Question

Figure \(14-4\) shows the \(\mathrm{pH}\) curves for the titrations of six different acids by NaOH. Make a similar plot for the titration of three different bases by 0.10 \(M\) HCl. Assume \(50.0 \mathrm{mL}\) of \(0.20 M\) of the bases and assume the three bases are a strong base (KOH), a weak base with \(K_{\mathrm{b}}=1 \times 10^{-5},\) and another weak base with \(K_{\mathrm{b}}=1 \times 10^{-10}\).

Short answer

To create the pH curve plot for the titration of three different bases (KOH, a weak base with Kb=1×10⁻⁵, and another weak base with Kb=1×10⁻¹⁰) by 0.10 M HCl, follow these steps: 1. Calculate the initial pH for each base using their respective concentrations and Kb values. 2. During titration, determine the amount of OH⁻ neutralized and calculate the new OH⁻ concentration and pH at each interval. 3. Calculate the pH at the equivalence point, when the moles of HCl added equal the initial moles of OH⁻. 4. Plot the pH curves with the volume of HCl added on the x-axis and pH value on the y-axis, resulting in three curves representing the titrations of the strong base (KOH), the weak base with Kb=1×10⁻⁵, and the weak base with Kb=1×10⁻¹⁰.

Step-by-step solution

Determine the initial pH of each base

First, calculate the initial pH values for each base before the titration begins. For the strong base KOH, the initial pH can be found by taking the negative logarithm of the hydroxide ion concentration: pH = 14 - pOH pOH = -log([OH^-]) pH = 14 - (-log(0.20)) pH ≈ 13.7 For the weak bases, use the Kb value and the initial concentration of the base to determine the OH- concentration: \[K_{b} = \frac{[OH^-][BH^+]}{[B]}\] Then, use this value to find the pH as shown above. Repeat this process for both weak bases.

Calculate the pH during titration

During the titration, we need to calculate the pH value as HCl is added to each base. To do this, we need to determine the amount of OH- neutralized at various points during the titration process as HCl is added. For each base, calculate the moles of OH- in the solution initially: moles OH- = (0.20 M)(50.0 mL) = 0.010 moles At various intervals, calculate the moles of HCl that have been added and determine the remaining moles of OH-. The moles of HCl and OH- will react in a 1:1 ratio. Once you have the remaining moles of OH-, calculate the new OH- concentration and find the pH at that point in titration.

Calculate the pH at the equivalence point

The equivalence point is when the moles of HCl added equal the moles of OH- initially in the solution. At this point, recalculate the OH- concentration and find the pH value. For the strong base, the salt formed will not affect the pH significantly and the OH- concentration will be close to zero. For the weak bases, the conjugate acid formed will affect the pH value. Use the Kb value of each weak base and the Henderson-Hasselbalch equation to find the pH value at the equivalence point.

Plot the pH curves

Now that you have calculated the initial pH, the pH values during titration, and the pH values at the equivalence point, you can plot the pH curves for each base. On the x-axis, plot the volume of HCl added, and on the y-axis, plot the pH value. The plot should show three different curves that represent the different titrations of the strong base (KOH), weak base with Kb=1*10^-5, and weak base with Kb=1*10^-10.

Key concepts

Titration Curves

Titration curves are graphical representations showing how the pH of a solution changes as a titrant is added. In the case of titrating bases with an acid, the curve typically starts at a higher pH and decreases as the titrant—the acid in this case—is added.

For a strong base like KOH, the curve starts at a high pH and drops sharply as you approach the equivalence point because strong bases fully dissociate in water. As the strong acid is added, it neutralizes the OH- ions, decreasing the pH dramatically. In contrast, titration curves for weak bases begin at a lower pH and show a more gradual decline. This is due to the partial dissociation of the weak base, resulting in less OH- available to be neutralized by the acid.

The shape of the titration curve provides vital information about the nature of the base being titrated and can be used to determine the pKa (for acidic titrations) or pKb (for basic titrations) of the substance in question.

pH Calculation

To calculate the pH during an acid-base titration, we need to understand the concentrations of the hydronium (H3O+) and hydroxide (OH-) ions present in the solution. The pH is the negative logarithm of the concentration of H3O+ ions:
\[\text{pH} = -\log([\text{H}_3\text{O}^+])\]
Similarly, the pOH is the negative logarithm of the OH- ion concentration, and pH and pOH are related by the equation:
\[\text{pH} + \text{pOH} = 14\]
During titration, these concentrations change as the titrant reacts with the substance in the solution. For the initial pH of a strong base, we can assume complete dissociation, while for weak bases, we apply the equilibrium constant (Kb) to find the initial OH- concentration.

Equivalence Point

The equivalence point is a crucial moment in titration, representing the point where the amount of titrant added is stoichiometrically equivalent to the quantity of substance present in the solution being titrated.

For acid-base titrations, the equivalence point occurs when the moles of H+ from the acid equal the moles of OH- from the base. At this point for a strong base like KOH titrated with a strong acid such as HCl, the solution is neutral with a pH of 7.

However, when titrating a weak base, the solution at equivalence point is slightly acidic due to the formation of the conjugate acid of the weak base, and thus, the pH at this point will be less than 7. The exact value depends on the strength of the weak base (indicated by its Kb) and requires a different calculation method, often involving the Henderson-Hasselbalch equation.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation elegantly connects the pH of a solution with the pKa or pKb of the dissolved substance and the ratio of the concentrations of the conjugate acid/base pair.

For a weak base (B) being titrated with a strong acid, the equation is written as follows:
\[\text{pH} = \text{pKb} + \log\left(\frac{[\text{BH}^+]}{[\text{B}]}\right)\]

Here, [BH+] is the concentration of the conjugate acid, and [B] is the concentration of the weak base. This equation is particularly useful for calculating the pH at the equivalence point during the titration of a weak base. Since at the equivalence point, the concentrations of [BH+] and [B] are equal, the log term becomes zero, and the pH directly equals the pKb of the base.

Understanding and utilizing the Henderson-Hasselbalch equation is essential for accurately predicting and analyzing the results of titration experiments involving weak acids or bases.