Problem 50 - Chemistry: Principles and Reactions

Question

Morphine, \(\mathrm{C}_{17} \mathrm{H}_{19} \mathrm{O}_{3} \mathrm{~N}\), is a weak base \(\left(K_{\mathrm{b}}=7.4 \times 10^{-7}\right) .\) Consider its titration with hydrochloric acid. In the titration, \(50.0 \mathrm{~mL}\) of a \(0.1500 \mathrm{M}\) solution of morphine is titrated with \(0.1045 \mathrm{MHCl}\). (a) Write a balanced net ionic equation for the reaction that takes place during titration. (b) What are the species present at the equivalence point? (c) What volume of hydrochloric acid is required to reach the equivalence point? (d) What is the \(\mathrm{pH}\) of the solution before any \(\mathrm{HCl}\) is added? (e) What is the \(\mathrm{pH}\) of the solution halfway to the equivalence point? (f) What is the \(\mathrm{pH}\) of the solution at the equivalence point?

Step-by-Step Solution

Answer

Question: Calculate the pH of the morphine solution at the equivalence point during titration with hydrochloric acid. Answer: The pH at the equivalence point is 5.63.

Step 1: (Write a balanced net ionic equation for the reaction)

The net ionic equation for the reaction between morphine (Morph) and hydrochloric acid (HCl) is: Morph\(\left(\mathrm{aq}\right) + \mathrm{H}^{+}\left(\mathrm{aq}\right) \rightarrow \mathrm{MorphH}^{+}\left(\mathrm{aq}\right)\)

Step 2: (Identify the species present at the equivalence point)

At the equivalence point, all the morphine has reacted with the HCl to form the conjugate acid, MorphH+. Therefore, the species present at the equivalence point is MorphH+.

Step 3: (Calculate the volume of HCl to reach the equivalence point)

First, find the moles of morphine in the initial solution: Moles of morphine = Molarity * Volume = 0.1500 M * 0.050 L = 0.00750 moles At the equivalence point, the moles of morphine equal the moles of HCl: Moles of HCl = 0.00750 moles Now, find the volume of HCl needed: Volume of HCl = Moles of HCl / Molarity of HCl = 0.00750 moles / 0.1045 M = 0.0718 L or 71.8 mL

Step 4: (Calculate the pH before adding any HCl)

Calculate the initial concentration of Morph: [\(\mathrm{Morph}\)] = \([\mathrm{OH}^{-}]\) = \(\sqrt{K_b \cdot [\mathrm{Morph}]_{initial}}\) = \(\sqrt{7.4 \times 10^{-7} \cdot 0.1500}\) = 0.00135 M Now, find the pOH and then the pH: pOH = -log([\(\mathrm{OH}^{-}\)]) = -log(0.00135) = 2.87 pH = 14 - pOH = 14 - 2.87 = 11.13

Step 5: (Calculate the pH halfway to the equivalence point)

At the halfway point, the Henderson-Hasselbalch equation can be used to calculate the pH: pH = pKa + log(\(\frac{[\mathrm{A}^{-}]}{[\mathrm{HA}]})\) Since \(K_b\) is provided, we first need to find the pKa: pKa = -log(\(K_a\)) = -log(\(\frac{K_w}{K_b}\)) = -log(\(\frac{1 \times 10^{-14}}{7.4 \times 10^{-7}}\)) = 6.87 At the halfway point, \([\mathrm{A}^{-}] = [\mathrm{HA}]\), and the ratio is 1: pH = 6.87 + log(1) = 6.87

Step 6: (Calculate the pH at the equivalence point)

At the equivalence point, the pH will be determined by the hydrolysis of the MorphH+ ion: \(K_a = \frac{[\mathrm{Morph}][\mathrm{H}^{+}]}{[\mathrm{MorphH}^{+}]}\) Rearrange the equation to solve for [\(\mathrm{H}^{+}\)]: [\(\mathrm{H}^{+}\)] = \(\frac{K_a \cdot [\mathrm{MorphH}^{+}]}{[\mathrm{Morph}]}\) = \(\frac{K_a}{[\mathrm{Morph}]}\) Calculate the concentration of Morph and MorphH+ ions at the equivalence point, which is \([\mathrm{Morph}] = [\mathrm{MorphH}^{+}] = \frac{0.0075}{0.050+0.0718} = 0.0573\,\mathrm{M}\). Now, find the concentration of [\(\mathrm{H}^{+}\)]: [\(\mathrm{H}^{+}\)] = \(\frac{K_a}{[\mathrm{Morph}]}\) = \(\frac{1.35 \times 10^{-7}}{0.0573}\) = \(2.36 \times 10^{-6}\) Finally, calculate the pH: pH = -log([\(\mathrm{H}^{+}\)]) = -log(\(2.36 \times 10^{-6}\)) = 5.63

Key Concepts

Net Ionic Equation

The net ionic equation represents the actual chemical change occurring in a reaction, minus the spectator ions that remain unchanged. It's a simplification showing the essentials of the chemical process. For the titration of a weak base with a strong acid, such as morphine and hydrochloric acid, the equation is straightforward:

\[\begin{equation}\text{Morph}\left(\text{aq}\right) + \text{H}^{+}\left(\text{aq}\right) \rightarrow \text{MorphH}^{+}\left(\text{aq}\right)\end{equation}\]This reveals that morphine, the weak base, accepts a proton from the strong acid (\(\text{HCl}\)) to form its conjugate acid (\(\text{MorphH}^{+}\)).

Equivalence Point

The equivalence point in a titration is the moment when the amount of titrant added exactly neutralizes the analyte solution. Here, each morphine molecule has reacted with a hydrogen ion, resulting in equal moles of morphine and hydrochloric acid. At this point, the solution contains primarily the conjugate acid of morphine (\(\text{MorphH}^{+}\)). Since the equivalence point involves the stoichiometric reaction of acid and base, calculating the volume of hydrochloric acid needed to reach it involves stoichiometry, leading to the understanding of the reaction's molar relationships.

pH Calculation

Calculating pH is crucial in understanding the acidity or basicity of a solution. Initially, the pH of the weak base morphine is found by calculating the concentration of \(\text{OH}^{-}\) ions, using the square root of the product of the base's concentration and its \(K_b\), and then using \(14 - \text{pOH}\) to find the pH. Halfway to the equivalence point, the concentrations of the weak base and its conjugate acid are equal, allowing us to use the Henderson-Hasselbalch equation to calculate the pH. Upon reaching the equivalence point, the pH is dependent on the hydrolysis of the conjugate acid, which defines the new acidic character of the solution.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a formula used to estimate the pH of a buffer solution composed of a weak acid and its conjugate base (or a weak base and its conjugate acid). For titrations, it's especially useful when the solution is halfway to the equivalence point, where the concentrations of the weak acid and base are equal. The equation is:\[\begin{equation}\text{pH} = \text{p}K_{a} + \log\left(\frac{[\text{A}^{-}]}{[\text{HA}]}\right)\end{equation}\]where \(\text{p}K_{a}\) is the negative log of the acid dissociation constant (\(K_a\)) of the conjugate acid. At the halfway point, the logarithmic term is zero as the concentration ratio is one, simplifying the pH to be equal to the \(\text{p}K_{a}\) of the weak acid.

Hydrolysis of Conjugate Acid

Hydrolysis occurs when the conjugate acid of the weak base reacts with water, producing hydronium ions (\(\text{H}_3\text{O}^{+}\)) and the original base. At the equivalence point, the pH is determined primarily by the hydrolysis of the conjugate acid rather than by the initial weak base or strong acid. The extent of hydrolysis, and thus the solution's acidity, is described by the acid dissociation constant (\(K_a\)) of the conjugate acid. As the concentration of conjugate acid is the same as that of the base at the equivalence point in this titration, you can calculate the resulting \(\text{H}^{+}\) ion concentration and therefore the pH.