TRIUMF Summer Institute 2011 Lecture 10 · Muonium reactions are pseudo-first order because… only...

Chemical Kinetics

Paul Percival

TRIUMF Summer Institute 2011

Lecture 10

Paul Percival TRIUMF Summer Institute, August 2011

2

Mu A products+ ⎯⎯→

M[Mu] [A][Mu] [Mu]d kdt

λ− = =

Muonium reactions are pseudo-first order because… only a few million Mu atoms are needed for an experiment.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Muo

n A

sym

met

ry

The Mu signal decays exponentially

time / µs

Muonium Kinetics

0[Mu] [Mu] e tt

λ−==

M[A]kλ =with

[A] is constant

second-order rate constantunits: M-1s-1


3

The Ergodic Principle

Complaint: What does [Mu] mean when we only have one Mu atom at a time?

Answer: It doesn’t matter if the atoms are present at the same time or spread over an interval.

The average of a parameter over time and the average over the statistical ensemble are the same

time

Mu survival probability

e–λt

37%

τ= 1/λ


4

7.7 G TF

Garner, Fleming, Arseneau, Senba, Reid and Mikula (1990)

N2 +1% C2D2

purenitrogengas

Chemical Decay of Muonium


5

Reid, Garner, Lee, Senba, Arseneau and Fleming (1987)

slope = kM

Muonium decay rate λ = λ0 + kM[X]

Extracting the Second-order Rate Constant


6

Experimental tests of reaction rate theory: Mu+H2 and Mu+D2Fleming et al, J. Chem. Phys. 1987

Fundamental Kinetic Studies: H + H2


7

Reaction Dynamics

reactants

transition state

products

Can the system tunnel through the potential barrier?

What is the probability that the system will move from reactants to products?

reaction coordinate

pote

ntia

l ene

rgy

What is the structure of the transition state?

How is this energy distributed amongst the reaction products?

How is energy supplied to overcome this barrier?


8

1-D Reaction Coordinate

The simplest reaction “surface” has 1 dimension, such as the interatomic distance in the dissociation of a diatomic. e.g. AB → A + B

The potential energy V is the internal energy U from thermodynamics.

In the Born-Oppenheimer Approximation the nuclear and electronic parameters are separable: product of wavefunctions, sum of energies.

The potential energy surface then corresponds to a plot of the energy of the system as a function of nuclear coordinates.

For r ≈ re the potential can be modelled by the simple harmonic oscillator.

21e e2( ) ( ) ( )V r V r k r r= + − V(r)

0rAB

Dissociation Energy

Bond length

( )e2

e e( ) 1 e r rV r D D−α −⎡ ⎤= − −⎣ ⎦

But extreme anharmonicity is needed to model dissociation:

The Morse Potential

e

2

2r r

d Vkdr =

⎛ ⎞= ⎜ ⎟

⎝ ⎠


9

2-D Potential Energy Surface

rA-B

A + BCAB + C

E

or a contour plot:

rB-C

rA-B

symmetric vibration of A-B-C

minimum energy path

A collinear triatomic reaction such as

A + BC → AB + C

needs a 3-D plot:

“collinear” = angle fixed


10

Views of a Potential Energy Surface

0.60

0.90

1.20

1.50

1.80

2.10

2.40

2.70

3.00

0.60 0.90

1.20 1.50

1.80 2.10

2.40 2.70

3.00

r (B-C)

r (A-B)

0.60

0.90

1.20

1.50

1.80

2.10

2.40

2.70

3.000.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00

r (B-C)

r (A-B)

0.600.901.201.501.802.102.402.703.00

0.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00 r (B-C)r (A-B)

E


11

Skewed Coordinate System

( )

( )

( )( )

12

12

ab bc

bc

a b c

c b a

a b c

2 a c

a b b c

X cosY sin

cos

r rr

m m mM

m m mM

M m m mm m

m m m m

= α + β θ= β θ

+⎡ ⎤α = ⎢ ⎥

⎣ ⎦

+⎡ ⎤β = ⎢ ⎥

⎣ ⎦= + +

θ =+ +

PE surfaces can be used for classical trajectory calculations as long as the effective mass of the reacting system (modelled by rolling ball) is constant.

A mass-weighted coordinate system diagonalizes the kinetic energy of the system.

X

Y

qvibrational excitation of product


12

The LEPS Surface

An analytic function is often more practical than a table of points − it is continuous and can have adjustable parameters.

London-Eyring-Polanyi-Sato (LEPS) surface

ab bc acab bc ac

ab bc ac1/ 22 2 2

ab bc bc ac ac ab

ab bc bc ac ac ab

( , , )1 1 1

1 1 1 1 1 1 12

Q Q QV r r rS S S

J J J J J JS S S S S S

= + ++ + +

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − + − + −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ + + + + +⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

2 212

2 212

( ) M( )(1 ) AM( )(1 )

( ) M( )(1 ) AM( )(1 )

Q r r S r S

J r r S r S

⎡ ⎤= + + −⎣ ⎦⎡ ⎤= + − −⎣ ⎦

( ) ( )

( ) ( )

e e

e e

2e

21e2

M( ) e 2e

AM( ) e 2e

r r r r

r r r r

r D

r D

− α − −α −

− α − −α −

⎡ ⎤= −⎣ ⎦⎡ ⎤= +⎣ ⎦

Q, J and S are derived from the Coulomb, exchange and overlap integrals of the Heitler-London valence-bond theory

Morse function

anti-Morse function


13

Tunnelling

Consider a particle of energy E striking a potential barrier of height V.

x

a

ikx ikxAe Be−ψ = +

( )122 /k mE=

( )122

x xCe De

m V E

κ −κψ = +

κ = −⎡ ⎤⎣ ⎦

ikx ikxA e B e−′ ′ψ = +

( )( )

122

2 116 1

a a

E EV V

e eAGA

−κ −κ⎧ ⎫−′ ⎪ ⎪= = +⎨ ⎬−⎪ ⎪⎩ ⎭

Application of boundary conditions gives the transmission probability:

the mass of the particleits energy (compared to the barrier)the width of the barrier

Tunnelling depends on:G(E)

Energy

classical

Q.M.

V0


14

Tunnelling in Chemical Reactions

G(E)

Energy

classical

Q.M.

V0

VE

quantum classical( ) ( ) ( )k T T k T= κThe transmission coefficient κ is the correction factor

The transmission probability G or permeability depends on energy.

B BB

B

// /quant quant quant /0 0 0

quant0//Bclass class0 0

( ) e e e( ) e( ) e e

E k TE RT V k TE k T

E k TE RT

V

k T dE G dE G dET G dE

k Tk T dE G dE dE

∞ ∞ ∞ −−∞ −

∞ ∞ ∞ −−κ = = = =∫ ∫ ∫

∫∫ ∫ ∫

The PE curve is often approximated by a standard function to get an analytic solution.

e.g the Eckart barrier gives2

B

1( ) 124

hTk T

⎛ ⎞νκ = + +⎜ ⎟

⎝ ⎠…

‡ imaginary frequency of reaction coordinate


15

Experimental tests of reaction rate theory: Mu+H2 and Mu+D2Fleming et al, J. Chem. Phys. 1987

Quasiclassical trajectory (and variational transition state theory) study of the rates and temperature-dependent activation energies of the reactions Mu+H2 (completely thermal) and H, D, and Mu+H2(v=0, j=2)Truhlar et al, J. Chem. Phys. 1983

Fundamental Kinetic Studies: H + H2


16

Hydrogen Isotopes

µ+

e−

p+

e−

HMu He+µ−

µ−

e−

He++

e−

D

p+n

0.11H 1H 2H 4.1H


17

Muonic Helium is a heavy isotope of hydrogen

Taken from a comment on a TRIUMF experiment by Don Fleming et al.

Published by AAAS

D G Fleming et al. Science 2011; 331: 448-450

Published by AAAS

D G Fleming et al. Science 2011; 331: 448-450

k0.11/k4.1


20

Muonium Isotope Effects

The chemistry of an atom depends primarily on

the ionization potential How easy is it to remove an electron?

the radius How are the electrons distributed?

For Mu these are almost the same as for H

However, for molecular vibrations involving Mu,

∴ = ≈υ υ υMuXHX

MuXHX HX

mm

3Xr

X

m mm m

m mμ

μμ

= ≈+

⇒ Vibrational frequencies involving Mu are higher than for H

if mX >> mµ

Mu―X


21

The Muon as Spectatorin a free radical rearrangement

Mu does not affect the reaction rate because it is remote from the reaction site

Burkhard, Roduner, Hochmann and Fischer (1983)

kR(Mu) = 9.3 × 106 M-1s-1 at 338 K

kR(H) = 8.8 × 106 M-1s-1 at 338 K

OMu

O

CH2Mu

H2C

O

CH2Mu

kR


22

Arrhenius Temperature DependenceThe Arrhenius “law” is an empirical description of the T dependence of the rate constant:

a /e E RTk A −=

aln ln Ek ART

= −

1/T

ln k

0

The pre-exponential factor is often interpreted as a collision rate. Collision theory predicts T½

dependence for A.Transition-state theory predicts linear T dependence for A.

The exponential factor describes the fraction of collisions with sufficient energy for reaction, as predicted by the Boltzmann distribution

n(E)

EEa

Curvature in the Arrhenius plot is often attributed to tunneling, but there are many other potential reasons.

1/T

ln k

0


23

time / µs0.0 0.5 1.0 1.5 2.0

350ºC

Muo

n A

sym

met

ry

375ºC

400ºC

Muonium Signals in1.4×10-4 M (Ni2+)aq at 250 atm

Negative Temperature Dependence!


24

Mu + Benzene

A fall-off of rate is common for reactions in high T water

0 100 200 300 400Temperature / °C

0.0

0.2

0.4

0.6

0.8

1.0

1.2

k Mu

/ 1010

M-1

s-1

250 bar> 310 bar

0 100 200 300 400Temperature / oC

107

108

109

1010

1011

k Mu

/ M-1

s-1

350 bar245 bar< 200 bar1 bar (lit.)

aqMu OH MuOH e− −+ → +


25

Non-Arrhenius Temperature Dependence

Rate Constants for Reaction of the Hydrated Electron in Water

Rat

e co

nsta

nt /M

-1s-1

1011

1010

109

1.7 2.2 2.7 3.2 3.71000K/T

N2O

NO3−

NO2−

Elliot, Buxton, et al., J C S. Far. Trans. (1990)


26

Diffusion-Reaction Kinetics

{ }Mu A MuA products+ ⎯⎯→

difo fbs act

1 1 1k k k

= +

For fast reactions in liquids the rate-determining step can be diffusion of the reactants to form the encounter pair.

diffusion reaction

actob

acs

di

f t

ff

di f

k kkk k

=+

( )( )Mu Aff Ai Mud 4k R R D D= π + +

or

( )act exp /ak A E RT= −

slow diffusion limit

fast diffusion limit, “reaction controlled”


27

Non-Arrhenius Temperature Dependence − 2

Example: Reaction of the Hydroxyl Radical with Hydroperoxyl

Elliot et al., AECL Report 11073 (1994)

1000K/T

Rat

e co

nsta

nt /M

-1s-1

1011

1010

2.0 2.5 3.0 3.5

diffobs react

1 1 1k k k

= +

negative activation energy?


28

Diffusion-Reaction Kinetics – Modified

{ }Mu A MuA products+ ⎯⎯→

difo fbs act

1 1 1k k k

= +

For fast reactions in liquids the rate-determining step can be diffusion of the reactants to form the encounter pair.

diffusion reaction

actob

acs

di

f t

ff

di f

k kkk k

=+

( )( )Mu Aff Ai Mud 4k R R D D= π + +

or

( )Ract exp /ak f A E RT= −

R collR -1

enc R coll

p Zfp Z

=τ +where

The reaction efficiency depends on the number of collisions of the reactant molecules per encounter.

slow diffusion limit

fast diffusion limit, “reaction controlled”

pR = orientation factor


29

Collisions per Encounter

2

liq 6dρη

τ =

calculated for Mu + hydroquinone

time

many collisions per encounter at low temperature

collision ≡ encounter for gas-like behaviour at high T

0 100 200 300 400

0.01

0.1

1

10

times

/ps

Temperature /°C

τcoll

τliq

τenc

0 100 200 300 400100

101

102

103

104

τ enc

/ τco

ll

Temperature /°C

1D

enc2

8[H O]

k −

τ = liq 1coll gas

gas

DZ

D−τ =


30

Diffusion-Reaction Kinetics − 3

0 100 200 300 400Temperature / °C

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

f RReaction Efficiency for Mu + Hydroquinone in water


31

0.1

1.0

10.0

100.0

0 100 200 300 400 500

Temperature / °C

k /

106 M

-1 s

-1

H abstraction from methanol by Mu (H)

H + CH3OHMezyk and Bartels, 1994 Percival et al., 2007


32

H + OH → H2O

0 100 200 300 400 500Temperature /°C

0.0

2.0

4.0

6.0

8.0

k / 1

010 M

-1s-1

M3

M1

AECL

M2

current PWR reactors

next generation reactorsData limited to 200°C

Buxton and Elliot, JCS Far. Trans. 89 (1993) 485

Ghandi and Percival, J. Phys. Chem. A 107 (2003) 3006


33

Supercritical-Water-Cooled Reactor

Canada is one of ten countries (the Generation IV International Forum) working together to lay the groundwork for fourth generation nuclear reactor systems.

The priority R & D areas for Canada include “improved understanding of radiolysis under supercritical water conditions and the effect of radiolysis products on corrosion and stress corrosion cracking”.

The Supercritical-Water-Cooled Reactor (SCWR) system is a high-temperature, high-pressure water cooled reactor that operates above the thermodynamic critical point of water (374°C, 22 Mpa)

The SCWR system is primarily designed for efficient electricity production.


34

Pressure Dependence of Reaction Rates

From classical thermodynamics T

G VP

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠dG VdP SdT= −

T

G VP

∂Δ⎛ ⎞ = Δ⎜ ⎟∂⎝ ⎠

For a reaction at equilibrium

ln 1T T

K G VP nRT P RT

∂ ∂Δ Δ⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠°

( ) ( )prod react react prod 1 1V V V V V V V V V−Δ − = − − − = Δ − Δ‡ ‡ ‡ ‡° = ° °

Since 1 1/K k k−= 1 1 1 1ln lnT T

k k V VP P RT RT

− −∂ ∂ Δ Δ⎛ ⎞ ⎛ ⎞− = − +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

‡ ‡° °

Most books state 1 1 1 1ln lnT T

k V k VP RT P RT

− −∂ Δ ∂ Δ⎛ ⎞ ⎛ ⎞= − =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

‡ ‡° °

This neglects the difference between K° and Kc

Since cK V K= ° ° for 1Δν = −

only different for 0Δν ≠

0ln

T

k VP RT

∂ Δ⎛ ⎞ = − − κ⎜ ⎟∂⎝ ⎠

‡ °compressibility of the solvent

volume of activation

ln 1

T T

V VP V P

∂ ∂⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠


35

Pressure Dependence − 2

Neglecting 0κ

0ln ln Vk k PRT

Δ= −

‡

P

0

ln kk

0

A BV V VΔ < +‡

A BV V VΔ > +‡If reactants..

combine

expect

dissociate

0VΔ <‡

0VΔ >‡

have like charges

unlike charges

0VΔ <‡

0VΔ >‡

⎫⎪⎬⎪⎭

similar effects for polar molecules (dipoles)Solvent effects are often dominant

Electrorestriction+

+

+

+

ordering of the solvent molecules reduces their effective volume


36

Volume of Activation for H Atom Reactions

AH A H HV V V V VΔ = − − ≈ −‡A AH+ H

A A+ H H ( )w HHV V VΔ ≈ −‡

( )HVΔ ‡ ( )MuVΔ ‡Comparison of and shows that V(Mu) > V(H) in water.


37

Parallel Reactions – Competition

Consider a molecule that can react by two different routes:A

C

Bkb

kcThe overall decay of A depends on both reactions:

( ) ( )b cb c b c 0e

k k tda k a k a k k a a adt

− +− = + = + =⇒

The rate of formation of each product depends on both rate constants:

Define a = [A], b = [B], c =[C].

( )

( )

b c

b c

b b 0b b

ccc c 0

e//e

k k t

k k t

db k a k a k adt kb db dtdtdc c k dc dtk adtk a k adt

− +

− +

⎫= = ⎪⎪ = = =⎬⎪= =⎪⎭

⇒ ∫∫

b

c

B yield of BC yie

[ ]ld of C[ ]

kk

= =

This is the basis for competition kinetics, whereby an unknown rate constant is determined from a known rate constant and the ratio of competitive products.


38

Muon Spin Dephasing During Reaction

Mu radical

ωM

ωR

N

Δt

Distribution of reaction times Dephasing of

muon spins


39

Muon Spin Dephasing During Reaction: Theory

P c sx ν12

12

12122

141221 1

a f =+

++

LNM

OQP

M2

M2

Δ ΔP c s

y ν1212

1212

12122

1212

141221 1

a f =+

++

LNM

OQP

M2

M2Δ

ΔΔΔ

P P Px y⊥ = +ν ν ν12 122

122 1/2a f a f a f

Δ1212 12 12= −ω ω λM Ra f /

Δ1214 12 14= −ω ω λM Ra f /

Δ4343 43M 43R= −ω ω λb g /Δ2343 43M 23R= −ω ω λb g /

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

100 1000 10000Magnetic Field / G

Muo

n P

olar

izat

ion

in R

adic

al

ω12R

ω34R

high field


40

Transfer of Muon Polarization as Mu → Radical

( )( ) ( ) ( ) ( )

1/ 22 22 2 2 2 2 2 2 2M R M M R R M R M M R R M R M M R R 1212 M R M M R R 14121

12 2 2 2 2 21212 1412 1212 14121 1 1 1

c c c s c s s c c s c s c c c s c s s c c s c sP

⎧ ⎫⎡ ⎤ ⎡ ⎤+ − + Δ − Δ⎪ ⎪⎢ ⎥ ⎢ ⎥ν = + + +⎨ ⎬+ Δ + Δ + Δ + Δ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

( )( ) ( ) ( ) ( )

1/ 22 22 2 2 2 2 2 2 2M R M M R R M R M M R R M R M M R R 2323 M R M M R R 43231

23 2 2 2 2 22323 4323 2323 43231 1 1 1

c c c s c s s c c s c s s s c s c s c s c s c sP

⎧ ⎫⎡ ⎤ ⎡ ⎤+ − + Δ − Δ⎪ ⎪⎢ ⎥ ⎢ ⎥ν = + + +⎨ ⎬+ Δ + Δ + Δ + Δ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

( )1214 12M 14Rwhere / etc.Δ = ω − ω λ

0.0

0.1

0.2

0.3

0 25 50 75 100 125 150 175 200

Magnetic Field / G

Muo

n P

olar

izat

ion

P 12

P 23

λ = 3.0 x 107 s-1

λ = 3.0 x 108 s-1

λ = 3.0 x 106 s-1


41

If the radical is formed via muonium there is loss of signal amplitude due to incoherent spin precession in the product. The muon polarization at the lower radical precession frequency is given by:

( )1/22

112 M2 2 2

12

RP h⎡ ⎤λ

= ⎢ ⎥λ + Δω⎣ ⎦

M

MM R

12 12 12

initial fraction of muon polarization in Mu[ethene] first-order reaction rate

change in precession frequency

hk=

λ = =

Δω = ω − ω =

Mu + CH2=CH2 → MuCH2-CH2·

0.0

0.1

0.2

0.3

0.4

0.5

0 2000 4000 6000 8000C2H4 Pressure / Torr

P12

(R)

10 -1 -1M 1 10 M sk = ×

10 -1 -1M 3 10 M sk = ×

11 -1 -1M 1 10 M sk = ×

11 -1 -1M 3 10 M sk = ×


42

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 2000 4000 6000 8000

C2H4 Pressure / Torr

A12

(R)

10 kG15 kG20 kG

The data for pure ethene can be described by a simple 1-step model in which Mu is converted to a radical.

Mu + CH2=CH2 → MuCH2-CH2·


43

( )1/ 22 2

1212 1412112 2 2 2 2 2

1212 1412 1212 14121 1 1 1C SC SP

⎧ ⎫⎡ ⎤ ⎡ ⎤Δ Δ⎪ ⎪ω = + + +⎨ ⎬⎢ ⎥ ⎢ ⎥+ Δ + Δ + Δ + Δ⎣ ⎦ ⎣ ⎦⎪ ⎪⎩ ⎭

( )1/ 22 2

4343 2343143 2 2 2 2 2

4343 2343 4343 23431 1 1 1C SC SP

⎧ ⎫⎡ ⎤ ⎡ ⎤Δ Δ⎪ ⎪ω = + + +⎨ ⎬⎢ ⎥ ⎢ ⎥+ Δ + Δ + Δ + Δ⎣ ⎦ ⎣ ⎦⎪ ⎪⎩ ⎭

2 2R1 R 2 R1 R1 R 2 R 2

2 2R1 R 2 R1 R1 R 2 R 2

C c c c s c s

S s c c s c s

= +

= −

( ) /klmn kl mnΔ = ω − ω λ

( )( )

1/ 2

e 2 21/ 222

0 e

1 , 1c s cμ

μ

⎧ ⎫ω + ω⎪ ⎪= + = −⎨ ⎬⎡ ⎤ω + ω + ω⎪ ⎪⎣ ⎦⎩ ⎭

( )( )

1/ 22

112 2 22

12R1 12R2

P⎡ ⎤λ

ω = ⎢ ⎥λ + ω − ω⎢ ⎥⎣ ⎦

Transfer of Muon Spin Polarization Between Radicals


44

107 108 109 1010

0.0

0.2

0.4

0.6

0.8

1.0

P12

/(P12

) max

λ /s-1

R1(hfc 722.5 MHz) → R2(hfc 235.4 MHz) at 14.5 kG

Measured polarization = 51% ⇒ λ = 8.9 × 108 s-1

⇒ k = 5.7 × 108 M-1s-1

( )( )

1/ 22

112 2 22

12R1 12R2

P⎡ ⎤λ

ω = ⎢ ⎥λ + ω − ω⎢ ⎥⎣ ⎦

McCollum, Brodovitch, Clyburne, Percival and West, Physica B, 404 (2009) 940-942.

Radical Coupling of a Silyl to Form a Disilanyl

NSi

N

tBu

tBuMu

N

N

tBu

tBu

SiN

SiN

tBu

tBuMu

N

N

tBu

tBu

Si+

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