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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.

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.