Langmuir Hinshelwood Mechanism​
​Authors: Benjamin J. McCoy, Housam BINOUS and Ahmed Bellagi

The strategy is to solve the rate equations with a mass balance on the total number of adsorption sites,
S
tot
.
One reaction is chosen as the rate limiting step, and the other reactions are at equilibrium.
Reaction A ⇌ B
Rate expressions when adsorption or desorption are rate limiting steps
adsorption is rate limiting step
​​
​A + S ⇌ As (rate limiting step, equilibrium constant KA and rate constants are k1 and k2)
​As ⇌ Bs (equilibrium constant K and rate constants are k5 and k6)
​Bs ⇌ B+S (equilibrium constant 1/KB and rate constants are k3 and k4)
​A ⇌ B (equilibrium constant KAB=K KA/KB)
​
Concentration of reactants are A and B. Concentration of adsorbed reactants are As and Bs.
Concentration of empty adsorption sites is S=Stot-As-Bs where the concentration of all sites (empty and occupied) is Stot.
​
​Rate expression when adsorption of A is rate limiting step
In[]:=
Solve[{r==
k
1
As-
k
2
As,​​
k
3
Bs-
k
4
Bs==0,​​
k
6
Bs-
k
5
As0,​​
S
tot
s+As+Bs},{r,s,As,Bs}]//Simplify
Out[]=
r
(A
k
1
k
4
k
5
-B
k
2
k
3
k
6
)
S
tot
k
4
k
5
+B
k
3
(
k
5
+
k
6
)
,s
k
4
k
5
S
tot
k
4
k
5
+B
k
3
(
k
5
+
k
6
)
,As
B
k
3
k
6
S
tot
k
4
k
5
+B
k
3
(
k
5
+
k
6
)
,Bs
B
k
3
k
5
S
tot
k
4
k
5
+B
k
3
(
k
5
+
k
6
)

In[]:=
(A
k
1
k
4
k
5
-B
k
2
k
3
k
6
)
S
tot
k
4
k
5
+B
k
3
(
k
5
+
k
6
)
/.
k
2

k
1
/
K
A
/.
k
4

k
3
/
K
B
/.
k
5

K
B
k
6
K
AB
/
K
A
//Simplify
Out[]=
-
k
1
(B-A
K
AB
)
S
tot
B
K
A
+
K
AB
(1+B
K
B
)
In[]:=
rate=
k
1
(-B/
K
AB
+A)
S
tot
1+
K
B
B(1+
K
A
/(
K
AB
K
B
))
Out[]=
k
1
A-
B
K
AB
S
tot
1+B1+
K
A
K
AB
K
B
K
B
In[]:=
rate=
k
1
A-
B
K
AB
S
tot
1+B1+
1
K

K
B
Out[]=
k
1
A-
B
K
AB
S
tot
1+B1+
1
K

K
B
desorption is rate limiting step
​​
​A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)
​As ⇌ Bs (equilibrium constant K and rate constants are k5 and k6)
​Bs ⇌ B+S (rate limiting step, equilibrium constant 1/KB and rate constants are k3 and k4)
​A ⇌ B (equilibrium constant KAB=K KA/KB)
​
Concentration of reactants are A and B. Concentration of adsorbed reactants are As and Bs.
Concentration of empty adsorption sites is S=Stot-As-Bs where the concentration of all sites (empty and occupied) is Stot.
​
​Rate expression when desorption of B is rate limiting step
In[]:=
Solve[{0==
k
1
As-
k
2
As,​​-
k
3
Bs+
k
4
Bsr,​​
k
6
Bs-
k
5
As0,​​
S
tot
s+As+Bs},{r,s,As,Bs}]//Simplify
Out[]=
r
(A
k
1
k
4
k
5
-B
k
2
k
3
k
6
)
S
tot
k
2
k
6
+A
k
1
(
k
5
+
k
6
)
,s
k
2
k
6
S
tot
k
2
k
6
+A
k
1
(
k
5
+
k
6
)
,As
A
k
1
k
6
S
tot
k
2
k
6
+A
k
1
(
k
5
+
k
6
)
,Bs
A
k
1
k
5
S
tot
k
2
k
6
+A
k
1
(
k
5
+
k
6
)

In[]:=
(A
k
1
k
4
k
5
-B
k
2
k
3
k
6
)
S
tot
k
2
k
6
+A
k
1
(
k
5
+
k
6
)
/.
k
2

k
1
/
K
A
/.
k
4

k
3
/
K
B
/.
k
5

K
B
k
6
K
AB
/
K
A
//Simplify
Out[]=
-
k
3
(B-A
K
AB
)
S
tot
1+A
K
A
+A
K
AB
K
B
In[]:=
rate=
k
3
K
AB
(-B/
K
AB
+A)
S
tot
1+A
K
A
(1+
K
AB
K
B
/
K
A
)
Out[]=
k
3
A-
B
K
AB
K
AB
S
tot
1+A
K
A
1+
K
AB
K
B
K
A
In[]:=
k
3
A-
B
K
AB
K
AB
S
tot
1+A
K
A
1+
K
AB
K
B
K
A
/.
K
AB
K
B
K
A
K/.
K
AB
k
3

k
4
K
A
K
Out[]=
K
k
4
K
A
A-
B
K
AB
S
tot
1+A(1+K)
K
A
Irreversible Reaction A + B  C
Rate expressions when reactants adsorb on same or different type of adsorption site with and without product adsorption
One site model. Reactants A and B adsorb on same type of site. Product C does not adsorb.
​
​A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)​
​B+S ⇌ Bs (equilibrium constant KB and rate constants are k3 and k4)​
​As+Bs  C+S+S (rate limiting step, rate constant is kp)
​
Concentration of reactants are A and B. Concentration of adsorbed reactants are As and Bs.
Concentration of empty adsorption sites is S=Stot-As-Bs where the concentration of all sites (empty and occupied) is Stot.​
​
​Rate expression when reactants adsorb on same type of site and product does not adsorb
In[]:=
Solve[{
k
1
A(
S
tot
-As-Bs)-
k
2
As==0,​​
k
3
B(
S
tot
-As-Bs)-
k
4
Bs==0,​​dPkpAsBs},{dP,As,Bs}]//Simplify
Out[]=
dP
ABkp
k
1
k
2
k
3
k
4
2
S
tot
2
(A
k
1
k
4
+
k
2
(B
k
3
+
k
4
))
,As
A
k
1
k
4
S
tot
A
k
1
k
4
+
k
2
(B
k
3
+
k
4
)
,Bs
B
k
2
k
3
S
tot
A
k
1
k
4
+
k
2
(B
k
3
+
k
4
)

In[]:=
AB
k
1
k
2
k
3
k
4
kp
2
S
tot
2
(B
k
2
k
3
+(A
k
1
+
k
2
)
k
4
)
/.
k
1
->
k
2
K
A
/.
k
3
->
k
4
K
B
//Simplify
Out[]=
ABkp
K
A
K
B
2
S
tot
2
(1+A
K
A
+B
K
B
)
One site model. Reactants A and B adsorb on same type of site with product C adsorption.
​​
A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)​
B+S ⇌ Bs (equilibrium constant KB and rate constants are k3 and k4)​
As+Bs  Cs+S (rate limiting step, rate constant is kp)​
Cs ⇌ C+S (equilibrium constant KC and rate constants are k5 and k6)
​
Concentration of reactants and product are A, B and C. Concentration of adsorbed reactants and adsorbed product are As, Bs and Cs.
Concentration of empty adsorption sites is S=Stot-As-Bs-Cs where the concentration of all sites (empty and occupied) is Stot.
​​
​Rate expression when reactants and product adsorb on same type of site
In[]:=
Solve[{
k
1
A(
S
tot
-As-Bs-Cs)-
k
2
As==0,​​
k
3
B(
S
tot
-As-Bs-Cs)-
k
4
Bs==0,​​
k
6
C(
S
tot
-As-Bs-Cs)-
k
5
Cs0,​​dPkpAsBs},{dP,As,Bs,Cs}]//Simplify
Out[]=
dP
ABkp
k
1
k
2
k
3
k
4
2
k
5
2
S
tot
2
(A
k
1
k
4
k
5
+
k
2
(B
k
3
k
5
+
k
4
(
k
5
+C
k
6
)))
,As
A
k
1
k
4
k
5
S
tot
A
k
1
k
4
k
5
+
k
2
(B
k
3
k
5
+
k
4
(
k
5
+C
k
6
))
,Bs
B
k
2
k
3
k
5
S
tot
A
k
1
k
4
k
5
+
k
2
(B
k
3
k
5
+
k
4
(
k
5
+C
k
6
))
,Cs
C
k
2
k
4
k
6
S
tot
A
k
1
k
4
k
5
+
k
2
(B
k
3
k
5
+
k
4
(
k
5
+C
k
6
))

In[]:=
AB
k
1
k
2
k
3
k
4
2
k
5
kp
2
S
tot
2
(B
k
2
k
3
k
5
+
k
4
(A
k
1
k
5
+
k
2
(
k
5
+C
k
6
)))
/.
k
1
->
k
2
K
A
/.
k
3
->
k
4
K
B
/.
k
6

k
5
K
C
//Simplify
Out[]=
ABkp
K
A
K
B
2
S
tot
2
(1+A
K
A
+B
K
B
+C
K
C
)
Two different sites S1 and S2 without product adsorption.
​
​A+S1 ⇌ As (equilibrium constant KA and rate constants are k1 and k2)​
​B+S2 ⇌ Bs (equilibrium constant KB and rate constants are k3 and k4)​
​As+Bs  C+S1+S2 (rate limiting step, rate constant is kp)​
​
​Concentration of reactants are A and B. Concentration of adsorbed reactants are As and Bs.
Concentration of empty adsorption sites is S1=S1tot-As and S2=S2tot-Bs where the concentration of all sites (empty and occupied) of type 1 and 2 are S1tot and S2tot.​
​
​Rate expression when reactants adsorb on different sites without product adsorption
In[]:=
Solve[{
k
1
A(
S1
tot
-As)-
k
2
As==0,​​
k
3
B(
S2
tot
-Bs)-
k
4
Bs==0,​​dPkpAsBs},{dP,As,Bs}]//Simplify
Out[]=
dP
ABkp
k
1
k
3
S1
tot
S2
tot
(A
k
1
+
k
2
)(B
k
3
+
k
4
)
,As
A
k
1
S1
tot
A
k
1
+
k
2
,Bs
B
k
3
S2
tot
B
k
3
+
k
4

In[]:=
AB
k
1
k
3
kp
S1
tot
S2
tot
(A
k
1
+
k
2
)(B
k
3
+
k
4
)
/.
k
1
->
k
2
K
A
/.
k
3
->
k
4
K
B
//Simplify
Out[]=
ABkp
K
A
K
B
S1
tot
S2
tot
(1+A
K
A
)(1+B
K
B
)
In[]:=
AB
K
A
K
B
kp
S1
tot
S2
tot
1+A
K
A
+B
K
B
+AB
K
A
K
B
//Factor
Out[]=
ABkp
K
A
K
B
S1
tot
S2
tot
(1+A
K
A
)(1+B
K
B
)
Two different site S1 and S2 with product adsorption on site S1
​
​A+S1 ⇌ As (equilibrium constant KA and rate constants are k1 and k2)​
​B+S2 ⇌ Bs (equilibrium constant KB and rate constants are k3 and k4)​
​As+Bs  Cs+S2 (rate limiting step, rate constant is kp)​
​Cs ⇌ C+S1 (equilibrium constant KC and rate constants are k5 and k6)​
​
​Concentration of reactants and product are A, B and C. Concentration of adsorbed reactants and adsorbed product are As, Bs and Cs.
Concentration of empty adsorption sites are S1=S1tot-As-Cs and S2=S2tot-Bs where the concentration of all sites (empty and occupied) of type 1 and 2 are S1tot and S2tot.​
​
​Rate expression when reactants adsorb on different sites with product adsorption
Reaction A + H2  AH2 (for example: hydrogenation reactions)
Rate expression when H2 follows dissociative adsorption
Dissociative adsorption with product adsorption​
​
​A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)
​H2+2S ⇌ 2Hs (dissociative adsorption) (equilibrium constant KH2 and rate constants are k3 and k4)
​As+2Hs  AH2s+2S (rate limiting step, rate constant is kp)
​AH2s ⇌ AH2+S (equilibrium constant KAH2 and rate constants are k5 and k6)
​
Concentration of reactants and product are A, H2 and AH2. Concentration of adsorbed reactants and adsorbed product are As, Hs and AH2s.
Concentration of empty adsorption sites is S=Stot-As-Hs-AH2s where the concentration of all sites (empty and occupied) is Stot.​
​​
​Rate expression when there is dissociative adsorption of H2 and product adsorption
Dissociative adsorption without product adsorption​
​
​A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)
​H2+2S ⇌ 2Hs (equilibrium constant KH2 and rate constants are k3 and k4)
​As+2Hs  AH2+3S (adsorption of AH2 is neglected because it is weak) (rate limiting step, rate constant is kp)
​
Concentration of reactants are A and H2. Concentration of adsorbed reactants are As and Hs.
Concentration of empty adsorption sites is S=Stot-As-Hs where the concentration of all sites (empty and occupied) is Stot.
​
​Rate expression when there is dissociative adsorption without product adsorption
Reaction A + B ⇌ C or A + B  C
Rate expressions ⇌or reversible and irreversible reactions involving three components
Eley-Rideal Mechanism (reactants A and B are such A is adsorbed and B stays in the gas phase).
​​
​One site model single site reaction
​​
​A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)
​As+B  Cs (rate limiting step, rate constant is kp) (equilibrium constant KAB)
​Cs ⇌ C+S (equilibrium constant 1/KC and rate constants are k5 and k6)
​
Equilibrium constants are not independent: KC C=KA KAB A B
​
Concentration of reactants and product are A, B and C. Concentration of adsorbed reactant and product are As and Cs.
Concentration of empty adsorption sites is S=Stot-As-Cs where the concentration of all sites (empty and occupied) is Stot.
​
​Rate expression ⇌or the Eley-Rideal mechanism
One site model with dual site reaction and inert component : D
A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)​
B+S ⇌ Bs (equilibrium constant KB and rate constants are k3 and k4)​
D+S ⇌ Ds (equilibrium constant KD and rate constants are k7 and k8)​
As+Bs  Cs+S (rate limiting step, rate constant is kp) (equilibrium constant KAB)​
Cs ⇌ C+S (equilibrium constant 1/KC and rate constants are k5 and k6)
​
Equilibrium constants are not independent: KC C=KA KB KAB A B
​
Concentration of reactants, inert and product are A, B, D and C. Concentration of adsorbed reactants, inert and product are As, Bs, Ds and Cs.
Concentration of empty adsorption sites is S=Stot-As-Bs-Cs-Ds where the concentration of all sites (empty and occupied) is Stot.
​
​Rate expression when there is an inert component D competing ⇌or adsorption sites
One type of site only with reversible reaction
​
​A+S ⇌ As (equilibrium constant KA and rate constants are k1 and k2)​
​B+S ⇌ Bs (equilibrium constant KB and rate constants are k3 and k4)​
​As+Bs ⇌ Cs+S (rate limiting step, rate constants are kp and kp') (equilibrium constant KAB=kp/kp’)​
​Cs ⇌ C+S (equilibrium constant 1/KC and rate constants are k5 and k6)
​
Equilibrium constants are not independent: KC C=KA KB KAB A B
A+B⇌C (equilibrium constant Keq is equal to KAB KA KB/KC )
​
Concentration of reactants and product are A, B and C. Concentration of adsorbed reactants and product are As, Bs and Cs.
Concentration of empty adsorption sites is S=Stot-As-Bs-Cs where the concentration of all sites (empty and occupied) is Stot.
​
​Rate expression ⇌or a reversible reaction involving three components