Linear Free Energy Rela.onships Idea: understand new behavior by comparing it to old behavior Examples: -­‐ Hamme= (electronic effects on rate) -­‐ Bronsted (acidity on rate) -­‐ TaC (sterics on rate) Defini1on: a linear log-­‐log correla.on between the rate/equilibrium constants in one series of reac.ons with the rate/equilbrium constants in another series of reac.on 2
1
m = slope, the strength of the effect b = intercept, the rate of the reference reac.on In general: -­‐  rates (kine.cs) are related to equilibria (thermodynamics) -­‐  these correla.ons only hold within a series of closely related compounds ΔG = mΔG + b
!
Hamme= Equa.on (1933) O
OH
K
O
+ H 2O
X
+ H 3O
X
O
O
OCH3 + NMe
3
X
reference equilibria:
O
k
O
X
+ NMe 4
acid-base dissociation
of benzoic acids
reaction rates of interest:
alkaline ester hydrolysis
The linear correla.on between log K and log k implies a linear correla.on between the free energies: −ΔG0
log K =
2.3RT
‡
kBT
−ΔG
logk =
+ log
2.3RT
h
!
The second equa.on is the Eyring rela.on. Hamme= Chem. Rev. 1935 17 125 Hamme= Equa.on O
OH
K
O
+ H 2O
X
+ H 3O
O
X
X
O
O
O
OCH3 + NMe
3
X
O
O
k
O
X
+ NMe 4
X
δ
O
NMe 3
Why is there a correla1on? (1)  The posi.on of the ground state acid-­‐base equilibrium is primarily determined by the stability of the benzoate anion. (2)  The rate of the hydrolysis reac.on depends on the stability of the transi.on state. If nucleophilic a=ack is rate limi.ng, then the benzoate and the transi.on state are similar in structure. Threfore, subs.tuents will have a similar effect on both. (3)  The correla.on is commonly interpreted as a measure of electronic effects. (For a more detailed analysis of benzoic acid acidity, please see J. Chem. Ed. 1971 48 338.) Hamme= Equa.on Tradi.onally, the Hamme= LFER is wri=en as: ⎛ kX ⎞
log ⎜ ⎟ = ρσ
⎝ kH ⎠
!
kX – rate of reac.on with subs.tuent X kH – rate of reac.on with no subs.tuent (hydrogen) σ – subs.tuent constant, rela.ve pKa of X-­‐subs.tuted benzoic acid vs. benzoic acid: KX
σ ≡ log
= pK a PhCO2H − pK a X,PhCO2H
KH
!
ρ – reac.on constant, the slope of the correla.on note: capital le=ers K represent equilibrium constants, lowercase le=ers k are rates (
)
(
)
Hamme= Subs.tuent Constants (σ) KX
(
)
(
)
σ ≡ log
= pK a PhCO2H − pK a X,PhCO2H
KH
!
σpara σmeta X– posi1ve = electron withdrawing CH3C(=O)O– CH3C(=O)– NH2– Br– t-­‐Bu– HOC(=O)– Cl– NC– F– H– HO– CH3O– CH3– NO2– Ph– CF3– 0.45 0.50 –0.66 0.23 –0.20 0.45 0.23 0.66 0.06 0 –0.37 –0.27 –0.17 0.78 0.01 0.54 0.37 0.38 –0.16 0.37 –0.10 0.37 0.37 0.56 0.34 0 0.12 0.12 –0.07 0.71 0.06 0.43 nega1ve = electron dona1ng -­‐ ortho not used because of sterics -­‐ meta is a measure of pure induc.on -­‐ para is induc.on and resonance (by defini1on) OR groups are e– dona1ng as para subs1tuents, but e– withdrawing as meta subs1tuents Hansch, Leo, TaC Chem Rev 1991 91 165 Hamme= Reac.on Constants benzaldehyde + semicarbazide (pH 1.75): ⎛ kX ⎞
log ⎜ ⎟ = ρσ
⎝ kH ⎠
!
-­‐ ρ measures the rela.ve sensi.vity of the reac.on to electronic effects -­‐ “rela.ve” means transi.on vs. ground state |ρ| > 1: more sensi.ve than benzoic acid dissocia.on
|ρ| < 1: less sensi.ve than benzoic acid dissocia.on
ρ > 0: electron withdrawing groups speed up the reac.on (same direc.on as the benzoic acid dissocia.on) ρ < 0: electron dona.ng groups speed up the reac.on J. Chem. Ed. 1971 48 103 Some examples: CH3
X
C
CH3
Cl + H2O
CH3
X
C
+ HCl
CH3
r = –4.48 (prototypical SN1) rate-limiting
OH
H 2O
‡
CH3
X
C
δ+
CH3
δ–
X
Cl
C
CH3
CH3
late transition state
large δ+ on C
δ+
OH2
X
H2
C
Cl + H2O
X
δ+
C
δ–Cl
‡
H
H2
C
X
H
little positive charge
build up in the transition state
+ HCl
r = –1.31 (prototypical SN2) OH
X
OH + H2O
X
O – + H 3O +
ρ = +2.26
CH2NH2 + H3O+
X
X
CH2NH3+ + H2O
X
NH3+ + H2O
ρ = –1.05
X
NH2 + H3O +
ρ = –3.19
Non-­‐Linear Hamme= Plots Concave down curvature indicates a change in rate-­‐determining step: O
OH
H
X
N
H
+ n-BuNH2
X
–H 2O
N
X
le9 hand por<on rds is addi.on, which is sped up by EWGs (slope is posi.ve) right hand por<on rds is elimina.on, which is sped up by EDGs (slope is nega.ve) Non-­‐Linear Hamme= Plots Concave up curvature indicates a change in mechanism: -­‐ reac.ons are performed in 99.9% H2SO4 -­‐ methyl ester hydrolysis is accelerated by electron dona.ng groups -­‐ EDGs assist the departure of methanol and stabilize the acylium ion -­‐ this corresponds to an AAC1 mechanism (unimolecular acyl-­‐carbon cleavage in acid) O
O
O
X
CH 3
+
O
H
H
X
O
CH 3
OH2
X
Non-­‐Linear Hamme= Plots Concave up curvature indicates a change in mechanism: -­‐ with ethyl esters and electron withdrawing groups, a new mechanism becomes available -­‐ this mechanism is accelerated by electron withdrawing groups -­‐ EWGs assist the departure of benzoic acid -­‐ this corresponds to an AAL1 mechanism (alkyl-­‐oxygen cleavage) -­‐ ethyl ca.on is metastable in superacid media O
O
X
O
O
+
OH
O
H
H
X
X
+
CH2CH 3
Non-­‐Linear Hamme= Plots Curved plots can indicate a gradual change in mechanism: Cl
N
+
PhH, 100 °C
X
Br
N
X
X
Cl
N
+
acetone, 20 °C
N
X
Br
SN1-­‐like, large slope SN2-­‐like, small slope Meta and para subs.tuents are on separate lines because para subs.tuents are able to stabilize posi.ve charge through resonance as well as induc.on. Swain JACS 1951 73 2813 Changing the Reference In some cases, the correla.on with Hamme= constants may be poor. For example, subs.tued phenol and benzoic acid pKa values are not well correlated for strongly electron dona.ng or withdrawing subs.tuents: Changing the Reference Benzoate is stabilized by resonance to a far smaller degree than is phenolate: O
O
O
O
O
O
N
N
N
N
O
O
O
O
O
O
O
O
benzoate reference
phenolate reference
Which reference is best will depend on the reac.on. We can define a modified subs.tuent constant using the phenolate reference: ⎛ kX ⎞
log ⎜ ⎟ = ρ −σ −
⎝ kH ⎠
KX
(
)
(
σ ≡ log
= pK a PhOH − pK a X*PhOH
KH
!
−
)
Jaffe Chem. Rev. 1953 53 191 Changing the Reference For example, the hydrolysis of phosphate dianions involves expulsion of phenolate: O +OO +O N
N
O
X
O- O
P
O
O
O
+
X
P
O
O-
The slope is 2.6 and the rate is highly correlated with leaving group pKa, sugges.ng bond-­‐breaking is highly advanced in the transi.on state. Kirby JACS 1967 89 415 Changing the Reference For reac.ons involving resonance-­‐stabilized carboca.ons, the canonical reference is the solvolysis rate of cumyl chlorides: Cl
X
X
X
Brown showed that this is a much be=er reference for many electrophilic aroma.c subs.tu.on reac.ons. This is benzhydryl chloride ethanolysis: Brown and Okamoto JACS 1957 79 1913, JOC 1957 22 486 Changing the Reference In the solvolysis of benzyl tosylates in aqueous acetone, Brown proposes a change in mechanism from SN1 (electron dona.ng) to SN2 (electron withdrawing): O
O
S
O
X
H 2O
HO
X
The clear break in this plot was used to argue that SN1 and SN2 are in fact, dis.nct mechanisms, rather than extremes along a con.nuum. Brown and Okamoto JACS 1957 79 1913, JOC 1957 22 486 Steric Effects: TaC Equa.on In ortho-­‐subs.tuted or alipha.c systems, no single reference, however chosen, is sufficient because of steric effects. Following a sugges.on by Ingold, TaC a=empted to separate electronic and steric effects by studying ester hydrolysis: basic conditions
R
O
O
O
O
R'
R
O
OH
R'
R
O
H
+ R'OH
H
+ R'OH
HO
neutral
anionic
acidic conditions
H
R
O
OH
O
O
R'
H 2O
cationic
R
O
OH2
R'
R
O
cationic
TaC proposed that under basic condi.ons, the substrate becomes anionic, so the rate is affected by both steric and electronic effects. add reference to review by Shorter (1970) Steric Effects: TaC Equa.on Under acidic condi.ons, the substrate remains ca.onic, so presumably the reac.on rate is dominated by steric effects. acidic conditions
H
O
OH
O
H + R'OH
R'
R'
R
O
R
O
R
O
OH2
H 2O
cationic
cationic
This is supported by the Hamme= rho values for benzoate ester hydrolysis: basic condi.ons: +2.2 to +2.8, acidic condi.ons: –0.2 to +0.5 TaC defined a new polar subs.tuent constant: ⎡ ⎛k⎞
⎛k⎞ ⎤
*
σ ≡ ⎢log ⎜ ⎟ − log ⎜ ⎟ ⎥ /2.48
⎝ k0 ⎠ A ⎥⎦
⎢⎣ ⎝ k0 ⎠ B
!
k is the rate for the hydrolysis of R-­‐CO2R’ k0 is the rate for the hydrolysis of Me-­‐CO2R’ B and A mean basic and acidic condi.ons, respec.vely 2.48 roughly equalizes the magnitude with that of the Hamme= constants Steric Effects: TaC Equa.on ⎡ ⎛k⎞
⎛k⎞ ⎤
σ ≡ ⎢log ⎜ ⎟ − log ⎜ ⎟ ⎥ /2.48
⎝ k0 ⎠ A ⎥⎦
⎢⎣ ⎝ k0 ⎠ B
!
*
The ra.os normalize the effect against a standard acetate. The first term contains steric and electronic effects, while the second term contains only steric effects. This means the result should only contain electronic effects. TaC called the second term the steric subs.tuent constant: ⎛k⎞
E S ≡ log ⎜ ⎟
⎝ k0 ⎠ A
!
Thus, the TaC–Ingold LFER is: ⎛k⎞
log ⎜ ⎟ = ρ *σ * + δ E S
k
! ⎝ 0⎠
where each reac.on now has two sensi.vity factors, one for electronic and one for sterics. Steric Effects: Charton and Sterimol Values Charton showed the the TaC steric constants are highly correlated to van der Waals radii, indica.ng that they are a true measure of steric effects. Charton modified the TaC constants so that they could be used with van der Waals parameters directly. The Verloop Sterimol parameters take this a step further by considering the fact that subs.tuents are non-­‐spherical. Charton JACS 1969 91 615, Sigman JOC 2009 74 7635, Nat Chem 2012 4 366 Applica.ons to Asymmetric Catalysis Sigman has shown that the enan.oselec.vity of NHK allyla.ons can be studied with Charton parameters. The breaks in the graph might suggest a change in the catalyst structure or mechanism. Sigman instead proposes that the Charton parameters are flawed because they consider the minimum radius of rota.on about the primary subs.tuent axis. Effec.vely, subs.tuents are considered spherical! The Charton values of these groups is the same: n-­‐Pr, n-­‐Bu, i-­‐Bu: v = 0.68 Sigman Nat Chem 2012 4 366 Applica.ons to Asymmetric Catalysis Re-­‐analysis of the same NHK allyla.on data with the Sterimol parameters shows a much be=er fit. Other systems have been analyzed with good results. Sigman Nat Chem 2012 4 366 Synergis.c Electronic and Steric Effects Regardless of the size of X and Y, the oxazoline-­‐proline catalysts are poorly selec.ve: This is simply a two-­‐dimensional free energy rela.onship: ‡
2
2
2
!ΔΔG = −1.87 +1.34X + 3.03Y − 0.99Y −1.19XY + 0.62XY − 0.73YX
Note that this is a quadra.c, rather than, linear func.on. This is called a response surface. The presence of linear and quadra.c cross-­‐terms indicates an interac.on between X and Y. Sigman Science 2011 333 1875, JACS 2013 130 2482 Synergis.c Electronic and Steric Effects The quinoline-­‐proline scaffold is much more selec.ve. A similar response surface was fi=ed to the data: ΔΔG ‡ = −1.20+1.22E + 2.84S −
!
0.85S 2 − 3.79ES +1.25ES 2
This equa.on describes the data very well: Sigman Science 2011 333 1875, JACS 2013 130 2482 Synergis.c Electronic and Steric Effects Electronic and steric factors interact strongly. Electron poor ligands are poorly sensi.ve to sterics, while electron rich ligands are improved greatly by larger subs.tuents. This argues for op.mizing variables together, rather than one at a .me. Sigman Science 2011 333 1875, JACS 2013 130 2482