Question

A \(2.5 \times 10^{-3} \mathrm{M}\) solution of an unknown acid has a pH of 3.80 at \(25^{\circ} \mathrm{C}\) (a) What is the hydronium ion concentration of the solution? (b) Is the acid a strong acid, a moderately weak acid \(\left(K_{\mathrm{a}} \text { of about } 10^{-5}\right)\) Or a very weak acid\(\left(K_{\mathrm{a}} \text { of about } 10^{-10}\right) ?\)

Short answer

(a) \([\text{H}_3\text{O}^+] = 1.58 \times 10^{-4} \text{ M}\) ;\n(b) The acid is a moderately weak acid.

Step-by-step solution

Calculate Hydronium Ion Concentration

The pH of a solution is given by the negative logarithm of the hydronium ion concentration: \[ \text{pH} = -\log[\text{H}_3\text{O}^+] \]Given, \( \text{pH} = 3.80 \), we can determine the hydronium ion concentration by rearranging the formula:\[ [\text{H}_3\text{O}^+] = 10^{-\text{pH}} \]Substituting the given pH:\[ [\text{H}_3\text{O}^+] = 10^{-3.80} \approx 1.58 \times 10^{-4} \text{ M} \]

Determine the Degree of Ionization

We compare the calculated hydronium ion concentration \([\text{H}_3\text{O}^+] = 1.58 \times 10^{-4} \text{ M}\) with the initial concentration of the acid \(2.5 \times 10^{-3} \text{ M}\).The degree of ionization is:\[ \frac{1.58 \times 10^{-4}}{2.5 \times 10^{-3}} \approx 0.0632 \]This indicates the proportion of the acid that ionizes, which is approximately \(6.32\%\).

Calculate the Acid Ionization Constant \(K_a\)

The ionization of the acid can be written as:\( \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \).From the acid equilibrium expression, we have:\[ K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \]Since \([\text{H}^+] = [\text{A}^-] = 1.58 \times 10^{-4} \text{ M}\) and the remaining acid concentration \([\text{HA}] = 2.5 \times 10^{-3} - 1.58 \times 10^{-4} = 2.342 \times 10^{-3} \text{ M}\), we calculate:\[ K_a = \frac{(1.58 \times 10^{-4})^2}{2.342 \times 10^{-3}} \approx 1.07 \times 10^{-5} \]

Classify the Acid's Strength

Compare the calculated \(K_a \approx 1.07 \times 10^{-5}\) to the given classifications:- A strong acid typically has a very high \(K_a\) value.- A moderately weak acid has a \(K_a \approx 10^{-5}\).- A very weak acid has a \(K_a \approx 10^{-10}\).The calculated \(K_a\) is closest to \(10^{-5}\), indicating the acid is moderately weak.

Key concepts

Hydronium Ion Concentration

The hydronium ion concentration reflects the amount of \( \text{H}_3\text{O}^+ \) ions in a solution, which is a direct measure of its acidity. When you know the pH of a solution, you can easily find this concentration using the formula:
\[ [\text{H}_3\text{O}^+] = 10^{-\text{pH}} \]This equation originates from the definition of pH, \( \text{pH} = -\log[\text{H}_3\text{O}^+] \).
In our example, with a pH of 3.80, we find that:
\[ [\text{H}_3\text{O}^+] = 10^{-3.80} \approx 1.58 \times 10^{-4} \text{ M} \]This process allows us to translate pH, a logarithmic scale, into a tangible concentration in moles per liter. Understanding hydronium ion concentration is crucial in evaluating the properties and behaviors of acidic solutions.

Acid Strength

Acid strength is a measure of an acid's ability to donate protons or hydrogen ions in a solution. This is usually represented by the acid ionization constant, \( K_a \).
A higher \( K_a \) value indicates a stronger acid, which means it donates protons more readily.
Here’s a simple way to classify acids based on \( K_a \):

• Strong acids have a very large \( K_a \), usually not calculated as they are fully ionized.
• Moderately weak acids have a \( K_a \) around \( 10^{-5} \).
• Very weak acids have a \( K_a \) around \( 10^{-10} \).

In our scenario, the calculated \( K_a \) is approximately \( 1.07 \times 10^{-5} \), which classifies the unknown acid as moderately weak.
Identifying acid strength helps us understand reaction dynamics and acid behavior in chemical processes.

Degree of Ionization

The degree of ionization is the proportion of acid molecules that dissociate into ions in solution. It’s given by the formula:
\[ \text{Degree of Ionization} = \frac{\text{Concentration of ionized acid}}{\text{Initial concentration of acid}} \]For acids, this tells us how many of the acid's molecules release \( \text{H}^+ \) ions when dissolved.
In our problem, the degree of ionization of the acid is calculated as:
\[ \frac{1.58 \times 10^{-4}}{2.5 \times 10^{-3}} \approx 0.0632 \]This simplifies to a 6.32% ionization degree.
A higher degree of ionization typically aligns with stronger acids as more molecules donate protons. This concept is important for predicting how an acid behaves in different solutions and understanding its reactivity.