Kinetics and Dynamics

Transition-state Theory (TST)

Video VII.viii

Transition-state Theory and Kinetics

Elementary Unimolecular Reactions

A B

€

−d A[ ]dt

= k1 A[ ]

€

k1 =kBThQ ‡

QA

e− U‡, 0 −UA, 0( ) / kBT

A + B C

Elementary Bimolecular Reactions

€

−d A[ ]dt

= k1 A[ ] B[ ]

€

k1 =kBTh

Q ‡

QAQB

QAoQB

o

Q ‡, o e− U‡, 0 −UA, 0 −UB, 0( ) / kBT

Reaction Coordinate

E

A‡kact

kdeact

k‡

A B

kB is Boltzmann’s constant, h is Planck’s constant, T is temperature, Q is the partition function, and U0 is the internal energy at 0 K (E + ZPVE)

standard-state again…

In general:

€

k =kBTh

Q‡

QR

QRo

Q‡,o e−ΔV ‡ / kBT =

kBThe−ΔG

o,‡ /RT

TST, Eyring, and Arrhenius Expressions

€

k =kBTh

Q‡

QR

QRo

Q‡,o e−ΔV ‡ / kBT =

kBThe−ΔG

o,‡ /RT =kBThe−ΔH

o,‡ /RTeΔSo,‡ /RTST

Eyring

€

ln kT⎛

⎝ ⎜

⎞

⎠ ⎟ = −

ΔH o, ‡

RT+ΔSo, ‡

R+ ln kB

h⎛

⎝ ⎜

⎞

⎠ ⎟ Plot ln(k / T) vs. 1/T

Arrhenius

€

k = Ae−Ea / RT Plot ln(k) vs. 1/T

€

Ea = ΔH o, ‡ + RT

€

A =kBThe 1+ΔS

o, ‡ / R( )

Be very careful making comparisons

What is a Block CoPolymer?Situation:

Consequence:If you want to design new materials that incorporate properties of both polymers on small length scales, you must keep the polymers from phase separating by covalently attaching chains of one type to chains of the other type, e.g., AAAAAAAAAAAAA–BBBBBBBBBBBBBBB

UsesThermoplastic elastomers (e.g., running shoe soles)Pressure sensitive adhesives (Post-It™ Notes)Viscosity modifiers for oilsCompatibilizers (the polymer equivalent of a soap)

Mixtures of two polymers—even seemingly very similar polymers—nearly always phase separate rather than "alloy"

Challenge:How can you synthesize a well-defined BCP (e.g., having low polydispersity)?

R2

F

F

R1

• mild• selective (no other insertion products)• quantitative• experimentally simple

R2

O

FF

FCF3

O

F CF3

FF

R1

+

n

polyisoprene H Mepolybutadiene H H

polydimethylbutadiene Me Me

R1 R2

n

180 °C

One Technique for Making Fluorinated BCPs

R2

CnF2n

CnF2n

R1 R2CnF2nF2nCn

R1

n

n

If One Fluorine is Good... (E. I. DuPont)

Are there concerns?

Carbene Rearrangements in Hydrocarbons

CH3

CH3

HH

CH3

CH3

HH

HHH

CH3

H

H

H

CH3

HCH3

1,2-H shift

ΔG‡ = 5.2 kcal/mol

1,3-H shift

ΔG‡ = 8.3 kcal/mol

1,2-CH3 shift

ΔG‡ = 18.1 kcal/mol

Kinetics 101Carbene additions typically proceed without an activation barrier. The rates of barrierless reactions in solution are typically "diffusion controlled". Over a reasonable range of viscosities, an appropriate rate expression is:

Ratebi (M sec–1) ≈ 1010 • [A] [B]

Unimolecular rearrangements typically follow a particularly simple rate law:

Rateuni (M sec–1) ≈ 1014 • [A] • exp(–ΔG‡ / RT)

We would like the ratio of bimolecular reaction to unimolecular rearrangementto be at least a factor of 100, i.e.,

Ratebi (M sec–1)

Rateuni (M sec–1)= 100 = 10–4 • [B] • exp(ΔG‡ / RT)

Given a realistic maximum [B] (molar concentration of double bonds) of about1 M, this implies the minimum activation energy for unimolecular rearrangement cannot be lower than 12.9 kcal/mol at 200 °C

Carbene Rearrangements in Fluorocarbons

CF3

F

FF

CF3

FF

F

FFF

F

F

F

F

FF

CF3

1,2-F shift

ΔG‡ = 25.9 kcal/mol

1,3-F shift

ΔG‡ = 36.4 kcal/mol

1,2-CF3 shift

ΔG‡ = 18.5 kcal/mol

Because fluorine holds electrons more "tightly" than hydrogen, it is muchharder to insert into C–F bonds than into C–H bonds. Interestingly, the

accessibility of C–C bonds is relatively unperturbed by H vs. F.

C

F

1.8191.723

1.963 2.091

1.897

1.561

Feasibility Study on Epoxide Cracking

Kinetics 102: Left path preferred by about 5,000,000 to 1 at 200 °C

‡ ‡

++

COF

ΔG‡ = 32.0kcal/mol

ΔG‡ = 46.7kcal/mol

Feasibility Study on Epoxide Cracking 2

Kinetics 103: Half-life for a unimolecular process (like cracking) is roughly

COF

ΔG‡ = 52.1kcal/mol

‡

+

t1/2 (sec) ≈ ln2 • 10–14 • exp(ΔG‡ / RT)

For above reaction at 200 °C, 50% cracking takes 317 years . . .(4.8 hours for previous example via its preferred path)

Cramer and Hillmyer J. Org. Chem. 1999, 64, 4850

Kinetics and Dynamics

Kinetic Isotope Effects

Video VII.ix

Kinetic Isotope Effects

Reaction Coordinate

E

hωlight/2hωheavy/2

ΔVheavy‡

ΔVlight‡

Primary KIE from difference in ZPVE

Reaction Coordinate

E

hωlight/2hωheavy/2

ΔVheavy‡

ΔVlight‡

hωheavy/2hωlight/2‡

‡

RR

Secondary KIE from difference in change in ZPVE

Very straightforward calculation

€

klightkheavy

=

Qlight‡

QR, light

e−ΔVlight‡ / kBT

Qheavy‡

QR, heavy

e−ΔVheavy‡ / kBT

=Qlight‡

Qheavy‡

QR, heavy

QR, light

e− ΔZPVElight‡ −ΔZPVEheavy

‡( ) / kBT

Can be very useful for validating quality of computed transition-state structures

Protein Prenylation

Farnesylation of ras protein key to carcinogenesis

cf = 0.057 for PFT + GPP

1° 13C KIE = 1.039 ± 0.003 2° 2H KIE = 1.068 ± 0.003

What is structure of transition state?

OP

OP

O–

O O

O– O–n

n = 1, Farnesyldiphosphaten = 0, Geranyldiphosphate

SH

SNH

HN

NH

HN

NH

O

O O

O

O

OHO

O

N

PFTase

S

S NH

HN N

H

HN N

H

O

O O

O

O

OHO

O

N

13C

N-Dansyl-GCVIA

N-Dansyl-GC(S-geranyl)VIA

Choice of Theoretical Model (Validation)

O

Cl

PPh3

Associative Model

ReactionPh3PTHF

Cl

O

Dissociative Model

ReactionPhCH2OH

DMF

SN2KIEexpt = 1.040±0.003

KIEtheor = 1.040

1° 13C KIE

SN1KIEexpt = 0.997±0.003

KIEtheor = 1.001

mPW1N/6-31+G(d) density functional theory

Modeling Prenylation mPW1N applied to GPP / ethanethiolate (aq)

1° 13C KIE = 1.039 ± 0.003 2° 2H KIE = 1.068 ± 0.003

OP

OP

O–

O O

O– O–n

n = 1, Farnesyldiphosphaten = 0, Geranyldiphosphate

SH

SNH

HN

NH

HN

NH

O

O O

O

O

OHO

O

N

PFTase

S

S NH

HN N

H

HN N

H

O

O O

O

O

OHO

O

N

13C

N-Dansyl-GCVIA

N-Dansyl-GC(S-geranyl)VIA

OPO

PO–

O O

O– O–

S

S

13C

SN2 process 1° 13C KIE = 1.067 2° 2H KIE = 1.150

Controlling Making and Breaking Bonds 1° 13C KIE = 1.039 ± 0.003 2° 1H KIE = 1.068 ± 0.003

OPO

PO–

O O

O– O–S

S

13C

+

Active Site Influence on TS Structure

Uncat. Cat.

Looser — more ionic Consistent with rate deceleration by electron-withdrawing groups

Kinetics and Dynamics

Tunneling, Variational Transition-state Theory (VTST), and Marcus Theory

Video VII.x

Quantum Effects on the Rate Constant Reaction Probability Through a Parabolic Barrier

Reactant Energy

0

1

dPdE

ΔV‡

non-classicalreflection

tunneling

Temperature-dependentBoltzmann distribution

P

€

k = κ T( )kBThQ ‡

QR

QRo

Q ‡, o e−ΔV ‡ / kBT

Tunneling and Eyring Plot Curvature

1 / T

ln k

light isotope

heavy isotope

no tunneling turnover regime all tunneling

Tunneling in a Nutshell •  Typically only significant for reaction coordinates

having large proton, H atom, or hydride motion •  Typically less significant at higher temperatures (but

demonstrated to be important in many biological systems at their temperatures!)

•  Accounting for tunneling is, frankly, hard, although the Skodje-Truhlar approximation is fairly straightforward

•  Beware of experimental data that may be interpreted incorrectly because of a failure to consider tunneling!

Skodje-Truhlar

Methane Metathesis in Lutetiocene (kcal/mol)

M CH3 MCH3

13CH3

H– 13CH4

13CH4

M 13CH3

– CH4

CH4

€

ΔH‡

mPWPW91/ECP 20.3

Eyring plot 11.6

Sherer and Cramer Organometallics 2003, 22, 1682

Fooled by Tunneling

€

ΔH‡ =11.6

€

ΔH‡ =19.2DFT 20.3

kcal/mol

Variational Transition-state Theory

Reaction Coordinate

G

A‡kact

kdeact

k‡

A B

E

€

kVTST T ,s( ) = mins

kBTh

Q‡ T ,s( )QR

QRo

Q‡,oe−ΔV

‡ s( ) / kBT

s may be different for H and D (because Q is)

Electron Transfer—A Very Hard KIE Problem

q

G

A– + B A + B–

λ

λ/4

q

q

G

G

–ΔGo = λ

ΔGo = 0

–ΔGo > λ

€

kET = Ze− ΔG o +λ( )2 / 4λRT Semiclassical effects in ΔGo

Quantum (tunneling) effects in Z

THANKS!!