Page 356 - Advanced Organic Chemistry Part A - Structure and Mechanisms, 5th ed (2007) - Carey _ Sundberg

Substituting K and k with the appropriate expression for free energy gives 337

m logK −logK = logk−logk 0 SECTION 3.6
0
Linear Free-Energy
‡

m − G/2 3RT + G /2 3RT = G /2 3RT + G /2 3RT Relationships for
o
0
Substituent Effects
‡
m − G+ G =− G + G ‡ 0 (3.43)
0
m G = G ‡
‡
The linear correlation therefore indicates that the change in G on introduction of
a series of substituent groups is directly proportional to the change in the G of
ionization that is caused by the same series of substituents on benzoic acid. The
correlations arising from such direct proportionality in free-energy changes are called
linear free-energy relationships. 129
‡
Since G and G are combinations of enthalpy and entropy terms, a linear
free-energy relationship between two reaction series can result from one of three
circumstances: (1) H is constant and the S terms are proportional for the series;
(2) S is constant and the H terms are proportional; or (3) H and S are linearly
related. Dissection of the free-energy changes into enthalpy and entropy components
has often shown the third case to be true.
The Hammett linear free-energy relationship is expressed in the following
equations for equilibria and rate data, respectively:
K
log = (3.44)
K 0
k
log = (3.45)
k
0
The numerical values of the terms and are defined by selection of the reference
reaction, the ionization of benzoic acids. This reaction is assigned the reaction
constant = 1. The substituent constant, , can then be determined for a series of
substituent groups by measurement of the acid dissociation constant of the substituted
benzoic acids. The values are then used in the correlation of other reaction series,
and the values of the reactions are thereby determined. The relationship between
Equations (3.44) and (3.45) is evident when the Hammett equation is expressed in
terms of free energy. For the standard reaction log K/K = :
0
− G/2 3RT + G /2 3RT = = (3.46)
0
since = 1 for the standard reaction. Substituting into Eq. (3.42):
‡
‡
m =− G /2 3RT + G /2 3RT
m = logk−logk 0
(3.47)
k
m = log
k
0

129
A. Williams, Free-Energy Relationships in Organic and Bio-Organic Chemistry, Royal Society of
Chemistry, Cambridge, UK, 2003.