1

Kinetics

• Definition: chemical kinetics is the basis for the quantitative description of the course of a chemical reaction with time, that is of the reaction rate.

• Goal: establishing a functional relation between the rate of the reaction and the parameters influencing the reaction rate (concentrations and temperature).

Microkinetics and macrokinetics:

The microkinetics is the kinetics of the pure chemical process. The macrokinetics includes heat and mass transport processes.

2

Scope of chemical microkinetics

The reaction mechanism is only a hypothesis (model, proposal). We cannot know the reaction mechanism, we can only test it.The fact that a mechanism explains the experimental results (kinetic data, detection of intermediates, etc.) is not a proof that the mechanism is correct.

1. If the kinetics of a chemical reaction is known, it is possible to postulate a reaction mechanism which is a sequence of elementary steps that explains how the overall reaction proceeds.

DCBA +→+

DCBA +→+ reaction

FEBA +→+DCFE +→+

mechanism

2. The knowledge of the reaction kinetics is an important factor for the choice of reactor type.

3

Outline• Reaction rate: IUPAC and other useful definitions• Parameters affecting the reaction rate: temperature and composition

• Effect of temperature: Arrhenius’ law• Interpretation of the activation energy.

• Effect of composition: power law expression• Elementary and composite reactions; molecularity and reaction order• Methods for reaction order determination: method of integration (0,

1st, 2nd, nth order) and differential method (two variations).• Method of isolation• Single and multiple reactions (consecutive and parallel)• Reversible and irreversible reactions

4

Reaction rate: IUPAC definition

• According to IUPAC-guidelines, the reaction rate is the change of number of moles, caused by the reaction, per unit of time, referred to the stoichiometric coefficient νi

νi < 0 for reactants

νi > 0 for products

• Here the reaction rate is an extensive magnitude (depends on the size sample).

44332211 AAAA νννν +→+

dtdnr i

iν1* = ⎥⎦

⎤⎢⎣⎡

smol

r* always >0 !

5

Reaction rate: useful definitions

⎟⎟⎠

⎞⎜⎜⎝

⎛===

dtdc

dtdn

VVrr i

i

i

i νν111*' ⎥⎦

⎤⎢⎣⎡

smmol

3

dtdn

mmrr i

iν11*'' == ⎥

⎦

⎤⎢⎣

⎡Kgsmol

dtdn

SSrr i

iν11*''' == ⎥⎦

⎤⎢⎣⎡

smmol

2

.''''''* etcSrmrVrr ====

etc.

• This yields to several definitions of reaction rate, all intensive (invariant regarding the size of the system) and all interrelated :

( )propertyextensiverr *

=

• It is advisable to refer the reaction rate to extensive properties characterizing the reaction system

(for homogeneous reactions the volume V or the mass m of the reaction mixture, for heterogeneous reactions the interfacial surface S, the volume of the solid Vs, the mass m of the solid ms etc.)

6

Rate of formation (consumption) vs. reaction rate

0>=dtdcr i

i(intensive) rate of product i formation

(intensive) rate of reactant i consumption0>−=dtdcr i

i

If we use ri, we must always specify the reaction component to which it is referred and remember to include the appropriate sign !

HBrBrH 222 →+

HBrBrH rrr21

22==

322 23 NHNH →+

322 21

31

NHHN rrr ==

44332211 AAAA νννν +→+

(intensive) rate of reaction, for V=costdtdc

Vrr i

iν1*' ==

νi < 0 for reactants

νi > 0 for products

7

[Ao]

-rA = slope of the curve [A] vs. time

t

Graphic representation of the rate

[A]

0 t

Typical concentration profile in a batch reactor

productsA→[ ]dtAdrA −=

8

Parameters which affect the reaction rate

Determination of thereaction orders

For many reactions, the rate expression can be written as a product of a temperature-dependent term and a composition-dependent term, or

)(*)( 21 ncompositiofetemperaturfr =

k = rate constantDetermination of the activation energy

)(* 2 ncompositiofk=

9

Effect of temperature on reaction rate:the Arrhenius’ law

The temperature-dependent term, the rate constant k, is well represented by the Arrhenius’ law:

RTEekk /0

−= k0 = pre-exponential factor

E = activation energy R = universal gas constant= 8,314 J/mol·K T = absolute temperature (K)

RTEkk /lnln 0 −=

I/T (K-1)

E= -Slope*R

ln k

Common range for E: 40-200 kJ/mol

10

Activation energy and temperature dependency1. Reaction with high activation energies are very temperature-

sensitive; reaction with low activation energies are relatively temperature-insensitive.

ln k

1/T (k-1)

high E

low E

1/T (k-1)

k

ln k

2000K 1000K 463K 376K

ΔT= 1000°for

doublingof rate

ΔT= 87°for

doublingof rate

2. Any given reaction is much more temperature-sensitive at low temperature than at high temperature.

11

Activation energyFollowing a thermodynamic approach, the activation energy can be interpreted as the energetic barrier between reactant and products.

Between the initial states (reactants) and the final states (products) the potential energy passes through a maximum (activated complex), which corresponds to the top of the barrier.

For reaction from left to right the barrier height is E1 and for the reaction from right to left it is E-1. These energies are related by the equation E1-E-1=ΔH

The reaction coordinatecorresponds to the pathway with minimum energy between reactants and products.

E

Reaction coordinate

reactants

products

Activated complex

E1 E-1

ΔH

Exothermic reaction (ΔH<0)

E

Reaction coordinate

reactantsproducts

Activated complex

E1E-1

ΔH

Endothermic reaction (ΔH>0)

12

Potential-energy surfaces and reaction coordinateIf the reaction occurs between two atoms A and B, the potential energy of the system can be described by a two-dimensional diagram (energy plotted vs. interatomic distance A-B).

The potential energy is zero for big interatomic distances (A and B are not interacting).As the interatomic distance decreases, the potential energy first decreases (the attractive interactions prevail), then it reaches a minimum corresponding to the bond distance. Finally, for interatomic distances below the bond distance the potential energy rapidly increases due to repulsive interactions.

Interatomic distance A-BP

oten

tial e

nerg

y

Bond distance

13

Potential-energy surfaces and reaction coordinateIf the reaction occurs between three atoms, as in the reactionA-B + C A-C + B we need three parameters to describe it, for example the A-B, B-C and A-C distances or two distances and an angle.To plot energy against these three parameters a four-dimensional diagram would be necessary. Since this diagram cannot be visualized, it is necessary to use a series of three-dimensional diagrams, in each of which one parameter (for example the angle) is fixed. All these diagrams are sections of the four-dimensional surface.

dA-B

dA-CBond

distance A-B

Bond distance A-C

Potential energy

Reaction coordinate

The reaction coordinate is the path corresponding the minimum energy with respect to all the possible paths between reactants and products.The section through the minimum-energy path is the potential-energy profile.

E

reactantsproducts

14

Parameters which affect the reaction rate

Determination of thereaction orders

k = rate constant =k0e-E/RT

For many reactions, the rate expression can be written as a product of a temperature-dependent term and a composition-dependent term, or

)(* 2 ncompositiofkr =

)(*)( 21 ncompositiofetemperaturfr =

15

Effect of composition on reaction rate:power law rate expression

k : rate constantmi : partial reaction order with respect to the species i

(may be positive, negative, zero, full number or a fraction,mi are purely experimental quantities)

: overall reaction order∑=i

imm

44332211 AAAA νννν +→+ εδγβα54321 ***** ccccckr =

(5 = hidden factor in reaction –e.g. catalyst, intermediate)mi

ii

ckr ∏= *or generally

[ ][ ][ ] [ ]22

2/1221/ BrHBrk

BrHkHBrr +=HBrBrH 222 →+

Sometimes the reaction rate does not follow a power law and the reaction order should not be used.Ex.

16

Molecularity vs. reaction orderIt is important to distinguish molecularity from reaction order:

The reaction order is an empirical quantity obtained from the experimental rate law.

The molecularity is the number of reactant particles (atoms, molecules, free radicals, ions) colliding together successfully in the microscopic chemical event.Molecularity must be ≤ 3: is not elementary,

could be elementary, but it is not

The molecularity is defined only for elementary reactions (reactions which occur in a single step, without intermediates), which are proposed as an individual step in a reaction mechanism. Non elementary reactions are referred as composite reactions.

Only for the so-called elementary reactions:the reaction order mi corresponds to the stoichiometric coefficient νi;the overall order m is also designed as molecularity.

322 23 NHNH →+HBrBrH 222 →+

17

Determination of reaction orderTwo main procedures exist:1. Method of integration

This method deals with the integrated form of the rate equation.First a tentative decision of the reaction order is made,the corresponding differential reaction is integrated, and aftermathematical manipulation, the plot of a certain concentration function versus time should yield a straight line. If the experimental data fit reasonably well the integrated equation, the suggested rate equation is accepted.

2. Differential methodThis method deals directly with the differential rate equation.First, the reaction rate is measured by determining the slopes of concentration-time experimental curves.Then, the ln(rate) is plotted vs. the ln(conc). The slope of this straight line gives the reaction order.

18

Determination of reaction order: integral methodzero-order reaction

Slope = -k

productsA→ (irreversible unimolecular reaction)

[ ] ∫∫ −=tA

A

dtkAd00

[ ] [ ]0AktA +−=

Separating and integrating:

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

t

[A]

If the reaction is zero order, the rate of reaction is independent from theconcentration:

[ ] [ ] kAkdtAdrA ==−= 0

19

Determination of reaction order: integral method1st order reaction

productsA→ (irreversible unimolecular reaction)

[ ][ ] kdtAAd

−=[ ][ ] ∫∫ −=

tA

A

dtkAAd

00[ ][ ] ktAA

−=0

ln

Separating and integrating:

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

t

[A]

[ ] [ ] kteAA −= 0

-5

-4

-3

-2

-1

0

1

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

t

ln[A

]

[ ] [ ] ktAA −= 0lnln

Slope = -k

If the reaction is of first order:[ ] [ ]1AkdtAdrA =−=

20

Determination of reaction order: integral method2nd order reaction

[ ] [ ] [ ]( )00 1/ AktAA +=

productsA→ (irreversible unimolecular reaction)

[ ][ ]

kdtAAd

−=2[ ][ ] ∫∫ −=

tA

A

dtkAAd

02

0

Separating and integrating:

0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

[A]

t0

1

2

3

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

1/[A

]

t

[ ] [ ] ktAA

+=0

11

Slope = k

If the reaction is of second order:[ ] [ ]2AkdtAdrA =−=

21

Comparison of reaction rates

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t

[A]

0 order1st order2nd order

The 2nd order reaction is slower than the 1st order reaction

Same initial concentration [A0] and same k

22

Determination of reaction order: integral methodnth order reaction

productsA→ (irreversible unimolecular reaction)

The order n cannot be found explicitly from this equation, so a trial-and-error solution must be found. Select a value for n and calculate k. The value of n which minimizes the variation in k is the desired value of n.

Separating and integrating:[ ][ ]

kdtAAd

n −=[ ][ ] ∫∫ −=

tA

An dtk

AAd

00

[ ] [ ]( ) ktAAn

nn −=−−

−− 10

1

)1(1

for n≠1

0

1

2

3

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

[A]1-

n

t

Slope = k(n-1)

If the reaction is nth order:[ ] [ ]nA AkdtAdr =−=

23

Determination of reaction order: differential method

Experimental data

Time, s

[A],

mol/lit

1. Plot [A] vs. time

Tangents

2. Determine the slope of this curve for selected concentration values. These slopes are the rates rA at these concentrations.

[ ]AnkrA lnlnln +=3. Plot ln(rA) vs. ln[A]. The slope give the reaction order n.

ln [A]

ln (rA)

Slope = n

productsA→

This method deals directly with the differential rate equation[ ] [ ]nA AkdtAdr =−=

24

Determination of reaction order: differential method:Method of the initial rate

The differential mode may be applied in a different way, which is often called “method of the initial rate”.

Time, s

[A]mol/lit

The reaction is run at different initial concentrations [A0] and the initial rates determined by measuring initial slopes.

ln[A0]

ln (rA0)

Slope = ni

A double-logaritmic plot gives the initial reaction order ni.

This procedure avoids possible complications due to interferenceby products.

25

Determination of reaction order: differential method:Comparison of the two variants

The reaction orders determined by the two variants of the differential method are not always the same for a given reaction.The fact that the reaction order with respect to time (n) is greater than the order with respect to initial concentration (ni), means that, as the reaction proceeds, the rate falls off more rapidly that if the order niapplied for the complete reaction. This means that some products are acting as inhibitors. Conversely, the products have a positive effect on the reaction rate and the reaction is said to be autocatalytic.

Slope = ni

ln [A]

ln rA Slope = n

The blue symbols correspond to rate obtained in individual runs;the green points correspond to the initial rates

26

Comparison between integral and differential methodsIn the integral method of analysis we guess a particular form of rate equation and we use its integrated form. In the differential method we test the fit of the rate expression to the data directly and without integration.

There are advantages and disadvantages to each method.

The integral method:+ is easy to use- creates a prejudice in favor of integral orders and deviation from such orders

might escape notice (if for example the reaction order is 1.8, the experimental data might fit the second-order integrated equation within the experimental error).+ it is recommended when testing specific mechanism or when the data are so

scattered that we cannot reliably determine the rate with the differential method

The differential method:+ It is used when there are no previous information about the kinetics- requires more accurate or larger amount of data+ It distinguishes between the two reaction order n and ni and therefore reveals

information about the influence of products on the rates.

27

Determination of reaction order:isolation method

• Up to now, we have assumed that the reaction rate only depends on the concentration of the reactant A and we have written:

• However, in general, the overall reaction rate is also affected by the concentration of other species, for example the B and C:

• How can we separate the effects of the different species on the overall reaction rate and determine the partial reaction orders? The “isolation method” or “excess method”.

[ ] [ ]nA AkdtAdr =−=

[ ] [ ] [ ] [ ]omnA CBAk

dtAdr =−=

28

• If a reaction is of order m, n, o with respect to A,B and C, and if B and C are in excess of A, the apparent order determined by any of the methods described, will correspond to m.

In fact:

Determination of reaction order:isolation method

CBA +→

[ ] [ ] [ ]CoBnAmkrA lnlnlnlnln +++=

[ ]AmkrA ln'lnln +=

The concentration of the components in excess will not change very much during the course of the reaction and can be considered constant.

[ ] [ ] [ ] [ ]omnA CBAk

dtAdr =−=

29

• Up to now, we have described just single reactions, for ex.

In these cases, a single stoichiometric equation and a single rate equation are sufficient to describe the progress of the reaction:

• When more than one stoichiometric reaction equation is needed todescribe the system, more kinetic expressions are needed to follow the changing composition.These reactions are called multiple reactions.Multiple reactions may be classified as:

consecutive reactions:

parallel reactions:

Single and multiple reactions

DCBA +→+

SRA →→

mii

ickr ∏= *

RA

S→→

30

Irreversible consecutive reactions

With the assumption that all the reactions are 1st order:SRA →→

k2k1

We will not consider the mathematical solution of this system of linear differential equations limiting ourselves to the qualitative/intuitive description of the concentration profiles of A, R and S.

k 1 = k 2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5t

[x]/[

A 0]1. [A] always decreases

2. [R] exhibits a maximum(two opposite processes, where in the

long run the second always wins)

3. [S] always increasesInitial slope of S is zero

The maximum of R corresponds to the inflection point of S

[ ] [ ]AkdtAdrA 1=−=

[ ] [ ] [ ]RkAkdtRdrR 21 −==

[ ] [ ]RkdtSdrs 2==

31

Irreversible consecutive reactions

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5t

[x]/[

A 0] k1 ≈ k2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5t

[x]/[

A 0]

k1 = k2 / 20

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5t

[x]/[

A 0] k1 = 20k2

A → R → Sk1 k2

The ratio k1/k2 affects the concentration profiles

32

Irreversible parallel reactions

With the assumption that all the reactions are 1st order:

We will not consider the mathematical solution of this system of linear differential equations limiting ourselves to the qualitative/intuitive description of the concentration profiles of A, R and S.

R→→A

S

k1

k2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5t

[x]/[

A 0]1. [A] always decreases

2. [R] and [S] always increase

3. . [ ][ ]

[ ][ ] 21 / kkSR

SdRd

==

[ ] [ ]AkkdtAdrA )( 21 +=−=

[ ] [ ]AkdtRdrR 1==

[ ] [ ]AkdtSdrS 2==

33

Irreversible and reversible reactions• Up to now, we have described just irreversible reactions, for ex.

However, any reaction is basically reversible. In practice a reaction is considered reversible only if the forward and reverse reaction can be measured.

DCBAk

+→+1

1−

←k

DCBAk

+→+1

34

Reversible reactions• We consider only the simplest case: reversible reaction of 1st order

in both directions:

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5t[x

]/[A 0

]1−

←k

PAk1

→

After a certain time the system reaches a dynamic equilibrium state and the overall reaction rate is zero.

[ ][ ] eq

eq

eq KAP

kk

==−1

1

[ P ] eq

[ A ] eq[ ] [ ] [ ]PkAkdtAdrA 11 −−=−=

35

Books• Octave Levenspiel

Chemical Reaction Engineering, third edition, Wiley (1999).

• Keith J. LaidlerChemical Kinetics, third edition, Harper & Row, New York (1987).

• http://www.ltc1.uni-erlangen.de/htdocs/e/index.htmChapter III-Kinetics of Chemical Reactions – Microkinetics.

36

Vocabulary

activated complex aktivierter Komplexactivation energy Aktivierungsenergiebatch reactor Satzreaktor, diskontinuierlich betriebener Rührkesselcomposite reactions nicht elementare Reaktionenconsecutive reactions sequentielle Reaktionen, Konsekutivereaktionendifferential method Differentialmethodeelementary reactions Elementarreaktionenenergetic barrier Energiebarriereexcess method Überschussesmethodeextensive magnitude Extensive Größeheat and mass transport processes Stoff- und Wärmetransportmethod of integration Integralmethodeintensive magnitude intensive Größeirreversible reactions irreversible Reaktionenisolation method IsolirmethodeMacrokinetics MakrokinetikMechanism Mechanismusmethod of the initial rate AnfangsreaktionsgeschwindigkeitmethodeMicrokinetics MikrokinetikMolecularity Molekularitätmultiple reactions Mehrstufenreaktionenoverall reaction order Gesamtordnungparallel reactions Parallelreaktionenpartial reaction order Teilordnungpower law rate expression Potenzansatzpre-exponential factor Stoßfaktor, Preexponentieller Faktorreaction coordinate ReaktionskoordinateReaction rate Reaktionsgeschwindigkeit, Reaktionsratereversible reactions reversible Reaktionensingle reactions Einstufereaktionslope Steigung

37

Exercises

1. Compare the formation and consumption rates for all the productsand reactants involved in the following reactions:A + 2B → P3A + 2B → 3C + D +2E

2. Write the reaction rate of the following equation assuming it to bean elementary reaction:A + 2B → P (irrev.)A + 2B ↔ P (rever.)

3. a) A zero order reaction is 50% complete in 20 minutes. Howmuch time will it take to complete 90%?b) How much time will take a first order reaction with the samerate contant to half the initial concentration of reactant?

38

Exercises4. Reactant A decomposes in a batch reactor: A→ products.The composition of A in the reactor is measured at various times. Find

a rate equation to represent the data.

1300

2180

3120

560

640

820

100

[A] (mol/liter)Time (sec)

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

time (sec)

[A] (

mol

lite

r-1)

39

Integral method: guess 1st order kinetics.This means that ln[A]0/[A] vs time should give a straight line

123568

10

[A] (mol/liter)

2.3031.6091.204

0.69310.511

0.22310

ln [A]0/[A]

3001801206040200

Time (sec)

[ ][ ] ktAA

=0ln

0

0,5

1

1,5

2

2,5

0 50 100 150 200 250 300 350

time (sec)

ln [A

] 0/[A

]

The data do not fall on a straight line:

1st order rejected

40

Integral method: guess 2nd order kinetics:This means that 1/[A] vs time should give a straight line

123568

10

[A] (mol/liter)

2.3031.6091.204

0.69310.511

0.22310

1/[A]

3001801206040200

Time (sec)

[ ] [ ] ktAA

+=0

11

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150 200 250 300 350

time (sec)

1/[A

]

The data do not fall on a straight line:

2nd order rejected

41

Differential method1. Draw a smooth curve to represent the data2. Determine the slopes rA= -d[A]/dt at different concentrations

12356810

[A] (mol/liter)

4,5/260=0,01746/240=0,02058/160=0,0500

8,5/140=0,060710/92=0,109710/92=0,1087

rA

3001801206040200

Time (sec)

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

time (sec)

[A] (

mol

lite

r-1)

rA=10/92

rA=6/240

42

0,01740,02050,05000,06070,10970,1087

rA

12356810

[A] (mol/liter

)

-4,0566-3,6889-2,9957-2,8016-2,4027-2,2192

ln rA

3001801206040200

Time (sec)

y = 1,186x - 4,921R2 = 0,996

-4,5

-4

-3,5

-3

-2,5

-2

-1,5

-1

-0,5

00,0000 0,5000 1,0000 1,5000 2,0000 2,5000

ln[A]ln

(r A)

3. Plot lnrA vs ln[A]: the slope corresponds to the reaction order, theintercept corresponds to lnk

nA=1,2k=e-4,921=0,007 (liter0,2/mol0,2 s)

rA= -d[A]/dt = 0,007 [A]1,2 (mol/liter s)