Chemistry Reference
In-Depth Information
Data are typically graphed as illustrated in Fig.
2
:
log (
K
'/
K
'
0
)
Figure 2:
Example of a graph for a linear free energy relationship.
K
0
and
K
'
0
represent equilibrium constants for unsubstituted
compounds and
K
or
K
' for substituted derivatives. Values for the abscissa are calculated from the dissociation constants of
unsubstituted and substituted benzoic acid. Values for the ordinate are obtained from phenylacetic acid with identical patterns of
substitution.
Because this relationship is linear, equation (1) can be written:
log (
K
/
K
o
) = log (
K
'/
K
'
o
)
(1)
where is the slope of the line. The values for the abscissa in Fig.
2
are always those for benzoic acid and are given
via the symbol . Therefore, the following can be written:
log (
K
/
K
o
) =
(2)
, the slope of the line, is a proportionality constant pertaining to a given equilibrium. It relates the effect of
substituents on that equilibrium to the effect of those substituents on the benzoic acid equilibrium. That is, if the
effect of substituents is proportionally greater than on the benzoic acid equilibrium, then > 1; if the effect is less
than on the benzoic acid equilibrium, < 1. By definition, for benzoic acid is equal to 1.
is a descriptor of the substituents. The magnitude of gives the relative strength of the electron-withdrawing or -
donating of the substituents. is positive if the substituent is electron-withdrawing and negative if it is electron-
donating.
These relationships as developed by Hammett are termed linear free energy relationships. Recall the equation
relating free energy to an equilibrium constant.
G
= -
RT
ln
K
(3)
That is, the free energy is proportional to the logarithm of the equilibrium constant. These linear free energy
relationships are termed “extrathermodynamic”. Although they can be stated in terms of thermodynamic parameters,
no thermodynamic principle states that the relationships should be true.
To develop a better understanding of these relationships, it is instructive to consider some values of and . Values
of are provided below:
NH
3
+
= 2.90
OH
= 2.23
= 0.49
CH
2
COOH
= 0.21
CH
2
CH
2
COOH
