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
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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
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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!!