Question

A solution of 0.23 mol of the chloride salt of protonated quinine \(\left(\mathrm{QH}^{+}\right)\) a weak organic base, in 1.0 \(\mathrm{L}\) of solution has \(\mathrm{pH}=4.58 .\) Find the \(K_{\mathrm{b}}\) of quinine \((\mathrm{Q}) .\)

Short answer

The base dissociation constant (\(K_{b}\)) of quinine (Q) is approximately \(3.32 \times 10^{-4}\).

Step-by-step solution

Write the ionization equation for quinine

To find the base dissociation constant (\(K_b\)) for quinine (Q), we first need to write the ionization reaction of its protonated form (\(QH^+\)) in water: \[QH^+ + H_2O \rightleftharpoons Q + H_3O^+\]

Convert pH to \(H_3O^+\) concentration

Using the pH value, calculate the concentration of hydronium ions (\(H_3O^+\) ions). The pH is the negative logarithm of the \(H_3O^+\) ion concentration: \[pH = -\log[H_3O^+]\] Solve for \(H_3O^+\) concentration: \[[H_3O^+] = 10^{-pH}\]

Calculate the concentration of \(H_3O^+\) ions

Plug in the given pH value to find the \(H_3O^+\) ion concentration: \[[H_3O^+] = 10^{-4.58} \approx 2.63 \times 10^{-5} \mathrm{M}\]

Write the expression for \(K_b\)

Write the equilibrium expression for the base dissociation constant \(K_b\) of quinine: \[K_b = \frac{[Q][H_3O^+]}{[QH^+]}\] Since \(QH^+\) reacts with water to produce \(Q\) and \(H_3O^+\), their concentrations are equal at equilibrium.

Relate \(K_a\) of \(QH^+\) to \(K_b\) of quinine

Since \(QH^+\) is the conjugate acid of the base quinine (\(Q\) ), we can use the relationship between \(K_a\) and \(K_b\) which is given by the equation \(K_w = K_a \times K_b\). Here, \(K_w\) is the water dissociation constant and is equal to \(1.0 \times 10^{-14} \).

Calculate \(K_a\) for \(QH^+\)

The \(K_a\) for \(QH^+\) is determined by the concentration of \(H_3O^+\) produced and the initial concentration of \(QH^+\): \[K_a = \frac{[Q][H_3O^+]}{[QH^+]_0 - [Q]}\] Since \( [QH^+]_0 = [Q] = [H_3O^+] \), we simplify the equation for \(K_a\): \[K_a = \frac{[H_3O^+]^2}{[QH^+]_0}\]

Identify the initial concentration of \(QH^+\)

The initial concentration of \(QH^+\) is given as 0.23 mol/L.

Calculate \(K_a\) of \(QH^+\)

Insert the known values into the expression for \(K_a\) to find its value: \[K_a = \frac{(2.63 \times 10^{-5})^2}{0.23}\] \[K_a = \frac{6.92 \times 10^{-10}}{0.23}\] \[K_a \approx 3.01 \times 10^{-11}\]

Calculate \(K_b\) of quinine

Now we can determine \(K_b\) using the relationship between \(K_w\), \(K_a\), and \(K_b\): \[K_b = \frac{K_w}{K_a}\] \[K_b = \frac{1.0 \times 10^{-14}}{3.01 \times 10^{-11}}\] \[K_b \approx 3.32 \times 10^{-4}\]

Key concepts

Acid-Base Equilibrium

Understanding acid-base equilibrium is essential for grasping many concepts in chemistry, including the behavior of weak bases. It refers to the state of balance between a conjugate acid-base pair where the forward and reverse reactions occur at equal rates. This equilibrium can be expressed in terms of an equilibrium constant
, for bases this is the base dissociation constant (Kb).

In the context of weak bases, the equilibrium lies more toward the un-ionized form because weak bases only partially dissociate in solution. This equilibrium plays a crucial role in determining the pH of the solution and the concentrations of the various species present in the solution.

pH Calculation

Calculating the pH of a solution is a fundamental skill in chemistry that reflects how acidic or basic a solution is. The pH is calculated as the negative logarithm of the concentration of hydronium ions (\(H_3O^+\)) in the solution.

To determine the pH, one needs to have knowledge of the concentration of the acid or base in solution and its dissociation constant (\(K_a\) or \(K_b\) respectively). From the pH, we can deduce information about the equilibrium state of the system and the extent of ionization of the solute involved.

Conjugate Acid-Base Pairs

Conjugate acid-base pairs are a pair of compounds that differ by the presence of one proton (\(H^+\)). In every acid-base reaction, an acid donates a proton to form its conjugate base, while a base accepts a proton to form its conjugate acid.

The strength of an acid or base is related to its conjugate partner; the weaker the acid, the stronger its conjugate base, and vice versa. This is significant because the reaction of bases, like the weak base ionization of quinine, can be understood better when considering the behavior of their conjugate acids in solution.

Weak Base Ionization

Weak base ionization describes the process where a weak base, such as quinine, partially dissociates in water to yield its conjugate acid and hydroxide ions. Unlike strong bases, which dissociate completely, weak bases establish an equilibrium in solution that is governed by the base dissociation constant (\(K_b\)).

The value of \(K_b\) is a measure of the base's strength and its tendency to accept a proton. It helps us calculate the pH of the solution and the concentrations of all species at equilibrium. In the ionization reaction, for instance, the equilibrium concentrations of the products and the reactants are related through the \(K_b\) expression. The process is an excellent example of how acids and bases interact with water and is foundational for understanding the concept of pH and the behavior of compounds in solutions.