Question

Calculate the \(K_{\mathrm{a}}\) of butyric acid if a \(0.025-\mathrm{M}\) butyric acid solution has a pH of 3.21 .

Short answer

The acid dissociation constant \(K_a\) for butyric acid is \(1.56 \times 10^{-5}\).

Step-by-step solution

Identify Given Information and Required Unknown

The problem provides that the concentration of butyric acid \([HA]\) is 0.025 M, and the pH of the solution is 3.21. We need to calculate the acid dissociation constant \(K_a\) for butyric acid.

Calculate the Hydronium Ion Concentration

Start by using the pH to find the concentration of hydronium ions \([H^+]\). The formula is: \([H^+] = 10^{-\text{pH}}\). Substitute the given pH value: \([H^+] = 10^{-3.21}\). Calculate this to get \([H^+] = 6.16 \times 10^{-4}\, \text{M}\).

Write the Ionization Equation of Butyric Acid

The ionization of butyric acid can be expressed as: \[ HA \rightleftharpoons H^+ + A^- \]. Initially, \([HA] = 0.025 \text{ M}\), and at equilibrium, \([H^+] = [A^-] = 6.16 \times 10^{-4} \text{ M}\).

Calculate the Equilibrium Concentration of Butyric Acid

At equilibrium, the concentration of undissociated butyric acid \([HA]\) can be calculated as: \([HA]_{eq} = [HA]_{initial} - [H^+]\). Substitute the known values: \([HA]_{eq} = 0.025 - 6.16 \times 10^{-4} \). This simplifies to \([HA]_{eq} = 0.024384 \text{ M}\).

Calculate the Acid Dissociation Constant \(K_a\)

The formula for \(K_a\) is \(K_a = \frac{[H^+][A^-]}{[HA]}\). Substitute the equilibrium concentrations determined: \(K_a = \frac{(6.16 \times 10^{-4})(6.16 \times 10^{-4})}{0.024384}\). Calculate \(K_a = 1.56 \times 10^{-5}\).

Key concepts

Butyric Acid

Butyric acid, also known as butanoic acid, is a carboxylic acid with the chemical formula \( C_4H_8O_2 \).
It is commonly found in butter and contributes to its distinctive aroma.
Understanding butyric acid is crucial for calculating its acid dissociation properties.When dissolved in water, butyric acid partially ionizes to release protons (\( H^+ \)).
This ionization process is essential to understand the acid's behavior in a solution and how it affects properties like pH and dissociation constant.

pH Calculation

pH is a measure of the acidity or basicity of a solution. It is calculated using the formula:
\[ \text{pH} = -\log_{10} [H^+] \]To find the pH of a solution, you need the concentration of hydronium ions \([H^+]\).
In our exercise, the given pH is 3.21, and the corresponding \([H^+]\) is calculated using:
\[ [H^+] = 10^{-3.21} = 6.16 \times 10^{-4} \text{ M} \]This calculation helps us determine how acidic the butyric acid solution is and plays a vital role in calculating the dissociation constant.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of reactants and products in a chemical reaction at equilibrium.
In this exercise, we calculate it for the ionization of butyric acid.Initially, we have a concentration of \([HA] = 0.025 \text{ M}\).
At equilibrium, the concentration of hydronium ions \([H^+]\) is \(6.16 \times 10^{-4} \text{ M}\).
Therefore, the equilibrium concentration of undissociated butyric acid is:
\[ [HA]_{eq} = [HA]_{initial} - [H^+] = 0.025 - 6.16 \times 10^{-4} = 0.024384 \text{ M} \]Knowing these concentrations allows us to accurately calculate the acid dissociation constant \(K_a\).

Ionization Equation

Ionization is the process by which a molecule, like butyric acid, dissociates into ions in water.
The ionization equation for butyric acid is:\[ HA \rightleftharpoons H^+ + A^- \]This shows that butyric acid \(HA\) dissociates into a proton \(H^+\) and conjugate base \(A^-\).
It’s essential to write this equation to solve for equilibrium concentrations and calculate the \(K_a\).With known concentrations of \([H^+]\), \([A^-]\), and \([HA]_{eq}\), we use the formula for \(K_a\):
\[ K_a = \frac{[H^+][A^-]}{[HA]_{eq}} \]Substituting the values yields accurate calculations of the dissociation properties of butyric acid.