Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical...

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Rates of chemical reactions

Transcript of Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical...

Page 1: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Rates of chemical reactions

Page 2: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Measuring rates of chemical reactions

• Experimental measuring progress of the reaction

– Monitoring pressure in the reaction involving gases

2 5 2 2

0

2 ( ) 4 ( ) ( )1(1 ) 22

3(1 )2

N O g NO g O g

n n n

p p

α α α

α

⎯⎯→ +

= +

– Absorption at particular wavelength (e.g. Br2 below)

2 2( ) ( ) 2 ( )H g Br g HBr g+ ⎯⎯→

– Conductance of the ionic solution

3 3 2 3 3( ) ( ) ( ) ( ) ( ) ( ) ( )CH CCl aq H O aq CH COH aq H aq Cl aq+ −+ ⎯⎯→ + +

Page 3: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Measuring rates of chemical reactions

•Flow method

•Stopped-flow method

•Flash photolysis

•Chemical quench flow

•Freeze quench method

Page 4: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Rates of chemical reactions2 3A B C D+ ⎯⎯→ +

Instantaneous rate of consumption of a reactant:

[ ] /d R dt−Instantaneous rate of formation of a product:

[ ] /d P dtFrom stoichiometry

[ ] 1 [ ] [ ] 1 [ ]3 2

d D d C d A d Bdt dt dt dt

= = − = −

Rate of the reaction:

1 1 1i

i

dn dvV dt V dt

ξν

= =

In case of heterogeneous reaction the rate will be defined as mol/m2s

0i i

i

n nξν−

=where the extent of the reaction

Page 5: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Reaction orderReaction rate is generally proportional to the concentration of the reagents to some power

[ ] [ ] ...a bv k A B=

Depends only on T and not on the concentration

The power is generally not equal to the stoichiometric coefficients!

Order of a reaction:

[ ][ ]v k A B= First order in A, first order in B, overall second order.

v k= Zero order.

1/ 2[ ] [ ]v k A B= Halforder in A, first order in B, overall tree-halves order.

Page 6: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Experimental determination of the rate law

• Isolation method

where all the components except one are present in large amounts (therefore their concentration is constant)

experiment repeated at several concentration

0[ ] [ ] [ ]a b av k A B k A′= =

A B P+ ⎯⎯→

Advantages:• easy interpretation of data• non-integral orders can be easily determinedDisadvantages• needs to be repeated for every reactant• physiological conditions can not always be used if high concentration are

required• high concentrations might affect reaction mechanism

Page 7: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Experimental determination of the rate law• Method of initial rates

several initial concentration of A measured (again assuming that the concentrations are constant)

0 0 0 0[ ] log log log[ ]av k A v k a A′= = +

Advantages:• easy interpretation of data• non-integral orders can be easily determined• not necessary to use large access of any reagentDisadvantages• needs to be repeated for many times for every reactant• reaction processes involving e.g. products can be missed

0 0 0[ ] [ ] [ ]a b av k A B k A′= =

A B P+ ⎯⎯→

Page 8: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Reaction order• Example 22 ( ) ( ) ( ) ( )I g Ar g I g Ar g+ ⎯⎯→ +

10

5.0

1.0

6.04.02.01.0[I0 ] x10-5mol dm-3

[Ar] x mmol dm-3

Page 9: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Integrated rate laws

• First order reaction.– Let’s find concentration of reagent A after time t

[ ] [ ]d A k Adt

= −0 0

[ ][ ]

A t

A

d A k dtA

= −∫ ∫

0[ ] [ ] ktA A e−=0

[ ]ln[ ]

A ktA

⎛ ⎞= −⎜ ⎟

⎝ ⎠

Slope -> k

Page 10: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Integrated rate laws• First order reaction

0[ ] [ ] ktA A e−=

– Half-life – time required for concentration to drop by ½.

0

1/ 2 1/ 20

1 [ ] ln 22ln ln 2[ ]

Akt t

A k

⎛ ⎞⎜ ⎟

= − = =⎜ ⎟⎜ ⎟⎝ ⎠

– Time constant – time required for concentration to drop by 1/e:

1k

τ =

Page 11: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Integrated rate laws• Second-order reactions

2[ ] [ ]d A k Adt

= −0

20

[ ][ ]

A t

A

d A k dtA

= −∫ ∫

0

0

[ ][ ]1 [ ]

AAkt A

=+0

1 1[ ] [ ]

ktA A

− =Second order

First order– Half-life – depends on initial concentration, long when concentration is low

00 1/ 2

1/ 2 0 0

[ ]1 1[ ]2 1 [ ] [ ]

AA tkt A k A

= =+

Page 12: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Integrated rate laws

Page 13: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Reaction approaching equilibrium

• Generally, most kinetics are made far from equilibrium where reverse reactions are not important. Close to equilibrium the amount of products is significant and reverse reaction should be considered.

[ ]'[ ]

A B v k AB A v k B⎯⎯→ =

⎯⎯→ =

0[ ] [ ] [ ] [ ] [ ]dA k A k B A B Adt

′= − + + =

0[ ] ([ ] [ ])dA k A k A Adt

′= − + −

( )

0[ ] [ ]k k tk keA A

k k

′− +′ +=

′+

0 0

[ ][ ] [ ] , [ ] [ ] ,

[ ]eq

eq eqeq

Bk k kA A B A Kk k k k A k

′= = = =

′ ′ ′+ +

Generally, for a multistage reaction: ...a b

a b

k kKk k

= × ×′ ′

Page 14: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Reaction approaching equilibrium• Relaxation denotes return of the system to equilibrium after

external influence, e.g. temperature jump (via electrical discharge, or microwave, or laser pulse)

/0

1ta bx x e k kτ

τ−= = +

x( [ ] ) ( [ ] )

( )

a eq b eq

a b

dA k x A k x Bdt

k k x

= − + + − + =

= − +

Indeed at the new condition:

• ka and kb can be found from the relaxation experiment (as K is known)

Page 15: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

The temperature dependence of reaction rate• The rate constants of most reaction increase with the

temperature (usually 2-4 times for every T raise 25 ->35 °C)• Experimentally for many reactions k follows Arrhenius

equation

ln ln aEk ART

= −

A – pre-exponential (frequency) factor,Ea – activation energy

High activation energy means that rate constants depend strongly on the temperature, zero would mean reaction independent on temperature

Page 16: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

The temperature dependence of reaction rate

• Stages of reaction:– Reagents

– Activation complex

– Product

/aE RTk Ae−=

Transition state

Activation energy is the minimum kinetic energy reactants must have to form the products.

Pre-exponential is rate of collisionsArrhenius equation gives the rate of

successful collisions.

Fraction of collision with required energy

Page 17: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Elementary reactions• Most reactions occur in a sequence of steps called elementary reactions.• Molecularity of an elementary reaction is the number of molecules coming

together to react (e.g. uni-molecular, bimolecular)

Uni-molecular: first order in the reactant

[ ] [ ]d AA P k Adt

⎯⎯→ = −

Bimolecular: first order in the reactant

[ ] [ ][ ]d AA B P k A Bdt

+ ⎯⎯→ = −

Proportional to collision rate

2H Br HBr Br+ ⎯⎯→ +

Page 18: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Consecutive elementary reactions

a bk kA I P⎯⎯→ ⎯⎯→

[ ] [ ]

[ ] [ ] [ ]

[ ] [ ]

a

a b

a

d A k Adt

d I k A k Idt

d P k Idt

= −

= −

=

Solution for A should be in a form: 0[ ] [ ] ak tA A e−=

0 0[ ] [ ] [ ] [ ] ( )[ ]a a bk t k t k ta

b ab a

kd I k I k A e I e e Adt k k

− − −+ = = −−

0 0[ ] [ ] [ ] [ ] [ ] 1 [ ]a bk t k t

a b

b a

k e k eA I P A P Ak k

− −⎛ ⎞−+ + = = +⎜ ⎟−⎝ ⎠

Reaction can proceed through formation of an intermediate:

Page 19: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Consecutive elementary reactions• The steady-state approximation

[ ] 0d Idt

Then [ ] [ ] [ ] 0a b

d I k A k Idt

= − ≈

[ ] [ ] [ ]b ad P k I k Adt

= ≈and

( ) 0[ ] 1 [ ]ak tP e A−≈ −

Assumption: after initial induction period the concentration of intermediate stays constant

Page 20: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Consecutive elementary reactions• The rate-limiting step

Suppose kb>>ka than if intermediate is formed it decays rapidly into P, so forming intermediate is a rate-limiting step.

( )0 0[ ] 1 [ ] [ ] 1 [ ]a b

a

k t k tk ta b

b a

k e k eP A P e Ak k

− −−⎛ ⎞−

= + = −⎜ ⎟−⎝ ⎠

Rate of formation of P doesn’t depend on decay rate of I.

Page 21: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Consecutive elementary reactions

• Kinetic and thermodynamic control of reactions– kinetic control: far from equilibrium

( )( )

1 1 1

2 2 2

[ ][ ]

[ ][ ]

A B P v P k A B

A B P v P k A B

+ → =

+ → =

[ ][ ]

2 2

1 1

P kP k

=

– thermodynamic control : concentration at equilibrium are defined by equilibrium constants (Gibbs energies)

Page 22: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Consecutive elementary reactions• Pre-equilibria

Then

[ ] [ ] [ ][ ]b bd P k I k K A Bdt

= =and

a b

a

k kk

A B I P′

+ ⎯⎯→

This condition arises when k’a >>kb.

[ ][ ][ ]

a

a

kIKA B k

= =′

Second order form with composite rate constant

k

Page 23: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

The kinetic isotope effectObservation of slowing down the reaction upon replacement an atom with heavier isotope facilitate determining the rate limiting step

Primary kinetic isotope effect: rate limiting requires breaking a bond involving the isotopeSecondary kinetic isotope effect: reduction of the reaction rate even when the rate limiting step doesn’t require breaking a bond involving the isotope

Page 24: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Unimolecular reactions

• As the molecule acquires the energy as a result of collision why the reaction is still a first order?

• Lindemann-Hinshelwood mechanism*

* 2

** *

** *

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

a

b

ka

ka

kb

d AA A A A k Adt

d AA A A A k A Adt

d AA P k Adt

+ ⎯⎯→ + =

′+ ⎯⎯→ + = −

⎯⎯→ = −

If the last step is rate-limiting the overall reaction will have first order kinetics

Page 25: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Lindemann-Hinshelwood mechanism

*2 * *

2*

2*

[ ] [ ] [ ][ ] [ ]

[ ][ ][ ]

[ ][ ][ ]

a a b

a

b a

b ab

b a

d A k A k A A k Adt

k AAk k A

k k AdP k Adt k k A

′= − −

=′+

= =′+

** 2

** *

** *

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

a

b

ka

ka

kb

d AA A A A k Adt

d AA A A A k A Adt

d AA P k Adt

+ ⎯⎯→ + =

′+ ⎯⎯→ + = −

⎯⎯→ = −

If the rate of deactivation is much higher that unimolecular decay than:

2[ ] [ ][ ]

b a b a

b a a

k k A k k AdPdt k k A k

= ≈′ ′+

The Lindemann-Hinshelwood mechanism can be tested by reducing the pressure (slowing down the activation step) so the reaction will switch to the second order.

Page 26: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Lindemann-Hinshelwood mechanism

• testing for the theory…

2[ ][ ]

1 1[ ]

b a

b a

a

b a a

k k AdPdt k k A

kk k k k A

=′+

′= +

isomerization of CHD=CHD

Page 27: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

RRK-model

• RRK (Rice-Ramsperger-Kassel) model is correction to Lindemann-Hinshelwood mechanism that takes into account energy distribution over the modes of motion

1*( ) 1s

b bEk E kE

−⎛ ⎞= −⎜ ⎟⎝ ⎠

s – number of modesE* - energy required to break the bond

Page 28: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

The activation energy of the composite reaction

Let’s consider Lindemann-Hinshelwood mechanism and apply Arrhenius-like temperature dependence to each rate constant

( )( )( )( ) / ( ) /

{ ( ) ( ) ( )}/( ) /

a a

a a a

a

E a RT E b RTa b E a E a E a RTa b a b

E a RTa aa

A e A ek k A Ak ek AA e

− −′− + −

′−= = =

′ ′′

( ) ( ) ( )

( ) ( ) ( )

a a a

a a a

E a E b E a

E a E b E a

′+ >

′+ <

Overall activation energy can be positive or negative

Page 29: Rates of chemical reactions - Aalborg Universitethomes.nano.aau.dk/lg/2007_Physical Chemistry_8.pdf · Rates of chemical reactions ... The temperature dependence of reaction rate

Problems• E22.6a At 518°C, the half-life for the decomposition of a sample of gaseous

acetaldehyde (ethanal) initially at 363 Torr was 410 s. When the pressure was 169 Torr, the half-life was 880 s. Determine the order of the reaction.

• E22.10a The second-order rate constant for the reaction

CH3COOC2H5(aq) + OH-(aq) → CH3CO2-(aq) + CH3CH2OH(aq)

is 0.11 dm3 mol-1 s-1. What is the concentration of ester after (a) 10 s, (b) 10 min when ethyl acetate is added to sodium hydroxide so that the initial concentrations are [NaOH] = 0.050 mol dm-3 and [CH3COOC2H5] = 0.100 mol dm-3?

• E22.16a The effective rate constant for a gaseous reaction that has a Lindemann–Hinshelwood mechanism is 2.50 × 10-4 s-1 at 1.30 kPa and 2.10 ×10-5 s-1 at 12 Pa. Calculate the rate constant for the activation step in the mechanism.