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Lecture 4

Chemical Kinetics

http://en.wikipedia.org/wiki/Chemical_kineticshttp://en.wikipedia.org/wiki/Stoichiometric_coefficient

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Why we need Chemical Kinetics

dN

dt= S

k

k

!d" k

dt

d! k

dt= r

k(p,T ,N j ) = rf

k(p,T ,N j ) " rb

k(p,T ,N j )

Enforcing atomic mass conservation yields the rate equations describing the time evolution of thecomposition:

The time rate of change of the progress variable ξk needs to be related to the state of the system(p,T,Nj), according to a constitutive law of the form:

Aim of chemical kinetics is to find explicit forms of the constitutive laws for the forward andbackward(reverse) rates of a reaction

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m A + n B

Kf

Kb! "!!# !!! p C + q D

rf = K f (T )[A]m[B]

n rb = Kb (T )[C]

p[D]

q

from: [Xj ] =pj

!T=p Xj

!T " r = r(T , p,Xj )

Irreversible Processes: Chemical Reactions

The symbolic formula describing a reaction between m moles of reactant A and n moles ofreactanct B, yieldings p moles of product C and q moles of reactant D, is written as:

where:1) Kf, Kb indicate the reaction constants of the forward (left arrow) and backward (reverse,

right arrow) reactions, respectively;2) The stoichiometric coefficients are conventionally assumed: m, n < 0 ; p, q > 0.

The main result of chemical kinetics is the explicit form of the reaction rates:

The exponents n and m are called reaction orders and depend on the reaction mechanism. Thestoichiometric coefficients and reaction orders are very often equal, but only in one step reactions,molecularity (number of molecules or atoms actually colliding), stoichiometry and reaction ordermust be the same.

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Molecularity

• Molecularity in chemistry is the number of colliding molecular entities that areinvolved in a single reaction step

• the molecularity is a theoretical concept and can only be applied to elementaryreactions

• In elementary reactions, the reaction order, the molecularity and the stoichiometriccoefficient are the same, although only numerically, because they are differentconcepts

A reaction involving one molecular entity is called unimolecular. A reaction involving two molecular entities is called bimolecular. A reaction involving three molecular entities is called termolecular.

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Reaction Order

• The Order of reaction with respect to a certain reactant is defined, in chemicalkinetics, as the power to which its concentration term in the rate equation is raised:

• For example, given a chemical reaction A + 2B → C with a rate equation r = k[A]1[B]2,the reaction order with respect to A would be 1 and with respect to B would be 2,the total reaction order would be 2+1=3.

• It is not necessary that the order of a reaction is a whole number - zero and fractionalvalues of order are possible - but they tend to be integers.

• Zero-order reactions are often seen for thermal chemical decompositions where thereaction rate is independent of the concentration of the reactant (changing theconcentration has no effect on the speed of the reaction)

dci

dt! cj

n n reaction order wrt to species j

dci

dt! cj

j

!"

#$%

&'

n

n total reaction order

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Reaction Pseudo-Order

• If the concentration of one of the reactants remains constant (because it is a catalystor it is in great excess with respect to the other reactants) its concentration can beincluded in the rate constant, obtaining a pseudo constant

• if B is the reactant whose concentration is constant then:

r = k[A][B] = k'[A]

• The second order rate equation has been reduced to a pseudo first order rateequation.

• Reaction orders can be determined only by experiment.• Their knowledge allows conclusions about the reaction mechanism.• The reaction order is not necessarily related to the stoichiometry of the reaction,

unless the reaction is elementary.

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Reaction Constants

• Kinetic theory of gases provides the result:

• Empirical formulas (Arrhenius type):

K(T ) = A NA (!A +!B)2 8"KBT

mA+ m

B

mAm

B

exp #Ea

$T

%

&'(

)*

A steric factor

!A,!B diameters of molecules of type A and B

mA,mB mass of molecules of type A and B

Ea activation energy

K (T ) = A T !e"Ea

#T

A steric factor

Ea activation energy

! temperature dependence

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Activation Energy, Transition State & Enthalpy of Combustion

• The activation energy is the threshold energy, or the energy that must be overcomein order for a chemical reaction to occur. Activation energy may otherwise bedenoted as the minimum energy necessary for a specific chemical reaction to occur.

• The transition state along a reaction coordinate is the point of maximum freeenergy, where bond-making and bond-breaking are balanced.

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!1

m

d[A]

dt= !

1

n

d[B]

dt= +

1

p

d[C]

dt= +

1

q

d[D]

dt=d"

dt= rf (T ,[A],[B]) ! rb (T ,[C],[D]){ }

d[A]

dt= !m (rf ! rb ) = !m K f (T )[A]

m[B]

n! Kb (T )[C]

p[C]

q( )

d[B]

dt= !n (rf ! rb ) = !n K f (T )[A]

m[B]

n! Kb (T )[C]

p[C]

q( )

d[C]

dt= p (rf ! rb ) = p K f (T )[A]

m[B]

n! Kb (T )[C]

p[C]

q( )

d[D]

dt= q (rf ! rb ) = q K f (T )[A]

m[B]

n! Kb (T )[C]

p[C]

q( )

Rates of Change of Species

For any reaction of the form:

the stoichiometric constraint yields:

enabling to find the rate of change of each species involved in that reaction:

m A + n B

Kf

Kb! "!!# !!! p C + q D

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dN

dt= Sk

k

! d" k

dt= Sk

k

! rk (p,T ,nj ) = Sk

k

! rfk # rb

k( )

limt$%

dNj

dt= 0 & rf

k # rbk ' 0

For the example, it holds:

K f (T )[A]m[B]n ' Kb (T )[C]p[D]q

(

K f (T )

Kb (T )'

[C]p[D]q

[A]m[B]n

Kc (T ) =[C]p[D]q

[A]m[B]n

)

*++

,++

-K f (T )

Kb (T )' Kc (T ) =:Kp (T ) .T( )

# / jk

j

!01

23

Equilibrium Condition as Asymptotic Solution of Kinetics

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Alternative Way to Compute Reverse Reaction Constants

Kb (T ) =K f (T )

Kc (T )=

K f (T )

Kp (T ) !T( )" # j

k

j

$=

K f (T )

Exp "%Go

k(T )

!T

&'(

)*+!T( )

" # jk

j

$

NB:Kf is a kinetic information,Kp is a thermochemical information

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Frozen and Equilibrium Limits

• An alternative form to express the reaction rate is:

• Two limit cases of chemical kinetics:

Frozen limit:

Equilibrium limit:

d[A]

dt= !m K f (T )[A]

m[B]

n ! Kb (T )[C]p[C]

q( )

= !m K f (T )[A]m[B]

n !K f (T )

Kc (T )[C]

p[C]

q"

#$%

&'

= !m K f (T ) [A]m[B]

n !1

Kc (T )[C]

p[C]

q"

#$%

&'

K f (T ) << 1!d[A]

dt" 0 ! [A] is frozen

[A]m[B]n !1

Kc (T )[C]p[C]q " 0 # Kc (T ) =

[C]p[C]q

[A]m[B]n# [A] in chem.eq.

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Global vs Elementary Reactions

• Elementary Reactions describe the actual process which originates productsfrom the collision between reactants; example:

• Global Reactions describe the overall relationship between reactants andproducts, without providing the details of the underlying processes

F + Ox! P

d[F]

dt= "K

Global(T )[F]m[Ox]n

m,n can be non-integer and found by curve fits of exp. data

H2 +O! H +OH

d[H2 ]

dt= "K f (T )[H2 ]1[O]1

Exponents are integers identifying the reaction order wrt to H2,O, resp.

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Classification of Reactions

• One-Step/Multi-Step

• Irreversible/Reversible

• Molecularity: uni-molecular, bi-molecular, ter-molecular

• Reaction Order: zero-order, first-order, second-order, third-order

• Consecutive

• Parallel

• Combined=Consecutive & Parallel

• Thermal

• Homogeneous/Heterogeneous

H2+O! H +OHH

2+O! H +OH

A + B! AB! C + D

A + B! AB

A + B! E + F

AB + Heat! A + B

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Example of Multi-Step Mechanism

d[H2]

dt= !K f 1[H2

][O2]+ Kb1[HO2

][H ]! K f 3[OH ][H2]+ Kb3[H2

O][H ]

d[O2]

dt= !K f 1[H2

][O2]+ Kb1[HO2

][H ]! K f 2[H ][O2]+ Kb2[OH ][O]

d[OH ]

dt= !K f 2[H ][O2

]+ Kb2[OH ][O]! K f 3[OH ][H2]+ Kb3[H2

O][H ]

H2+O

2! HO

2+ H R#1

H +O2!OH +O R#2

OH + H2! H

2O + H R#3

Given the tree-step mechanism involving only bi-molecular, second-order, reversible reactions:

the corresponding rate equations for each of the species involved read:

If the system is isothermal, each K is constant and the only unknowns are the molarconcentration of the 3 species

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Notation for a generic Multi-Step Mechanism

(!k

j )'j=1,Nreac

" Mj! (!k

j )''j=1,Nprod

" Mj k=1,Nr

! ' stoichiometric coefficient of reactant (negative sign)

! " stoichiometric coefficient of product (positive sign)

Mj chemical species

A mechanism involving Ns species and Nr reactions can be formally written as:

The set of Ns rate equations can be written as:

d Ni!" #$

dt= (%k

i)''& (%k

i)'( )

k=1,Nr

' rfk[T , N

j!" #$]& rbk[T , N

j!" #$]( ) i=1, NS

= (%k

i)''& (%k

i)'( )

k=1,Nr

' K f

k(T ) N

j!" #$(%k

j) '

j=1,Nreac

( & Kb

k(T ) N

j!" #$(%k

j)"

j=Nprod

()

*+,

-. i=1, N

S

= (%k

i)''& (%k

i)'( )

k=1,Nr

' K f

k(T ) N

j!" #$(%k

j) '

j=1,Nreac

( &1

Kc

k(T )

Nj!" #$

(%kj)"

j=Nprod

()

*+,

-. i=1, N

S

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!! j[T , p,Xj ] = ("k

j )''# ("k

j )'( )k=1,Nr

$ rfk[T ,

p Xj

%T]# rb

k[T ,p Xj

%T]

&'(

)*+

j=1,Ns

!! j= Sk

j

k=1,Nr

$ (rfk # rb

k ) j=1,Ns

!! j{ }g(Ns x 1)"

= S1# S

Nr;#S

1# #S

Nr[ ]S (Ns x 2Nr)

$ %&&&&& '&&&&&

rfk

rbk

,-.

/01

r(2Nr x 1)( )**

= S r[T , p,Xj ]

Reaction Rates in Matricial Form

It is convenient to cast the reaction rates in matricial notation:

NB: S is a sparse matrix of constants !NB: r depends on the state

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d[Y j ]

dt=

Wj

!(p,T ,Yj )!" j[T ,

p WYj

#WjT] j=1,Ns !(p,T ,Yj ) =

p

T RjYj

j

$

du

dt= g(u) u(0) = u0

u = Yj{ } g =

Wj

!(p,T ,Yj )!" j[T ,

p WYj

#WjT]

%&'

('

)*'

+'

How a Detailed Chemical Mechanism Evolve in Time

For an adiabatic, isobaric, and isothermal system, the evolution equationsinvolve Ns rate equations for the species:

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Initial Value Cauchy Problem

du

dt= g u( )=Q u( )Sr u( )

u(t0 ) = u0

!"#

$#

u,g %&Ns+1

r %&2Nr

u %&Ns+1 g' (' g u( )%&Ns+1

u %&Ns+1 r' (' ) = r u( )%&2Nr QS' (' g = Q u( )S )( )%&Ns+1

The Kinetics Problem as an ODE’s Problem

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Characteristic Kinetic Time Scale

d[A]

dt= !K[A]

[A](0) = [A]0

"

#$

%$

Whose exact solution is:

Consider an irreversible, one-step, uni-molecular reaction:

[A](t) = [A]0e!Kt

By definition, the characteristic time scale of the reaction is the value τch at which the ratio:

The exact solution can be written as:

Log[A](t)

[A]0

!

"#$

%&= 'Kt

which defines a line on a Log-Lin plot having slope K = - 1/ τch

[A](!ch)

[A]0

=1

e"!

ch=1

K Log[A]

[A]0

!

"#$

%&

t

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First-Order Consecutive Reactions

AK

1

! "! B R#1

BK

2

! "! C R#2

Consider two irreversible, uni-molecular, first-order, consecutive reactions:

d[A]

dt= !K

1[A]

d[B]

dt= +K

1[A]! K

2[B]

"

#$$

%$$

&

[A] = [A]0e!K1t

d[B]

dt= +K

1[A]

0e!K1t ! K

2[B]

"

#$

%$

[A](0) = [A]0

[B](0) = 0

!

[A] = [A]0e"K1t

[B] = [A]0

K1

K2" K

1

e"K1t " e"K2 t( )

#

$%

&%

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Fast and Slow Reactions

K1 >> K2

!K1

K2 " K1

# "1 $

t>1/K1

e"K1t! 0

! [B] ! [A]0e"K2 t

![A] decays with the fast scale of R#1

[B] decays with the slow scale of R#2

K1 << K2

!K1

K2 " K1

# 0

! [B] ! 0

![A] decays with the slow scale of R#1

[B] is always small

[A] = [A]0e!K1t

[B] = [A]0

K1

K2! K

1

e!K1t ! e!K2 t( )

"

#$

%$

K1= 1,K

2= 0.1 K

1= 0.01,K

2= 1

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Consecutive Reactions with One Bi-Molecular Reaction

A + BK

1

! "! AB R#1

ABK

2

! "! C R#2

Consider two irreversible, bi-molecular consecutive reactions:

d[A]

dt= !K

1[A][B]

d[B]

dt= !K

1[A][B]

d[AB]

dt= +K

1[A][B]! K

2[AB]

d[C]

dt= K

2[AB]

"

#

$$$$

%

$$$$

[A](0) = [A]0

[B](0) = [B]0

[AB](0) = 0

[C](0) = 0

K1= 1,K

2= 0.1