Handout: Linear Free Energy Relationships (LFER) (see also A.&D. p.445-453)
1. Correlations between rates and equilibria of reactions of compounds containing substituted phenylgroups was first noted by L. Hammett in 1937.
Hammett found that the following equation (1) applies for the rates of alkaline hydrolysis ofsubstituted ethyl benzoates and the acidities (dissociation constants) of the corresponding benzoicacids.
Hammett equation(LFER)XX

OOH-OEtkO

H2OOHm log

K= logKoXOkko+ EtOHOXKOOKo = K of benzoic acid X=HK = K of subst. Benzoic acid X≠Hko = k of ethyl benzoate X=Hk = k of subst. Ethyl benzoat X≠H
Note: Bronsted equations are LFERs)+ H3O(1)2. From (1) :m (logK – logKo) = logk – logko
andm (-∆G/2.3RT + ∆Go/2.3RT) = -∆G‡/2.3RT + ∆Go‡/2.3RT (2)thusm (-∆G + ∆Go) = -∆G‡ + ∆Go‡
andm ∆∆G = ∆∆G‡ (3)
In (2) we make use of definition of equilibrium constant K = exp(- ∆G/RT) or logK = -∆G/2.3RTand transition state theory: k = κkT/h × exp(-∆Go‡/RT) or logk = C - ∆Go‡/2.3RT; C = log(κkT/h)
Equation (3) means that changes in ∆G‡ on introducing a series of substituents are directly proportional
to the changes in ∆G of ionization that is caused by the same series of substituents.
This makes sense:
r.d. step is addition of OH- to benzoate with the following TS:OXδHO OEtδ

TS will decrease, the more X can aid instabilizing δ- charge at O

Dissociation constant largly depends on stability of conj. Base:XOOAcidity will increase, the more X can aid instabilizing - charge at O
Thus, inductive and resonance effects that increase acidity of the subst. Benzoic acid will also have astabilizing effect on the transition state for the hydroysis of the ester.

3. Hammett equation is typically expressed as follows:
Log(k/ko) = σ × ρ (4)Log(K/Ko) = σ × ρ (5)ρ is the reaction constantρ = 1 for ionization of benzoic acid in water (standard reaction)

σ is the substituent constantcan be empirically determined from dissociation constants ofsubstituted benzoic acids

It follows that σ = 0 for X = H.
Note: Swain-Scott Nucleophilicity constants are derived from a LFER of the type (4). ko in (4)corresponds to the rate for the reference nucleophile (CH3OH) and σ to the nucleophilicity constantn. The constant S in the Swain-Scott relationship corresponds to ρ above and allows application ofthe Swain-Scott relationship to other nucleophilic reactions. Again, S = 1 for the standard reaction(Nu + CH3I)
4. Determination of ρ values
σ values defined for different X with standard
reaction are used to determine empirically ρfor other reactions (e.g. hydrolysis of ethylbenzoate) see Table to the right5. Meaning of ρ: The value of  ρ  is a measure of how sensitive the reaction is towards the nature ofnearby substituents. E.g. in SN1 reaction,  ρ  is related to the amount of δ+ in the TS, which is larger
for a late TS.
ρ is positive, if substituents stabilize/destabilize a negative or partial negative charge.
ρ is negative, if substituents stabilize/destabilize a positive or partial positive charge.
6. σ values are typically listed for meta and para substituents (σm, σp; see Table next page)
There are no σ values for ortho substituents, because the different steric interactions of an ortho
substituent with the reaction center would prevent good Hammett relationships.
7. Different electronic effects contribution to the σ values:
Resonance Effects:
OCH3

O

CH3

Field Effects: the dipole of the C-X bond will
have an effect on a δ+ elsewhere in the molecule.
This effect is transmitted through space.

CNCNY

δ+stabilizes δ+ at Ydestabilizes δ+ at YXY

Inductive Effects: the dipole of the C-X bondis transmitted through polarization of theintervening sigma bonds.

δ+Ystabilizes δ+ at YX

δ+XY

δ+

δ-δ+

δ-δ+
X

destabilizes δ+ at Y

8. Table of σ values
for common organicsubstituents

9. The dual-substituent-parameter equation (6), tries to separate resonance and inductive effects.
σR and σI are the reaction constants, which refect the sensitivity of the system towards resonance andinductive effects.
The sum of σR and σI is approximately σp: σp = σR + σI
If there is direct resonance between the substituent and a cationic or anionic reaction center, theresonance component strongly increases and σp and σm fails to correlate the reaction series. In thiscase σ+ and σ- are used, which take into account the enhanced resonance component in these systems.
E.g.:
OO
H3COCH2

H3COCH2

NOO

NOO

10. Application of Hammett Equation:
1. Calculate how much faster p-bromobenzyl chloride will solvolyze in water than will pnitrobenzyl chloride.
Answer: kBr/kNO2 = 5.25.
2. The r value for alkaline hydrolysis of substituted methyl benzoates is 2.38. The rate forsaponification of methyl benzoate is 2 x 10-4 M-1s-1. Calculate the rate constant for methyl mnitrobenzoate
Answer: 98 x 10-4 M-1s-1
3. The pKa of p-chlorobenzoic acid is 3.98, that of benzoic acid is 4.19. Calculate σ for p-Cl
Answer: 0.21