Question

Figure \(15.4\) shows the pH curves for the titrations of six different acids by \(\mathrm{NaOH}\). Make a similar plot for the titration of three different bases by \(0.10 M\) HCl. Assume \(50.0 \mathrm{~mL}\) of \(0.20 \mathrm{M}\) of the bases and assume the three bases are a strong base \((\mathrm{KOH})\), 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 generate the pH curves for the titration of three different bases by 0.10 M HCl, follow these steps: 1. Identify the stages of titration: before titrant addition, during titrant addition (before equivalence point), at equilibrium, and after equivalence. 2. Calculate initial pH of each base considering their concentrations and \(K_b\) values. 3. Calculate pH during titration before the equivalence point using an ICE table. 4. Determine the pH at the equivalence point using the Henderson-Hasselbalch equation for weak bases. 5. Calculate the pH after the equivalence point considering excess titrant reacting with water. 6. Compile the pH values obtained in steps 2-5 for each base and plot the pH curves for all three bases: KOH (strong), and the two weak bases with \(K_b = 1 \times 10^{-5}\) and \(K_b = 1 \times 10^{-10}\).

Step-by-step solution

Identify the stages of titration

For each base, we have to identify the stages of titration. In general, there are four stages: 1. Before the addition of the titrant (HCl) 2. During the addition of the titrant, before the equivalence point 3. At the equivalence point 4. After the equivalence point

Calculate initial pH of the bases

First, calculate the initial pH values of each base by considering their concentrations and \(K_b\) values. For KOH (Strong base), the initial pH can be calculated using the following formula: \[pH = -log_{10} [H^+] \] As it's a strong base, the initial concentration of OH- ions is 0.20 M. Weak bases: For the weak bases with \(K_b = 1 \times 10^{-5}\) and \(K_b = 1 \times 10^{-10}\), we can use the base dissociation constant to find the concentration of OH- ions in the initial solution. \[K_b = \frac{[OH^-][BH^+]}{[B]} \]

Calculate pH during the titration (before the equivalence point)

As we add HCl, the formation of water and the acid-base reaction will affect the pH values. We can set up an ICE table to find the concentrations of the species during the titration process before the equivalence point based on the moles of the reactant and the titrant. For example, for a \(0.20 \ \text{M}\) weak base with \(K_b = 1 \times 10^{-5}\) that has reacted with \(x \ \text{M}\) of HCl: \[K_b = \frac{(0.20-x)(x)}{(0.20-x)}\] Solve for \(x\) to find the concentration of the products and reactants at this stage.

Calculate the pH at the equivalence point

At the equivalence point, equal moles of the titrant (HCl) and the base have reacted. Calculate the number of moles for both titrant and base: \[moles = C × V\] Next, calculate the pH at the equivalence point using the Henderson-Hasselbalch equation for weak bases: \[pH = pK_w + log_{10}\left(\frac{[B]}{[BH^+]}\right)\]

Calculate the pH after the equivalence point

Once the equivalence point is reached, any excess titrant (HCl) will react with water and determine the pH value of the solution. Therefore, calculate the excess \(H^+\) concentration in the solution after the equivalence point and then calculate the pH using: \[pH = -log_{10} [H^+]\]

Plot the pH curves

Now that you have the data points for the pH values at various stages of titration for every base, plot the pH curves of each base titrated by \(0.10 \ \text{M}\) HCl. The final plot should show three curves for the strong base and the two weak bases with their corresponding \(K_b\) values.

Key concepts

Acid-Base Titration

Acid-base titration is a laboratory procedure used to determine the concentration of an unknown acid or base by reacting it with a base or acid of known concentration. During the titration, a titrant is slowly added to a reaction mixture until the chemical reaction is complete. This point of completion is known as the equivalence point, where the number of moles of acid equals the number of moles of base. Indicators or pH meters are typically used to detect the equivalence point, which is characterized by a sudden change in pH.

To visualize this process, a pH curve can be plotted, which shows the change in the pH of the solution as titrant is added. Three key stages on the pH curve are the initial pH, the buffering region before reaching the equivalence point, and the vertical section which represents the rapid pH change at the equivalence point.

In the titration of bases with hydrochloric acid (HCl), the pH will generally decrease as the strong acid reaction progresses. For strong bases like KOH, the initial pH is high and drops sharply at the equivalence point. However, this transition may be more gradual with weak bases, reflecting their partial dissociation and buffering capacity.

pH Calculation

Calculating the pH is an essential part of acid-base chemistry, determining the acidity or basicity of a solution. The pH scale ranges from 0 to 14, with lower values indicating acidic solutions and higher values indicating basic solutions. The pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration, mathematically represented as:
\[pH = -\text{log}_{10} [H^+].\]

For strong acids and bases, pH calculation is straightforward because they fully dissociate in water. However, weak acids and bases only partially dissociate, creating an equilibrium that makes pH calculation more complex. In these cases, the pH depends on the equilibrium constant for dissociation (\(K_a\) for acids, and \(K_b\) for bases) and the concentrations of the acid or base and its conjugate counterpart. During a titration, the pH changes with the addition of the titrant, and these changes require different calculations at each stage of the titration process.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is invaluable for estimating the pH of buffer solutions, and it can also be applied near the equivalence point of a titration involving weak acids or bases. The equation relates pH to \(pK_a\) (the negative logarithm of the acid dissociation constant) or \(pK_b\) (the negative logarithm of the base dissociation constant) and the ratio of the concentrations of the conjugate base (\([A^-]\)) and conjugate acid (\([HA]\)) in the solution:
\[pH = pK_a + \text{log}_{10}\left(\frac{[A^-]}{[HA]}\right).\]

In the context of titrating a weak base, the modified form of the equation using \(pK_w\) (the negative logarithm of the water dissociation constant) becomes relevant:
\[pH = pK_w + \text{log}_{10}\left(\frac{[B]}{[BH^+]}\right).\]

This equation aids in predicting the pH at the equivalence point and can help plan the appropriate indicator for a given titration.

Base Dissociation Constant (\(K_b\))

The base dissociation constant, \(K_b\), is a measure of how readily a base dissociates into its constituent hydroxide ions (\(OH^-\)) and the corresponding conjugate acid. It's an equilibrium constant for the reaction where a base (\(B\)) reacts with water to form a hydroxide ion and the conjugate acid (\(BH^+\)):

\[B + H_2O \leftrightarrows OH^- + BH^+\]
\[K_b = \frac{[OH^-][BH^+]}{[B]}.\]

The value of \(K_b\) varies for different bases; a larger \(K_b\) value indicates a stronger base. Understanding \(K_b\) is crucial for calculating the pH of weak bases, as seen in the step by step solution where \(K_b\) is used to find the concentration of OH- ions in the initial solution. This knowledge also assists with predicting and explaining the different shapes of the pH curves for weak bases during titration.

Equivalence Point

The equivalence point of a titration is the moment when the quantity of titrant added is chemically equivalent to the quantity of the substance in the sample being titrated. It signifies the completion of the specific acid-base reaction in a titration curve. Determining the equivalence point is crucial because it provides the information necessary to calculate the concentration of the unknown substance.

For strong acids and strong bases, the equivalence point occurs at a pH of 7. However, the equivalence point for the titration of weak acids or bases does not occur at neutral pH due to the presence of the conjugate base or acid. At the equivalence point during the titration of a weak base with a strong acid, for instance, the pH will be less than 7. To find the pH at this juncture, the Henderson-Hasselbalch equation can be employed, as detailed in a previous section.

Understanding the equivalence point also helps scientists choose the right indicator, a chemical that changes color at a particular pH range, to visually signal the end of the titration process.