Chapter 3: Rate Laws and Stoichiometry

CHEMICAL REACTION ENGINEERING (SKF3223)

WAN NORHARYATI WAN SALLEH

hayati@petroleum.utm.my

RAFIZIANA MD. KASMANI rafiziana@petroleum.utm.my

TYPES OF REACTION

• Homogenous reaction – one phase reaction

• Heterogeneous reaction – more than one phase

• Irreversible reaction:

• Reversible reaction:

direction) (one

D C B A

backward) or (foward

D C B A

dD cC bB aA

Relative rates of reaction

Da

d C

a

c B

a

b A

For every mole of A that is consumed/reacted, c/a moles of C appear

or

d

r

c

r

b

r

a

r DCBA

d

r

c

r

b

r

a

r DCBA

The rate law:

• Order with respect to A = α

• Order with respect to B = β

• Overall reaction order,

n = α + β

• k = (Concentration)1-n

Time

BAAA CCkr

RATE LAWs

1233

32

A

1-

A

3

AA

.)/(dm{k}:

./(dm){k} : r-

order -

s{k} : r-

order-First

.smol/(dm){k} :kr-

:orderZero

smolCkr

orderThird

smolCk

Second

Ck

AAA

AA

AA

Elementary rate laws

• The stoichiometry coefficients are the same as the

individual reaction order of each species.

• For the reaction:

• The rate law would be:

DCBA 2

2

BAAA CCkr

• The stoichiometric coefficients are not the same as

the individual reaction order of each species.

For the reaction:

The rate law would be:

NonElementary rate laws

DCBA

BAAA CCkr 2

Then the reaction is said to be 2nd order in A, 1st order in B,

and 3rd order overall.

REVERSIBLE REACTIONS

dDcCbBaA

reverseAforwardAnetAA rrrr ,,,

• For general reaction:

• The net rate of formation A

• At equilibrium, rnet=0. Thus,

d

D

c

CA

b

B

a

A CCkCCkA

K = thermodynamic equilibrium constant

d

D

c

CAreverseA

b

B

a

AforwardA CCkrCCkrA ,, and

mequilibriu

A

A Kk

k

d

D

c

CA

b

B

a

AA CCkCCkrA

reverseAforwardAnetAA rrrr ,,,

• Multiplying both sides of equation by (-1), we obtain the rate law for the rate of

disappearance of A:

d

D

c

CA

b

B

a

AA CCkCCkrA

d

D

c

C

A

Ab

B

a

AAA CCk

kCCkr

mequilibriu

A

A Kk

k• Replacing with K

Kk

k

A

A 1

K

CCCCkr

d

D

c

Cb

B

a

AAA

0K

CCCCk

d

D

c

Cb

B

a

AA

• At equilibrium, -rA = 0:

0K

CCCC

d

D

c

Cb

B

a

A

K

CCCC

d

D

c

Cb

B

a

A

b

B

a

A

d

D

c

C

CC

CCK

k • Specific reaction rate or the rate constant

• k is temperature dependent, described by Arrhenius

equation:

RTE

A AeTk /)(

where

A = pre-exponential factor or frequency

factor

E = activation energy, J/mol or cal/mol

R = gas constant= 8.314 J/mol.K

= 1.987 cal/mol.K

T = absolute temperature, K

The activation energy is a measure of the minimum energy a that the

reacting molecules must have in order for the reaction to occur.

From Arrhenius equation:

RTE

A AeTk /)(

TR

EAkA

1lnln

Activation energy is

determined experimentally:

R

ESlope

In kA

1/T

High E

Low E

1a

b

a

c

a

d

Species Initially (mol) Change (mol)

Remaining (mol)

A NA0

B NB0

C NC0

D ND0

I (inerts) NI0 -

TOTALS NT0 ,

)( 0 XN A

)( 0 XNa

bA

)( 0 XNa

cA

)( 0 XNa

dA

XNNN AAA 00

XNa

bNN ABB 00

XNa

cNN ACC 00

XNa

dNN ADD 00

XNNN ATT 00

0II NN

STOICHIOMETRY TABLE - BATCH SYSTEMS

Da

d C

a

c B

a

b A

* -ve = disappearing from the system

1. To express the concentration of each

component in terms of the conversion X:

2. Therefore,

V

XN

V

XNN

V

NC AAAA

A

)1(000

V

XNabN

V

NC ABB

B00 )/(

V

XNacN

V

NC ACC

C00 )/(

V

XNadN

V

NC ADD

D00 )/(

3. Simplify,

V

Xa

b

N

NN

C A

BA

B

)(0

00

V

Xa

c

N

NN

C A

CA

C

)(0

00

V

Xa

d

N

NN

C A

DA

D

)(0

00

0

0

A

BB

N

N

V

NC A

A

0

0

0

0

0

0

A

i

A

i

A

ii

y

y

C

C

N

N

CONSTANT-VOLUME BATCH SYSTEMS

0VV

)1()1()1(

0

0

0000 XCV

XN

V

XN

V

XNN

V

NC A

AAAAAA

Xa

bC

V

XabN

V

XNabN

V

NC BA

BAABBB 0

0

000 ))/(()/(

Xa

cC

V

XacN

V

XNacN

V

NC CA

CAACCC 0

0

000 ))/(()/(

Xa

dC

V

XadN

V

XNadN

V

NC DA

DAADDD 0

0

000 ))/(()/(

EXAMPLE

Xa

bCC BAB 0

BAA CkCr

)1(0 XCC AA

Xa

bCXkCr BAAA 00 )1(

Xa

bXkCr BAA )1(

2

0

1a

b

a

c

a

d

Species

Feed rate to

reactor (mol/time) Change within

reactor (mol/time)

Effluent rate from reactor

(mol/time)

A

B

C

D

I (inerts) -

TOTALS ,

)( 0 XFA

)( 0 XFa

bA

)( 0 XFa

cA

)( 0 XFa

dA

)1(0 XFF AA

Xa

bFF BAB 0

XFFF ATT 00

IAI FF 0

STOICHIOMETRY TABLE - FLOW SYSTEMS

Da

d C

a

c B

a

b A

00 ABB FF

00 ACC FF

00 ADD FF

00 AII FF

Xa

cFF CAC 0

Xa

dFF DAD 0

0AF

0TF

To express the concentration of each component in terms of the

entering molar flow rate, F, the conversion X, and the volumetric

flow rate, v:

liter

moles

timeliters

timemoles

v

FC A

A/

/

0

0

0

0

00

00

0

0

A

B

A

B

A

B

A

BB

y

y

C

C

vC

vC

F

F

v

XF

v

XFF

v

FC AAAA

A

)1(000

v

XFabF

v

FC ABB

B00 )/(

v

XFacF

v

FC ACC

C00 )/(

v

XFadF

v

FC ADD

D00 )/(

)1()1()1(

0

0

00 XCv

XF

v

XF

v

FC A

AAAA

Xa

bCC BAB 0

Xa

cCC CAC 0

Xa

dCC DAD 0

0vv

LIQUID-PHASE REACTIONS

*similar with constant-volume batch systems

RTZNPV T

P=total pressure (atm)

V=volume

Z=compressibility factor

NT=total number of moles

R=gas constant=0.08206

dm3.atm/mol.K

T=temperature (K)

Equation of state:

BATCH REACTORS with VARIABLE VOLUME

00000 RTNZVP T

At t=0:

000

00

T

T

N

N

Z

Z

T

T

P

PVV

At any time t, the volume of gas (Z0=Z):

0

00 )1(

T

TX

P

PVV

XN

N

T

T 10

0

0

01 A

T

A yN

N

a

b

a

c

a

d

FLOW REACTORS with VARIABLE VOLUMETRIC

FLOW RATE

ZRT

P

v

FC T

T

00

0

0

00

RTZ

P

v

FC T

T

0

0

0

0T

T

P

P

F

Fvv

T

T

0

00 )1(

T

T

P

PXvv

At t=0:

At any time t, the volume of gas (Z0=Z):

Total concentration found from

the gas law:

v

FC

j

j

0

00 )1(

T

T

P

PXvv)(0 XvFF jjAj

CONCENTRATION IN TERMS

OF THE CONVERSION X

0

00

0

)1(

)(

T

T

P

PXv

XvFC

jjA

j

T

T

P

P

X

XvCC

jjA

j0

0

0

)1(

)(

, ,

We now have –rA as a function of X and can use the methods in Chapter 2 to design

reactors.

Expressing concentration in terms other than conversion

• Membrane reactors and gas-multiple reaction:

IDCBAT FFFFFF

For j = A,B,C,D,I

T

T

P

P

F

FCC

T

j

Tj0

0

0

Main Reference:

1. Fogler,H.S., “Elements of Chemical Reaction Engineering”, 4th

Edition,Prentice Hall, New Jersey, 2006.

Other References:

1. Davis, M.E and Davis, R.J, “Fundamentals of Chemical Reaction

Engineering”, Mc-Graw-Hill, New York, 2003

2. Schmidt, L.D, “The Engineering of Chemical Reactions”, Oxford,

New York, 1998

3. Levenspiel,O., “Chemical Reaction Engineering”, 3rd Edition,

Wiley,New York, 1998

4. Smith,J., “Chemical Engineering Kinetics”, 3rd Edition, McGraw-

Hill, New York, 1981

REFERENCES