2. The Rxn Order and the Rate Law VIII

2. The Rxn Order and the Rate Law VIII

o Reversible rxns 1

- Gas-phase rxn, elementary & reversible 1 or in symbolic

• benzene (B) is depleted by the forward rxn

• the rate of disappearance

• benzene (B) is produced by the reverse rxn

• the rate of production

2 10

, 12 6

6H C H H

2C  B B 

k

k

, D H2

2B B B 

k

k

2 10

12 6

6H C H H

2C ^kB 

2 B B forward

-rB,  k C

6 6 2

10

12H H 2C H

C  ^k-B



2. The Rxn Order and the Rate Law IX

o Reversible rxns 2

- Gas-phase rxn, elementary & reversible 2

• net rate of formation = (forward + reverse) rate

• or the rate of disappearance of benzene

• or with in equilibrium constant

reverse B,

forward B,

net B,

B r -r r

r   

H2

D B - 2

B B

B k C k C C

r   



 



 







 2 D H2

B B 2 -

B B

H D B - 2

B B

B C C

k C k

k C

C k C

k r

const eqlm

Conc where

,

B - H B

2 D B B

B

2

C C

k K k K

C C C

k

r  



 



 





2. The Rxn Order and the Rate Law X

o Rate Law Test

- What is the reaction rate law for the reaction A + ½B → C

if the reaction is elementary?

What is r_B? What is r_C?

Calculate the rates of A, B, and C in a CSTR where the concentrations are C_A = 1.5 mol/dm³, C_B = 9

mol/dm³ and k_A = 2 (dm³/mol)^(1/2)(1/s).

1/2 B A A

A k C C

r 



1/2 B A A A

B 2

1 2

1 r k C C

r   

1/2 B A A A

C r k C C

r   

9 mol )

9 )(

5 . 1 )(

2

( ^1/2

1/2  



 r k C C

3. Rate Constant, k (p 91)

o Specific reaction rate

- k is the specific reaction rate (constant) and is given by the Arrhenius Equation:

Where: E = activation energy (cal/mol) R = gas constant (cal/mol*K) T = temperature (K)

A = frequency factor (units of A, and k, depend on overall reaction order)

RT

Ae E

k  ^{ /}

1013

0 k

0 T

A k

T











 A

3. Rate Constant II

o Activation energy 1

- The height of the energy barrier separating two minima of potential energy of the reactants and products of a reaction)

) E

E ( E

E_B  ^_fA-B-C  ^_fA  ^_fB-C

3. Rate Constant III

o Activation energy 2 - Collision theory

• T ↑, E_k of reactant ↑ , molecular collision ↑

⇒ internal energy (stretching, bending) ↑

⇒ causing them to reach an activated state

⇒ bond breaking & rxn

3. Rate Constant IV

o Activation energy 3

- Rearrangement k  Ae^E/RT



 



 

 R T

A E

k_A 1

ln ln



 



 

 ^{R T T}

E

e T k T

k

1 1 0

) 0

( )

(

3. Rate Constant V

o Activation energy 4 (p 132 p 3-4)

- The following sound of chirping crickets was

recorded in the woods during the summer evening T(^◦C) 14.2 20.3 27.0

Chirps/min 80 126 200

3. Rate Constant VI

o Activation energy 5

- Consider the following elementary reactions

- The higher activation energy?

- The same activation energy?

- Virtually temperature insensitive?

- Which reaction will dominate (i.e. take place the U

A 4)

Y A

3)

D A

2)

B A

1)

4 k

3 k k k

2 1

















3. Rate Constant VII

o Examples of Rate Laws - First order reactions

3. Rate Constant VIII

o Examples of Rate Laws - Second order reactions

This reaction is first order in ONCB, first order in ammonia and overall second order

4. Reactor Sizing & Design

Reactor Design Equations

Batch

CSTR

PFR

PBR

V dt r

N_A0 dX   _A ^



A

A r V

N dX

t 0 0

A A

r X V F

 

A

A r

dV

F ₀ dX  

^ 

_

A

A

r

F dX V

A

r

dW

F

dX  

^{W  F}



^dX _r

5. Batch Systems (p.100)

o Limiting reactant A - Basis of calculation

- Define

D C

B

A a

d a

c a

b  



1





 a

b a

c a



X N

N

N_T  _T0 δ _A0

5. Batch Systems II

Species Initial (mol)

Change (mol)

Remaining (mol) A

B C D Inert Totals

NA0

NB0

NC0

ND0

NI0

NT 0

) (N_A0X



) (N_A0X a

 b

) (N_A0X a

c

) (N_A0X a

d

X N

N

N_A  _A0  _A0 X a N

N b

N_B  _B0  _A0 X a N

N c

N_C  _C0  _A0 X a N

N d

N_D  _D0  _A0

I0

I N

N 

X a N

b a

c a

N d

N_{T T 0} 1 _A0



 



   











5. Batch Systems III

o Eqns for batch conc’n

- Concentration of batch system (= number/volume)

- Define V C_A  N^A

V

X N

V

C N_{A A0}(1 )

A

 



V

X N

a b N

V

C_B N^{B B0} ( / ) ^A0





V

X N

a c N

V

C_C N^{C C0} ( / ) ^A0





X N

a d N

C N^{D D0} ( / ) ^A0





A0 0 A0

0 A0

0

y y C

C N

N_{i i i}

i   



V

X a b

C N_A0[ _B ( / ) ]

B



 

V

X a c

C N_A0[ _C ( / ) ]

C



 

X a d

C N_A0[_D ( / ) ]



5. Batch Systems IV

o Constant Volume Batch

- if the reaction occurs in the liquid phase or

if a gas phase reaction occurs in a rigid (e.g., steel) batch reactor

Then V = V₀



 



 

 

  X

a C b

V

X a b

C_B N^A0[ ^B ( / ) ] _{A0 B}

) 1

) ( 1

(

A0 A0

A C X

V

X

C N   





 



 

 

  X

a C c

V

X a c

C_C N^A0[ ^C ( / ) ] _{A0 C}



 



 

 

  X

a C d

V

X a d

C_D N^A0[ ^D ( / ) ] _{A0 D}