Amos J. Leffler Villanova University Villanova, Pennsylvania 19085
A Criterion for the Simple Approximation in Dissociation Equilibria
In the teaching of weak acid and weak base dissociation equilibria in general chemistry i t is common to use the simplifying assumption that the amount of the acid or base dissociated is small compared with the total concentration. This permits the neglect of the amount of dissociated species in the denominator of the dissociation equation and makes its solution relatively trivial. Most texts point out that this assumption is not always valid and cannot be used in some circumstances, but rarely is there any attempt to state any quantitative criterion for its validity. In the present note such a criterion is presented which depends on the amount of error permitted in the final proton or hydroxyl ion concentration. A simple extension of the method to salt hydrolysis is shown. For the dissociation of a weak acid, HA, we have the equilihrium HA=Ht+A-
*0
3o ? !
--.-
~ncremsing Error
_
\
\
\
0
\
:d -
\
(1)
where we are specifically neglecting the proton concentration from the solvent. Calling the total acid concentration, co, and the concentration of H+ and A-, x , we have the usual dissociation equation
In the approximation the concentration of x in the denominator is neglected and we have the relation
{L?: PUOH)
where we have designated the approximate concentration with a prime. Rearranging eqns. (2) and (3) and subtracting the difference of the square terms we find %l'
a-
9 = Ko x
,;
(4)
The left-hand side of the equation can he factored so that
and since x'
( x ' - X ) =KJ2 (6) which shows that the difference between the exact and approximate values only becomes significant when it is of the order of K.12. I t should be evident that eqn. (6) offers a convenient approximation to the exact solution by rearranging to give x in terms of x' and K,. Equation (6) is instructive but not very useful since we are more interested in determining what error will result in the approximate calculation a t a particular acid concentration. This is clearly a function of KO and can readily he derived in the following way. Assume that we are interested in a 10% error. Thus from eqn. (6)
- X) = 0.10~'
(7)
where we are neglecting the second-order difference between 10%of x or x'. Using eqn. (3) we have 0 . 1 0 ~ '= O . ~ O ( K . C ~ )= '/~ Ka/2
(8)
or more conveniently
460 / Journal of Chemical Education
For other allowed error values, the constant in eqn. (10) will vary. Equation (10) can conveniently be plotted as shown in the figure. In the figure lines are drawn for 5 and 10%errors as a function of pK, or pKb. The derivation in the latter case is identical except for the change in species. It is evident from the figure that for larger dissociation constant values the value of co will increase so that in extreme cases such as the second dissociation of sulfuric acid an approximate calculation is almost impossible. At the other end of the scale care must be taken to include the proton or hydroxyl ion concentration from the solvent. This region is indicated hy a line drawn at H+or OH-concentrations of 1.0 X 1 0 - W which is about the point a t which the solvent contribution becomes important. The method as descrihed can he applied to any equilibrium of the form of eqn. (1). A simple example is the hydrolysis of an acidic or basic salt. An acid salt hydrolysis has the form BC
or by simple rearrangement co = 5K,'I2
pKa or pKb
Plot of neglect of ionization compared with concentration far weak acids and bases-5% and 10% enor as a function of pK. and -log c.
x we finally obtain
(x'
,;
(9)
+ Hz0 = BOH + Ht
with the exact equilibrium expression having the form Km- x 2 Ka co- x and the approximate one
(11)
Q - x,Z Ks eo (I3) where the symbols x and x' are again the exact and approxi-
mate H+concentrations. Again the difference between the approximate and exact H+ values becomes (x' - X ) = K,/2Kb
(14)
with an analogous equation for the hydrolysis of a basic salt.
Again assuming that the difference between the approximate and exact values is 10%of the approximate one we find (15)
and following the same method as before we obtain -log co = 12.301 - pKb
(16)
a t 25'C. Again a figure similar to the figure can be drawn.
Volume 53, Number 7, July 1976 / 461
