Page 143 - Mechanism and Theory in Organic Chemistry

If it were possible to obtain the activity coefficients, Equation 3.33 would
provide a way of obtaining KaADH+. In dilute aqueous solution the Debye-Huckel
theory, which is based on calcuiation of interionic forces in a medium containing
dissociated ions, provides a method for estimating activity coefficients of ions.18
However, ,even for ionic strengths as low as 0.01 there are significant deviations
from the theory.lg In the strong acid-water mixtures under consideration here,
the concentration of ionic species (H,O+, HS0,-) is of necessity high; thus,
even if the concentrations of the acids and bases under study are kept small (as they
must in any case be in order for the spectrophotometric measurements to be
reliable), the Debye-Huckel theory is of no help. It is possible, however, to make
the following qualitative argument. The departure of the activity coefficients from
unity is the result of some nonideal behavior of the species involved. Departures
from ideality therefore depend on the structure, and probably particularly on the
charges, of the components. If A, and A, (and thus also A,H+ and A,H + ) are
sufficiently close in structure, one might guess that in a given solvent the ratio
yAIIYAIH+ would be approximately the same as yA,/yA,,+. If this were the case,
the ratio of activity coefficients in Equation 3.33 would equal unity and the
equation would become

An experimental check on this assumption about activity coefficients is
possible over a limited range of solvent acidity. If the composition of water-
sulfuric acid mixtures is varied over the range in which all four species, A,, A,,
AIH+, and A,H+ are present in appreciable concentration, then, since
KaA.,+ /KaAmH+ (by definition) constant, a constant ratio [A,TJ [A,H +I /
is
[A,-~[A,H;] implies that the assumption of the ratio of y's being constant is
correct in this range of solvents. Experimentally, for bases that are substituted
anilines this test is fairly successful, a result that supports the validity of the
method. The question of how similar two compounds must be to be "sufficiently
close in structure" will be considered later.
Proceeding with our analysis, we find that if we can assume that Equation
3.34 is valid, we know all quantities necessary to obtain KaAnH+, the equilibrium
constant for the second base. This base is now used in conjunction with a third
base, A,, in a solvent system containing a larger proportion of strong acid, and the
procedure is continued until equilibrium constants are established for the whole
range of bases.
Having found equilibrium constants for the series of bases, we may now use
them to characterize the proton-donating ability of any mixture of sulfuric acid
and water. Rearranging Equation 3.30, we have
[AH+] - as,+ y.4
Ka- - -
[A] as YAH+
The quantity on the right side of Equation 3.35 is defined as h,; it gives the

l8 See note 8, p. 127.
lo Hammett, Physical Organic Chemisty, p. 192.