but the rate decreases with [I], which is to be expected. Bimolecular reactions: We now consider reactions where there are two species that react together. The absorption/reaction processes can occur either independently or competitively. Case 1 – the Langmuir-Hinshelwood mechanism: In this mechanism, there is competitive adsorption, then the reaction rate between A and B is proportional to the probability that A and B are on adjacent sites: The rate of this reaction is A Brate kθ θ= Since we know Aθ and Bθ (from competitive adsorption:

( )2

[ ][ ]1 [ ] [ ]

A B

A B

kK K A BrateK A K B

=+ +

Consider this graphically, when [B] is held constant and [A] is varied:

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When [A] is low, the reaction is first order in [A]. When [A] is large, the reaction is (-1)st order in [A]. At intermediate concentrations, there is a region where the reaction rate is invariant to [A].

Case II – the Langmuir-Rideal mechanism: In this mechanism, only B adsorbs to the surface. The bound B then reacts with unbound A to form product. The reaction rate between A and B is then proportional to fraction of sites occupied by B, and the pressure of unbound A: The rate of this reaction is [ ] Brate k A θ= Since we know Bθ (from ideal chemisorption):

[ ][ ]1 [ ] [ ]

B

A B

kK A BrateK A K B

=+ +

This expression assumes there is competition between A and B for sites, even though bound B reacts with unbound A. If A never binds, the KA[A] term vanishes in the denominator. Consider this graphically:

Again, [A] is varied with constant [B]. At low concentrations of [A], the reaction is first order in [A]. At large concentrations of [A], the surface becomes saturated with [A] and the rate becomes zero order in [A].