Week 2. Chemical Kinetics Analysis of Rate Equation

8/13/2019 Week 2. Chemical Kinetics Analysis of Rate Equation

1/31

Reaction Engineering

Chemical Kinetics /

Analysis of rate equations

1


2/31

Chemical Kinetics - outcomes

Understand the significance of kinetics

Aware of concentration dependencies

Understand the difference elementary / non-elementary reactions

Calculate the order of reaction

Aware of temperature dependencies Arrhenius, activation energy

2


3/31

The Rate Equation / Stoichiometry

aA bB cC dD

dtdN

Vr AA 1 rate of disappearance of A

dt

dN

Vr CC

1 rate of appearance of C

d

r

c

r

b

r

a

r DCBA

Relationship

between rates of

reaction

3


4/31

Concentration / Temp. dependency terms

.).,( tempconcfr

,...)],()][([ BAA CCfTkr

Reaction rate

(constant)

Concentration

dependent terms

4


5/31

Concentration-dependent term

,...)],([ BAA CCfkr Almost without exception determined by experimental observation!!!

Most common expression:

DBAA CCkCr ....

Order of reaction: AinorderBinorder

orderoverall...n

5


6/31

Elementary / Non-elementary Reactions

Elementary reactions

iff stoichiometric coefficients are the same as theindividual reaction order of each species

Non-elementary reactions

stoichiometry does not match the kinetics

DCBA

BAA CkCr

stoichiometric coeff: 1 dcba

1 a

BAA CkCr 2 2

1

BAA CkCr

1 b

A sequence of elementary reactions 6


7/31

Elementary reactions / Molecularity

= the number of molecules involved in a reaction

refers ONLY to elementary reaction!!!

Elementary reactions

PX

PX2

PYX PX3

PYX 2

PZYX

Rate Law

XkC

2

XkC

YXCkC3

XkC

2

YXCkC

ZYX CCkC

Molecularity Comments

unimolecular

bimolecular

bimolecular

termolecular

termolecular

termolecular

X decomposes

X collides with X

X collides with Y

X+X+X collide

X+Y+Y collide

X+Y+Z collide

tymoleculariorder

7


8/31

Non-elementary reactions

Overall reaction:

)(2)(2)()(4 222 gBrgOHgOgHBr kexp

Sequence of reactionsproposed mechanism:

)()()( 2 gHOOBrgOgHBr

)(2)()( gHOBrgHBrgHOOBr

)()()()( 22 gBrgOHgHBrgHOBr

)()()()( 22 gBrgOHgHBrgHOBr

(1)

(2)

(3)

(4)

k1

k2

k3

k4

elementary

reactions(bimolecular)

(slow)

(fast)

(fast)

(fast)

kexp=func(k1,k2,k3,k4)

kexp=k1 8


9/31

Concentration / Temp. dependency terms

.).,( tempconcfr

,...)],()][([ BAA CCfTkr

Reaction rate

(constant)

Concentration

dependent terms

9


10/31

Rate constant (k)

,...)],([ BAA CCfkr

Dimensions vary with order of reaction (n):

nionconcentrattime 11 )()(

Reaction order

(mol/m3)

CA

(mol/m3)

(mol/m3)

(mol/m3.s)

(mol/m3.s)

(mol/m3.s)

-rA Rate law

(mol/m3)/skrA

AA kCr

2

AA kCr

1/s

(m3/mol)/s

zero

1st

2nd

k

10


11/31


12/31

Temperature dependency

Activation energy

RTEekk /0

Activation energy (E) calculation:

data for same concentration different temperatures

TR

Ekk 1

lnln 0

TR

Ekk

1

3.2loglog 0or

211

2 11ln

TTR

E

k

k

T1,T2

ln, log

211

2 11

3.2

log

TTR

E

k

k

12


13/31

Activation energy calculation

example 1

First order reaction: CBA

Data: k (1/s) 0.00043 0.00103 0.0018 0.00355 0.00717T(K) 313 319 323 328 333

211

2 11

3.2log

TTR

E

k

k

21

12

/1/1

)/log()3.2(

TT

kkRE

00303.0/1005.0 11 Tk

00319.0/10005.0 22 Tk

Hint: for 1)/log(1.0 2112 kkkkmolkJE /12013

0

0.5

1

1.5

2

2.5

3

3.5

4

0.00295 0.003 0.00305 0.0031 0.00315 0.0032 0.00325

1/T Vs log k

Log k

1/T


14/31


example 2

To obtain pasteurised milk:

heating at 63C for 30 min

heating at 74C for 15 s

KTt 336min30 11

212

1

1

2 11lnlnTTR

E

t

t

k

k

Reaction rate is inversely

proportional with reaction time

KTt 347min25.0sec15 22

347

1

336

1

314.825.0

30ln

E

molJE /422000

Find the activation energy of the process

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15/31

Data on the tenebrionid beetle whose body mass is 3.3 g show that it can

push a 35-g ball of dung at 6.5 cm/s at 27 C, 13 cm/s at 37 C, and 18 cm/s

at 40 C. How fast can it push dung at 41.5 C? [B. Heinrich. The Hot-

Blooded Insects (Cambridge, Mass.: 1993).]


example 3

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16/31

The behavior of the beetle can be modeled by the Arrhenius equation.

The beetle's speed (k) increases exponentially with temperature.

Arrhenius equation:

To calculate how fast the beetle can push the ball determine:

the activation energy (E)

the Arrhenius coefficient (k0) of the beetle


example 3

RTEekk /0

16


17/31

k1 = 6.5 T1 = 300K

k2=13 T2 =310 K

k3= 18 T3=313 K

k4 = ? T4 = 314.5 K

ln(k) for T=314.5 is equal to ~2.95

This corresponds to a k4value of 19.1 cm/s.


example 3

Plot ln(k) vs. 1/T

Predict ln(k) for T=314.5 K

Rate (cm/s) Temp (K)

How would you solve the

problem analytically?17


18/31

Analysis of rate equations - outcomes

Be able to use experimental data to get empirical rate equations

Necessity of a rate equation for design purposes

Able to analyse different types of reactions

Expand the analysis to multiple reactions

Chain reactions and their interpretation & analysis

18


19/31

Analysis of rate equations general

issues

dt

dN

Vr ii

1The rate equation:

The form might be suggested by:Theoretical considerations

Empirical curve-fitting procedure

The value of equation constants experimental only!

2 step-procedure:

Find concentration dependency at Temp=const.

Find temperature dependency of the rate constants

19


20/31

dt

dN

Vr ii

1The rate equation:

Equipment used for getting the empirical data

batch reactor container that holds the reactants usually for

homogeneous reactions

flow reactor continuous flow in / flow out - usually for

heterogeneous reactions

Measurements:

concentration of a component

change in physical properties (conductivity, refractive index)

change in total pressure when V=constant.

change in volume when P=constant.


issues (contd.)

20


21/31

dt

dN

Vr ii

1The rate equation:

Procedures for analysing the kinetic data:

integral method:

guess a form of rate equation

integration & mathematical manipulation

predict the plot of C=f (time) as a straight line

plot the data and if good fit accept the equation

differential method:

find (1/V)(dN/dt) from the data

test the fit directly (no integration)


issues (contd.)

21


22/31

0

0

A

AAA

N

NNX

Conversion of A:

Constant volume batch reactor

conversion

= the fraction of A

reacted away

0000

0 1/

/11

A

A

A

A

A

A

A

AAA

C

C

VN

VN

N

N

N

NNX

The volume of reaction mixture = constant

= constant density reaction system dt

dCr AA

22


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24/31

A. Integral method- irreversible, unimolecular, first order rxs. -

sProductA

A

A

A kCdt

dC

r

Test the first-order rate equation form:

using concentration

using conversion

0

1A

AA

CCX

0A

AA

C

dCdX

AAAA

A kCdt

dXC

dt

dCr 0

0A

AA

C

Ck

dt

dX

)1( AA

A Xkdt

dXr

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25/31


tX

A

A dtk

X

dXA

00

1

Separate & integrate:

tC

CA

A dtkC

dCA

A 00

ktC

C

A

A 0

ln

using concentration

using conversion

ktXA )1ln(

AA kC

dt

dC

)1( AA Xkdt

dX

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26/31


Plot the experimental data & the proposed equation line:

-ln(CA

/CA0

)orl

n(1-XA

)

t

If the data fit Accept first-order rate equation

kslope

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27/31


28/31

t

A

X

AA

A dtkCXMX

dXA

00

0 1

Separate & integrate:

A

B

AB

AB

A

A

A

B

MC

C

CC

CC

XM

XM

X

Xlnln

)1(ln

1

1ln

0

0

B1. Integral method- irreversible, bimolecular, second order rxs. -

ktCCktMC ABA )()1( 000

1MCondition:00 BA

CC or

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29/31


30/31


sProductA2

22

0

2

)1( AAAA

A XkCkCdt

dCr

The second-order rate equation form:

ktX

X

CCC A

A

AAA 1

111

00

t

AC

1

kslope

0

1

ACerceptint

t

kCslope A0

A

A

X

X

1

30


31/31


sProductBA 2

)2)(1(20 AAABAAA XMXkCCkCdtdCr

The second-order rate equation form:

kt

X

X

CCC A

A

AAA

2

1

111

00

ktMCXM

XM

CC

CCA

A

A

AB

AB )2()1(

2lnln 0

0

0

2M

2M31

Week 2. Chemical Kinetics Analysis of Rate Equation

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