CK-1Chemical Kinetics

Chemical kinetics is the study of time dependence of the change in the concentration

of reactants and

products.

“The field of chemical kinetics has not yet matured to a point where a set of unifying principles has been identified…There are many different theoretical models for describing how chemical reactions occur.” (M&S, 1137)

Generally, we want to understand the rate of reaction:

CK-2Recall from Slide CEq-2…

)()()()( gZvgYvgBvgAv ZYBA +→+

dttdv

dttdn

AA )()( ξ

−=

)()0()( tvntn AAA ξ−=

)()0()( tvntn BBB ξ−= )()0()( tvntn ZZZ ξ+=

)()0()( tvntn YYY ξ+=Reactants Products

For each constituent…

dttd

Vv

dtAd

dtdn

VAA )(][1 ξ

−==

CK-3Rate of reaction, v(t)

dtZd

vdtYd

vdtBd

vdtAd

vdttd

Vtv

ZYBA

][1][1][1][1)(1)( ==−=−==ξ

v(t), the rate of reaction, is defined as the rate of change in ξ(t) with time per unit volume

Note all quantities are positive. What are units of v(t)?

2NO(g) + O2 (g) 2NO2 (g)Examples:

H2 (g) + I2 (g) 2HI (g)

==dt

tdV

tv )(1)( ξ

==dt

tdV

tv )(1)( ξ

CK-4The integrated rate law

nm BAkdtd

Vtv ][][1)( ==

ξ

)()()()( gZvgYvgBvgAv ZYBA +→+

nm BAktv ][][)( =

Rate laws must be determined experimentally

and, generally, cannot

be deduced from the balanced reaction!!

The rate can be expressed as a function of the reactant concentrations. Most common function is of form:

2NO(g) + O2 (g) 2NO2 (g)

]O[]NO[)( 22ktv =

General Example

The Order:

CK-5Units of k, rate constant

nm BAkdtd

Vtv ][][1)( ==

ξ

Rate Law Order Units of kv = k 0 mol·dm-3·s-1

v =k[A] 1 s-1

v = k[A]2 2 dm3·mol-1·s-1

v = [A][B] 1 in [A], [B] 2 overall

dm3·mol-1·s-1

v = k[A]1/2 1/2 dm-3/2·mol1/2·s-1

concentrationtime (concentration)m

(concentration)n

CK-6Rate laws can be complicated

]I][H[)( 22ktv =H2 (g) + I2 (g) 2HI (g)

H2 (g) + Br2 (g) 2HBr (g) 12

2/122

]Br][HBr[1]Br][H[2)( −′′+

′=

kktv

These rate laws suggest that these two reactions occur via different mechanisms (sets of individual steps).

The first may be a elementary reaction (one step) whereas the latter is certainly a multistep

process.

We will soon explore how to obtain complicated rate laws from suggested mechanisms.

EX-CK1

CK-7Elementary Rxns

vs. Complex Reactions

products→A

products→++ CBA

products→+ BA

Elementary ReactionsComplex Reactions

Reactants Products

Reactants Intermediates Products

Molecularity

of Elementary Reactions:

Unimolecular

Bimolecular

Termolecular

CK-8Finding rate laws experimentally

nBkv ][′=

Method of isolationSet up reaction so one reactant is in excess. Any change in rate

will be due to changes in other reactant. Repeat for other reactant.

Method of initial ratesMeasure concentration change as a function of time, ~v(t),

for a

series of experimental conditions. (Conditions must include sets where the reactant A has the same initial concentration but B

changes and vice versa).

There are two common methods for determining rate laws:

mAkk ][=′

EX-CK2

where

CK-9First order reactions

products→+ BA ][][)( AkdtAdtv =−=

kdtAAd

−=][][

∫∫ −=tA

AkdtAd

At

0

][

][ 0

][][

1

ktAA t −=

0][][ln ktAA t −= 0]ln[]ln[

ktt eAA −= 0][][

The reaction: has rate law:

Let’s integrate…

Solution:

First order reactions decay exponentially.

CK-10Ozone decays via first order kinetics

)(O)(O)(O 23 ggg +→

ktt −=03

3

]O[]O[lnkt

t e−= 033 ]O[]O[

k

= 1.078 ×

10-5

s-1

at 300 K

What is slope?

CK-11What happens as k

increases?

ktAA t −=

0][][lnkt

t eAA −= 0][][

k = 0.0125 s-1

k = 0.0250 s-1

k = 0.0500 s-1

k = 0.1000 s-1

CK-12Half-life of a first order reaction

2/1tt =

2/1

21

][][

0

2/1 kteA

A −==

0][21][

2/1AA t =

Figure 28.3

kt )2ln(

2/1 =

The half-life, t1/2

,

is the time it takes to fall to ½ of the starting concentration:

At ,

CK-13Other order reactions…

products2 →A products→+ BA

2][][21)( Ak

dtAdtv =−=

Second order reaction:

Second order rate:

Integrated rate law:

]][[][)( BAkdtAdtv =−=

ktAA t

2][

1][

1

0

+= ( ) ktBAAB

BA t

t =⎟⎟⎠

⎞⎜⎜⎝

⎛− ][][

][][ln][][

1

0

0

00

Zero order reaction:

Zero order rate:

Integrated rate law:

products→A

kdtAdtv =−=

][)(

ktAA t −= 0][][

CK-14Pseudo-first order reactions

You can “overload” the other reactants to determine the order with respect to one individual reactant (method of isolation).

products→+ BA

]][[][)( ABkdtAdtv =−=

For , what happens if [B] >> [A]?

CK-15Reversible reactions (small Δr G)

( ) tkkBkAk

ABAkAk tt )(][][

][][][][ln 110101

0011−

−

− +−=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+−

][][][11 BkAk

dtAd

−−=−

( )][][][][][0011 ABAkAk

dtAd

−+−=− −

][][][][ 00 BABA +=+][][][][ 00 ABAB −+=

A Bk1

k-1Assume first order, elementary rxn

in both directions

Rate:

Conservation of Mass:

Integrate:

CK-16At equilibrium

0][=−

dtAd

eqeq BkAk ][][ 11 −=

][][][11 BkAk

dtAd

−−=−A Bk1

k-1

At equilibrium…

What is the equilibrium constant for this reaction?

The forward rate equals the reverse at equilibrium.

In terms of rate constants?

CK-17Temperature Dependence of k

Svante

ArrheniusWinner of the 3rd

Nobel Prize in Chemistry

The rate constant can vary in different ways with T.

Most typicalfollows Arrhenius

Eq Enzyme KineticsExploisive

Rxn

2

lnRTE

dTkd a=

Differential form of the Arrhenius

Equation:

CK-18Arrhenius

Parameters

RTEAk a−= lnln RTEaAek /−=

Ea is the activation energy. This is the energy required to get over a barrier (at the activated or transition state) between the reactants and products. Ea has units of energy and is T independent.

Integrated forms of Arrhenius

equation:Activated (or transition) state

A is the pre-exponential or Arrhenius factor

and is also T independent. A is

a measure of rate at which collisions occur

(and takes lots of things into

acct such as orientation, molecular size, number of molecules per volume, molecular velocity, etc). 2HI(g)→I2

(g) + H2

(g)

CK-19Transition-State Theory

CB KhTkk =

RTGeK /Δ−=

AB‡

is the transition state (or activated complex.)

Transition state theory assumes that the transition state and reactants are in equilibrium with each other, and uses concepts from chemical equilibrium and statistical mechanics to find kinetic info such as rate constants!

‡

‡

Eyring

Equation (key to transition-state theory)

From CEq:

So…

‡

CK-20Relating Ea to thermodynamics!

RTEAk a−= lnln

2

lnRTE

dTkd a=

dTkdRTEa

ln2=

2

lnRT

UdT

Kd C Δ=

dTKd

TdTkd Cln1ln

+=

Arrhenius

Equation:

Differentiate wrt

T: or

From Eyring

Equation:

van’t

Hoff Equation (for Kc

):

Putting it all together…

Necessary Pieces…

‡

CK-21What about A, the pre-exponential?

gnRTHU Δ−Δ=ΔURTEa Δ+=

nRTHRTEa Δ−Δ+=

and

so

A A‡ Products

A+B AB‡ Products

Unimolecular

Gas Phase Reaction

Bimolecular Gas Phase Reaction

so

so

‡ ‡ ‡ ‡

‡ ‡

CK-22Transition State Theory and NMR Lab

SΔHΔ

In the NMR/N,N-DMA Paper, Gasparro

et al. found an activation energy of 70.3 kJ/mol and a pre-factor of 1.87 ×

1010

s-1. Using these values find…

GΔ‡ ‡ ‡

Why is TST important?