Page 152 - Mechanism and Theory in Organic Chemistry

Strengths of Weak Bronsted Acids 141

to set up the H- scale. Furthermore, the slope of the line correlating AHD with
pKa is nearly the same as the slope of the correlation between heat of protonation
in HS0,F and pKa for the weak bases. This latter result increases confidence in
the heat of protonation method as a valid way of measuring acid strength over a
very wide range.

The Brsnsted Catalysis Law
The experimental work described up to this point has been limited to those carbon
acids that are more acidic than pKa about 33. Most of these compounds owe their
acidity to some structural feature that allows the negative charge of the conjugate
base to be delocalized. We turn now to a brief discussion of a method by which
measurements can be extended, at least in a semiquantitative way, into the region
of still weaker acids.
In the acid-base reaction 3.46, it would seem reasonable that if the rate
(k,) at which a proton is removed by a particular base Bn+ were compared for

various acids AHm+, the base might remove the proton more rapidly from the
stronger acids. Relationships between rate of an acid-base reaction and an
equilibrium have been observed in many cases, and are frequently found to obey
an equation known as the Brernsted catalysis law:
k = CKaa

log k = a log Ka + log C
where k is the rate constant for the reaction, Ka is the acid dissociation constant,
and C is a constant of proportionality. If such a relationship could be shown to
hold between acid strength and rate of transfer of the proton to some particular
base, a means would be available to find equilibrium acidities through kinetic
measurements.
An appreciation for the form of the catalysis law may be gained by con-
sideration of the energy relationships involved. In Figure 3.4 is plotted schematic-
ally the free energy (AG) vs. reaction coordinate for proton transfer reactions
between a series of acids, AnH, and a single base, B. The differing pK,-values
of the acids are reflected in the different free-energy changes in going from re-
actants to products, AG,", AG;, . . ., AG;,. . ., and are caused by structural
differences among the acids AnH and among the conjugate bases A,-. If one
assumes that the factors that cause these free-energy differences also cause the
differences in the transition-state free energies, it is reasonable to suppose as a
first approximation that the activation free energy for proton transfer, AG:,
might be related to the AG; in a linear fashion. This relationship is expressed in
Equation 3.49, where we have arbitrarily chosen the first acid, A,H, as a reference
compound for the series.

We have from equilibrium thermodynamics Relation 3.50 between standard
free-energy change, AGO, and equilibrium constant, K, and from transition-state