Question

Amantadine, \(\mathrm{C}_{10} \mathrm{H}_{15} \mathrm{NH}_{2}\), is a weak base used in the treatment of Parkinson's disease. Its conjugate acid has \(K_{\mathrm{a}}=\) \(7.9 \times 10^{-11}\). Calculate the pH of a 0.0010-M aqueous solution of amantadine at \(25^{\circ} \mathrm{C}\).

Short answer

The pH of the solution is approximately 10.55.

Step-by-step solution

Write the Equilibrium Expression for the Weak Acid

The conjugate acid of amantadine has a given acid dissociation constant, \(K_a\). Write the dissociation equation in water: \[ BH^+ \rightleftharpoons B + H^+ \] where \(BH^+\) is the conjugate acid and \(B\) is amantadine. The expression for \(K_a\) is: \[ K_a = \frac{[B][H^+]}{[BH^+]} \]

Relate Ka to Kb

Use the relationship between \(K_a\), \(K_b\), and \(K_w\): \[ K_w = K_a \cdot K_b \] at \(25^{\circ}C\), \(K_w = 1.0 \times 10^{-14}\). Solve for \(K_b\): \[ K_b = \frac{1.0 \times 10^{-14}}{7.9 \times 10^{-11}} \] Calculate \(K_b\): \[ K_b = 1.27 \times 10^{-4} \]

Set Up the Base Dissociation Equation

Amantadine \((C_{10}H_{15}NH_2)\) dissociates in water: \[ B + H_2O \rightleftharpoons BH^+ + OH^- \] The equilibrium expression is: \[ K_b = \frac{[BH^+][OH^-]}{[B]} \]

Develop the ICE Table for Base Dissociation

Define the initial concentrations: - Initial \([B] = 0.0010 \, \text{M}\), \([BH^+] = 0\), \([OH^-] = 0\). Change in concentration: - \(-x\) for \([B]\), \(+x\) for \([BH^+]\) and \([OH^-]\). Equilibrium concentrations: - \([B] = 0.0010 - x\), \([BH^+] = x\), \([OH^-] = x\).

Solve for x Using the Kb Expression

Substitute equilibrium concentrations into the \(K_b\) expression: \[ 1.27 \times 10^{-4} = \frac{x^2}{0.0010 - x} \] Assuming \(x \ll 0.0010\), simplify: \[ 1.27 \times 10^{-4} = \frac{x^2}{0.0010} \] Solve for \(x\): \[ x^2 = 1.27 \times 10^{-7} \] \[ x = 3.56 \times 10^{-4} \]

Calculate pOH and then the pH

\(x\) represents \([OH^-]\), so \([OH^-] = 3.56 \times 10^{-4} \, \text{M}\). Calculate \(pOH\): \[ pOH = -\log(3.56 \times 10^{-4}) \approx 3.45\] Use the relation \( pH + pOH = 14 \) to find \(pH\): \[ pH = 14 - 3.45 = 10.55 \]

Key concepts

Acid-Base Equilibrium

Acid-base equilibrium is an essential concept in chemistry that describes the balance between acids and bases in a solution. It is particularly relevant when dealing with weak acids and bases, such as amantadine, which is used medically. In an equilibrium, the rate of the forward reaction (dissociation) is equal to the rate of the reverse reaction (reassociation).

For weak bases like amantadine, the equilibrium involves its conjugate acid. The equilibrium in water can be depicted as the equation: \[ BH^+ \rightleftharpoons B + H^+ \] here, \(BH^+\) represents the conjugate acid. Amantadine itself acts as the weak base, \(B\). The statement of equilibrium helps chemists predict how the concentrations of entities in the solution change under perturbations.

Key points include:

• The position of equilibrium is constant for a given temperature.
• Weak acids and bases don't fully dissociate, unlike strong acids and bases.
• The equilibrium can shift based on the concentration and presence of other substances.

pH Calculation

Calculating the pH of a solution that contains weak acids or bases involves understanding the relationship between hydrogen ions and hydroxide ions. The pH provides insight into the acidity or basicity of the solution.

For the base amantadine, the concentration of hydroxide ions \([OH^-]\) is pivotal. By determining this, one can first calculate the \(pOH\) (since pOH is a direct function of hydroxide concentration), and then convert that into the pH value:

• Calculate \([OH^-]\) from the equilibrium expression and ICE table.
• Find \(pOH\) using the formula \(-\log [OH^-]\).
• Relate \(pH\) to \(pOH\) with \(pH + pOH = 14\).

The calculated pH helps evaluate how alkaline or acidic the solution is, which is especially important for biological and chemical processes.

Equilibrium Expressions

Equilibrium expressions are mathematical equations used to describe chemical reactions that reach a state of balance. They are crucial for understanding acid-base reactions because they permit prediction of ion concentrations at equilibrium.

In the case of amantadine's conjugate acid, the acid dissociation constant, \(K_a\), is provided. The expression for \(K_a\) is:\[K_a = \frac{[B][H^+]}{[BH^+]}\]This equation shows the relationship between the acid's species at equilibrium.

To relate \(K_a\) to the base dissociation constant \(K_b\), use the expression:\[K_w = K_a \cdot K_b\]Where \(K_w\) is the ion-product of water at the specified temperature. Knowing \(K_a\), one can find \(K_b\) and thus explore the behavior of the base in water. This helps anticipate how the weak base amantadine will behave in solution, providing predictive power for chemical behavior.