Question

The \(K_{\mathrm{a}}\) for benzoic acid is \(6.5 \times 10^{-5} .\) Calculate the \(\mathrm{pH}\) of a \(0.10-M\) aqueous solution of benzoic acid at \(25^{\circ} \mathrm{C}\).

Short answer

The pH of the solution is approximately 2.59.

Step-by-step solution

Write the Ionization Equation

Benzoic acid, represented as \( \text{C}_6\text{H}_5\text{COOH} \), dissociates in water to form benzoate ions \( \text{C}_6\text{H}_5\text{COO}^- \) and hydrogen ions \( \text{H}^+ \). The equation is: \( \text{C}_6\text{H}_5\text{COOH} \rightleftharpoons \text{C}_6\text{H}_5\text{COO}^- + \text{H}^+ \).

Set Up the Expression for Ka

The equilibrium expression for the acid dissociation constant \( K_a \) is: \[ K_a = \frac{[\text{C}_6\text{H}_5\text{COO}^-][\text{H}^+]}{[\text{C}_6\text{H}_5\text{COOH}]} \] Given that \( K_a = 6.5 \times 10^{-5} \).

Define Initial Concentrations

The initial concentration of benzoic acid is \( 0.10 \: M \), and the initial concentrations of benzoate ions and hydrogen ions are \( 0 \) since no dissociation has occurred yet.

Define Change in Concentration

Let \( x \) represent the change in concentration due to dissociation. The concentration of benzoic acid becomes \( 0.10 - x \), while the concentrations of benzoate ions and hydrogen ions become \( x \).

Substitute into Ka Expression

Substitute the equilibrium concentrations into the \( K_a \) expression to get: \[ 6.5 \times 10^{-5} = \frac{x^2}{0.10 - x} \]

Assume \( x \ll 0.10 \) and Simplify

Assume \( x \) is small compared to \( 0.10 \), so \( 0.10 - x \approx 0.10 \). The equation simplifies to: \[ 6.5 \times 10^{-5} = \frac{x^2}{0.10} \]

Solve for x

Rearrange and solve for \( x \): \[ x^2 = 6.5 \times 10^{-5} \times 0.10 \] \[ x^2 = 6.5 \times 10^{-6} \] \[ x = \sqrt{6.5 \times 10^{-6}} \approx 2.55 \times 10^{-3} \]

Calculate pH

The \( \text{H}^+ \) concentration is \( x \), so \( \text{pH} = -\log(2.55 \times 10^{-3}) \approx 2.59 \).

Key concepts

Benzoic Acid

Benzoic acid is a simple aromatic carboxylic acid with the chemical formula \( \text{C}_6\text{H}_5\text{COOH} \). It is widely used in food preservation and manufacturing. Understanding its behavior in an aqueous solution is key to grasping the basics of acid-base chemistry.
In water, benzoic acid partially dissociates into its ions: benzoate (\( \text{C}_6\text{H}_5\text{COO}^- \)) and hydrogen ions (\( \text{H}^+ \)). This process is called ionization. Ionization is the splitting of a molecule into its ions, and it's important because it directly affects the acidity of a solution—linked to the solution's pH. By controlling how much benzoic acid dissociates, we can predict and understand its acidic properties, which is essential for applications ranging from chemistry lab experiments to food industry practices.

Equilibrium Expression

The equilibrium expression for an acid like benzoic acid helps in determining the concentrations of its ions when an equilibrium is reached in a solution. Equilibrium means that the rate of the forward reaction (dissociation) equals the rate of the reverse reaction (reassociation).
This balance gives rise to the equilibrium constant expression, which is crucial for understanding acid dissociation.
For benzoic acid, the equilibrium expression is formulated as follows:
\[ K_a = \frac{[\text{C}_6\text{H}_5\text{COO}^-][\text{H}^+]}{[\text{C}_6\text{H}_5\text{COOH}]} \]
In this expression:

• \( [\text{C}_6\text{H}_5\text{COO}^-] \) is the concentration of benzoate ions.
• \( [\text{H}^+] \) is the concentration of hydrogen ions.
• \( [\text{C}_6\text{H}_5\text{COOH}] \) is the concentration of undissociated benzoic acid.

By inputting the concentrations into the expression, we can find the acid's dissociation constant \( K_a \), helping predict behavior in the solution.

pH Calculation

The pH of a solution gives a numeric scale from 0 to 14 that tells us how acidic or basic a solution is. Calculating the pH of an acidic solution like that of benzoic acid involves several steps.
In our example, we assume initial concentrations and define changes mathematically using \( x \) for simplicity:

• Initial concentration of benzoic acid is 0.10 M.
• Initial concentrations of benzoate ions and hydrogen ions are 0.

The changes in concentration lead us to an equilibrium concentration expression, allowing us to find \( x \), which stands for \( [\text{H}^+] \).
The pH is then calculated using the equation:
\[ \text{pH} = -\log([\text{H}^+]) \]
After finding \( x \), which equals \( 2.55 \times 10^{-3} \), substitute it into the \( \text{pH} \) formula to get approximately 2.59. This value indicates that the solution is acidic.

Acid-Base Chemistry

Acid-base chemistry is a branch of chemistry that studies the reactions where acids and bases interact. An acid, like benzoic acid, donates \( H^+ \) ions, while a base accepts them.
Core ideas in this field revolve around dissociation, equilibrium, and pH. When acids are placed in water, they dissociate to form ions. This dissociation meets a point of balance called equilibrium, which is a significant focus in acid-base reactions.
In our case, studying the ionization of benzoic acid shows us about how acids behave and helps calculate the impact of acids on pH. Acid-base theories often classify substances based on their ability to donate or accept protons (like the Brønsted-Lowry theory). Understanding these concepts is pivotal in laboratory settings, where precise control and prediction of chemical behavior are required.
Engaging with such exercises not only furthers comprehension of theoretical chemistry but also enhances problem-solving skills applicable in real-world chemical applications.