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)

Reactant Productdt

dRate

]reactant[

Generally, we want to understand the rate of reaction:

CK-2Recall from Slide CEq-2…

)()()()( gZvgYvgBvgAv ZYBA

)()0()( tvntn AAA

)()0()( tvntn BBB )()0()( tvntn ZZZ

)()0()( tvntn YYY Reactants Products

For each constituent…

dt

td

V

v

dt

Ad

dt

dn

VAA )(][1

CK-3Rate of reaction, v(t)

dt

Zd

vdt

Yd

vdt

Bd

vdt

Ad

vdt

td

Vtv

ZYBA

][1][1][1][1)(1)(

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

td

Vtv

)(1)(

dt

td

Vtv

)(1)(

CK-4The integrated rate law

nm BAkdt

d

Vtv ][][

1)(

)()()()( 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

mth order in Anth order in B

(m+n)th order overall

The Order:

Determined experimentally!

CK-5Finding rate laws experimentally

nBkv ][

Method of isolation Set 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-6Units of k, rate constant

nm BAkdt

d

Vtv ][][

1)(

timeBA

tvk

nm

nm

1)ionconcentrat(

][][

)(

Rate Law Order Units of k

v = k 0

v =k[A] 1

v = k[A]2 2

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

2 overall

v = k[A]1/2 1/2

concentrationtime (concentration)m

(concentration)n

CK-7Rate 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)(

k

ktv

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-8Elementary Rxns vs. Complex Reactions

productsA

products CBA

products BA

Elementary Reactions

Complex Reactions

Reactants Products

Reactants Intermediates Products

Mechanism of a complex reaction is a sequence of

elementary reactions.

Molecularity of Elementary Reactions:

Unimolecular

Bimolecular

Termolecular

CK-9First order reactions

products BA ][][

)( Akdt

Adtv

kdtA

Ad

][

][

tA

AkdtAd

A

t

0

][

][ 0

][][

1

ktA

A 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?

ktA

A 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

2

1

][

][

0

2/1 kteA

A

0][2

1][

2/1AA t

Figure 28.3

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][][

2

1)( Ak

dt

Adtv

Second order reaction:

Second order rate:

Integrated rate law:

]][[][

)( BAkdt

Adtv

ktAA t

2][

1

][

1

0

ktBA

AB

BA t

t

][][

][][ln

][][

1

0

0

00

Zero order reaction:

Zero order rate:

Integrated rate law:

productsA

kdt

Adtv

][)(

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

]][[][

)( ABkdt

Adtv

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

Similar to SN2 Lab

CK-15Reversible reactions (small rG)

tkk

BkAk

ABAkAk tt )(][][

][][][][ln 11

0101

0011

][][][

11 BkAkdt

Ad

][][][][][

0011 ABAkAkdt

Ad

][][][][ 00 BABA ][][][][ 00 ABAB

A Bk1

k-1Assume first order, elementary rxn in both directions

Rate:

Conservation of Mass:

Integrate:

dt

Ad ][

CK-16At equilibrium

0][

dt

Adeqeq BkAk ][][ 11

eq

eqeq A

BK

][

][

][][][

11 BkAkdt

AdA B

k1

k-1

1

1

k

kKeq

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.

2

ln

RT

E

dT

kd a

Differential form of the Arrhenius Equation:

CK-18Arrhenius Parameters

RT

EAk alnln 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 T dependent. 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 Kh

Tkk

RTGeK /

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!

‡

‡

RTGB eh

Tkk /

RTHRSB eeh

Tkk //

Eyring Equation (key to transition-state theory)

‡

From CEq:

So…‡‡STHG

‡

Change in Gibbs energy from reactants to TS

Entropy of activation

Enthalpy of activation

CK-20Relating Ea to thermodynamics!

RT

EAk alnln

2

ln

RT

E

dT

kd adT

kdRTEa

ln2

2

ln

RT

U

dT

Kd C

dT

Kd

TdT

kd Cln1ln

dT

Kd

TRTE C

a

ln12 URTEa

Arrhenius Equation:

Differentiate wrt T: or

From Eyring Equation:

van’t Hoff Equation (for Kc):

Putting it all together…

or

Necessary Pieces…

‡

‡‡

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

gnRTHU

HRTEa

RTHEa 2

URTEa

nRTHRTEa

and

so

AA‡Products

A+BAB‡Products

Unimolecular Gas Phase Reaction

Bimolecular Gas Phase Reaction

RTERSB Aeeh

Tkek //2

RTERSB aeeh

Tkek //

so

so

What is A?

‡ ‡ ‡ ‡

‡ ‡

‡

‡

‡

‡

Same for reactions in solution

RTHRSB eeh

Tkk //

‡‡

CK-22Transition State Theory and NMR Lab

SH

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, and a temperature of 298 K, find…

G‡ ‡ ‡

Why is TST important?

1. Provides details of a reaction on the molecular scale.2. Connects quantum mechanics and kinetics.3. Currently used for many computational studies on reaction rates.

EX-CK3

CK-23

Chapter 29: Reaction Mechanisms

CK-24Always remember….

• One can never prove a reaction mechanism, although evidence may disprove a mechanism.

• Verifying proposed mechanisms requires extensive experimental verification of each proposed step!

CK-25Let’s examine a reaction …

PA obsk

IA k 1PA k 1

Reaction could progress in multiple ways… How can we distinguish?

Case 1: One elementary step Case 2: Two step reaction

PI k 2

EX-CK4

CK-26Now let’s focus on the intermediate…

PIA kk 21

How do k1 and k2 relate in case a? in case b?

(a) I forms quickly but decays slowly… k1 is fast relative to k2.

(b) I builds up to a constant, nearly negligible, concentration until near end of reaction. … k1 is slow relative to k2.

Steady state approximation... Valid only if k2 > k1. 0][

dt

Id

EX-CK5

CK-27Rate Laws do not yield unique mechanisms

)(NO2)(O)(NO2 22 ggg obsk

EX-CK6

An empirically determined rate law does not imply a unique reaction mechanism!

]O[]NO[)( 22

obsktv

Consider reaction:

Experimentally, it was determined that the rate is given by:

Researchers proposed two possible mechanisms. They need to determine if one of them is correct.

So how would researchers distinguish between the mechanisms?

CK-28Remember the Chain Rxn from CK-6?

)()(2)()( 12 gMgBrgMgBr k

)()()()( 22 gHgHBrgHgBr k

)()()()( 32 gBrgHBrgBrgH k

H2(g) + Br2 (g) 2HBr (g)1

2

2/122

]Br][HBr[1

]Br][H[2)(

k

ktv

)()()()( 22 gHgBrgHgHBr k

)()()()(2 21 gMgBrgMgBr k

Proposed Mechanism

Initiation:

Propagation:

Inhibition:

Termination:

EX-CK7

CK-29What about the solution kinetics lab?!

• Now that we’ve explored reaction mechanisms and rate laws, let’s try to derive the rate laws from the solution kinetics lab…

EX-CK8

CK-30Catalysis

productsA catalystproductscatalyst A

]][[][][

catalystAkAkdt

Adcat

Catalyst: A substance that participates in the chemical reaction but is not consumed. Provides a new mechanism for reaction and can cause reaction to occur faster.

In an experiment involving a catalyst, there are two competing reactions:

If both reactions are elementary, overall rate is given by:

EX-CK9