How do we calculate the hydronium ion
concentration during an acid-base titration or when a weak
acid/conjugate base solution is prepared? To do this properly, we
must take into account all the competing equilibria as well as
activity coefficients.

During an acid-base titration of the sort
dealt with in our Titration I and II lab experiments, activity
coefficients are ignored, since the position of the end point in a
titration is not affected by activity considerations. However, we
do make efforts to correctly deal with equilibria that affect the
relative hydronium concentration during the titration.

**Titration of a
Weak Acid with Strong Base**

There are four separate regions in the
titration of a weak acid by a strong base. There is no single
analytical expression that deals with these four regions. This
matter is well described in Daniel Harris', "Quantitative Chemical
Analysis," Chapter 12 (Acid-Base Titrations). Our approach involves
a branching computer program which employs a different analytical
expression to calculate the pH in each of these four
regions. Beginning with a solution of the weak
acid in water, a solution of strong base is added drop-wise, with
stirring, and the pH is calculated and plotted vs. mL of added
base. Since the addition of the titrant will result in significant
volume increase during the titration, we must take into account the
total volume of the solution.

*Region 1: Before base is added*

In this region, we can calculate the
[H^{+}] using the "systematic method" described in the
Harris text, Chapter 9. The usual quadratic equation resulting from
the solving of the K_{a} expression
provides sufficient accuracy in this case.

*Region 2: Before the Equivalence
Point*

Throughout this region, we
apply the Henderson-Hasselbalch equation, using the full
systematic solution for accuracy (that is, we consider charge
balance and mass balance for all species, requiring the
solution of four simultaneous equations). This region deals
with the formation of the weak acid buffer as the weak acid is
converted to its conjugate base. Midway to the equivalence point,
when the number of moles of strong base added equals
half the number of moles of weak acid originally present, the
point of maximum buffering is reached, and the pH of the solution
equals the pK_{a} of the weak acid.

*Region 3: At the Equivalence Point*

This is the point at which as many moles
of strong base have been added as moles of weak acid originally in
the solution. This pH calculation is the same as that
for a solution of the conjugate base alone, and is evaluated
using the K_{b} expression for the conjugate base.
Note that the pH of this solution will not be 7.00. While
the position of the equivalence point is not identical to
that of the end point (which is the inflection point of
the titration plot), these are indistinguishable in these
acquired data sets because of the steep curve observed in this
region.

*Region 4: After the Equivalence
Point*

Additions of strong base to the solution
of the conjugate base produces a solution whose pH is dominated by
the excess concentration of the strong base. Hydrolysis of the
conjugate base of the weak acid is suppressed and has negligible
effect on solution pH. The usual calculation of pH of in a solution
of a strong base is sufficient in this region.

**Titration of a
Strong Acid with Strong Base**

There are three separate regions during the
titration of a strong acid by a strong base. Again, there is no
single analytical expression that deals with these three regions.
As before, Harris', "Quantitative Chemical Analysis," Chapter
12 (Acid-Base Titrations) provides an excellent reference. Our
approach again involves a branching computer program with different
calculations in applied in three
regions.

*Region 1: Before base is added*

The simple calculation of the hydronium
concentration due to the moles of strong acid remaining after
subtracting the moles of added base, divided by the combined
volumes, is sufficient.

*Region 2: At the Equivalence Point*

The point at which as many moles of strong
base have added as moles of strong acid originally in the solution
is the equivalence point of the titration. This
situation is equivalent to a solution of pure water, since the
hydrolyses of the conjugate species is negligible (these
are "spectator ions"), and the pH of the solution at this
point is 7.00. The position of the equivalence point will be
identical to the end point of the titration, which is given by the
inflection point of the plot of pH vs. mL of OH^{-}.

*Region 3: After the Equivalence
Point*

Additions of strong base to the solution
of the conjugate base produces a solution whose pH is dominated by
the excess concentration of the strong base. The usual
calculation of pH of in a solution of a strong base is sufficient
in this region.

**Calculation of pH for Various
Solutions in the Weak Acid Equilibrium
Lab**

The same branching computer program used to
calculate the pH for the two titrations just described can be more
generally applied to calculate pH's for solutions in our
Weak Acid Equilibrium Lab. Rather than calculating the pH after
each drop-wise addition of strong base during a titration, the
program inputs are solution compositions, and the pH is
calculated and displayed on the pH meter readouts. Within
limits, students can prepare solutions with widely varying
compositions, by combining the solid and solution reagents with
distilled water, and the pH changes can be correctly
determined.

*Case 1: Adding reagents to
pure water *

In this case, the flask of pure water will
be further diluted when the student adds water from a pipette. The
pH is continuously displayed. The student may also add small
measured amounts of a strong base solution to observe the lack of
buffering. The manner in which the pH of water is strongly altered
by adding the base will be contrasted with the cases which
follow.

*Case 2: Solution of Weak Acid*

The student prepares a solution of a weak
acid of known molecular weight but unknown K_{a}. The pH of
this solution is calculated using the branching equation described
above. The student then may dilute the solution with a known amount
of water to see the effect on pH, followed by adding known volumes
of a strong base. For each situation, the pH is calculated and
displayed.

*Case 3: Solution of Conjugate Base*

The student prepares a solution of the
conjugate base of the same weak acid of known molecular weight but
unknown K_{a}. The pH of this solution is calculated using
the branching equation described above. The student then may dilute
the solution with a known amount of water to see the effect on pH,
followed by adding known volumes of a strong base. For each
situation, the pH is calculated and displayed. The difference in
behavior of the solutions of the weak acid

*Case 4: Buffer Solution*

Finally, the student prepares a solution
containing a known number of moles of a weak acid and its conjugate
base, and the pH is displayed. The student may then dilute this
buffer solution, observing that the pH of a diluted buffer does not
change. In addition, measured amounts of strong base may be added
to see how well the buffer resists a change in pH. After all these
cases, the student can see clearly that the composition of the
solution is critical to the buffer capacity.

**Adding Noise and
Error**

After making the calculations according to
the above discussion, we add random error (and can add
systematic error) to the resulting data. This is similar to all our
data simulations, and is further discussed in Technical
Report #1.