ME606 Lect 7-9

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Chemical Kinetics

Chemical kinetics gives information about how fast thereaction proceeds. Global reactions, as given below,

are employed to describe the initial and final states.

22222188 4798)76.3(5.12 NOHCONOHC p

In reality, these global reactions take place in several

steps. An elementary reaction is one which can takeplace in a single step.

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Molecularity of a Reaction

Molecularity of a reaction is the number of moleculesor atoms taking part in each act leading to chemical

reaction. It is defined for an elementary reaction only.

N2O5 N2O4 + O2, a unimolecular reaction2HIH2 + I2, is a bimolecular reaction

CO2 + H2 CO + H2O, is a bimolecular reaction

2NO + O2

2NO2, is a termolecular reactionMost reactions are bimolecular, since there is higher

probability for two molecules to collide.

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Order of a Reaction

Order of a reaction is the power index of concentrationthat determines the reaction rate.

Molecularity gives insight into the mechanism of a

reaction.For the uni-molecular reaction N2O5 N2O4 + O2, the

rate is given by d[N2O5]/dt = -k1 [N2O5], indicating that

the reaction is first order.In 2NO22NO + O2, the rate is given by d[NO2]/dt = -

k2[NO2]2, indicating that the reaction is second order.

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Order of a Reaction

The reaction between methylamine and ethyl bromine,

(C2H5)3N + C2H5Br (C2H5)4NBr is second order; its

rate is d[C2H5Br]/dt = -k2 [C2H5Br] [(C2H5)3N].

The reaction 2NO + O2 2NO2, is third order as the

reaction rate depends on concentrations of two NO

molecules and one oxygen molecule.

The order for global reaction steps may be fractional;

for example, d[CH3CHO]/dt = -k [CH3CHO]1.5.

For elementary reactions order and molecularity are

usually the same.

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Order of a Reaction

There are also some zero order reactions. The ratesof these reactions do not depend upon the quantity of

the unconsumed reactants.

For hydro-carbons burning in oxygen, the overall order

of the reaction is known to lie between 1.7 to 2.2; the

order with respect to oxygen is near unity.

The order of a reaction being equal to 3 or more is

very rare, as the probability of 3 molecules to collide at

the same location is very less.

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Collision theory of reactions

A

BC

VA

(dA+ dB)

A + B C

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)(BAABddd !

BAABAB

AB

ABCCTnnV

dZ E

T

!

4

2

Consider a cylinder ofdiameterdAB around an A particle

A

AB Vd

4

2TThe volume covered by the A particle in unit time

BA

AB nd

4

2T

!No. of collisions for one A particle with B per sec.

A

A

m

TV

O3!Velocity of an A particle

No. of collisions between A andB per unit vol. per sec.

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? A ? A? ABAA CCk

dtCd !

All the ZAB collisions are not fruitful. Only those whichcan provide activation energy for creating active radicals

will be fruitful.

RT

Ea

AB expNumber of fruitful collisions =

Finally the rate of a reaction A + B C is given as

The rate constant k E

R

T

EaexpT1/2

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Law ofMass Action: The rate of a reaction isproportional to the product of reactant

concentrations.

The proportionality constant (k) is known asthe Rate constant. It is actually a strong

function of the temperature T.

BABA

aACCkCC

RT

ETa

dt

dC!

! exp

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Rate Constant From experimental data, k is usually evaluated in a

form k = a Tm exp{-Ea/RT}

Here a is called the pre-exponential factor, m is the

temperature power index and Ea is the activation

energy

Ea

Qcomb

E

t

H2 H + H

O2 O + O

Such active species are

required for combustionreactions to proceed

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Rate Constant

In Arrhenius type reactions, k = a exp{-Ea/ RT}. Theterm Tm is usually unimportant and its effect can be

absorbed in the stronger exponential term.

Only in low temperature pre-combustion reactions,

the Tm term may have to be considered.

ln k

1/T

Intercept = ln(a)

Slope = -Ea/R

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Reaction Rate

DCBADCBA

RRRR p

Consider the reaction

? A ? A BABA CCk

RR

[ !

y

? A ? A BABA

D

D

C

C

B

B

A

A CCkRR

R

[

R

[

R

[

R

[

[ !!!

!

!

yyyy

y

For the reactants, the reaction rate is negative and

for the products it is positive

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Concentration variation with time

For a first order reaction A B, the rate of

consumption of A

? A ? AAkdtAd ! ? A

? Adtk

A

Ad !

? A ? A }exp{0 ktAA !

kkt

693.02ln2/1 !!

t

A0

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For a second order reaction A + B C, let

there be x moles of C formed. Also, let a

and b be the initial number of moles of A

and B present. Then we can write

dtkxbxa

dx!

))((

? A? A? A )()( xbxakBAk

dt

dx

dt

Cd!!!

dtkdxxb

Cdx

xa

C!

)()(21

Where C1 = -C2 = 1/(b-a)

321 )ln()ln( CktxbCxaC !

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For a third order reaction A + B + C D, let

there be x moles of D formed. Also, let

a,b and c be the initial number of moles

of A,B and C present. Then we can write

dtkxcxbxa

dx ! ))()((

? A? A? A? A ))(()( xcxbxakCBAk

dt

dx

dt

Dd!!!

The above equation can be solved using the

method of partial fractions

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Multi-step Reactions

OHHOH

OHHO

OOO

HHH

k

k

kk

kk

bf

bf

2

,

2

,2

4

3

22

11

p

p

pn

pn

? A? A ? A ? A? A ? A? AHOHHOHbHf

H CCkCCkCkCkdt

Cd43

2

11 22 2 !

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For multi-step reactions, a particular species may be

consumed or produced from more than one reaction

step and so it is necessary to find the net rate of

production of each species. For the set of reaction steps shown for the hydrogen-

oxygen system, six species (H2, O2, H, O, OH and

H2O) are present and hence a set of six coupled

ordinary differential equations can be formulated. The coupled set of ODEs can be solved numerically

by time- marching methods like the Runge- Kutta

schemes.

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Quasi- steady state assumption

ReactantsProducts

Intermediates

C

t

Many intermediate species

have a short life time and

they may reach a quasi-

steady state. That is, thenet reaction rate is equal

to zero for these species.

With quasi- steady assumption, the temporal variation of an

intermediate species can be determined in terms of the other

species concentrations, by taking the net rate as zero.

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A multi- step reaction scheme can be categorized

as chain- initiation, chain- branching or chain-

propagation and chain- termination steps.

Steps like H2 2H are chain initiation steps.

Steps like H + O OH are chain- branching.

Steps like OH + H are chain- terminating.

If the rate at which active species are formed isfaster than chain- termination, an explosive reaction

will result.

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Chemical Equilibrium

Consider HHHbf kk pn 11 ,2

Forward reaction rate ? A2,1 HkR ff !

Backward reaction rate ? A2

,1 HkR bb !

At chemical equilibrium, forward rate = backward rate

? A2,1 HkR ff ! = ? A2

,1 HkR bb !

? A

? A2

2

,1

,1

H

H

k

kK

b

f

eq !! Keq = f(T) only

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ME606 Lect 7-9